index int64 0 18.8k | text stringlengths 0 826k | year stringdate 1980-01-01 00:00:00 2024-01-01 00:00:00 | No stringlengths 1 4 |
|---|---|---|---|
1,800 | Interfacing Sound Stream Segregation to Recognition - Preliminar Several Sounds Si Hiroshi G. Okuno, Tomohiro Nakatani and Takeshi Kawabata NIT Basic Research Laboratories Nippon Telegraph and Telephone Corporation 3-l Morinosato-Wakamiya, Atsugi, Kanagawa 243-01, JAPAN okun@nue.org nakatani@horn.brl.ntt.jp kaw@idea.brl.ntt.jp Abstract This paper reports the preliminary results of experiments on listening to several sounds at once. ‘Ike issues are addressed: segregating speech streams from a mixture of sounds, and interfacing speech stream segregation with automatic speech recognition (AD). Speech stream segregation (SSS) is mod- eled as a process of extracting harmonic fragments, grouping these extracted harmonic fragments, and substituting some sounds for non-harmonic parts of groups. This system is implemented by extending the harmonic-based stream segre- gation system reported at AAAI-94 and IJCAI-95. The main problem in interfacing SSS with HMM-based ASR is how to improve the recognition performance which is degraded by spectral distortion of segregated sounds caused mainly by the binaural input, grouping, and residue substitution. Our solution is to re-train the parameters of the HMM with train- ing data binauralized for four directions, to group harmonic fragments according to their directions, and to substitute the residue of harmonic fragments for non-harmonic parts of each group. Experiments with 500 mixtures of two women’s utterances of a word showed that the cumulative accuracy of word recognition up to the 10th candidate of each woman’s utterance is, on average, 75%. Introduction Usually, people hear a mixture of sounds, and people with normal hearing can segregate sounds from the mixture and focus on a particular voice or sound in a noisy environment. This capability is known as the cocktailparty efSect (Cherry 1953). Perceptual segregation of sounds, called auditory scene analysis, has been studied by psychoacoustic re- searchers for more than forty years. Although many obser- vations have been analyzed and reported (Bregman 1990), it is only recently that researchers have begun to use computer modeling of auditory scene analysis (Cooke et al. 1993; Green et al. 1995; Nakatani et al. 1994). This emerging re- search area is called computational auditory scene analysis (CASA) and a workshop on CASA was held at IJCAI-95 (Rosenthal & Okuno 1996). One application of CASA is as a front-end system for automatic speech recognition (ASR) systems. Hearing impaired people find it difficult to listen to sounds in a noisy environment. Sound segregation is expected to improve the performance of hearing aids by reducing background noises, echoes, and the sounds of competing talkers. Similarly, most current ASR systems do not work well in the presence 1082 Perception of competing voices or interfering noises. CASA may provide a robust front-end for ASR systems. CASA is not simply a hearing aid for ASR systems, though. Computer audition can listen to several things at once by segregating sounds from a mixture of sounds. This capability to listen to several sounds simultaneously has been called the Prince Shotoku efSect by Okuno (Okuno et al. 1995) after Prince Shotoku (574-622 A.D.) who is said to have been able to listen to ten people’s petitions at the same time. Since this is virtually impossible for humans to do, CASA research would make computer audition more powerful than human audition, similar to the relationship of an airplane’s flying ability to that of a bird. At present, one of the hottest topics of ASR research is how to make more robust ASR systems that perform well outside laboratory conditions (Hansen et al. 1994). Usually the approaches taken are to reduce noise and use speaker adaptation, and treat sounds other than human voices as noise. CASA takes an opposite approach. First, it deals with the problems of handling general sounds to develop methods and technologies. Then it applies these to develop ASR systems that work in a real world environment. In this paper, we discuss the issues concerning interfacing of sound segregation systems with ASR systems and report preliminary results on ASR for a mixture of sounds. Sound Stream Segregation Sound segregation should be incremental, because CASA is used as a front-end system for ASR systems and other appli- cations that should run in real time. Many representations of a sound have been proposed, for example, auditory maps (Brown 1992) and synchrony strands (Cooke et al. 1993), but most of them are unsuitable for incremental processing. Nakatani and Okuno proposed using a sound stream (or simply stream) to represent a sound (Nakatani et al. 1994). A sound stream is a group of sound components that have some consistent attributes. By using sound streams, the Prince Shotoku effect can be modeled as shown in Fig. 1. Sound streams are segregated by the sound segregation system, and then speech streams are selected and passed on to the ASR systems. Sound stream segregation consists of two subprocesses: 1. Stream fragment extraction - a fragment of a stream that has the same consistent attributes is extracted fi-om a mixture of sounds. From: AAAI-96 Proceedings. Copyright © 1996, AAAI (www.aaai.org). All rights reserved. sound stream m$uwds 71 w 1 Automatic Speech Recognition 1 Tracer generator -Tracer- I Sound Stream - t-! Automatic Speech Recognition Segregation i *I. Automatic Speech Recognition Figure 1: Modeling of the Prince Shotoku Effect or of Listening to Several Sounds Simultaneously 2. Stream fragment grouping - stream fragments are grouped into a stream according to some consistent attributes. Most sound segregation systems developed so far have limitations. Some systems assume the number of sounds, or the characteristics of sounds such as voice or music (e.g., (Ramalingam 1994)). Some run in a batch mode (e.g., (Brown 1992; Cooke et al. 1993)). Since CASA tries to manipulate any kind of sound, it should be able to segregate any kind of sound from a mixture of sounds. For that reason, sound segregation systems should work primarily with the low level characteristics of sound. Once the performance of such systems has been assessed, the use of higher level characteristics of sounds or combining bottom-up and top-down processing should be attempted. Nakatani et al. used a harmonic structure ’ and the direction of the sound source as consistent attributes for segregation. They developed two systems: the harmonic- based stream segregation (HBSS) (Nakatani et al. 1994; Nakatani et al. 1995a), and the binaural harmonic- based stream segregation (Bi-HBSS) systems (Nakatani et al. 1996). Both systems were designed and implemented in a multi-agent system with the residue-driven architecture (Nakatani et al. 1995b). We adopted these two systems to extract stream fragments from a mixture of sounds, since they run incrementally by using lower level sound charac- teristics. This section explains in detail how HBSS and Bi-HBSS work. Harmonic-based Sound Segregation The HBSS uses three kinds of agents: an event-detector, a tracer-generator, and tracers (Fig. 2) (Nakatani et al. 1994; Nakatani et al. 1995a). It works as follows: 1. 2. 3. An event-detector subtracts a set of predicted inputs from the actual input and sends the residue to the tracer- generator and tracers. If the residue exceeds a threshold value, the tracer- generator searches for a harmonic structure in the residue. If it finds a harmonic structure and its fundamental stream, it generates a tracer to trace the harmonic structure. Each tracer extracts a harmonic stream fragment by tracing the fundamental frequency of the stream. It also composes a predicted next input by adjusting the segregated stream fragment with the next input and sends this prediction to the event-detector. ‘A harmonic structure consists of a fundamental frequency and its integer multiples or overtones. Tracer.-, - Stream fragment Figure 2: Harmonic-based Stream Segregation (HBSS) Since tracers are dynamically generated and terminated in response to the input, a HBSS system can manipulate any number of sounds in principle. Of course, the setting of various thresholds determines the segregation performance. The tracer-generator extracts a fundamental frequency from the residue of each time frame. For that purpose, the harmonic intensity Et(w) of the sound wave Zt (7) at frame t is defined as w4 = c II &,kW II*, where r is time, k is the index of harmonics, Zt (7) is the residue, and Ht,k (0) is the sound component of the kth overtone. Since some components of a harmonic structure are destroyed by other interfering sounds, not all overtones are reliable. Therefore, only a valid overtone for a harmonic structure is used. An overtone is defined as valid if the intensity of the overtone is larger than a threshold value and the time transition of the intensity can be locally approximated in a linear manner. The valid harmonic intensity, Ei (w ), is also defined as the sum of the II Ht,k(w) II of valid overtones. When a (harmonic) tracer is generated, it gets the initial fundamental frequency from the tracer-generator, and at each time frame it extracts the fundamental frequency that maximizes the valid harmonic intensity Ei (w). Then, it calculates the intensity and the phase of each overtone by evaluating the absolute value and that of Ht,k(W) and extracts a stream fragment of the time frame. It also creates a predicted next input in a waveform by adjusting the phase of its overtones to the phase of the next input frame. If there are no longer valid overtones, or if the valid harmonic intensity drops below a threshold value, it terminates itself. Binaural Harmonic-based Sound Segregation When a mixture of sounds has harmonic structures whose fundamental frequencies are very close, HBSS may fail to segregate such sounds. For example, consider two har- monic sounds; one’s fundamental frequency is increasing and the other’s fundamental frequency is decreasing. When both fundamental frequencies cross, the HBSS cannot know whether two fundamental frequencies are crossing or ap- proaching and departing. To cope with such problems and improve the segregation performance, binaural harmonic- based stream segregation (SLHBSS), which incorporates di- Perception 1083 rection information into the HBSS, was proposed (Nakatani et al. 1996). The Bi-HBSS takes a binaural input and extracts the direction of the sound source by calculating the interaural time difference (ZTD) and interaural intensity difference (IID). More precisely, the Bi-HBSS uses two separate HBSS’s for the right and left channels of the binaural input to extract harmonic stream fragments. Then, it calculates the ITD and IID by using a pair of harmonic stream fragments segregated. This method of calculating the ITD and IID reduces the computational costs, which is an important advantage since these values are usually calculated over the entire frequency region (Blauert 1983; Bodden 1993; Stadler & Rabinowitz 1993). The Bi-HBSS also utilizes the direction of the sound source to refine the harmonic structure by incorporating the direction into the validity. Thus, Bi-HBSS extracts a harmonic stream fragment and its direction. Internally, direction is represented by ITD (msec) and fundamental frequency is represented by cent. The unit, cent, is a logarithmic representation of frequency and 1 octave is equivalent to 1,200 cent. The Bi-HBSS improves the segregation performance of the HBSS (Nakatani et al. 1995b; Nakatani et al. 1996). In addition, the spectral distortion of segregated sounds became very small when benchmarking was used with various mixtures of two women’s utterances of Japanese vowels and interfering sounds (Nakatani et al. 1996). However, the usage of binaural inputs may cause spectral distortion, because the spectrum of a binaural input is not the same as that of the original sound due to the shape of the human head. Such transformation is called the head-related transferfunction (HRTF) (Blauert 1983). Due to the HRTF, the power of lower frequencies is usually decreased while that of higher frequencies is increased. Thus, it may make it difficulty to segregate a person’s speech. The literature mentioned above did not examine this possibility. Design of Speech Stream Segregation Neither HBSS nor Bi-HBSS can segregate a speech stream, because it contains non-harmonic structures (e.g., conso- nants, especially unvoiced consonants) as well as harmonic structures (e.g., vowels and some voiced consonants). In this paper, we propose a simple method to extract a speech stream. First, the harmonic structures (vowels and some voiced consonants) of each stream are extracted by HBSS or Bi-HBSS and reconstructed by grouping. This process is called harmonic grouping. Second, non-harmonic struc- tures (or most consonants) are reconstructed by substituting the residue. This process is called residue substitution. These processes also work incrementally, like the stream fragment extraction process. Note that in this scheme, consonants are extracted implicitly. Harmonic Grouping Suppose that a new harmonic stream fragment 4 is to be grouped. Let f+ be the funda- mental frequency of 4. The harmonic part of a stream is reconstructed in one of the following three ways (Nakatani et al. 1996; Rosenthal & Okuno 1996): 1084 Perception . F-grouping - according to the nearness of the funda- mental frequencies. Find an existing group, say \k, such that the difference 1 fb - f\~ I< 6. The value of 6 is 300 cent if other new stream fragments exist at the same time with 4, 600 cent otherwise. If more than one existing group is found, q5 is grouped into the group that is the closest to f4. If only one existing group is found, 4 is grouped into 9. Otherwise, 4 forms a new group. D-grouping- according to the nearness of the directions of the sound source. The range of nearness in ITD is 0.167 msec, which corresponds roughly to 20”. The algorithm is the same as the F-grouping. B-grouping - If a stream fragment, 4, satisfies the above two conditions for a group, Q, it is grouped into Q. However, if 4 has more than one such group,the group of minimum combined nearness is selected. ‘Ihe combined nearness, K, is defined as follows: Cf ’ ’ cd where cf = 300 cent, and cd = 0.167 msec. The current value of the normalized factor, cy, is 0.47. The grouping is controlled by the gap threshold; if the time gap between two consecutive stream fragments is less than the gap threshold, they are grouped together with information about the missing components. The current value of the gap threshold is 500 msec, which is determined by the maximum duration of the consonants in the utterance database. Note that since HBSS extracts only harmonic structures, only F-grouping is applicable. Residue substitution The idea behind the residue substi- tution is based on the observation that human listeners can perceptually restore a missing sound component if it is very brief and replaced by appropriate sounds. This auditory mechanism of phonemic restoration is known as auditory induction (Warren 1970). After harmonic grouping, har- monic components are included in a segregated stream or group, while non-harmonic components are left out. Since the missing components are non-harmonic, they cannot be extracted by either HBSS or Bi-HBSS and remain in the residue. Therefore, the missing components of a stream may be restored by substituting the residue produced by HBSS or Bi-HBSS. The residue substitution, or which part of the residue is substituted for missing components, may be done by one of the following methods: 1. All-residue substitution - All the residue is used. 2. Own-residue substitution - Only the residue from the direction of the sound source is used. In this paper, the former method is used, because the latter requires a precise determination of the sound source direction and thus the computational cost of separation is higher. In addition, the recognition performance of the latter is lower than that of the former, as will be shown later. Issues in Interfacing SSS with ASR We use an automatic speech recognition system based on a hidden Markov model-based (HMM). An HMM usually uses the three characteristics in speech recognition; a spec- tral envelop, a pitch or a fundamental frequency, and a label or a pair consisting of the onset and offset times of speech. Since the input is a mixture of sounds, these characteristic, in particular the spectral envelop, are critically affected. Therefore, the recognition performance with a mixture of sounds is severely degraded by spectral distortion caused by interfering and competing sounds. The segregation of the speech streams is intended to reduce the degradation, and is considered effective in re- covering spectral distortion from a mixture of sounds. However, it also introduces another kind of spectral distor- tion to segregated streams, which is caused by extracting the harmonic structure, the head-related transfer function, or a binaural input, and the grouping and residue substitu- tion. In the next section, the degradation of the recognition performance caused by segregation will be assessed and methods of recovery will be proposed. The pitch error of Bi-HBSS for simple benchmarks is small (Nakatani et al. 1996). However, its evaluation with larger benchmarks is also needed. The onset of a segregated stream is detected only from the harmonic structures in HBSS. Since the beginning and end of speech are usually comprised of non-harmonic structures, the onset and offset times are extended by 40 msec for sounds segregated by HBSS. Since Bi-HBSS can detect whether a leading and/or trailing sound exists according to the directional information, the onset and offset is determined by this. Influence of SSS on ASR In this section, we assess the effect of segregation and propose methods to reduce this effect. on ASR The ASR system used in this paper The “HMM-LR” developed by ATR Inc. (Kita et al. 1990) is used system in this paper. The HMM-LR is a continuous speech recognition system that uses generalized LR parsing with a single discrete codebook. The size of the codebook is 256 and it was created from a set of standard data. The training and test data used in this paper were also created by ATR Inc. Since the primitive HMM-LR is a gender- dependent speech recognition system, HMM-LRs for male speakers (the HMM-m) and for female speakers (the HMM- j) were used. The parameters of each system were trained by using 5,240 words from five different sets of 1,048 utterances by each speaker. The recognition performance was evaluated by an open test, and 1,000 testing words were selected randomly from non-training data. The evaluation was based on word recognition. Therefore, the LR grammar for the HMM-m/f consists of only rules that the start symbol derives a terminal symbol directly. The evaluation measure used in this paper is the cumulative accuracy up to the 10th candidate, which specifies what percentage of words are recognized up to the 10th candidate by a particular E & 95.0 J 90.0 8 - k! 85.0 E 3 m.0 ii 75.0 8 70.0 65.01 0 1 2 3 4 5 6 7 6 9 oR&R Figure 3: Influence of the Harmonic Structure Extraction (Experiment 1) HMM-LR. This measurement is popular for evaluating the actual speech recognition performance in a whole speech understanding system, because the top nth recognition candidates are used in successive language understanding. Influence of the Harmonic Structure Extraction To assess the influence of harmonic structure extraction on the word recognition performance, we have defined a new operation called harmonic structure reconstruction, which is done as follows: 1. The HBSS extracts harmonic stream fragments utterance of a word by a single speaker. 2. All the extracted harmonic into the same stream. stream fragments are grouped from an 3. All the residue is substituted in the stream for the frames where no harmonic structure was extracted. time Experiment 1: Harmonic structure reconstruction and word recognition was performed using the HMM-m for over 1,000 utterances of a word by a male speaker. The cumulative accuracy of the recognition is shown in Fig. 3. In Fig. 3, the curve denoted as the original data indicates the recognition rate for the same original utterances by the same speaker. The word recognition rate was lower by 3.5% for the first candidate when the HMM-m was used, but was almost equal in cumulative accuracy for the 10th candidate. This demonstrates that the harmonic structure reconstruction has little effect on the word recognition performance. We tried to improve the recognition rate by re-training the parameters of the HMM-LR by using all the training data provided through harmonic structure reconstruction. The resulting HMM-LR, however, did not improve the recognition rate as shown in Fig. 3. Therefore, we did not adopt any special treatment for harmonic structure reconstruction. Influence of the Head-related Transfer Function As we mentioned, a binaural sound is equivalent to its orig- inal sound transformed by a head-related transfer function (HRT’ with a particular direction. Experiment 2: To evaluate the influence of the HRTF, all the test data were converted to binaural sounds as follows, and then recognized by the HMM-m. Perception 1085 6 0’ 90.0. B J m.0. % - !l o&o. c 9 80.0. g 75.0 - CJ 70.0 - 0‘50 - 00.0 - 55.0 - 50.0 - 40.0 - 40.0 - 35.0 - Jo.0 - 25,ol ’ ’ 8 ’ ’ ’ ’ ’ c I 0 12 3 4 5 0 7 0 0 ORboER Figure 4: Influence of the Head-related Transfer Function (Experiment 2) HRTFs in four directions (0”) 30”) 60”) and 90’) 2 were applied to each test utterance to generate a binaural sound. For each binaural sound, the monaural sound was ex- tracted from the channel with the larger power, in this case, the left channel. The power level was adjusted so that its average power was equivalent to that of the original sound. This operation is called power adjustment. The resulting monaural sounds (the HRTF’ed test data) were given to the HMM-m for word recognition. The cumulative recognition accuracy for the HRTF’ed test data is shown in Fig. 4. The original data is also shown for comparison. The decrease in the cumulative accuracy for the 10th candidate ranged from 11.4% to 30.1%. The degradation depended on the direction of the sound source and was the largest for 30” and the smallest for 90”. Recovering the Performance Degradation caused by the HRTF Two methods to recover the decrease in recognition accu- racy caused by HRTF have been tried: 1. Re-training the HMM-LR parameters with the HTRF’ed training data, and 2. Correcting the frequency characteristics of the HRTF. Re-training of the parameters of the HMM-LR We converted the training data for the HMM-LR parameters by applying the HRTF in the four directions to the training data with power adjustment. We refer to the re-trained HMM-LR for male speakers as the HMM-hrtfm. The cumulative recognition accuracy of the HRTF’ed test data by the HMM-hrtf-m is shown in Fig. 5. The decrease in the cumulative accuracy was significantly reduced and *The angle is calculated counterclockwise from the center, and thus O”, 90°, and -90” mean the center, the leftmost and the rightmost, respectively. 0 12 3 4 5 6 7 6 9 *l&R Figure 5: Recovery by Re-trained HMM-LR Original Data under HMMm 0 0 &gmo under HMM-m 0-e 36 degree under HMh+m e- - -0 60 dogma under HMM-m A-A 90 &groa under HMM-m 65.01 a ’ 8 ’ 9 a a ( I 0 1 2 3 4 5 6 7 6 9 OR%? Figure 6: Recovery by Correcting the F-char of HRTF almost vanishes for 90”. However, the degradation still depended on the direction of the sound source. Frequency Characteristics (F-Char) Correction The effect of the HRTF is to amplify the higher frequency region while attenuating the lower frequency region. For example, the Japanese word “aji” (taste) sounds like “ashi” (foot) if an HRTF of any degree is applied. To recover the spectral distortion caused by the HRTF, we corrected the frequency characteristics (F-Char) of the HRTR’ed test data through power adjustment. After this correction, the test data were recognized by the HMM-m (Fig. 6). The variance in the recognition rate due to different directions was resolved, but the overall improvement was not as great as with the HMM-hrtf-m. Since the latter method requires a precise determination of the directions, though, it cannot be used when the sound source is moving. In addition, the size of HRTF data for the various directions is very large and its spatial and computational cost is significant. Therefore, we used the HMM-hrtf-m/f to recognize binaural data. Influence of the Harmonic Grouping and Residue Substitution Experiment 3: The influence of harmonic grouping by the F-grouping, D-grouping, and B-grouping was evaluated by the following method: 1. The Bi-HBSS extracted harmonic stream fragments from binaural input in four directions (0”) 30”) 60”) and 90’) for a man’s utterance. 2. Sound stream fragments were grouped into a stream by one of the three groupings and the non-harmonic components of the stream are filled in through the all- residue substitution. 1086 Perception c $ 95.0 2 5 go.0 8 q L 85.0 = s 80.0 P 75.0 03 70.0 6.50 W.0 550 .---. sod,grN .---I wdqpw t 0 o 0 hgw wllh F-Gtvuplng 0 .a sodoww Figure 7: Influence of Harmonic Grouping (Experiment 3) c & 05.0 - L 3 w.o- 8 q k? 65.0- E 1 Lao- 75.0 - 70.0 - 65.0 - 60.0 - 55.0 - Figure 8: Influence of Residue Substitution (Experiment 4) 3. Power adjustment was applied to the segregated sound streams. 4. The resulting sounds were recognized with the HMM- hrtf-m. The recognition rate is shown in Fig. 7. The best performance was with the D-grouping, while the worst was with the F-grouping. The recognition with the F-grouping was poor because only the previous state of the fundamental frequency was used to group stream fragments. This also led to poor performance with the B-grouping. Longer temporal characteristics of a fundamental frequency should be exploited, but this remains for future work. Therefore, we adopted the D-grouping for the experiments described in the remainder of this paper. Experiment 4: We evaluated the effect of residue sub- stitution by either all-residue substitution or own-residue substitution in the same way as Experiment 3. The resulting recognition rates are shown in Fig. 8. The recognition rate was higher with the all-residue substitution than with the own-residue substitution. This is partially because the sig- nals substituted by the own-residue were weaker than those by the all-residue. Therefore, we will use the all-residue substitution throughout the remainder of this paper. Experiments on Listening to a Sound Mixture Our assessment of the effect of segregation on ASR suggests that we should use Bi-HBSS with the D-grouping and the all-residue substitution and that segregated speech streams should be recognized by the HMM-hrtf-m/f. We also evaluated monaural segregation by HBSS with the all- residue substitution and the HMM-m/f. The experiments on recognizing a mixture of sounds were done under the following conditions: The first speaker is 30’ to the left of the center and utters a word first. The second speaker is 30” to the right of the center and utters a word 150 msec after the first speaker. There were 500 two-word testing combinations. The power adjustment was not applied to any segregated sound, because the system cannot determine the original sound that corresponds to a segregated sound. The utterance of the second speaker was delayed by 150 msec because the mixture of sounds was to be recognized directly by the HMM-m/f. Note that the actual first utterance is sometimes done by the second speaker. Listening to Two Sounds at the Same Time Since the HMM-LR framework we used is gender- dependent, the following three benchmarks were used (see Table 1). The cumulative accuracies of recognition of the original data for Woman 1, Woman 2, Man 1, and Man 2 by the HMM-m/f were 94.19%, 95.10%, 94.99%, and 96.10%, respectively. The recognition rate was measured without segregation, with segregation by HBSS, and with segregation by Bi- HBSS. The recognition performance in terms of cumulative accuracy up to the 10th candidate is summarized in Tables 2 to 4. The recognition performance of speech segregated by Bi-HBSS was better than when HBSS was used. With Bi-HBSS, the decrease in the recognition rate of the second woman’s utterance from that of the original sound was 21.20%. Since these utterances could not be recognized at all without segregation, the error rate was reduced by 75.60% on average by the segregation. Without segregation, the utterances of the first speaker could be recognized up to 37% if the label (the onset and offset times) was given by some means. In this experiment, the original labels created by human listeners at ATR were used. However, the recognition rate falls to almost zero when another sound is interfering (see the following experiments and Table 6 and 7). The Bi-HBSS reduces the recognition errors of HBSS by 48.1%, 22.7%, and 23.1% for benchmarks 1, 2, and 3, re- spectively. The improvement for benchmark 1 is especially large because the frequency region of women’s utterances is so narrow that their recognition is prone to recognition errors. Men’s utterances, in particular, the second man’s ut- terances of benchmark 3, are not well segregated by HBSS or Bi-HBSS. The fundamental frequency (pitch) of the sec- ond man is less than 100 Hz while that of the first man is Perception 1087 Table 1: Benchmark sounds l-3 c Table 2: Recognition Rate of Benchmark 1 Table 3: Recognition Rate of Benchmark 2 Table 4: Recognition Rate of Benchmark 3 about 110 Hz. A sound of lower fundamental frequency is in general more difficult to segregate. Listening to Three Sounds at the Same Time Our next experiment was to segregate speech streams from a mixture of three sounds. Two benchmarks were com- posed by adding an intermittent sound to the sounds of benchmark 1 (see Table 5). The intermittent sound was a harmonic sound with a 250 Hz fundamental frequency that was repeated for 1,000 msec at 50 msec intervals. Its di- rection was O”, that is, from the center. The signal-to-noise ratio (SNR) of the woman’s utterance to the intermittent sound was 1.7 dB and - 1.3 dB, respectively, for benchmark 4 and 5. The actual SNR was further reduced, because the other woman’s utterance was also an interfering sound. The recognition performance in terms of 10th cumulative accuracy are summarized in Tables 6 and 7. The degradation with HBSS and Bi-HBSS caused by the intermittent sound of benchmark 4 was 7.9% and 23.3%, respectively. When the power of the intermittent sound was amplified and the SNR of the woman’s utterances decreased by 3 dB as in benchmark 5, the additional degradation with HBSS and Bi-HBSS was 1.5% and 5.8%, respectively. Segregation by either HBSS or Bi-HBSS seems rather robust against an increase in the power level of interfering sounds. Discussion and Future work In this paper, we have described our experiments on the Prince Shotoku effect, or listening to several sounds simulta- Table 5: Benchmark sounds 4-5 Table 6: Recognition Rate of Benchmark 4 10th Cumulative Accurac Table 7: Recognition Rate of Benchmark 5 neously. We would like to make the following observations. (1) Most of the sound stream segregation systems devel- oped so far (Bodden 1993; Brown 1992; Cooke et al. 1993; Green et al. 1995; Ramalingam 1994) run in batch. How- ever, HBSS and Bi-HBSS systems run incrementally, which is expected to make them easier to run in real time. (2) Directional information can be extracted by binau- ral input (Blauert 1983; Bodden 1993) or by microphone arrays (Hansen et al. 1994; Stadler & Rabinowitz 1993). Our results prove the effectiveness of localization by us- ing a binaural input. However, this severely degrades the recognition rate due to spectral distortion; this has not been reported in the literature as far as we know. Therefore, we are currently engaged in designing a sophisticated mech- anism to integrate HBSS and Bi-HBSS to overcome the drawbacks caused by a binaural input. (3) The method to extract a speech with consonants is based on auditory induction, a psychacoustical observation. This method is considered as the first approximation for speech stream segregation, because it does not use any characteristics specific to human voices, e.g., formants. In addition, we should attempt to incorporate a wider set of the segregation and grouping phenomena of psychoacoustics such as common onset, offset, AM and FM modulations, formants, and localization such as elevation and azimuth. (4) In HMM-based speech recognition systems, the lead- ing part of a sound is very important to focus the search and if the leading part is missing, the recognition fails. Ex- amination of the recognition patterns shows that the latter part of a word or a component of a complex word is often clearly recognized, but this is still treated as failure. (5) Since a fragment of a word is more accurately segre- gated than the whole word, top-down processing is expected to play an important role in the recognition. Various meth- ods developed for speech understanding systems should be 1088 Perception incorporated to improve the recognition and understanding. (6) In this paper, we used standard discrete-type hidden Markov models for an initial assessment. However, HMM technologies have been improved in recent years, especially in terms of their robustness (Hansen et al. 1994, Minami & Furui 1995). The evaluation of our SSS in sophisticated HMM frameworks remains as future work. (7) Our approach is bottom-up, primarily because one goal of our research is to identify the capability and limita- tions of the bottom-up approach. However, the top-down approach is also needed for CASA, because a human lis- tener’s knowledge and experience plays an essential role in listening and understanding (Handel 1989). (8) To integrate bottom-up and top-down processes, sys- tem architecture is essential. The HBSS and Bi-HBSS systems are modeled on the residue-driven architecture with multi-agent systems. These systems can be extended for such integration by using subsumption architecture (Nakatani et al. 1994). A common system architecture for such integration is the black board architecture (Cooke et al. 1993; Lesser et al. 1993). The modeling of CASA represents an important area for future work. Conclusions This paper reported the preliminary results of experiments on listening to several sounds at once. We proposed the segregation of speech streams by extracting and grouping harmonic stream fragments while substituting the residue for non-harmonic components. Since the segregation sys- tem uses a binaural input, it can interface with the hidden Markov model-based speech recognition systems by con- verting the training data to binaural data. Experiments with 500 mixtures of two women’s utter- ances of a word showed that the 10th cumulative accuracy of speech recognition of each woman’s utterance is, on average, 75%. This performance was attained without us- ing any features specific to human voices. Therefore, this result should encourage the AI community to engage more actively in computational auditory scene analysis (CASA) and computer audition. In addition, because audition is more dependent on the listener’s knowledge and experience than vision, we believe that more attention should be paid to CASA in the research of Artificial Intelligence. Acknowledgments We thank Kunio Kashino, Masataka Goto, Norihiro Hagita and Ken’ichiro Ishii for their valuable discussions. References Blauert, J. 1983. Spatial Hearing: the Psychophysics of Human Sound Localization. MIT Press. Bodden, M. 1993. Modeling human sound-source localization and the cocktail-party-effect. Acta Acustica 1:43-55. Bregman, AS. 1990. Auditory Scene Analysis - the Perceptual Organization of Sound. MIT Press. Brown, G.J. 1992. Computational auditory scene analysis: A representational approach. Ph.D diss., Dept. of Computer Science, University of Sheffield. Cherry, E.C. 1953. Some experiments on the recognition of speech, with one and with two ears. Journal of Acoustic Society of America 25:975-979. Cooke, M.P.; Brown, G.J.; Crawford, M.; and Green, P. 1993. Computational Auditory Scene Analysis: listening to several things at once. Endeavour, 17(4): 186 190. Handel, S. 1989. Listening -An Introduction to the Perception of Auditory Events. MIT Press. Hansen, J.H.L.; Mammone, R.J.; and Young, S. 1994. Editorial for the special issue of the IEEE transactions on speech and audio processing on robust speech processing”. Transactions on Speech and Audio Processing 2(4):549-550. Green, P.D.; Cooke, Ml?; and Crawford, M.D. 1995. Auditory Scene Analysis and Hidden Markov Model Recognition of Speech in Noise. In Proceedings of 1995 International Conference on Acoustics, Speech and Signal Processing, Vol. 1: 40 l-404, IEEE. Kita, K.; Kawabata, T.; and Shikano, H. 1990. HMM continuous speech recognition using generalized LR parsing. Transactions of the Information Processing Society of Japan 3 1(3):472-480. Lesser, V.; Nawab, S.H.; Gallastegi, 1.; and Klassner, F. 1993. IPUS: An Architecture for Integrated Signal Processing and Signal Interpretation in Complex Environments. In Proceedings of the Eleventh National Conference on Artificial Intelligence, 249-255. Minami, Y, and Furui, S. 1995. A Maximum Likelihood Procedure for A Universal Adaptation Method based on HMM Composition. In Proceedings of 1995 International Conference on Acoustics, Speech and Signal Processing, Vol. 1: 129- 132, IEEE. Nakatani, T.; Okuno, H.G.; and Kawabata, T. 1994. Auditory Stream Segregation in Auditory Scene Analysis with a Multi- Agent System. In Proceedings of the Twelfth National Conference on Artificial Intelligence, lOO- 107, AAAI. Nakatani, T.; Kawabata, T.; and Okuno, H.G. 1995a. A compu- tational model of sound stream segregation with the multi-agent paradigm. In Proceedings of 1995 International Conference on Acoustics, Speech and Signal Processing, Vol.4267 l-2674, IEEE. Nakatani, T.; Okuno, H.G.; and Kawabata, T. 1995b. Residue- driven architecture for Computational Auditory Scene Analysis. In Proceedings of the th International Joint Conference on Artificial Intelligence, Vol.1: 165-172, IJCAI. Nakatani, T.; Goto, M.; and Okuno, H.G. 1996. Localization by harmonic structure and its application to harmonic sound stream segregation. In Proceedings of 1996 International Conference on Acoustics, Speech and Signal Processing, IEEE. Forthcoming. Okuno, H.G.; Nakatani, T.; and Kawabata, T. 1995. Cocktail- Party Effect with Computational Auditory Scene Analysis - Preliminary Report -. In Symbiosis of Human and Artifact - Proceedings of the Sixth International Conference on Human- Computer Interaction, Vo1.2:503-508, Elsevier Science B.V. Ramalingam, C.S., and Kumaresan, R. 1994. Voiced-speech analysis based on the residual interfering signal canceler (RISC) algorithm. In Proceedings of the International Conference on Acoustics, Speech, and Signal Processing, Vol.I:473-476, IEEE. Rosenthal, D., and Okuno, H.G. eidtors 1996. Auditory Scene Analysis, LEA. Forthcoming. Computational Stadler, R.W., and Rabinowitz, W.M. 1993. On the potential of fixed arrays for hearing aids. Journal of Acoustic Society of America 94(3) Pt.1: 1332-1342. Warren, R.M. 1970. Perceptual restoration of missing speech sounds. Science 167: 392-393. Perception 1089 | 1996 | 161 |
1,801 | Motion and Color Analysis for Ani Tamer F. Rabie and Demetri Terzopoulos Department of Computer Science, University of Toronto 10 King College Road, Toronto, Ontario, M5S 3G4, Canada e-mail: {tamerldt}@cs.toronto.edu Abstract We propose novel gaze control algorithms for active percep- tion in mobile autonomous agents with directable, foveated vision sensors. Our agents are realistic artificial animals, or animats, situated in physics-based virtual worlds. Their active perception systems continuously analyze photoreal- istic retinal image streams to glean information useful for controlling the animat eyes and body. The vision system computes optical flow and segments moving targets in the low-resolution visual periphery. It then matches segmented targets against mental models of colored objects of interest. The eyes saccade to increase acuity by foveating objects. The resulting sensorimotor control loop supports complex behaviors, such as predation. Introduction Animals are active observers of their environment (Gibson 1979). This fact has inspired a trend in the computer vi- sion field popularly known as “ active vision” (Bajcsy 1988; Ballard 1991; Swain & Stricker 1993). Unfortunately, ef- forts to create active vision systems for physical robots have been hampered by hardware and processor limita- tions. The recently proposed animat vision paradigm (Ter- zopoulos & Rabie 1995) offers an approach to developing biomimetic active vision systems that does not rely on robot hardware. Instead of physical robots, animat vision pre- scribes the use of virtual robots that take the form of arti- ficial animals, or animats, situated in physics-based virtual worlds. Animats are autonomous virtual agents possess- ing mobile, muscle-actuated bodies and brains with motor, perception, behavior and learning centers. In the percep- tion center of the animat ’ brain, computer vision algo- rithms continually analyze the incoming perceptual infor- mation. Based on this analysis, the behavior center dis- patches motor commands to the animat ’ body, thus form- ing a complete sensorimotor control system. Animat vision, implemented entirely in software, has several important ad- vantages over conventional “ hardware vision”, at least for research purposes (refer to (Terzopoulos & Rabie 1995; Terzopoulos 1995) for a discussion). In many biological eyes, the high-acuity foveacovers only a small fraction of a visual field whose resolution decreases monotonically towards the periphery. Spatially nonuniform retinal imaging provides opportunities for increased compu- 1090 Perception Figure 1: Artificial fishes swimming among aquatic plants in a physics-based virtual marine environment. tational efficiency through economization of photoreceptors and focus of attention, but it forces the visual system to solve problems that do not generally arise with a uniform field of view. A key problem is determining where to redirect the fovea when a target of interest appears in the periphery. In this paper we present a solution to this problem through the exploitation of motion and color information. Motion and color play an important role in animal per- ception. Birds and insects exploit optical flow for obstacle avoidance and to control their ego-motion (Gibson 1979). Some species of fish are able to recognize the color signa- tures of other fish and use this information in certain piscene behaviors (Adler 1975). The human visual system is highly sensitive to motion and color. We tend to focus our attention on moving colorful objects. Motionless objects whose col- ors blend in to the background are not as easily detectable, and several camouflage strategies in the animal kingdom rely on this fact (Cedras & Shah 1995). Following the animat vision paradigm, the motion and color based gaze control algorithms that we propose in this paper are implemented and evaluated within artificial fishes in a virtual marine world (Fig. 1). The fish animats are the result of research in the domain of artificial life (see (Ter- zopoulos, Tu, & Grzeszczuk 1994) for the details). In the present work, the fish animat serves as an autonomous mo- bile robot situated in a photorealistic, dynamic environment. From: AAAI-96 Proceedings. Copyright © 1996, AAAI (www.aaai.org). All rights reserved. Our new gaze control algorithms significantly enhance the prototype animat vision system that we implemented in prior work (Terzopoulos & Rabie 1995) and they support more robust vision-guidednavigation abilities in the artificial fish. We review the animat vision system in the next section be- fore presenting our new work on integrating motion and color analysis for animat perception in subsequent sections. A Prototype Animat Vision System The basic functionality of the animat vision system, which is described in detail in (Terzopoulos & Rabie 1995), starts with binocular perspective projection of the color 3D world onto the animat’s 2D retinas. Retinal imaging is accom- plished by photorealistic graphics rendering of the world from the animat’s point of view. This projection respects occlusion relationships among objects. It forms spatially variant visual fields with high resolution foveas and progres- sively lower resolution peripheries. Based on an analysis of the incoming color retinal image stream, the visual center of the animat’s brain supplies saccade control signals to its eyes to stabilize the visual fields during locomotion, to attend to interesting targets based on color, and to keep moving tar- gets fixated. The artificial fish is thus able to approach and track other artificial fishes visually. Fig. 2 provides a block diagram of the active vision system showing two main mod- ules that control retinal image stabilization and foveation of the eyes. Eyes and Retinal Imaging The artificial fish has binocular vision. The movements of each eye are controlled through two gaze angles (6,4) which specify the horizontal and vertical rotation of the eyeball, respectively. The angles are given with respect to the head coordinate frame, such that the eye is looking straight ahead when 0 = 4 = 0’. Each eye is implemented as four coaxial virtual cameras to approximate the spatially nonuniform, foveal/peripheral imaging capabilities typical of biological eyes. Fig. 3(a) shows an example of the 64 x 64 images that are rendered by the coaxial cameras in each eye (rendering employs the GL library and graphics pipeline on Silicon Graphics work- stations). The level 1 = 0 camera has the widest field of view (about 120’) and the lowest resolution. The resolution increases and the field of view decreases with increasing 1. The highest resolution image at level I = 3 is the fovea and the other images form the visual periphery. Fig. 3(b) shows the 5 12 x 5 12 binocular retinal images cornposited from the coaxial images at the top of the figure. To reveal the retinal image structure in the figure, we have placed a white border around each magnified component image. Vi- sion algorithms which process the four 64 x 64 component images are 16 times more efficient than those that process a uniform 5 12 x 5 12 retinal image. Foveation by Color Object Detection The brain of the artificial fish stores a set of color models of objects that are of interest to it. For instance, if the fish is by habit a predator, it would possess models of prey fish. --s 0 --I Model II I=0 8 I I- #+ c Stabilization Module Foveation Module Figure 2: The animat vision system. The flow of the gaze control algorithm is from right to left. A: Update gaze angles (0,+) and saccade using these angles, B: Search current level for model target and if found localize it, else search lower level, C: Select level to be processed (see text), F: Reduce field of view for next level and render, M: Compute a general translational displacement vector (u, v) between images I(t - 1) and I(t), S: Scale the color histogram of the model for use by the current level. The mental models are stored as a list of 64 x 64 RGB color images. To detect and localize any target that may be imaged in the low resolution periphery of its retinas, the animat vision system of the fish employs an improved version of a color indexing algorithm proposed by Swain (Swain & Ballard 1991).’ Since each model object has a unique color his- togram signature, it can be detected in the retinal image by histogram intersection and localized by histogram backpro- jection. Saccadic Eye Movements When a target is detected in the visual periphery, the eyes will saccade to the angular offset of the object to bring it within the fovea. With the object in the high resolution fovea, a more accurate foveation is obtained by a second pass of histogram backprojection. A second saccade typically centers the object accurately in both left and right foveas, thus achieving vergence. Module A in Fig. 2 performs the saccades by incrementing ‘Our improvements, which include iterative model histogram scaling and weighted histograms, make the technique much more robust against the large variations in scale that occur in our ap- plication. The details of the improved algorithm are presented in (Terzopoulos & Rabie 1995). Perception 1091 _-.___----..- _-_. -- _ I=0 l=l 1=2 1=3 l=O I=1 1=2 1=3== (4 Left eye (W Right eye Figure 3: Binocular retinal imaging (monochrome versions of original color images). (a) 4 component images; I = 0, 1,2, are peripheral images; I = 3 is fovea1 image. (b) Cornposited retinal images (borders of cornposited component images are shown in white). the gaze angles (6,4) required gaze direction. in order to rotate the eyes to the Visual Field Stabilization using Optical Flow It is necessary to stabilize the visual field of the artificial fish because its body undulates as it swims. Once a target is verged in both foveas, the stabilization process (Fig. 2) assumes the task of keeping the target foveated during lo- comotion. Stabilization is achieved by computing the overall transla- tional displacement (u, V) of intensities between the current fovea1 image and that from the previous time instant, and updating the gaze angles to compensate. The displacement is computed as a translational offset in the retinotopic coor- dinate system by a least squares minimization of the optical ff ow between image frames at times t and t - 1 (Horn 1986). The optical flow stabilization method is robust only for small displacements between frames. Consequently, when the displacement of the target between frames is large enough that the method is likely to produce bad estimates, the foveation module is invoked to re-detect and re-foveate the target as described earlier. Each eye is controlled independently during foveation and stabilization of a target. Hence, the two retinal images must be correlated to keep them verged accurately on the target. Referring to Fig. 4, the vergence angle is 0~ = (0~ - 0~) and its magnitude increases as the fish comes closer to the target. Therefore, once the eyes are verged on a target, it is Fixation I- Left Eye I Right Eye Figure 4: Gaze angles and range to target geometry. straightforward for the vision system to the target from the gaze angles. to estimate the range Vision-Guided Navigation The artificial fish can also employ the gaze direction (i.e., the gaze angles) while the eyes are fixated on a target to navigate towards the target. The t9 angles are used to com- pute the left/right turn angle 0~ shown in Fig. 4, and the 4 angles are similarly used to compute an up/down turn angle c$p. The fish turn motor controllers are invoked to exe- cute a left/right turn- right-turn-MC for an above-threshold positive t!?p and left-turn-MC for negative BP-with IBp 1 as parameter. Up/down turn motor commands are issued to the fish pectoral fins, with an above-threshold positive 1092 Perception where (u, V) is the computed optical flow and q5p interpreted as “up” and negative as “down”. The motor controllers are explained in (Terzopoulos, Tu, & Grzeszczuk 1994). on The remainder of integrating color and the paper presents motion analysis in our new work active vision. Integrating Motion and Color for Attention Selective attention is an important mechanism for dealing with the combinatorial aspects of search in vision (Tsotsos et al. 1995). Deciding where to redirect the fovea can involve a complex search process (Tsotsos et al. 1995; Rimey & Brown 1992; Maver & Bajcsy 1990). In this section we offer an efficient solution which integrates motion and color to increase the robustness of our animat’s perceptual functions. Motion and color have been considered extensively in the literature in a variety of passive vision systems, but rarely have they been integrated for use in dynamic perception systems. The conjunction of color and motion cues has recently been exploited to produce more exact segmenta- tions and for the extraction of object contours from natural scenes (Dubuisson & Jain 1993). Color and motion fea- tures of video images have been used for color video image classification and understanding (Gong & Sakauchi 1992). Integrating motion and color for object recognition can improve the robustness of moving colored object recogni- tion. Motion may be considered a bottom-up alerting cue, while color can be used as a top-down cue for model-based recognition (Swain, Kahn, & Ballard 1992). Therefore, in- tegrating motion and color can increase the robustness of the recognition problem by bridging the gap between bottom- up and top-down processes, thus, improving the selective attention of dynamic perceptual systems such as the animat vision system that we are developing. Where to Look Next Redirecting gaze when a target of interest appears in the periphery can be a complex problem. One solution would be to section the peripheral image into smaller patches or focal probes (Burt et al. 1989) and search of all the probes. The strategy will work well for sufficiently small images, but for dynamic vision systems that must process natural or photorealistic images the approach is not effective. We choose a simple method based on motion cues to help narrow down the search for a suitable gaze direction (Cam- pani, Giachetti, & Torre 1995). We create a saliency image by initially computing a reduced optical flow field between two stabilized peripheral image frames (an advantage of the multiresolution retina is the small 64 x 64 peripheral im- age). Then an affine motion model is fitted to the optical flow using a robust regression method that will be described momentarily. The affine motion parameters are fitted to the dominant background motion. A saliency map is de- termined by computing an error measure between the affine motion parameters and the estimated optical flow as follows: %4x, Y) = a + bx + cy, %h Y) = d+ex+ffy (2) is the affine flow at retinal image position (x, y). The saliency image S is then convolved with a circular disk of area equal to the expected area of the model object of interest as it appears in the peripheral image.2 The blurring of the saliency image emphasizes the model object in the image. The maximum in S is taken as the location of the image probe. The image patches that serve as probes in consecutive peripheral frames form the image sequence that is processed by the motion segmentation mod- ule described later. Fig. 5 shows four consecutive peripheral images with the image probes outlined by white boxes. The blurred saliency image is shown at the end of the sequence in Fig. 5. Clearly the maximum (brightness) corresponds to the fast moving blue fish in the lower right portion of the peripheral image. Robust Optical Flow A key component of the selective attention algorithm is the use of optical flow. Given a sequence of time-varying images, points on the retina appear to move because of the relative motion between the eye and objects in the scene (Gibson 1979). The vector field of this apparent motion is usually called optical flow (Horn 1986). Optical flow can give important information about the spatial arrangement of objects viewed and the rate of change of this arrangement. For our specific application, however, we require effi- ciency, robustness to outliers, and an optical flow estimate at all times. Recent work by Black and Anandan (Black & Anandan 1990; 1993) satisfies our requirements. They propose incremental minimization approaches using robust statistics for the estimation of optical flow which are geared towards dynamic environments. As is noted by Black, the goal is incrementally to integrate motion information from new images with previous optical flow estimates to obtain more accurate information about the motion in the scene over time. A detailed description of this method can be found in (Black 1992). Here we describe our adaptation of the algorithm to the animat vision system. Ideally optical flow is computed continuously” as the ani- mat navigates in its world, but to reduce computational cost and to allow for new scene features to appear when no in- teresting objects have attracted the attention of the animat, we choose to update the current estimate of the optical flow every four frames. The algorithm is however still “con- tinuous” because it computes the current estimate of the optical flow at time t using image frames at t-3, t-2, t-l, and t in a short-time batch process. Fig. 6 shows this more 2Reasonably small areas suffice, since objects in the 64 x 64 peripheral image are typically small at peripheral resolution. Methods for estimating appropriate areas for the object, such as Jagersand’s information theoretic approach (Jagersand 1995), may be applicable. 3By continuously, we mean that there is an estimate of the optical flow at every time instant. Perception 1093 283 284 285 286 Saliency Image Figure 5: Four consecutive peripheral images with image probes outlined by white squares. Saliency image (right), with bright areas indicating large motions. w UPDATE ROF ROF(k4) Figure 6: Incremental estimation of robust optical flow over time. clearly. This arrangement requires storage of the previous three frames for use by the estimation module. The flow at t + 1 is initialized with a predicted flow computed by forward warp of the flow estimate at t by itself4 and then the optical flow at t + 4 is estimated by spatiotemporal regression over the four frames. We compute our optical flow estimate by incrementally minimizing the cost function E(u, v) = XDED(% q+k&+, V)+~TJ%(U, 4, (3) where EEL> is the data conservation constraint and is given in terms of the intensity constraint equation as ED = Pa&-& + VIP + b), (4) and Es is the spatial coherence constraint and is given as Es = x Epus(u-u(m,n))+po,(v-v(m,n))l, (5) m,nEN where N is the 4-connected neighbors of the current pixel position. We formulate our temporal continuity constraint ET by imposing some coherence between the current flow estimate and its previous and next estimate: ET =PuT(U-UBW)+puT(U-UFW), (6) where u = (u, V) is the current optical flow estimate at time t, UBW is the previous estimate at t - 1 obtained by setting it to the most recent estimate, and UFW is a prediction of what the optical flow will be at t + 1 and is computed by forward warp of the current estimate by itself.5 The X parameters in (3) control the relative importance of the terms, and the pa functions in the above equations are taken to be the Lorentzian robust estimator: Pu(4 = hl (1 + ; (x, ) (7) and its influence function, $a (2)) is the first derivative with respect to 2. This function characterizes the bias that a particular measurement has on the solution (Hampel 1974; Black & Anandan 1993). This robust formulation of our cost function E causes it to be non-convex. A local minimum can, however, be obtained using a gradient-based optimization technique. We choose the successive over relaxation minimization technique. The iterative equations for minimizing E are i+1- i u - I-J dE u -T,m (8) where 1 < ~1 < 2 is an over-relaxation parameter that con- trols convergence. A similar iterative equation for v is obtained by replacing u with v in (8). The terms T,, TV are upper bounds on the second partial derivatives of E, and can be given as T, = (9) and similarly for T.. by replacing u with v and x with v. The partial derivative in (8) is dE - = ~,~, + VIy + It)+ dU As c 1clus(u - U(T 4) + m,nEN AT [&T (u - WV) + ‘ - WV)], (10) and similarly for dE/dv. The above minimization will generally converge to a local minimum. A global minimum may be found by construct- ing an initially convex approximation to the cost function 4The flow estimate is being used to warp itself, thus predicting what the motion will be in the future. 5Note that UBW can also be estimated by backward warping of u by itself. 1094 Perception COLOR OBJECTMODELS Figure 7: The robust optical flow vectors estimated for the four image probe sequence (Fig. 5). Large vectors indicate large motion of the fish object. by choosing initial values of the 0 parameters to be suf- ficiently large (equal to the maximum expected outlier in the argument of pi (*)), so that the Hessian matrix of E is positive definite at all points in the image. The minimum is then tracked using the graduated non-convexity (GNC) con- tinuation method (Blake & Zisserman 1987) by decreasing the values of the 0 parameters from one iteration to the next, which serves to gradually return the cost function to its non-convex shape, thereby introducing discontinuities in the data, spatial, and temporal terms. These discontinuities are, however, dealt with by the robust formulation and are rejected as outliers, thus producing more accurate optical flow estimates. The values of the X parameters are deter- mined empirically (typically AD = 10, As = XT = 1). To deal with large motions in the image sequence, we per- form the minimization using a coarse-to-fine flow-through strategy. A Gaussian pyramid (Burt & Adelson 1983) is constructed for each image in the sequence, and minimiza- tion starts at the coarsest level and flows through to the finest resolution level. Our flow-through technique is based on the assumption that displacements which are less than 1 pixel are estimated accurately at each individual level and thus need not be updated from a coarser level’s estimate, while estimates that are greater than 1 pixel are most prob- ably more accurately computed at the coarser level, and are updated by projecting the estimate from the coarser level. This incremental minimization approach foregoes a large number of relaxation iterations over a 2 frame sequence in favor of a small number of relaxation iterations over a longer sequence. Fig. 7 shows the optical flow estimated for the sequence of four image probes of Fig. 5. The figure clearly shows the complex motion of the target fish. It is a non-trivial task to segment such motions. Motion Segmentation and Color Recognition For the animat to recognize objects moving in its periphery it must first detect their presence by means of a saliency map as described earlier. Once it detects something that might be worth looking at, it must then segment its region of support out from the whole peripheral image and then match this segmentation with mental models of important ROF I*; Figure 8: Incremental motion segmentation and object recognition using multi-resolution robust optical flow (ROF) estimation, affine parametric motion segmentation and color object recognition. objects. Fig. 8 shows the steps involved in an incremental segmentation of the detected object over the duration of the four probe images as explained above. Segmentation of the optical flow at each time instant is performed by fitting an affine parametric motion model to the robust optical flow (ROF) estimated so far at the current time instant. This is done by incrementally minimizing the cost function given as E(a, h c, 4 e, f) = Ex(a, h c) + E,(d) e, f), (11) where (a, b, c, d, e, f) are the affine motion parameters. Ez and E, are formulated using robust estimation to account for outliers E, = c p&x(x, Y) - 4x, Y)L xc,yER E, = x P&y(X,Y) - V(X,Y)), (12) x,YER where R is the current region of support of the segmented object (initially equal to the full frame image size). v, and vy are horizontal and vertical affine motion flow vectors according to (2). ( u, v) is the ROF estimated at the cur- rent instant, and p0 (x) is taken to be the Lorentzian robust estimator. We use successive over relaxation and GNC to minimize this cost function by using a small number of it- erations over a sequence of four image probes and updating the segmentation at every time instant. The estimated affine motion parameters at the current time instant are then used to update the segmentation by calculating an error norm between the affine flow estimate (vx, vY) and the ROF estimate as in (1). This norm is then thresholded by an appropriate threshold taken to be the minimum outlier in the affine fit. The updated segmentation serves as the region of support R for the next frame’s affine minimization step. Perception 1095 If more than one moving object is present in the probe sequence, the current segmentation is subtracted from the image, and another affine motion model is fitted to the re- maining pixels thus segmenting other moving objects. To clean up the segmentation (in case some pixels where mis- classified as outliers) a 9 x 9 median filter is passed over the segmentation mask to fill in missing pixels and remove mis- classified outliers. Fig. 9 shows the segmented background (showing two objects as outliers) and the segmentation of the outlier pixels into the object of interest (a blue fish). At the end of the motion segmentation stage, the seg- mented objects are matched to color models using the color histogram intersection method. If a match occurs, the cur- rent estimate of the ROF is set to zero thus accounting for the dynamic changes in the system, otherwise the ROF is used to initialize the optical flow at the next time step as Segmented Background Segmented Object Figure 9: Results of incremental motion segmentation mod- ule. shown in Fig. 6. If the model object matches the peripheral segmented re- gion, the animat localizes the recognized object using color histogram backprojection and foveates it to obtain a high- resolution view. It then engages in appropriate behavioral responses. Behavioral Response to a Recognized Target physics-based, virtual marine world inhabited by lifelike artificial fishes that emulate the appearance, motion, and behavior of real fishes in their natural habitats. We have successfully implemented a set of active vision algorithms for artificial fishes that integrate motion and color analy- sis to improve focus of attention and enable the animat to better understand and interact with its dynamic virtual en- vironment. The behavioral center of the brain of the artificial animal assumes control after an object is recognized and fixated. If the object is classified as food the behavioral response would be to pursue the target in the fovea with maximum speed until the animat is close enough to open its mouth and eat the food. If the object is classified as a predator and the animat is a prey fish, then the behavioral response would be to turn in a direction opposite to that of the predator and swim with maximum speed. Alternatively, an object in the scene may serve as a visual frame of reference. When the animat recognizes a reference object (which may be another fish) in its visual periphery, it will fixate on it and track it in smooth pursuit at an intermediate speed. Thus, the fixation point acts as the origin of an object-centered reference frame allowing the animat to stabilize its visual world and explore its surroundings. Fig. 10 shows a sequence of retinal images taken from the animat left eye. The eyes are initially fixated on a red reference fish and thus the images are stabilized. In frame 283 to 286 a blue fish swims close by the animat right side. The animat recognizes this as a reference fish and thus saccades the eyes to foveate the fish. It tracks the fish around, thereby exploring its environment. By foveating different reference objects, the animat can explore different parts of its world. Fig. 11 shows a plot of the (6~) 0~) gaze angles and turn angle between frames 200 and 400. It is clear from the figure that the animat was first fixated on the red fish which was to the left of the animat (negative gaze angles), and at frame 286 and subsequent frames the animat is foveated on the blue fish which is to its right (positive gaze angles). Conclusion and Future Work We have presented computer vision research carried out within an animat vision framework which employs a 1096 Perception In future work we will endeavor to increase the arsenal of active vision algorithms to support the whole behavioral repertoire of artificial fishes. The animat approach allows us to do this step by step without compromising the com- plete functionality of the artificial fish. It is our hope that the vision system that we are developing will also provide insights relevant to the design of active vision systems for physical robotics. Acknowledgements We thank Xiaoyuan Tu for developing and implementing the artificial fish model, which made this research possible. The research described herein was supported by grants from the Natural Sciences and Engineering Research Council of Canada. References Adler, H. E. 1975. Fish Behavior: Why Fishes do What They Do. Neptune City, NJ: T.F.H Publications. Bajcsy, R. 1988. Active perception. Proceedings of the IEEE 76(8):996-1005. Ballard, D. 1991. Animate vision. Artificial Intelligence 48:57- 86. Black, WI., and Anandan, P. 1990. A model for the detec- tion of motion over time. In Proc. Inter. Con$ Computer Vision (lCCV 33-37. Black, M., and Anandan, P. 1993. A framework for the robust estimation of optical flow. In Proc. Intec Con5 Computer Vision (ICCV 23 l-236. Black, M. 1992. Robust incremental optical flow. Technical Re- port YALEU/DCS/RR-923, Yale University, Dept. of Computer Science. Blake, A., and Zisserman, A., eds. 1987. Visual Reconstruction. Cambridge, Massachusetts: The MIT Press. Burt, P., and Adelson, E. 1983. The laplacian pyramid as a com- pact image code. IEEE Trans. on Communications 31(4):532- 540. Figure 10: Retinal image sequence from the left eye of the predator (top) and overhead view (bottom) of the predator as it pursues a red reference fish (frames 283-285). A blue reference fish appears in the predator right periphery and is recognized, fixated, and tracked (frames 286-300). The white lines indicate the gaze direction. 6O.oil - ss.ocl - so.00 - 45.00 - 40.00 - 35.00 - 3o.M) - 25.00 - 20.00 - 15.00 - 10.00 - 5.00 - 0.00 - -5.00 - -10.00 - -15.00 - -20.00 -25.00 -30.00 -35.00 -40.00 i I I I I I I I frames 200.00 250.00 300.00 350.00 4cmMO . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Right Theta --__----______. Turn Command Figure 11: Gaze angles as the animat changes reference points at frame 286 from left (negative angles) to right (pos- itive angles). Burt, P.; Bergen, J.; Hingorani, R.; Kolczynski, R.; Lee, W.; Le- ung, A.; Lubin, J.; and Shvaytser, H. 1989. Object tracking with a moving camera: An application of dynamic motion analysis. Proc. IEEE Workshop Ksual Motion 2 - 12. Campani, M.; Giachetti, A.; and Torre, V. 1995. Optic flow and autonomous navigation. Perception 24:253-267. Cedras, C., and Shah, M. 1995. Motion-based recognition: A survey. Image and Vision Computing 13(2): 129-155. Dubuisson, M., and Jain, A. 1993. Object contour extraction using color and motion. In Proc. Computer Vision and Pattern Recognition Co& (CVPR ’ 471-476. Gibson, J. J. 1979. The Ecological Approach to Visual Perception. Boston, MA: Houghton Mifflin. Gong, Y., and Sakauchi, M. 1992. An object-oriented method for color video image classification using the color and motion features of video images. In 2nd. Intel: Conj on Automation, Robotics and Computer Vision. Hampel, F. 1974. The influence curve and its role in robust estimation. J. Amer. Statistical Association 69(346):383-393. Horn, B. K. P. 1986. Robot Vision. Cambridge, MA: MIT Press. Jagersand, M. 1995. Saliency maps and attention selection in scale and spatial coordinates: An information theoretic approach. In Proc. Inter. Conjf Computer Vision (ICCV 195-202. Maver, J., and Bajcsy, R. 1990. How to decide from from the first view where to look next. In Proc. DARPA Image Understanding Workshop. Rimey, R., and Brown, C. 1992. Where to look next using a bayes net: Incorporating geometric relations. In Proc. Euro. Co@ Computer Ksion (ECCV 542-550. Swain, M., and Ballard, D. 1991. Color indexing. Inter. J. Computer Ksion 7: 11 - 32. Swain, M., and Snicker, M. 1993. Promising directions in active vision. Inter. J. Computer Vision 1 l(2): 109 - 126. Swain, M.; Kahn, R.; and Ballard, D. 1992. Low resolution cues for guiding saccadic eye movements. In Proc. Inter. ConJ Computer Ksion (ICCV 737-740. Terzopoulos, D., and Rabie, T. 1995. Animat vision: Active vision in artificial animals. In Proc. Fifth Intel: ConJL: Computer Vision (ICCV 801- 808. Terzopoulos, D.; ‘ X.; and Grzeszczuk, R. 1994. Artifi- cial fishes: Autonomous locomotion, perception, behavior, and learning in a simulated physical world. Artijicial Life 1(4):327- 351. Terzopoulos, D. 1995. Modeling living systems for computer vision. In Proc. Int. Joint Con$ Artijcial Intelligence (IJCAI 1003-1013. Tsotsos, J.; Culhane, S.; Wai, W.; Lai, Y.; Davis, N.; and Nuflo, F. 1995. Modeling visual attention via selective tuning. ArtiJicial Intelligence 78:507-545. Perception 1097 | 1996 | 162 |
1,802 | ise and the C mmon Sense rmatic Sit r a Mobile Robot Murray Shanahan Department of Computer Science, Queen Mary and Westfield College, Mile End Road, London El 4NS, England. mps@dcs.qmw.ac.uk Abstract Any model of the world a robot constructs on the basis of its sensor data is necessarily both incomplete, due to the robot’s limited window on the world, and uncertain, due to sensor and motor noise. This paper supplies a logical account of sensor data assimilation in which such models are constructed through an abductive process which hypothesises the existence, locations, and shapes of objects. Noise is treated as a kind of non-determinism, and is dealt with by a consistency-based form of abduction. Introduction The aim of Cognitive Robotics is to design and build mobile robots based on the idea of logical representation [Lesperance, et al., 19941. By reinstating the ideals of the Shakey project [Nilsson, 19841, Cognitive Robotics has reinvigorated a research programme that has been largely dormant for the past twenty years. This has been made possible by recent advances in the field of common sense reasoning: formalisms now exist for reasoning about action which incorporate robust solutions to the frame problem, and which can represent a wide variety of phenomena, including concurrent action, non- deterministic action, and continuous change. The Cognitive Robotics approach is in marked contrast to that of Brooks and his followers. According to Brooks, work in the style of Shakey is flawed because, it [relies] on the assumption that a complete world model [can] be built internally and then manipulated. [Brooks, 199la] A complete model of the world is hard for a robot to construct because, as Brooks points out, the data delivered by sensors are not direct descriptions of the world as objects and their relationships [and] commands to actuators have very uncertain effects. [Brooks, 199lb] This sort of incompleteness and uncertainty is a feature of what McCarthy 119891 calls the common sense informatic situation, and is dealt with extremely well by 1098 Perception predicate logic. This paper supplies a logical account of the common sense informatic situation for a small mobile robot with very poor sensors, and thereby defends the Cognitive Robotics approach from arguments along the lines of the oneadvancedabove. The paper is organised as follows. Section 1 presents a generic formalism for reasoning about action and space. Section 2 applies this formalism to the mobile robot under consideration, and presents an abductive characterisation of sensor data assimilation. A more detailed presentation of the material in these two sections is to be found in [Shanahan, 1996b]. The next two sections focus on the issue of noise. Section 3 shows how noise can be considered as a form of non-determinism, and Section 4 shows how the abductive characterisation of Section 2 can be modified to handle this non-determinism. epresenting Action and Space The proposed framework is the product of three steps. 1. Develop a formalism for representing action, continuous change, space and shape. 2. Using this formalism, construct a theory of the robot’s interaction with the world. 3. Consider the process of sensor data assimilation as a form of abduction, following [Sham&an, 19891. This section concerns the first of these steps. For more details, see [Shanahan, 1996b]. To begin with, we have a formalism for reasoning about action and continuous change, based on the circumscriptive event calculus of [Shanahan, 1995b]. A many sorted language is assumed, with variables for jluents, actions (events), and time points. We have the following axioms, whose conjunction will be denoted CEC. Their main purpose is to constrain the predicate HoldsAt. HoldsAt(f,t) represents that fluent f holds at time t. HoldsAt(f,t) t Initially(f) A l Clipped(O,f,t) HoldsAt(f,t2) t Happens(a,tl) A Initiates(a,f,tl) A tl < t2 A 1 Clipped(t1 ,f,t2) (Ecu (EC2) From: AAAI-96 Proceedings. Copyright © 1996, AAAI (www.aaai.org). All rights reserved. 1 HoldsAt(f,t2) t (EC3) Happens(a,tl) A Terminates(a,f,tl) A tl < t2 A 1 Declipped(t 1 ,f,t2) Clipped(t 1 ,f,t2) t) (EC4) 3 a,t [Happens(a,t) A [Terminates(a,f,t) v Releases(a,f,t)] A tl < t A t < t2] Declipped(t1 ,f,t2) tj (EC% 3 a,t [Happens(a,t) A [Initiates(a,f,t) v Releases(a,f,t)] A tl < t A t < t2] HoldsAt@,t2) t Happens(a,tl) A Initiates(a,fl,tl) h tl < t2 A t2 = t 1 + d A Trajectory(f 1 ,t 1 ,f2,d) A 1 Clipped(t 1 ,f 1 ,t2) (EW A particular domain is described in terms of Initiates, Terminates, Releases, and Trajectory formulae. Initiates(a,f,t) represents that fluent f starts to hold after action a at time t. Conversely , Terminates(a,f,t) represents that f starts to not hold after action a at t. Releases(a,f,t) represents that fluent f is no longer subject to default persistence after action a at t. The Trajectory predicate is used to capture continuous change. Trajectory(f1 ,t,f2,d) represents that f2 holds at time t+d if fl starts to hold at time t. A particular narrative of events is represented in terms of Happens and Initially formulae. The formula Initially(f) represents that fluent f holds at time 0. Happens(a,t) represents that action a occurs at time t. In rough terms, if E is a domain description and N is a narrative description, then the frame problem is overcome by considering, CIRC[N ; Happens] A CIRC[E ; Initiates, Terminates, Releases] A CEC. However, care must be taken when domain constraints and triggered events are included. The former must be conjoined to CEC, while the latter are conjoined to N. A detailed account of this solution is to be found in [Shanahan, 1996a, Chapter 161. To construct a theory of the robot’s interaction with the world, a means of representing space and shape is required. Space is assumed to be Iw2. Objects occupy regions, which are open, path-connected subsets of Iw2. The following axioms will be required, defining the functions Disc, Distance, Bearing, and Line. The intended meaning of these functions should be obvious. p E Disc(z) t) Distance(p,(O,O)) < z (SPU Distance((xl,yl),(x2,y2)) = 4 (x1-~2)~ + (yl-~2)~ (Sp2) Bearing((xl,yl),(x2,y2)) =r t (SP3) z = Distance(( xl,y l), (x2,y2)) A z # 0 A x2-x 1 Sin(r) = - A Cos(r) = y2-yl Z z p E Line(p1 ,p2) # Bearing(pl,p) = Bearing(p1 ,p2) A Distance@ 1 ,p) I Distance(p1 ,p2) (SP4) Spatial occupancy is represented by the fluent Occupies. The term Occupies(w,g) represents that region g is the smallest region which covers all the space occupied by object w. Objects cannot overlap, and can only occupy one region at a time. [HoldsAt(Occupies(w,gl),t) A HoldsAt(Occupies(w,g2),t)] + gl = g2 HoldsAt(Occupies(w 1 ,g l),t) A (SP5) (SPY) HoldsAt(Occupies(w2,g2),t) A wl f w2 + ljP[PE glAPE 821 The Displace function will be used to capture the robot’s continuous motion through space. Displace(g,(x,y )) denotes the region obtained by displacing g by x units east and y units north. (xl,yl ) E Displace(g,(x2,y2)) t) (SP7) (xl-x2,yl-y2) E g For reasons set out in [Shanahan, 1995a], a means of default reasoning about spatial occupancy is required. This is done by minimising the predicate AbSpace in the presence of the following axiom. AbSpace(w) t Initially(Occupies(w,g)) (Sp8) Let 0 denote the conjunction of (Spl) to (Sp8). As we’ll see in the next section, 0 is included in a separate circumscription describing the initial situation, in which AbSpace is minimised. In the present context, this is a description of the initial locations and shapes of objects, in other words a map. obot’s Relationship to the Worlcl Shortly, the abductive process whereby maps are constructed out of the robot’s sensor data will be defined. First, a theory has to be constructed which captures the robot’s relationship to the world: the effects of its actions on the world, and the effect of the world on its sensors. This theory is constructed using the formalism of the previous section. The robot under consideration is based on the Rug Warrior described by Jones and Flynn [ 19931. This is a circular robot with two drive wheels. The three actions it can perform are to rotate by a given number of degrees, to start moving forwards, and to stop. It has two forward bump switches, which can detect collisions. First we have a pair of uniqueness-of-names axioms for the three fluents Occupies, Facing and Moving, and for the robot-performed actions Rotate, Go and Stop, and the triggered events Bump, Switch1 and Switch2. UNA[Occupies, Facing, Moving, @W Blocked, Touching] UNA[Rotate, Go, Stop, Bump, Switchl, Switch21 (B2) Next we have a Trajectory formula which describes the continuous variation in the Occupies fluent as the robot moves through space, and a collection of domain constraints. The robot moves one unit of distance in one unit of time. Blocked(w1 ,w2,r) means that object w 1 cannot move in direction r because it is obstructed by Perception 1099 object w2. Touching(w1 ,w2,p) means that objects wl and w2 are touching at point p. Trajectory(Moving,t,Occupies(Robot,g2),d) t- (B3) HoldsAt(Occupies(Robot,gl),t) A HoldsAt(Facing(r),t) A g2 = Displace(gl,( d.Sin(r),d.Cos(r))) HoldsAt(Facing(rl),t) A (B4) HoldsAt(Facing(r2),t) + rl=r2 HoldsAt(Blocked(w 1 ,w2,r),t) ti (B5) 3 gl,g2 [HoldsAt(Occupies(wl,gl),t) A HoldsAt(Occupies(w2,g2),t) A WlfW2A3Zl [Zl>oA!fZ2[Z2<Zl + gP[P E &’ p E Displace(g1 ,(z2.Sin(r),z2.Cos(r)))]]] HoldsAt(Touching(w 1 ,w2,p),t) f3 (B6) HoldsAt(Occupies(w 1 ,g l),t) A HoldsAt(Occupies(w2,g2),t) A wl # w2 A 3pl,p2 [p E Line(plp2) Ap#pl Ap#p2 A V p3 [ [p3 E Line(pl ,p) A p3 # p] * P3 E 811 A V p3 [[p3 E Line(p,p2) A p3 # p] + P3 E @II Let B denote the conjunction of CEC with Axioms (Bl) to (B6). Next we have a collection of Initiates, Terminates and Releases formulae. Initiates(Rotate(rl),Facing(rl+r2),t) t HoldsAt(Facing(r2),t) (El) Releases(Rotate(rl),Facing(r2),t) t HoldsAt(Facing(r2),t) A rl # 0 Initiates(Go,Moving,t) Releases(Go,Occupies(Robot,g),t) Terminates(a,Moving,t) t a=Stop va= Bump v a = Rotate(r) Initiates(a,Occupies(Robot,g),t) t @a (E3) WI (E5) (E6) [a = Stop v a = Bump] A HoldsAt(Occupies(Robot,g),t) Let E denote the conjunction of Axioms (El) to (E6). The final component of the theory describes the conditions under which Bump, Switch1 and Switch2 events are triggered. Happens(Bump,t) t (HI) [HoldsAt(Moving,t) v Happens(Go,t)] A HoldsAt(Facing(r),t) A HoldsAt(Blocked(Robot,w,r),t) Happens(Switch1 ,t) t ma Happens(Bump,t) A HoldsAt(Facing(r),t) A HoldsAt(Occupies(Robot,Displace(Disc(z),pl)),t) A HoldsAt(Touching(Robot,w,p2),t) A r-90 < Bearing(p 1 ,p2) < r+ 12 Happens(Switch2,t) t (H3) Happens(Bump,t) A HoldsAt(Facing(r),t) A HoldsAt(Occupies(Robot,Displace(Disc(z),pl)),t) A HoldsAt(Touching(Robot,w,p2),t) A r-l 2 < Bearing(p 1 ,p2) < r+90 We’re now in a position to supply an abductive characterisation of the task of sensor data assimilation, where the robot’s sensor data is a stream of Switch1 and Switch2 events. First we have a definition which permits the exclusion of certain anomalous explanations, by ensuring that only the sensor data the robot actually receives is abductively explained. Definition 2.1. COMP[ ‘I’] G&f [Happens(a,t) A [a = Switch1 v a = Switch211 + v [a=ar\t=z] (a J)E I- where F = {(a ,z) I Happens(a ,z) E Y }. cl Sensor data assimilation is the task of finding explanations of the sensor data in terms of hypothesised objects. Given an Initially formula Ml describing the initial location of the robot, a collection N2 of Happens formulae describing the robot’s actions, and a collection Y of Happens formulae describing the sensor data received by the robot, we’re interested in finding conjunctions M2 of formulae in which each conjunct has the form, 3 g [Initially(Occupies( 0,g)) A V p [p E g e Ill] where o is an object constant and II is any formula in which p is free, such that 0 A Ml A M2 is consistent and, CIRC[O A Ml A M2 ; AbSpace ; Initially] A CIRC[Nl A N2 ; Happens] A CIRC[E ; Initiates, Terminates, Releases] A B I= y A com[Y]. This definition is very liberal, and the full paper utilises a boundary-based representation of shape to render the space of possible explanations more manageable (see [Davis, 1990, Chapter 61). 3 Noise as Non-Determinism The hallmark of the common sense informatic situation for a mobile robot is incomplete and uncertain knowledge of a world of spatio-temporally located objects. Incompleteness is a consequence of the robot’s limited window on the world, and uncertainty results from noise in its sensors and actuators. This section deals with noise. Both noisy sensors and noisy actuators can be captured using non-determinism. (An alternative is to use probability [Bacchus, et al., 19951). Here we’ll only look at the uncertainty in the robot’s location that results from its noisy motors. The robot’s motors are “noisy” for various reasons. For example, the two wheels might rotate at slightly different speeds when the robot is trying to travel in a straight line, or the robot might be moving on a slope or a slippery surface. Motor noise of this kind can be captured using a non- deterministic Trajectory formula, such as the following replacement for Axiom (B3).’ Note that, while objects l The Rotate actio n could also have been made non-deterministic. 1100 Perception occupy open subsets of W2, regions of uncertainty are closed. 3 p [Trajectory(Moving,t, Occupies(Robot,Displace(g,p)),d) A Distance(p, (d.Sin(r),d.Cos(r))) I d.E] t HoldsAt(Occupies(Robot,g),t) A HoldsAt(Facing(r),t) (B7) In effect, Axiom (B7) constrains the robot’s location to be within an ever-expanding circle of uncertainty centred on the location it would be in if its motors weren’t noisy. The constant E determines the rate at which this circle grows. Axiom- (BS) below ensures that there are no discontinuities in the robot’s trajectory. Without this axiom the robot would be able to leap over any obstacle which didn’t completely cover the circle of uncertainty for its location. The term Abs(d) denotes the absolute value of d. Trajectory(f,t,Occupies(x,Displace(g,p l)),dl) + 038) Vz[z>O+ 3 d ‘d d2,p2 [[d2 > 0 A Abs(d2dl) < d A Trajectory(f,t,Occupies(x,Displace(g,p2)),d2)] + Distance(p 1 ,p2) < z]] Figure 1: The Robot Explores a Corner Figure 1 shows the robot exploring the corner of an obstacle. Figure 2 shows the evolution of the corresponding circle of uncertainty, highlighting the points where the robot changes direction. The Evolution of the Circle of Uncertai nty Figure 2: Figure 2 is somewhat misleading, however. Consider Figure 3. On the left, the evolution of the circle of uncertainty for the robot’s location is shown. In the middle, three potential locations are shown for the three changes of direction. Although these locations all fall within the relevant circles of uncertainty, the robot could never get to the third location from the second. This is because, as depicted on the right of the figure, in any given model the circle of uncertainty for the robot’s location at the end of a period of continuous motion can only be defined relative to its actual location at the start of that period. This can be verified by inspecting Axioms (B7) and (B8). The relative nature of the evolution of the circle of uncertainty means that the robot can acquire a detailed knowledge of some area Al of its environment, then move to another area A2 which is some distance from A 1, and acquire an equally detailed knowledge of A2. The accumulated uncertainty entails only that the robot is uncertain of where Al is relative to A2. This natural feature of the formalisation conforms with what we would intuitively expect given the robot’s informatic situation. Figure 3: The Circle of Uncertainty Is Relative Not Absolute In the presence of non-determinism, the abductive account of sensor data assimilation presented in Section 2 will not work. The next section presents a modified characterisation which overcomes the problem. 4 Non-Determinism and Abduction Non-determinism is a potential source of difficulty for the abductive approach to explanation. Even with a precise and complete description of the initial state of the world, including all its objects and their shapes, a non- deterministic theory incorporating a formula like (B7) will not yield the exact times at which collision events occur. Yet the sensor data to be assimilated has precise times attached to it. Fortunately we can recast the task of assimilating sensor data as a form of weak abduction so that it yields the required results. Intuitively what we want to capture is the fact that without the hypothesised objects, the sensor data could not have been received. This is analogous to the consistency-based approach to diagnosis proposed by Reiter [ 19871. Definition 4.1. Given, 0 the conjunction B of CEC with Axioms (Bl), (B2), and (B4) to (B8), e the conjunction E of Axioms (El) to (E6), e the conjunction 0 of Axioms (Spl) to (Sp8), Perception 1101 . a conjunction Ml of Initially formulae describing the initial locations, shapes, and orientations of known objects, including the robot itself, . the conjunction Nl of Axioms (Hl) to (H3), . a conjunction N2 of Happens formulae describing the robot’s actions, and . a conjunction Y of formulae of the form Happens(Switchl,z) or Happens(Switch2,2), an explanation of Y is a conjunction M2 of formulae in which each conjunct has the form, 3 g [Initially(Occupies( 0,g)) A V p [p E g t) II]] where u) is an object constant and II is any formula in which p is free, such that 0 A Ml A M2 is consistent, and, CIRC[O A Ml A M2 ; AbSpace ; Initially] A CIRC[Nl A N2 ; Happens] A CIRC[E ; Initiates, Terminates, Releases] A B If 1 [y A coh@[Y/]]. q There will, naturally, be many explanations for any given Y which meet this definition, even using the boundary-based representation of shape adopted in the full version of the paper. A standard way to treat multiple explanations in abductive knowledge assimilation is to adopt their disjunction [Shanahan, 1996a, Chapter 171. This has the effect of smothering any explanations which are stronger than necessary, such as those which postulate superfluous obstacles. The disjunction of all explanations of Y is the cautious explanation of Y. A variety of preference relations over explanations can also be introduced. For example, it might be reasonable to assume that nearby collision points indicate the presence of a single object. Such preference relations are a topic for further study. The following theorem establishes that the above definition of an explanation is equivalent to the deterministic specification offered in Section 2 when E is 0. Let B u be the conjunction of CEC with Axioms (Bl) to VW. Definition 4.2. A formula M is a complete spatial description if the region occupied by each object mentioned in M is the same in every model of, CIRC[O A M ; AbSpace ; Initially]. 0 Theorem 4.3. If E = 0 and Ml is a complete spatial description, then M2 is an explanation of Y if and only if 0 A Ml A M2 is consistent and, CIRC[O A Ml A M2 ; AbSpace ; Initially] A CIRC [Nl A N2 ; Happens] A CIRC[E ; Initiates, Terminates, Releases] A &jet k y A com[Y]. Proof. See full paper. cl To illustrate the new definition, suppose the robot behaves as illustrated in Figure 4. Let N2 be the conjunction of the following formulae, which represent the robot’s actions up to and including the time it bumps into obstacle A. R .obot 0 1 2 3 4 Figure 4: The Robot Collides with an Obstacle Happens(Go,O) (4.4) Happens(Stop,2.1) (4.5) Let Ml be the conjunction of the following formulae. Initially(Facing(0)) (4.6) Initially(Occupies(Robot, Displace(Disc(O.5),( 2,1)))) (4.7) Let M2 be the following formula. 3 g [Initially(Occupies(A,g)) A (4.8) v X,)’ [(XJ) E g t) 1 < X < 3 A 3.5 < J’ < 4.511 In the noise-free case, the robot would collide with A at time 2.0. However, let’s assume the collision takes place at time 2-l. Let Y be the conjunction of the following formulae. Happens(Switch1,2.1) (4.9) Happens(Switch2,2.1) (4.10) Let E be O-25. The following proposition says that M2 is indeed an explanation of Y according to the new definition. Proposition 4.11. CIRC [0 A Ml A M2 ; AbSpace ; Initially] A CIRC[Nl A N2 ; Happens] A CIRC[E ; Initiates, Terminates, Releases] A B # l[‘u A COMP[Y]J. Proof. See full paper. q Concluding Remarks A considerable amount of further work has been carried out, which is reported in the full version of the paper, but which it is only possible to present in outline here. Two further theorems have been established which characterise the abductive explanations defined above in terms which appeal more directly to the information available to any map-building algorithm which might be executed on board the robot. These theorems have been used to prove the correctness, with respect to the abductive specification given, of an algorithm for sensor data assimilation which constructs an occupancy array [Davis, 1990, Section 6.2.11. 1102 Perception This algorithm forms the core of an implementation in C, which runs on data acquired by the robot in the real world. Some preliminary experiments have been conducted in which the robot, under the control of a behaviour-based architecture [Brooks, 19861, explores an enclosure, and makes a record of its actions and sensor data for subsequent processing using the algorithm. In the paper accompanying his 1991 Computers and Thought Award Lecture, Brooks remarked that, [The field of Knowledge Representation] concentrates much of its energies on anomalies within formal systems which are never used for any practical task. [Brooks, :991a] The work presented in this paper and in [Shanahan, 1996b] should be construed as an answer to Brooks. According to the logical account given in this paper, a robot’s incoming sensor data is filtered through an abductive process based on a framework of innate concepts, namely space, time, and causality.1 The development of a rigorous, formal account of this process bridges the gap between theoretical work in Knowledge Representation and practical work in robotics, and opens up a great many possibilities for further research. The following three issues are particularly pressing. . The assimilation of sensor data from moving objects, such as humans, animals, or other robots. Movable obstacles should also be on the agenda. e The assimilation of richer sensor data than that supplied by the Rug Warrior’s simple bump switches. . The control of the robot via the model of the world it acquires through abduction. Future implementation is expected to adopt a logic programming approach. Existing work in the Cognitive Robotics vein is likely to be influential here [Lesperance, et al., 19941, [Kowalski, 19951, [Poole, 19951. Acknowledgements The inspiration for Cognitive Robotics comes from Ray Reiter and his colleagues at the University of Toronto. Thanks to Neelakantan Kartha and Rob Miller. The author is an EPSRC Advanced Research Fellow. References [Bacchus, et al., 19951 F.Bacchus, J.Y.Halpren, and H.J.Levesque, Reasoning about Noisy Sensors in the 1 This is somewhat reminiscent of Kant, according to whom, “the natural world as we know it . . . is thoroughly conditioned by [certain] features: our experience is essentially experience of a spatio-temporal world of law-governed objects conceived of as distinct from our temporally successive experiences of them” [Strawson, 1966, page 211. Situation Calculus, Proceedings IJCAI 95, pages 1933- 1940. [Brooks, 19861 R.A.Brooks, A Robust Layered Control System for a Mobile Robot, IEEE Journal of Robotics and Automation, ~012, no 1 (1986), pages 14-23. [Brooks, 1991a] R.A.Brooks, Intelligence Without Reason, Proceedings IJCAI 91, pages 569-595. [Brooks, 1991 b] R.A.Brooks, Artificial Life and Real Robots, Proceedings of the First European Conference on Artificial Life (1991), pages 3-10. [Davis, 19901 E.Davis, Representations of Commonsense Knowledge, Morgan Kaufmann (1990). [Jones & Flynn, 19931 J.L.Jones and A.M.Flynn, Mobile Robots: Inspiration to Implementation, A.K.Peters (1993). [Kowalski, 19951 R.A.Kowalski, Using Meta-Logic to Reconcile Reactive with Rational Agents, in Meta-Logics and Logic Programming, ed. K.R.Apt and F.Turini, MIT Press (1995), pages 221-242. [Lesperance, et al., 19941 Y.Lesp&ance, H.J.Levesque, F.Lin, D.Marcu, R.Reiter, and R.B.Scherl, A Logical Approach to High-Level Robot Programming: A Progress Report, in Control of the Physical World by Intelligent Systems: Papers from the I994 AAAI Fall Symposium, ed. B.Kuipers, New Orleans (1994), pages 79-85. [McCarthy, 19891 J.McCarthy, Artificial Intelligence, Logic and Formalizing Common Sense, in Philosophical Logic and Artificial Intelligence, ed. R.Thomason, Kluwer Academic (1989), pages 161-190. [Nilsson, 19841 N.J.Nilsson, ed., Shakey the Robot, SRI Technical Note no. 323 (1984), SRI, Menlo Park, California. [Poole, 19951 D.Poole, Logic Programming for Robot Control, Proceedings IJCAI 95, pages 150- 157. [Reiter, 19871 R.Reiter, A Theory of Diagnosis from First Principles, Artificial Intelligence, vol 32 (1987), pages 57- 95. [Shanahan, 19891 M.P.Shanahan, Prediction Is Deduction but Explanation Is Abduction, Proceedings IJCAI 89, pages 1055-1060. [Shanahan, 1995a] M.P.Shanahan, Default Reasoning about Spatial Occupancy, Artificial Intelligence, vol 74 (1995), pages 147-163. [Shanahan, 1995b] M.P.Shanahan, A Circumscriptive Calculus of Events, Artificial Intelligence, vol 77 (1995), pages 249-284. [Shanahan, 1996a] M.P.Shanahan, Solving the Frame Problem: A Mathematical Investigation of the Common Sense Law of Inertia, MIT Press (1996), to appear. [Shanahan, 1996b] M.P.Shanahan, Robotics and the Common Sense Informatic Situation, Proceedings ECAI 96, to appear. [Strawson, 19661 P.F.Strawson, The Bounds of Sense, Methuen ( 1966). Perception 1103 | 1996 | 163 |
1,803 | A HYBRID LEARNINGAPPROACHFORBETTERRECOGNITION OF VISUALOBJECTS Ibrahim F. Imam* SRA International 4300 Fair Lakes Court Fairfax, VA 22033 iimam@verdi.iisd.sra.com Abstract Real world images often contain similar objects but with different rotations, noise, or other visual alterations. Vision systems should be able to recognize objects regardless of these visual alterations. This paper presents a novel approach for learning optimized structures of classifiers for recognizing visual objects regardless of certain types of visual alterations. The approach consists of two phases. The first phase is concerned with learning classifications of a set of standard and altered objects. The second phase is concerned with discovering an optimized structure of classifiers for recognizing objects from unseen images. This paper presents an application of this approach to a domain of 15 classes of hand gestures. The experimental results show significant improvement in the recognition rate rather than using a single classifier or multiple classifiers with thresholds. 1 Introduction Recently, there has been a great interest in developing multimedia applications in wide-ranging fields. Communicative applications including audio and video systems require visual information about different objects to be able to recognize these objects, and communicate among each other (Maggioni, 1995; Freeman & Weissman, 1995; Kjeldsen & Kender, 1995). Such applications should be able to recognize objects within any environmental conditions. The reasons for such limitations include learning object classifiers using standard, noise free, and normalized objects; and using a non-adaptive strategy for recognizing new objects. This paper introduces a new approach for learning optimized structures of classifiers for recognizing visual objects. In this paper, the term “o&x&” denotes an image of one of the visual classes (e.g., hand poses). A scfllcture qf C&@&-J is a tree where each non-leaf node contains a classifier (e.g., a neural network), branches correspond to different outcomes or quantized intervals of the outcomes of each classifier, and leaves represent different classes of visual objects. The set of classifiers, used for building the decision structures, represents different visual alterations to the standard image of each object. * Also a Research Affiliate of the MLI Laboratory at GMU. 1104 Perception Srinivas Gutta Computer Science Department George Mason University Fairfax, VA 22030 sgutta@cs.gmu.edu The main goal of this approach is to recognize a set of visual objects regardless of visual alterations of these objects in the corresponding images. This is done in two phases. The first phase is concerned with generating a set of classifiers (e.g., neural networks), one for each combination of visual alteration of the standard object and parameter settings of the classifiers. Only two alterations were considered, in this paper, by applying a set of geometrical transformations and noise to the original image of each object. The outputs from the alteration processes are called &g-ed o&g&. In the second phase, another set of training images are tested by all classifiers. For each image, the set of values obtained from all classifiers along with the correct recognition are used for discovering an optimized structure of classifiers for recognizing objects in testing (unseen) images. To perform this task, we used a system, called AQDT-2 (Imam dz Michalski, 1993), for learning task-oriented decision structures from examples. An ~~ti??ud decision Structure is a StrUCW that contains the minimum number of nodes (classifiers), the minimum number of leaves, and correctly classifies the maximum number of testing examples (images of similar and different visual objects). The methodology was applied on a hand gesture database created by the authors. The hand gesture database contains 15 different gestures. For testing, 9 cycles of two cross-fold testing method were applied to test the recognition rate. The results obtained in this paper show a significant improvement in the recognition r&e when using the proposed approach over using single classl$ier or ensemble of classifiers. 2 Related Work Carpenter et al. (1992) proposed a Fuzzy system, called ARTMAP, which achieves a synthesis of the Adaptive Resonance Theory (ART) between neural network and fuzzy logic by exploiting a close formal similarity between the computations of ART category choice and fuzzy membership functions. Greenspan, Goodman, ti Chellappa (1994) proposed an architecture for the integration of neural networks and rule-based methods using unsupervised and superv&l learning. This approach was used for pattern recognition tasks. Also, Towel1 and Shavlik (1994) presented a methodology for transferring From: AAAI-96 Proceedings. Copyright © 1996, AAAI (www.aaai.org). All rights reserved. symbolic knowledge into a neural network and for extracting rules from a trained neural network. The proposed approach defers from the above ones in the fact that it uses an adaptive methodology to combine multiple neural networks with decision tree approach. An early example of using ensembles of neural networks, called Meta-Pi, was presented by Hamshire and Waibel (1992). The Me&Pi classifier is a connection&t pattern classifier that consists of a number of sub-networks. These sub-networks are integrated Delay Neural Network (TDNN) superstructure. The TDNN combines the outputs of the modules, tmined independently, in order to provide a global classification. Lincoln and Skrzypek (1990) proposed a clustering multiple back propagation networks to improve the performance and fault tolerance. Following training, a ‘cluster’ is created by computing the average of the outputs generated by the individual networks. The output of the ‘cluster’ is used as the desimd output during training by feeding it back to the individual networks. The basic behind using such a strategy is based on the assumption that while it is possible to ‘fool’ a single back propagation network all the time one cannot mislead all of them at the same time. Bat&i and Colla (1994) proposed an approach to combine the outputs of different neural network classifiers to improve the rejection-accuracy rates and to make the combined performance better than that obtained from the individual components. Decisions am made based on the majority rule (concept of democracy). The concept of democracy is analogous with the human way of reaching a pondered decision query by consensus. Several approaches to the problem of recognizing hand gestures have been proposed recently. One can distinguish between methods which assume a physically valid model of the hand (e.g., Quam, 1990; Sturman & Zeltzer, 1994) and those which do not extract or impose these type of 3- D constraints (e.g., Huang & Pavlovic, 1995; Lee & Kunii, 1993; Downton & Drouet, 1991). 3 The Approach This paper presents a novel approach for learning optimized structures of classifiers for improving image recognition. The proposed approach consists of two phases. In the first phase, all images of visual objects are converted into appropriate format (e.g., digital) and segmented. Then for each image a set of altered objects ate generated. An gltered obiect is a on of the standard object such that both should be recognized as members of the same class (e.g., same gesture). For each alteration process, a classifier is used for learning classification for all object classes. The second phase is concerned with determining an optimized structure of classifiers for recognizing unseen or new images. Figure 1 illustrates the proposed approach. All images am segmented and normalized. Then for each image, two identical copies were produced one by adding noise and the other by rotating it. This process three times using new classifiers with different ettings. All nine classifiers were trained using a portion of the training examples (set #I). Then, all classifiers are tested against remaining portion of the training examples (set #2). output values from testing the nine classifiers and the correct recognition (i.e., decision class) provided by the user are combined into a data vector, call& a record qf . . recomlttoq . All records of recognition are combined together to form a set of examples used later to build the optimized structure of classifiers. Building the optimized structure of classifiers is done by the program AQDT-2 (Imam & Michalski, 1993). This program is selected over other decision tree programs because it can optimize the obtained structures using a variety of cost functions. ase I: Learning Classification of Visua Objects The first step of capturing an object from an image is to locate the horizontal and vertical boundaries of the object. The method utilized here uses a simple algorithm that operates on the edge image using the Sobel’s edge extraction method. All images are passed to the object normalization process. Each object (or an image) is normal&d before starting the object recognition phase. This rest of this subsection illustrates the process of Classification values Object Recognition Figure I: An Illustration of the proposed approach. Vision 1105 building, training and testing a single RBF classifier and an Ensemble of RBF classifiers (ERBF) and using both for object recognition. The ERBF classifiers are used later for acquiring the records of recognition. Single Classifier: The construction of an RFJF network is similar to the construction of any neural network. the number of input nodes in each RBF neural network was always set equal to the number of input images. The number of output nodes was set equal to the number of decision classes. The number of hidden nodes are optimally found, in each single RBF, by varying the number of clusters from fifth to equal number of the input nodes. At each stage of the variation, two additional parameters, the overlap factors and the proportionality constants were changed. These changes are repeakd until all training examples (images) were classified correctly by the classifier. The same process is repeated for the cases when training on the original images with Gaussian noise and original images with geometrical transformation. The reason for using RBF network is because of its ability to cluster similar images before classifying them. A new object is assigned to the class with the highest value. To compare the performance of the RBF with the proposed approach, the training set #l is used for training the classifier and the testing set is used for testing it. The proposed ERBF model integrates three RBF components, Cl, C2 and C3, as shown in Figure 2. Each RBF component is further defined in terms of three RBF nodes, each of which specified in terms of the number of clusters and the overlap factors. The three RBF components have been trained on the original images, the original images after adding Gaussian noise, and the original images after geometrically rotating the objects with certain degree a, respectively. To recognize a new object, the image is tested by all the nine networks. Each network produces an output value for each class, called -cation value. 7Be sum of all 9 classification values for each class is called the recognition rate, R, for that class. An object X is a member of class Ci, if the recognition rate of this class is greater than the rate of all other classes, equation (1). Figure 2: EREIF Architectuz where O(Ni, CJ is the value assigned by classifier Ni (i= 1, . . . . 9) t0 ClaSS Cj (i= 1, *SS* m; m is the number of decision classes) when testing the image X. To generate the records of classification, The classifications provided by different classifiers and the correct recognition are grouped together to form one record. Figure 3 shows a description of the method. Note that the training set #2 is used in this phase as a testing set. 3.2 Phase II: Learning Optimized Structures of Classifiers The second phase in the proposed approach is CoIlcemed with leaming an optimized structure of classifiers for recognizing different objects from new images. m Situations to determine if the class@cation should be used for reconnition or for selectinn the best classifier for reconnit ’ lo?&. Figure 4 illustrate the methodology used to determine an optimized structures for object recognition. To obtain such a structure, the AQDT-2 learning system (Imam & Michalski, 1993) is used to learn a task-oriented structure of classifiers that recognizes any new object using the minimum number of classifiers. Input: Training (set #l.l) and testing (set #2.1) sets of segmented and normalized images. Output: Classifications of a set of objects in unseen images. Step 1: For all images (in set #l.l and set #2.1), generate two identical sets of images (sets #1.2, #1.3, #2.2, and #2.3) . Step 2: For all image in sets #1.2 and #2.2 add Gaussian noise. For all images in sets #1.3 and #2.3 add geometric transformation. Note: All images in sets “#lx” are used for training the classifiers, and those in sets “#2x” are used for testing the classifiers. Step 3: Create three identical copies of each set of images (e.g., #l.l.l, #1.1.2, #1.1.3 for set #l.l). Create a set of 9 RBF networks with different k-mean clustering one for each training set (start with #l). For each classifier, steps 4 to 5 are repeated: Step 4: Each classifier is trained using the corresponding set of images. Step 5: When the training process of each classifier is finished, the corresponding set of testing images is used for testing the network (e.g., testing on set #2.a for training set #l.a. where a is any number or combination of numbers). Step 6: The classification values obtained from testing each image using the 9 classifiers along with the correct recognition of the object were combined into a data vector. Figure 3: Learning object classifiers. 1106 Perception Input: A set of training examples (records of recognition) and testing unseen images. Output: An optimized decision structure that recognizes any new gesture (with or without alterations). Step I: Quantize all attribute values in the records of recognition. Step 2: (Optional) Determine a set of disjoint rules describing the training examples. Step 3: Specify the learning task for AQDT-2 (in this case, the decision structure should have minimum number of classifiers and minimum number of levels). Step 4 to Step 6 are repeated until learning task is satisfied. Step 4: Run AQDT-2 (initially with its default settings) to obtain a decision structure. Step 5: Compare the information of the obtained decision structure with the optimal one. If the new structure is more optimal, store the values of the cost functions and the optimizing criteria. If the stopping criteria is satisfied, exit. Step 6: Change the tolerance of the cost functions in AQDT-2 to give high preferences for some attributes. Go to step 4. Figure 4: The method for learning optimized structure for shape recognition. The advantages of using AQDT-2 over other decision tree learning programs is that it has adaptive capabilities including forcing partial attribute ranking, restructuring the decision structure according to a set of criteria and cost functions, etc. To select an attribute, AQDT-2 used the disjointness criterion which ranks attributes according to how their values discriminate the decision classes. For the experiments presented in this paper, the learning task was set to minimize the number of neural networks used in the obtained structure, and to maximize the recognition ZKYXlXCy. 4 The Experiment This section introduces an application of the proposed approach to a database of hand gestures. These images were taken by a KODAK Quick Take 100 Camera. The database contains 150 images of hand gestures. Images were taken from 5 different persons (2 sets of 15 images per each, Figure 5). The second set of images has been taken after a time gap of 30 minutes. The experiment was performed 3 times. For each time, a set of five different gestures (third second, then first five gestures respectively) were considered as one decision class. This was done to increase the complexity of the problem. One Two Four Five -------” ._-- ----- ----- StOD Fist -scout Love Star Trek Devil Duck Right Cross Hook Flat Two Figure 5: Images of hand gestures used in the experiment. For each testing combination, a set of 9 testing cycles were performed. A testing cycle uses a two cross-fold method to evaluate each method. This is done by splitting all gestures, first, into two sets. The first set is labelcd training set #l. The second set is divided into two subsets with ratio of 1:8. The smaller one is labeled training set #2, and the larger is labeled testing. The sizes of the training sets #l in the 9 cycles are 7%, 13%, 20%, 27%, 33%, 40%, 47%, 53%, and 60%, respectively. Original images were of size 240x320. All images were converted from RGB format to gray scale images of intensity varying from O-255. All 150 hand gesture images were segmented and normalized, then all images were visually verified. Images were normalized to standard size of 64x116 pixels. Two visual alterations arc used in this paper: 1) A gaussian noise with 0 mean and a variance of 10 was added to each image to generate new set of images; 2) A geometric transformation of 10’ was applied to the original images to generate a different set of images. The Recognition Rate of RBF and ERBF: To compare the recognition rate of using single RBF classifier and ensemble of classifiers ERBF, the training set #1 is used for training the classifiers and the testing set was used for reporting their performances (Note: training set #2 was not used). Table 1. shows results obtained from testing a single RBF classifier. Each row in this table presents the results obtained from single combination. Table 2 shows resuhs of testing the ERBF. The recognition process is done using equation (1). Table I: RBF Results. 2 Testing method 1 2 2 ‘able 2: ERBF Results. False Positive 27.2% 31.8% 22.7% False Positive 18.9% 17.8% 22.2% Vision 1107 Learning an Optimized Structure of Classifiers: To obtain the records of recognition from each experiment, the training set #l was used for training the classifiers, while the training set #2 was used for testing. All outputs from each classifier were quantized according to a uniform distribution of the length of the interval (e.g., 10 intervals each of length 0.1). A discriminant descriptions of the decision classes of gestures were obtained by the AQl5 learning system (Michalski, et al, 1986). This intermediate process usually improves the overall performance. AQDT-2 used these rules to determine an optimized decision structure to classify any new gesture using the minimum number of networks. The program parameters were set to run 10 iterations with variable costs for all attributes, variable degme of generalizations, and minimizing the number of nodes and levels in the structure. Each iteration uses random setting for the costs of the nine attributes. Each attribute represents one classifier. The program first learns a decision structure with its default settings. Then it changes the cost of one classifier at a time and re-learns anew structure. If the size of the structure is df2uawd or changed within a given tolerance, or the number of classifiers is decreased or changed within another tolerance, the system keeps the cost of that attribute and changes another attribute cost. The paper reports the results obtained from the default settings and the best 5 iterations. Figure 6 shows a decision structure that is learned for one of the cycles of the testing combination #l. This decision structure contains 161 tests (e.g. the sum of number of tests from the root to any leaf node) divided over 56 paths (a path is a connection from the root to a leaf node). Thus, jhe average (integer) number qf class@ers needed to classfi m unseen hmi&ature 1 ‘s 3 . The total number of networks needed to classify any gesture belongs to the given group is 6 networks. About 72% of all possible gestures can be classified using only four networks (N4, N5, N7, and N8). For the different testing combinations described above, the AQDT-2 program was very successful in discovering optimized decision structures using subset of the neural networks used in Figure 6. In the second testing combination, only five networks were used to derive a 2..6/ a decision (N3 was excluded). The average (integer) number of networks needed to derive a decision was also 3. In the third testing experiment, the same set of networks were used to obtain a decision. Table 3 shows the error rate when testing the structure obtained by AQDT-2 with its default settings. It also shows the median and mean of error rates of the best five iterations (structures with minimum number of classifiers and have minimum number of average tests). Figure 7 shows a comparison of the error rate of using one RBF network (Baysian classifier), using a combination of nine different classifiers @RBF) (majority voting), and using the proposed approach for image recognition. The results show a significant improvement in the recognition rate using the combination of the proposed approach. Table 3: The error rate of using combination of ERBF apld Comparing the Three Classifiers First Seccmd Third Testing Combinathm Figure 7: The error rate of using RBF, ERBF, and ERBF+AQDT-2 for gesture recognition. Figure 6: An optimized structure of classifiers for object recognition. 5 Conclusion The paper introduces a new approach for improving the recognition rate of visual objects. The approach separates the process of learning classifications of a set of visual objects from the process of recognizing new objects. For learning object classifications, the proposed methodology learns classification of all objects from the original images and from different sets of altered images. In this research, only two alterations were 1108 Perception considered by either geometric rotating them or adding noise to the original image. For each original or altered set of images, three classifiers with different parameter settings were used to learn classification of different classes of objects. To recognize a new object, a structure of classifiers is obtained by the AQDT-2 learning system. This structure is used as a plan for recognizing objects in unseen images. The maximum classification value obtained by this classifier is used to either assign a class to the object in the image or select another classifier to test the image. The role of AQDT-2 is to determine the minimum set of classifiers needed for recognition and in which order the testing should take place. The paper presented an application on a database of hand gestures. Three experimental combinations were performed on the data to analyze the performance of the methodology. In each combination, a subset of classes were grouped together as one class to increase the complexity of the problem. The results show a significant improvement in the recognition rate of new objects using the new approach against using a single RBF or ensemble of RBFs for recognition. The method allows more flexibility in recognizing visual objects. More analysis is to study the relationship between the number of alterations used and the complexity of obtained structures and the performance of object recognition. Acknowledgments: The authors thank Zoran Duric, Mark Maloof, and Nirmal Warke for reviewing an earlier draft of this paper. This research was supported partially by the Forensic Lab. at GMU through the US Army Research Lab under Contract DAALOl-93-K-0099; and partially by the ML1 Laboratory at GMU through the Advanced Research Projects Agency under grant No. NO001491-J- 1854 administered by the office of Naval Research, in part by the Advanced Research Projects Agency under grants F49620-92-J-0549 and F49620-95-I-0462 administered by the A3 Force Office of Scientific Research. References Battiti, R., and Colla, A. M., 1994. Democracy in Neural Nets: Voting Schemes for Classification, Neural Networks, Vol. 7, No. 4, pp. 691-707. Carpenter, G.A., Grossberg, S., Markuzon, N., Reynolds, J.H., and Rosen, D.B ., 1992. Fuzzy ARTMAP: A Neural Network Architecture for Incremental Supervised Learning of Analog Multidimensional Maps, IEEE Trans. on Neural Networks, Vol. 3, No. 5, pp. 698-713. Downton, A.C., and Drouet, H., 1991. Image analysis for model-based sign language coding, In Proceedings of the 6th International Conference on Image Analysis and Processing, pp. 637-644. Freeman, W.T., and Weissman, CD., 1995. Television control by hand gestures, In Proceedings of the International Workshop on Automatic Face- and Gesture- Recognition (ZWAFGR), pp. 179-181, Zurich. Greenspan, H., Goodman, R., and Chellappa, R., 1991.Texture Analysis via Unsupervised and Supervised Learning, In Proceedings of the International Joint Conference on Neural Networks, Vol. I, pp. 639-644. Hampshire, J.B ., and Waibel, A., 1992. The Me&Pi Network: Building Distributed Knowledge Representations for Robust Multisource Pattern Recognition, IEEE Trans. on Pattern Analysis and Machine Intelligence, Vol. 14, No. 7, pp. 751-769. Huang, T.S ., and Pavlovic, V.I., 1995. Hand Gesture Modelling, Analysis and Synthesis, In Pmce&ings of the International Workshop on Automatic Face- and Gesture- Recognition (IWAFGR), pp. 73-79, Zurich. Imam, I.F. and Michalski, R.S., 1993. Learning Decision Trees from Decision Rules: A method and initial results from a comparative study. The International Journal of Intelligent Information Systems JIIS, Ml. 2, No. 3, pp. 279-304, Kluwer Academic Pub., MA. Kjeldsen, R., and Kender, J., 1995. Visual Hand Gesture Recognition for Window System Control, International Workshop on Automatic Face- and Gesture-Recognition (ZWAFGR), pp. 184-188, Zurich. Lee, J., and Kunii, T.L., 1993. Constraint-based hand modeling and tracking, Models and Techniques in Computer Animation, pp. 110-127, Tokyo, Springer Verlag. Lincoln, W.P., and Skrzypek, J., 1990. Synergy of Clustering Multiple Back Propagation Networks, Advances in Neural Znformation Processing Systems (NIPS), Tour-et&y, D.S., (Ed.), Vol. 2, pp. 650-657, Morgan Kaufmann Publishers, San Francisco, CA. Maggioni, C., 1995. GestureComputer-New Ways of Operating a Computer, International Workshop on Automatic Face- and Gesture-Recognition (ZWAFGR), pp. 166-171, Zurich. MichaIski, R.S., Mozetic, I., Hong, J., and Lavrac, N., 1986. The Multi-Purpose Incremental Learning System AQIS and its Testing Application to Three Medical Domains, Proceedings of AAAZ-86, pp. 1041-1045, Philadelphia, PA. Quam, D.L., 1990. Gesture Recognition with a Data- glove, Proceedings of the IEEE National Aerospace Electronics Conference, Vol. 2. Sturman, D.J., and Zeltzer, D., 1994. A Survey of glove- based input, IEEE Computer Graphics and Applications, Vol. 14, pp. 30-39. Towell, G.G., and Shavlik, J.W., 1994. Refining Symbolic Knowledge Using Neural Networks, Machine Learning: A Multistrategy Approach, Vol. IV, pp. 405- 438, Michalski, R.S. and Tecuci, 6. (Eds.), Morgan Kaufrnann Publishers, San Francisco, CA. Vision 1109 | 1996 | 164 |
1,804 | Using Elimination Methods to Compute Thermop sical Algebraic In-variants from Infrared Imagery J.D. Michelt, N. Nandhakumart, Tushar Saxena*, Deepak Kapd t Dept of Electrical Engineering, Univ. of Virginia, Charlottesville, VA 22903 $ Inst. for Logic and Programming, Dept. of Computer Science, State Univ. of New York, Albany, NY 12222 {michel, nandhu}@virginia.edu,(saxena, kapur)@cs.albany.edu Abstract We describe a new approach for computing in- variant features in infrared (IR) images. Our ap- proach is unique in the field since it considers not just surface reflection and surface geometry in the specification of invariant features, but it also takes into account internal object composi- tion and thermal state which affect images sensed in the non-visible spectrum. We first establish a non-linear energy balance equation using the principle of conservation of energy at the sur- face of the imaged object. We then derive fea- tures that depend only on material parameters of the object and the sensed radiosity. These fea- tures are independent of the scene conditions and the scene-to-scene transformation of the “driving conditions” such as ambient temperature, and wind speed. The algorithm for deriving the in- variant features is based on the algebraic elim- ination of the transformation parameters from the non-linear relationships. The elimination ap- proach is a general method based on the extended Dixon resultant. Results on real IR imagery are shown to illustrate the performance of the fea- tures derived in this manner when used for an object recognition system that deals with multi- ple classes of objects. Introduction A very popular and increasingly affordable sensor modality is thermal imaging - where non-visible ra- diation is sensed in the long-wave infrared (LWIR) spectrum of 8pm to 14pm. The current generation of LWIR sensors produce images of contrast and res- olution that compare favorably with broadcast televi- sion quality visible light imagery. However, the images are no longer functions of only surface reflectance. As the wavelength of the sensor transducer passband in- creases, emissive effects begin to emerge as the dom- inant mode of electromagnetic energy exitance from object surfaces. The (primarily) emitted radiosity of LWIR energy has a strong dependence on internal com- position, properties, and state of the object such as specific heat, density, volume, heat generation rate of internal sources, etc. This dependence may be 1110 Perception exploited by specifying image-derived invariants that vary only if these parameters of the physical proper- ties vary. Here, we describe the use of the principle of con- servation of energy at the surface of the imaged ob- ject to specify a functional relationship between the object’s thermophysical properties (e.g., thermal con- ductivity, thermal capacitance, emissivity, etc.), scene parameters (e.g., wind temperature, wind speed, so- lar insolation), and the sensed LWIR image gray level. We use this functional form to derive invariant fea- tures that remain constant despite changes in scene parameters/driving conditions. In this formulation the internal thermophysical properties play a role that is analogous to the role of parameters of the tonics, lines and/or points that are used for specifying geometric invariants when analyzing visible wavelength imagery. Thus, in addition to the currently available techniques of formulating features that depend only on external shape and surface reflectance discontinuities, the phe- nomenology of LWIR image generation can be used to establish new features that “uncover” the composition and thermal state of the object, and which do not de- pend on surface reflectance characteristics. A general approach is described that enables the specification of invariant features that are satisfacto- rily justified in a thermophysical sense. The energy balance equation is inherently a non-linear form. We choose the variable labeling such that a polynomial is formed whose variables are the unknowns of the image formation and the coefficients are the object parame- ters. The choice of labels for the variables determines the form of the transformations from scene to scene. Consideration of the variable inter-dependencies spec- ifies the set of transformation to be a subgroup of the general linear group. A method based on elimination techniques is used to specify the features. Elimination methods eliminate a subset of variables from a finite set of polynomial equations to give a smaller set of polynomials in the remaining variables while keeping the solution set the same. Invariants can be computed using these meth- ods in three steps - (1) Set up the transformation equa- From: AAAI-96 Proceedings. Copyright © 1996, AAAI (www.aaai.org). All rights reserved. Figure 1: The vehicles used to test the object recogni- tion approach, (from top left clockwise) car, van, truck 1, and tank. The axis superimposed on the image show the object centered reference frames. The numbered points indicate the object surfaces used to form the measurement matrices. These points are selected such that there are a variety of different materials and/or surface normals within the set. tions relating the generic coefficients of the polynomial form before and after the action of the transformation subgroup, (2) Eliminate the transformation parame- ters from the transformation equations using any of the elimination methods, and finally, (3) Extract the invariants from the result of elimination from step 2. Using Elimination Methods for Computing Invariants Elimination methods are a general class of algorithms designed to eliminate a given set of variables from a finite system of polynomial equations. Some of the most general elimination methods are the Grijbner ba- sis method, characteristic set method, and various re- sultant methods see (Kapur & Lakshman 1992) for a survey. Such methods find applications in many areas of science and engineering and can be used to solve sys- tems of polynomial equations. They can also be used to automatically compute invariants of a given configura- tion (or quintic) under various transformation groups see (Kapur, Lakshman, & Saxena 1995). An absolute invariant is a rational function of the configuration parameters whose value remains con- stant under the action of a transformation group on this configuration. As a consequence, absolute invari- ants are very useful (Mundy & Zisserman 1992) in recognizing objects from images and building model- based object recognition libraries. Let p and q be the object and image parameters. Each absolute in- variant f/g generates a separable invariant relation, h(p, q) = f (p)g(q) - f (q)g(p). In other words, if these separable invariant relations can somehow be derived, then it may be possible to extract absolute invariants (which generate them) from them. The process of computing invariants using elimina- tion methods can be organized in three phases as fol- lows: Phase 1: Set up the transformation equations re- lating the image parameters to the object via the transformation parameters. Phase 2: Eliminate transformation parameters from the transformation equations to derive sepa- rable invariant relations. Phase 3: Extract the absolute invariants which generate the separable invariant relations. This is known as the separability problem. In phase 2, elimination methods such as Grobner basis algorithms, and in certain cases see (Kapur, Lak- shman, & Saxena 1995) resultant computations can be used to derive separable invariant relations. Given a separable invariant relation h(p, q), there exist many (algebraically dependent) invariants L f 12 which generate them, ie.: (f, g, z 9 c(PMq) - c(q)d(p) = h(P, q), f (p)g(q) - f WY(P) = h(P, a), k(p)l(q) - k(q)Kp) = h(P, a). But for a given ordering on the object parameters, there is a unique invariant I = f/g such that the: 1. leading term of g is strictly larger than the leading term of f, 2. leading term of f has zero coefficient in g and 3. leading coefficient of g is 1 (ie. g is manic). To extract the absolute invariant from separable in- variant relations, the algorithm in (Kapur, Lakshman, & Saxena 1995) fixes an ordering on the object and image parameters, and targets this unique invariant as follows. Let pef and pe 9 be the leading terms of f(p) and g(p) respectively, and cf , the leading coefficient of f(p). Then, using the above properties of this unique invariant, the separable invariant relation can be ex- pressed as h(P, q) = f (p)g(q) - Y(P)f (9) = f(P) (q”g +. * 3 - Y(P) (cfqeJ + * * *> = f(P) qeg - Cf Y(P) qej + . * *. As is evident from the above expansion of the separable invariant relation as a polynomial in q, the numerator f(p) of the absolute invariant is the coefficient of the leading term qeg . Once f(p) is known, and the de- nominator g(p) is the coefficient of the term -cf qef and can be easily read off from h(p, q) once it has been sorted according to a predetermined ordering. Vision 1111 a S Figure 2: Energy exchange at the surface of the im- aged object. Incident energy is primarily in the visible spectrum. Surfaces loses energy by convection to air, via radiation to the atmosphere, and via conduction to the interior of the object. The elemental volume at the surface also stores a portion of the absorbed energy. A Thermophysical Approach to LWI Image Analysis At the surface of the imaged object (figure 2) energy absorbed by the surface equals the energy lost to the environment. w abs - - west Energy absorbed by the surface is given by (1) W abs = WI cos& CN, ) (2) where, WI is the incident solar irradiation on a hor- izontal surface, & is the angle between the direction of irradiation and the surface normal, and cys is the surface absorptivity which is related to the visual re- flectance ps by as = 1 - ps. Note that it is reasonable to use the visual reflectance to estimate the energy absorbed by the surface since approximately 90% of the energy in solar irradiation lies in the visible wave- lengths (Incropera & Dewitt 1981). The energy lost by the surface to the environment was given by west = Wcv + Wrad + Wend + Wst (3) The energy convected from the surface to the ambient air is given by WC,, = h(Ts - Tama) where, Tamb is the ambient air temperature, Ts is the surface temperature of the imaged object, and h is the average convected heat transfer coefficient for the imaged surface, which depends on the wind speed, thermophysical properties of the air, and surface geometry (Incropera & Dewitt 1981). We note that surface temperature may be esti- mated from the thermal image based on an appropriate model of radiation energy exchange between the sur- face and the infrared camera. The radiation energy loss is computed from W rad = w(T,4 - T2mb), where u denotes the Stefan- Boltzmann constant. The energy conducted to the in- terior of the object is given by Wend = -E dT/dx, 1112 Perception where X: is the thermal conductivity of the material, and x is distance below the surface. Here, we as- sume that lateral energy conduction is insignificant compared to conduction along the direction normal to the surface. The increase in the stored, internal en- ergy of an elemental volume at the surface is given by wst = CT%, where CT denotes the lumped thermal capacitance of the object and is given by CT = DVc, D is the density of the object, V is the volume, and c is the specific heat. In the following section we use the energy conservation model described above to derive invariant features using ideas in algebraic elimination theory. Thermophysical Algebraic Invariants (TAI’s) The balance of energy expression, W abs = Wrad f Wcv + Wst + Wend (4 is the governing equation in our approach for comput- ing invariant features. Each term in the above equation can be expanded, which results in equation 4 being ex- pressed as a polynomial. The choice of labels for the variables determines both the form of the polynomial and transformation form. Since an absolute invariant feature value is not affected by transformations of the variables, the variables of the form are chosen to be the unknown parameters of the image formation. The coefficients are, then, the known/hypothesized object parameters and sensed measurements. An Algebraic Invariance Formulation The balance of energy expression, equation 4, may be written in the non-linear form where the variables and coefficients are labeled as al = CT: a2 = CT a3 = k a4 =TA a5 = -cosB a6 = -u a7 = A Xl = E ;;I 2 R” =- x4 a x5 = WI% 26 = Tamb (6) Thus, the polynomial chosen to represent equation 4 is a quintic form in six variables. Any pixel in a LWIR image of an object will yield a 7-D measurement vector, a. The image measurement (gray value) specifies al and ad. The values for a2, a3, and a5 are known when the identity and pose of the object are hypothesized. The coefficient a7, related to the convection term, h, is explained in greater detail in the discussion section. The driving conditions, xi, i = (1. . .6} are the unknown scene parameters that change from scene to scene. Consider two different LWIR images of a scene ob- tained under different scene conditions and from differ- ent viewpoints. Consider N points on the object that are visible in both views. Assume (for the nonce) that the object pose for each view, and point correspon- dence between the two views are available (or hypothe- sized). A point in each view yields a measurement vec- tor 8. The ;th component of the vector is denoted ai, wherei= l,..., 7 as defined by eqn (6). Let the collec- tion of these vectors be denoted by ai,k, k = 1, . . . , N for the first scene/image and u:,k, k = 1, . . . , N for the second scene. In the same vein, consider an associ- ated set of driving condition vectors for the first scene. We express the collection as a$$ where k = 1, . . . , N and i = l,..., 6 as defined in eqn (6). Similarly, the driving condition vector from the second scene is de- noted x;,$. Thermop hysical Transformat ion Consider a set of N 5 6 points imaged from the surface of an object. This creates a set of N vectors xi,k, k = 1 . ..N.i= l,..., 6 which define the driving conditions on the surface of the object in a scene at time t,. This forms a variable matrix of dimension 6 x N, call it X. These points are transformed from their values at time t, to their value at time t,+i, tn+l > t,, by a GL transformation, M, MX = X’. The transformation matrix M is 6 x 6. In order to determine the form of the transformation we view the components of a driving condition vector in terms of the inter-dependencies of the parameters. By doing so, superfluous parameters are eliminated. The dependency of the value of a variable at the cur- rent instance on other variables at a previous instance is established by the physical phenomena that cause scene-to-scene change in the different parameter val- ues. The dependencies are shown below (and explana- tions follow): variable . . xi = E 2; = CL l& =ds =h The change in emissivity is independent of dependency Xl (4 x2,x3,x4,x5(= %I &‘& x2, x3,54, x5(-& 2’ $4 (h) . r x5 (WI%> 26 (T,ma) g, WI%) h, WI%) (7) the values of any of-the variables. Hence, it is dependent only on itself. The second component, x2, is the temporal derivative of the surface temperature. Its value at t,+l will be affected by all of the parameters at t, except emissivity and the ambient temperature. Physically, the temporal derivative is independent of the ambient temperature and the emissivity of the surface; however, it is dependent on (1) its previous value, (2) the spatial derivative of the temperature in the material, (3) the convection coefficient- (the surface patches propensity to convect into the air), (4) incident solar irradiation and surface absorptivity. The spatial derivative, x3, has the same dependencies that 22 has. The remaining variables, x4, x5, and 26 depend, physically, only on their own previous values. The variable inter-dependencies determine the ther- mophysical transformation. Thus the transformation of the variables of equation 5 can be represented by a subgroup of the GL group of the form ml1 0 0 0 0 0 0 m22 m23 m24 m25 0 M= i ; m32 m33 m34 m35 0 0 0 - m44 0 0 (8) 0 0 0 0 m55 0 0 0 0 0 0 m66 Consider four points to compose X. Further explana- tion of the thermophysical behavior of these points is included in the discussion section. Each of the four points has seven components. Thus, the transforma- tion induced on the coefficients, ai, gives 28 constrain- ing equations. Since there are 12 parameters of the transformation, every additional constraining equation that is added to a set of 12 constraining equations gives rise to an invariant relationship. Thus, for a configura- tion of four points in the thermophysical space and a transformation consisting of 12 parameters, there are 28-12=16 invariant functions; however, a subset of these relations are physically trivial invariant rela- tionships. Given X, consisting of four copies of the equation 5, the elimination technique described in section 2 was applied to the algebraic configuration. This results in the following non-trivial invariants: a2,1 a2,2 a2,3 a3,l a3,2 a3,3 I1 = I a4,1 a4,2 a4,3 I a2,2 a2,3 a2,4 (9) a3,2 a3,3 a3,4 a4,2 a4,3 a4,4 a2,1 a2,2 a2,3 a3,l a3,2 a3,3 12 = a5,1 a5,2 z;‘; a2,2 a2,3 a3,2 a3,3 a314 a5,2 a5,3 a5,4 where ai,k is the ith component of the kth point. Employing TAPS for 0 b ject Recognition The feature computation scheme formulated above is suitable for use in an obiect recognition system that employs a hypothesize-and-verify-strategy.” The scheme would consist of the following steps: 1. extract geometric features, e.g., lines and tonics. Vision 1113 Figure 3: The truck 2 vehicle used in the recognition tests. The object centered coordinate axis is superim- posed on the image. The numbered points correspond to the point sets given in table 1. These points are selected such that there are a variety of different ma- terials and/or surface normals within the set. 2. for image region, r, hypothesize object class, k, and pose using, for example, geometric invariants as proposed by Forsyth, et al (Forsyth et al. 1991), 3. use the model of object k and project visible points la- beled i = 1,2,... onto image region r using scaled ortho- graphic projection, 4. for point labeled a in the image region, assign thermo- physical properties of point labeled i in the model of object k, 5. use the gray levels at each point and the assigned ther- mophysical properties, to compute the measurement vec- tors, qk, and hence compute the feature 11 or 12, and fin&5 6. compare feature the hypothesis. f ’ with model prototype Fk to verify Experimental Results object Recognition using TAIs The method of computing thermophysical algebraic in- variants discussed above was applied to real LWIR im- agery acquired at different times of the day. Five types of vehicles were imaged: a van, two types of trucks, a military tank, and a car (figures 1). Several points were selected (as indicated in the figures) on the surfaces of different materials and/or orientation. The measure- ment vector given by eqn (6) was computed for each point, for each image/scene. ’ The features described in section 4 require four points. Given a model of an object that has some & number of points defined, there is the possibility of forming Q different features. q=(f)(t) 1114 Perception (11) Point Set Mean 72 1.000 c4W,9~ 1.000 {%3,4,8) 4.757 {2,3,4,7) 4.746 {8,%W~ 0.983 {OW,fi~ 0.7361 U%%W) 0.0795 @,6,V) 1.057 STD Quality o.o02(j o.0026 0.0061 0.0061 0.0352 0.0074 0.0280 0.0059 0.1951 0.1984 0.1445 0.1963 0.0146 0.1836 0.0443 0.0419 Table 1: Intra-class variation over time of the feature, Il, defined by equation 9 applied with the point sets given in column 1 for truck type 2. The features were evaluated at five time instances over two consecutive days, Day 1 - llam, 12pm, lpm, Day 2 - 9am, loam. Column 2 is the mean of the feature over the five time instances and column 3 shows the feature stability in terms of standard deviation. Column 4 shows the qual- ity factor defined as std divided by the mean. The points correspond to the points labeled in figure 3. The first criterion for finding a useful feature is stable intra-class behavior. Nearly all of the point choices had low variation in intra-class tests; tests where the same object is viewed under different scene conditions. For example, a test was performed on the truck in figure 3. Table 1 shows the results for ten different features evaluated from truck 1. Although the performance of only ten features are shown, the performance is repre- sentative of the feature stability over all of the distinct point choices. As mentioned in section 4, one must consider inter- class behavior as well as intra-class behavior for an ob- ject recognition application of the features. To inves- tigate this we adopted the following procedure. Given an image of a vehicle, (1) assume the pose of the vehicle is known, then (2) use the front and rear wheels to es- tablish an object centered reference frame. The center of the rear wheel is used as the origin, and center of the front wheel is used to specify the direction and scaling of the axes. The coordinates of the selected points are expressed in terms of this 2D object-centered frame. For example, when a van vehicle is hypothesized for an image actually obtained of a car or some unknown ve- hicle, the material properties of the van are used, but image measurements are obtained from the image of the car at locations given by transforming the coordi- nates of the van points (in the van-centered coordinate frame) to the image frame computed for the unknown vehicle. Table 2 shows inter-class and intra-class variation when truck 1 is hypothesized. The data are gathered and images obtained at nine times during the daylight hours over a period of two days. The results show good inter-class separation and reasonable intra-class stability. Note that in the cases of wrong hypotheses, the feature values tend to be either indetermined or Hypothesis: Data From: 11 am 12 pm 1 pm 2 pm 3 pm 4 Pm 5 pm 9 am 10 am Truck 1 Van 4.62 1.00 1.00 1.00 7.50 1.00 2.95 1.00 4.00 Truck 1 Truck 1 Truck 1 Car Truck 2 Tank 1.00 -0.693 0.882 1.00 15.74 -1.00 NaN 1.00 2.846 1.00 2.20 -1.00 -1nf 1.00 1.00 19.0 13.67 1.00 51.0 1.71 4.20 1.20 3.00 -1.00 1.10 6.33 2.20 Table 2: Mistaken hypothesis feature values shows inter-class variation for feature A-l, consisting of point set {1,2,3,7}. Th e model for truck one is hypothe- sized. The feature value is formed using the model of truck 1 and the data from the respective other vehicles. When this feature is applied to the correctly hypoth- esized data of truck 1 it has a mean value of 0.0159 and a standard deviation of 0.0022. Thus feature, A-l, shows good separability when compared to the incor- rect hypothesis feature value listed in the table. unitary. This is a result of using the object centered coordinate system where the mistaken points fall on similar material types when dissimilar material types were expected. Discussion The approach described above is promising in that it makes available features that are (1) invariant to scene conditions, (2) able to separate different classes of ob- jects, and (3) b ased on physics based models of the many phenomena that affect LWIR image generation. Two aspects of the approach require further expla- nation. First, the factor, a7, from equation 6 was used in this formulation to expand the number of degrees of freedom in the algebraic expression of the balance of energy equation. Although it is not interpreted di- rectly as a physical parameter, it allows for the cre- ation of a proper form and has no effect on the phys- ical model. The motivation for including UT is that it is desirable to label as unknown variables both the convection parameter, h, and the ambient tempera- ture, Tama. These factors appear together in one of the terms of the balance of energy equation. With both factors labeled as variables, the coefficient can then only be unity, a7 = 1. The resulting labeling produces a form that loses important degrees of free- dom in the formation of invariant relations. Including a7 = A, implies that there is a scale of the temperature measurement, Ts, in the term a4 = Ts A. The transfor- mation, M, of the variables induces a transformation on the coefficients. For the coefficient in question the induced transformation can be written ai = m44u4. Since the features found in section 4 are invariant to transformations of the form 8 it is invariant to an addi- tional scale as in the action of the A parameter. Thus the term does not affect the relation of the physical model to the invariant feature. In addition, because a7 does not appear in the feature there is no need to physically interpret its value. Next, we consider the thermophysical justification of the transformation defined in the equation X’ = MX, (12) where X is a 6 x 4 collection of thermophysical vari- able vectors as defined in 6 at a time instance, t,, and X’ is the collection at a later time/scene &+I. The transformation M is defined in (8). The physical implication of such a transformation is that the four points in the thermophysical configuration are acted upon in the “same manner” by the environment. This is a reasonable assumption for the classes of objects under consideration. Note that if different types of surfaces are chosen (or points on surfaces with differ- ent surface orientations) the measurement vectors will, in general, be linearly independent. In other words, it is easy to select points such that the collection of mea- surement vectors span R6. Then the existence of a non-singular transformation of the form of, M, for any pair of scenes and for a subset of four such points is guaranteed. Physically, the effect of the convection co- efficient, solar irradiation and ambient temperature is consistent for the set of surface points. This fact taken with the fact that the emissivity can be considered rel- atively constant over time implies that it is reasonable to assume that equation (12) has physical justification. References Forsyth, D.; Mundy, J.; Zisserman, A.; Coelho, C.; HeIIer, A.; and Rothwell, C. 1991. Invariant descriptors for 3d object recognition and pose. IEEE Transactions on PAM 13( 12). Incropera, F., and Dewitt, D. 1981. Fundamentals of Heat Transfer. New York, NY: John Wiley and Sons. Kapur, D., and Lakshman, Y. 1992. Elimination meth- ods: an introduction. In Donald, K., and Mundy., eds., Symbolic and Numerical Computation for Artificial Intel- ligence. Academic Press. Kapur, D.; Lakshman, Y.; and Saxena, T. 1995. Com- puting invariants using elimination methods. In Proc of IEEE International Symposium on Computer Vision, 97- 102. Coral Gables, Florida: IEEE. Mundy, J., and Zisserman, A. 1992. Geometric Invariance in Computer Vision. MIT Press. Vision 1115 | 1996 | 165 |
1,805 | Approximate World Models: Incorporating Qualitative and Linguistic Informat ion into Vision Systems Claudio S. Pinhanez and Aaron F. Bobick Perceptual Computing Group - MIT Media Laboratory 20 Ames St. - Cambridge, MA 02139 pinhanez - bobick@media.mit .edu Abstract Approxhate world models are coarse descriptions of the elements of a scene, and are intended to be used in the selection and control of vision routines in a vision system. In this paper we present a con- trol architecture in which the approximate mod- els represent the complex relationships among the objects in the world, allowing the vision routines to be situation or context specific. Moreover, be- cause of their reduced accuracy requirements, ap- proximate world models can employ qualitative information such as those provided by linguistic descriptions of the scene. The concept is demon- strated in the development of automatic cameras for a TV studio - SmartCams. Results are shown where SmartCams use vision processing of real imagery and information written in the script of a TV show to achieve TV-quality framing. Introduction It has been argued - e.g. in (Strat & Fischler 1991) - that in any given situation most visual tasks can be performed by a relatively simple visual routine. For example: finding the ground reduces to finding a large (body-relative) horizontal plane if the observer is ver- tical and there are no other large horizontal planes. The difficulty in general, of course, is how to know the current state of the world without having to do all the detailed visual tasks first. The numerous possibilities for the fundamental relationships between objects in the scene is as much responsible for the complexity of vision as is the difficulty of the visual routines them- selves. The goal of this paper is to argue that vision systems should separate these two sources of complexity by us- ing coarse models of the objects in the scene called approximate world models. This proposal is based on the observation that the real world does not need to be fully and accurately understood to detect many situa- tions where a specific vision method is likely to succeed or fail. For instance, full and precise 3D reconstruction of the human body is not necessary to detect occlusion if the objective of the system is just to recognize faces of people walking through a gate. 1116 Perception A main feature of approximate world models is that their imprecision facilitates the use of incomplete and inaccurate sources of information such as linguistic de- scriptions of the elements and actions in a scene. In fact, we show in this paper that linguistic information can play a pivotal role in providing the contextual in- formation needed to simplify the vision tasks. We are employing approximate world models in the development of SmartCams, automatic TV cameras able to frame subjects and objects in a studio accord- ing to the verbal requests from the TV director. Our SmartCams are tested in the domain of a cooking show. The script of the show is available to SmartCams (in a particular format), and the cameras are shown to be able to produce TV-quality framing of subjects and objects. Approximate World Models Approximate world models are coarse descriptions of the main elements of a scene to be used in the selec- tion and control of vision routines. These models are to be incorporated into vision-based systems built from a collection of different, simple, task-specific vision rou- tines whose application is controlled according to the conditions described by the approximate world models. This proposal comes from the observation that a common reason for the failure of vision routines - especially, view-based methods - is related to the complex geometric relationships among objects in the world. For example, often template-based tracking routines produce wrong results due to partial occlu- sion. In such situations, a crude 3-D reconstruction of the main objects in the scene can determine if the tracked object is in a configuration where occlusion is probable. The advantages of using approximate models are at least three-fold. First, coarse reconstruction of the 3-D is arguably within the grasp of current computer vision capabilities. Second, as shown in this paper, control of task-specific vision routines can be based on inaccu- rate and incomplete information. And third, as we will demonstrate, reducing the accuracy requirements enables the use of qualitative information which might From: AAAI-96 Proceedings. Copyright © 1996, AAAI (www.aaai.org). All rights reserved. be available to the vision system. Coarse and/or hierarchical descriptions have been used before in computer vision (Bobick & Bolles 1992; Marr & Nishihara 1978). Particularly, Bobick and Bolles employed a multi-level representational system where different queries were answered by different rep- resentations of the same object. Part of the novelty of our work is related to the use of the models in the dy- namic selection of appropriate vision methods accord- ing to the world situation. And compared to other ar- chitectures for context-based vision systems, like Strat and Fischler’s Condor system, (Strat & Fischler 1991), approximate world models provide a much more clear distinction between the vision component and the 3-D world component. It is interesting to situate our scheme in the ongo- ing debate about reconstructionist vs. purposive vision discussed in (Tarr & Black 1994) and in the replies in the same issue. Our proposal falls between the strictly reconstructionist and purely purposive strategies. We are arguing that reconstruction should exist at the ap- proximate level to guide purposive vision routines: by building an approximate model of the scene vision sys- tems can use task-specific, purposive vision routines which work reliably in some but not all situations. Using Approximate World Models in Vision Systems To formalize the control exercised by the approximate world models in a vision system, we define applicability conditions for each vision routine: the set of assump- tions, that, if true in the current situation, warrants faith in the correctness of the results of that routine. The idea is to have each vision routine encapsulated in an applicability rule, which describes pre-conditions (IF portion of the rule), application constraints (THEN portion), and post-conditions (TEST IF), in terms of general properties about the targeted object, other ob- jects in the scene, the camera’s point of view, and the result of the vision routine. cording to the instructions in the THEN portion of the applicability rule which may also include information about routine parameters. The RESULT is then checked in the TEST IF portion of the rule, reducing the possi- bility that an incomplete specification of pre-conditions generates incorrect results. For instance, in the case of “extract-narrowest-moving-blob”, often the lack of actual object movement makes the routine return tiny, incorrectly positioned regions which are filtered out by the post-conditions. As an example, fig. 1 depicts the applicability rule of a vision routine, “extract-narrowest-movingblob”, and how the rule is applied in a given situation. The routine “extract-narrowest-moving-blob” is designed to detect moving regions in a sequence of two consecutive frames using simple frame differenc- ing, and then to divide the result into two areas, of which the narrowest is returned. It is important to differentiate the concept of ap- plicability rules from rule-based or expert-system ap- proaches to computer vision (Draper et al. 1987; Tsotsos 1985). Although we use the same keywords (IF, THEN), the implied control structure has no re- semblance to a traditional rule-based system: there is no inference or chaining of results. More examples of applicability rules can be found in (Bobick & Pinhanez 1995). A Working Example: SmartCams To use such a rule, the vision system consults the ap- proximate model of TARGET (in the example case, the head of a person) to obtain an estimation of its pro- jection into the image plane of the camera, and also to confirm if TARGET is moving. Moreover, the system also looks for other moving objects close to TARGET, constructing an approximate view model of the cam- era’s view. Our approach of using approximate world models is be- ing developed in a system we are constructing for the control of TV cameras. The ultimate objective is to develop a camera for TV that can operate without the cameraman, changing its attitude, zoom, and position to provide specific images upon human request. We call these robot-like cameras SmartCams. A “cooking show” is the first domain in which we are experiment- ing with our SmartCams. If the conditions are satisfied by the approximate The basic architecture of a SmartCam is shown in view model, the routine is applied on the imagery ac- fig. 2. Considering the requirements of this application, Approximate moving View Model object TARQET inside view THEiN APPLY vision routine Vuctract-narrowemt- moving-blob" TEST IF Final Result Figure 1: Example of an applicability rule for a vision rou- tine and how the information from the approximate world model is used. Vision 1117 0 TV director SmartCam --- APPROXIMATE WORLD MODEL scri wide-angle images Figure 2: The architecture of a SmartCam. The bottom part of the figure shows the structure of the modules re- sponsible for maintaining the approximate world models. the approximate world model represents the subjects and objects in the scene as 3-D blocks, cylinders, and ellipsoids, and uses symbolic frame-based representa- tions (slots and keywords). The symbolic description of an object includes information about to which class of objects the object belongs, its potential use, and its roles in the current actions. The 3-D representations are positioned in a 3-D vir- tual space corresponding to the TV studio. The cam- eras ’ calibration parameters area also approximately known. The precision can be quite low, and in our system the position of an object might be off by an amount comparable to its size. For example, if there is a bowl present in the studio, its approximate world model is composed by a 3-D geo- metric model of the bowl and by a frame-like symbolic description. The 3-D geometric model approximates the shape of the bowl, and is positioned in the vir- tual space according to the available information. The objects in the approximate world model belong to dif- ferent categories. For example, a bowl is a member of the “handleable objects” category. As so, its frame Figure 3: Example of response to the call “close-up chef” by two different cameras, side and center. The left images show the projection of the approximate models on the wide-angle images. The right images display the re- sult of vision routines as highlighted regions, compared to the predicted position according to the approximate model of the head and the trunk, shown as rectangles. includes slots which describe whether the bowl is being handled by a human, and, if so, there is a slot which explicitly points to him/her. The system which produced the results shown in this paper does not use real moving cameras, but simulates them using a moving window on wide-angle images of the set. Several performances of a 5-minute scene as viewed by three wide-angle cameras were recorded and digitized. The SmartCam output image is generated by extracting a rectangular window of some size from the wide-angle images. In fig. 3 we can see a typical result in the Smart- Cam domain where the inaccuracy of the approxi- mate world models does not affect the final results obtained by the vision routines. Two SmartCams, side and center, were tasked to provide a close-up of the chef. Although the geometric model correspond- ing to the chef is quite misaligned, as can be seen by its projection into the wide-angle images of the scene (left side), both S martcams, using routines similar to “ produce correct results (the highlighted areas on the right of fig.3). Other examples of applicability rules and results can be found in (Bobick & Pinhanez 1995). Building and Maintaining Approximate World Models Having shown how the information contained in ap- proximate world models can be exploited by a vision system performing tasks in a dynamic environment, a fundamental issue remains: how does one construct an initial model and then maintain such a model as time progresses and the scene changes. 1118 Perception Contextual and semantic information is rarely em- ployed in model construction because of its inability to provide accurate geometric data. If the geometric ac- curacy requirements are relaxed, as it is in the case of approximate world models, semantic information can be used to predict possible positions and attitudes of objects. Furthermore, we view one of the roles of contextual knowledge to be that of providing the basic relation- ships between objects in a given configuration of the world. For example, while pounding meat the chef re- mains behind the table, positioned near the cutting board, and the meat mallet is manipulated by the hands. As long as such a context remains in force, these relationships hold, and the job of maintaining the approximate world model reduces to, predominantly, a problem of tracking incremental changes within a known situation. Occasionally, though, there are major changes in the structure of the world which require a substantial up- date in the structure of the approximate models. For example, a new activity may have begun, altering most expectations, including, for example, the possible loca- tion of objects, which objects are possibly moving, and which objects are likely to be co-located making visual separation unlikely (and, more interestingly, unneces- sary). In our view, the intervals between major con- textual changes correspond to the fundamental actions performed by the subjects in the scene; the contex- tual shifts themselves reflect the boundaries between actions. Thus, maintaining approximate world models re- quires two different methods of updating: one re- lated to the tracking of incremental changes within a fixed context, and the other responsible for perform- ing the substantial changes in the context required at the boundaries between different actions. Of course, it is also necessary to have methods to obtain the initial approximate model. The three required mechanisms are briefly discussed below. Initializing Approximate World Models Whenever a vision system is designed using approxi- mate models, it is necessary to face the issue of how the models are initialized when the system is turned on. Current computer vision methods can be employed in the initialization process, if the system is allowed a reasonable amount of time to employ powerful vision algorithms without contextual information. In our current version of the SmartCams the 3-D models of the subjects and objects are determined and positioned manually in the first frame of the scene. All changes to the model after the first frame are accom- plished automatically using vision and processing the linguistic information as described later in this paper. Tracking Incremental Changes As proposed, while a particular action is taking place it is presumed that the basic relationships among the subjects and objects do not change. During such pe- riods most of the updating of the approximate mod- els can be accomplished by methods able to detect in- cremental changes, such as visual tracking algorithms. Though simple, those small updates are vital to main- tain the approximate model in an useful state, since approximate position information is normally an es- sential part of the applicability conditions of the task- specific vision routines. In the SmartCam domain, the update of the incre- mental changes in the approximate world model, and especially in its 3-D representations, is accomplished by vision tracking routines able to detect movements of the main components of the scene, as shown in the bot- tom part of the diagram in fig. 2. The two-dimensional motions of an object detected by each of the wide-angle cameras are integrated to determine the movement of the object in the 3-D world. More details can be found in (Bobick & Pinhanez 1995). Note that the use of an approximate world model may require additional sensing and computation which might not be performed to directly address current per- ceptual tasks: e.g. the position of the body of the chef is maintained even though the current task may only involve framing the hands. We believe that this additional cost is compensated by the increase in the competence of the vision routines. Actions and Structural Changes Tracking algorithms are likely to fail whenever there is a drastic change in the structure of the scene, as, for example when a subject leaves the scene. If this situation is recognized, the tracking mechanism of that subject should be turned off avoiding false alarms. It is not the objective of this paper to argue about the different possible meanings of the term ac- tion. Here, actions refer to major segments of time which people usually describe by single action verbs as discussed by (Newtson, Engquist, & Bois 1977; Thibadeau 1986; Kalita 1991). A change in the action normally alters substantially the relationships among subjects and objects. For ex- ample, we have a situation in the cooking show domain where the chef first talks to the camera, and then he picks up a bowl and starts mixing ingredients. While the action “talking” is happening, there is no need for the system to maintain explicit 3-D models for the arms and the hands of the chef: the important ele- ments involved are the position and direction of the head and body. When the chef starts mixing ingredients, it is es- sential that the approximate world model includes his hands, the mixing bowl, and the ingredients. Fortu- nately, “mixing” also sets up clear expectations about the positions of the hands in relation to the trunk of the Vision 1119 chef, to the mixing bowl, to the ingredients’ contain- ers, and to table where they are initially on. As this example illustrates, known contextual changes actually improve the robustness of the system since with each such event the system becomes “grounded” : there is evidence independent of, and often more reliable than, the visual tracking data about the state of the scene. Linguistic Sources of Action Information Without constraints from the domain, determining the actions which might occur can be difficult. However, in many situations the set of actions is severely restricted by the environment or by the task, as, for example, in the case of recognizing vehicle maneuvers in a gas- station as described in (Nagel 1994 5). The SmartCam domain exemplifies another type of situation, one in which there is available a linguistic description of the sequence of actions to occur. In a TV studio, the set of occurring actions - and, even, the order of the actions - is determined by the script of the show. We find the idea of having vision systems capable of incorporating linguistic descriptions of ac- tions very attractive: linguistic descriptions of actions are the most natural to be generated by human beings. In particular, this form of is suitable in semi-automated vision-based systems. The use of linguistic descriptions requires their auto- matic translation into the system’s internal represen- tational of actions. This is mostly a Natural Language Processing issue, although the feasibility is certainly dependent on the final representation used by the vi- sion system. In the SmartCam domain the final ob- jective is to employ TV scripts in the format they are normally written (see figure 4). Representing Actions Many formalisms have been developed to represent ac- tion, some targeting linguistic concerns (Schank 1975), computer graphics synthesis (Kalita 1991), or com- puter vision recognition (Siskind 1994 5). Currently we employ a simple representation based on Schank’s conceptualizations as described in (Schank 1975). In spite of its weaknesses - see (Wilks 1975) - Schank’s representation scheme is interesting for us because the reduced number of primitive actions helps the design of both the translation and the inference procedures. Our representation for actions uses action frames, a frame-based representation where each action is rep- resented by a frame whose header is one of Schank’s primitive actions - PROPEL, MOVE, INGEST, GRASP, EXPEL, PTRANS, ATRANS, SPEAK, ATTEND, MTRANS, MBUILD - plus the attribute indexes HAVE and CHANGE, and an undetermined action DO. Figure 5 contains two examples of action frames. The figure contains the representation for two ac- tions of the script shown in fig. 4, ‘ t chef wraps chicken with a plastic bag” and “chef pounds 1120 Perception Cooking-show scenario with a table on which there are bowls, ingredients, and different kitchen u tens&. A mi- crowave oven is in the back. Cam1 is a centered camera, cam2 is a left-sided camera, cam3 is a camera mounted in the ceiling. Chef is behind the table, facing caml. . . . Chef turns back to cam1 and mixes bread-crumbs, parsley, paprika, and basiI in a bowl. ‘Stir together 3 tablespoons of fine dry bread crumbs, 2 teaspoons snipped parsley, l/4 teaspoon paprika, and l/8 teaspoon dried basil, crushed. Set aside.” Chef wraps chicken with a plastic bag. “Place one piece of chicken, boned side up, between two pieces of clear plastic wrap.” Chef puts the chicken on the chopping-board and shows how to pound the chicken. “Working from the center to the edges, pound lightly with a meat maldet, forming a rectangle with this thickness. Be gentle, meat become as soft as you treat it.” Chef pounds the chicken with a meat-m&et. . . . Figure 4: The script of a TV cooking show. the chicken with a meat-mallet”. Each action frame begins with a primitive action and contains dif- ferent slots which supply the essential elements of the action. Undetermined symbols begin with a question mark (?); those symbols are defined only by the re- lationships they have with other objects. The actor slot determines the performer of the action while the object slot contains the object of an action or the owner of an attribute. The action frames resulting from an action are specified in the result slot, and action frames which are part of the definition of an- other action frame are contained in instrument slots. In the first example, “wrapping” is translated as some action (unspecified by DO) whose result is to make chicken be both contained in and in physical contact with a plastic bag. In the second example, “pound- ing” is represented as an action where the chef propels a meat mallet from a place which is not in contact with the chicken to a place which is in contact, and whose result is an increase in the flatness of the chicken. The current version of our SmartCams translates a simplified version of the TV script of fig. 4 into the action frames of fig. 5 using a domain-specific, very simple parser. Building a translator able to handle more generic scripts seems to be clearly a NLP prob- lem, and, as such, it is not a fundamental point in our research. Part of our current research is focused on designing a better representation for actions than the action frames . . ii0 "chef wraps chicken with a plastic bag" (actor chef) (result (change (object chicken) (to (and (contained plastic-bag) (physical-contact plastic-bag)))))) . . "chef pounds the chicken with a meat-mallet" ipropel (actor chef) (object meat-mallet) (from (location ?not-in-contact)) (to (location ?-in-contact)) (result (change (object chicken) (from (flatness ?X)) (to (flatness (greater ?X))>)) (instrument (have (object ?not-in-contact) (attribute (negation (physical-contact chicken))))) (instrument (have (object ?in-contact) (attribute (physical-contact chicken)))) (instrument (have (object chicken) (attribute (physical-contact chopping-board))))) Figure 5: Action frames corresponding to two actions from the script shown in fig. 4. described in this paper. We are still debating the con- venience of using Schank’s primitives to describe every action. Also, action frames need to be augmented by incorporating at least visual elements, as in (Kalita 1991), and time references, possibly using Allen’s in- terval algebra, (Allen 1984). Using Action Frames Obtained From a Script From the examples shown above, it is clear that lin- guistic descriptions of actions obtained from scripts do not normally include detailed information about the position, attitude, and movement of the persons and objects in the scene. Linguistic accounts of actions nor- mally describe only the essential changes in the scene, but not the implications of those changes. Therefore, to use information from scripts it is nec- essary to have an inference mechanism capable of ex- tracting the needed details from the action frames. In particular, to use the action frames generated from the TV script in the SmartCam domain, it was necessary to implement a simple inference system. It is impor- tant to make clear that our inference system is ex- tremely simple and designed only to meet the demands of our particular domain. The system was designed to infer position and movement information about hu- man beings’ hands, and physical contact and proximity among objects. The inference system is based on Rieger’s inference system for Schank’s conceptualizations, (Rieger III 1975). The inferred action frames are sub-actions, or instrument actions of the actions from which they are produced. To guarantee termination in a fast time, the inference rules are applied in a pre-determined se- quence, in a l-pass algorithm. As a typical case, the system uses as its input the action frame corresponding to the sentence ’ ‘chef wraps chicken with a plastic bag’ ’ (as shown in fig. 5) and deduces that the chef’s hands are close to the chicken. The appendix depicts a more complex example where from the action of pounding the sys- tem obtains the fact that the hands are close to the chopping board. From the PROPEL action frame shown in fig. 5, the inference system deduces some contact relations between some objects, which imply physical proximity. The SmartCam’s inference system is certainly very simple and works only for some scripts. However, the ability of approximate models to handle inaccurate in- formation helps the system to avoid becoming useless in the case of wrong inferences. For instance, in the “pounding” example, only one of the hands is in fact close to the chopping board: the hand which is grasp- ing the meat mallet is about 1 foot from the board. But, as we have seen above, such errors in positioning are admissible in the approximate world model frame- work. etermining the Onset of Actions In the examples above, the information from the script was represented and augmented by simple inference procedures. However, to use script information it is necessary to “align” the action frames with the on- going action, that is, the vision-based system need to recognize which action is happening in any given mo- ment in time. For the work presented here we have relied on man- ual alignment of the action frames to the events in the scene. All the results shown in this paper use a timed script, an extension of the script of fig. 4 which includes information about when each of the action is happen- ing. This is simplified by the fact that the we are using the simulated version of the SmartCams where the vi- sual data is pre-recorded, enabling manual annotation. The problem of visual recognition of actions is also a current object of our research. The alignment problem mentioned above can be viewed as a sub-problem of the general action recognition problem where the order of the actions is known in advance. Vision 1121 SmartCams in Action The current version of our SmartCams handles three types of framing (close-ups, medium close shots, medium shots) for a scenario consisting of the chef and about ten objects. All the results obtained so far employ only very simple vision routines similar to “ 7 based on move- ment detection by frame differencing. Figure 6 shows typical framing results obtained by the system. The leftmost column of fig. 6 displays some frames generated in response to the call “ chef . The center column of fig. 6 contains another sequence of frames, showing the images provided by the SmartCam tasked to provide “ hands The rightmost column of fig. 6 is the response to a call for a “ hands In this situa- tion, the action “ pounds the chicken with a meat-mallet 2 ) is happening. As shown above, this action determines that the hands must be close to the chopping board. This information is used by the sys- tem to initialize expectations for the hands in the be- ginning of the action (both in terms of position and movement), enabling the tracking system to detect the hands position based solely in movement information. One 80-second long video sequence is shown in the videotape distributed with these proceedings. It is also available on the WWW-web at: http://www-white.media.mit.edu/ vismod/demos/smartcas/smartcams.html The web-site also contains another performance of the same script where the chef is wearing glasses, and the actions are performed in a faster speed. The sequences were obtained by requesting the SmartCams to per- form specific shots (displayed as subtitles); the cuts be- tween cameras were selected manually. Both sequences clearly show that acceptable results can be obtained by our SmartCams in spite of the simplicity of the vision routines employed. Conclusion Approximate world models made the development of SmartCams feasible. Using the information about ac- tions from the script of the show and the control in- formation in the approximate world model, it has been possible to employ simple, fast - sometimes unreliable - vision routines to obtain the information required by TV framing. One of the major accomplishments of our research is the end-to-end implementation of a system able to deal with multiple levels of information and process- ing. A SmartCam is able to use contextual informa- tion about the world from the text of a TV script, and to represent the information in a suitable format (ap- proximate world models); updating the world model is accomplished through visual tracking, and the ap- proximate world models are used in the selection and control of vision routines, whose outputs control the movement of a simulated robotic camera. The system Figure 6: Responses to the calls “close-up chef”, “close-up hands”, and “close-up hands”. Refer to back- ground objects to verify the amount of correction needed to answer those calls appropriately. The grey areas to the right of the last frames of the first “close-up hands” se- quence correspond to areas outside of the field of view of the wide-angle image sequence used by the simulator. processes real image sequences with a considerable de- gree of complexity, runs only one order of magnitude slower than real time, and produces an output of good quality in terms of TV standards. 1122 Perception Appendix References Inferences in the “Pounding” Example The following is a manually commented printout of the action frames generated by the SmartCam’s inference system, using as input the action frame corresponding to the sentence "chef pounds the chicken with a meat-mallet ’ ‘. Only the relevant inferences are shown from about 80 action frames actually generated. The transitive rule used for the inference of proximity is sensitive to the size of the objects, avoiding its use if one of the objects is larger than the others. Allen, J. F. 1984. Towards a general theory of action an time. Artificial Intelligence 23:123-154. Bobick, A. F., and Bolles, R. C. 1992. The represen- tation space paradigm of concurrent evolving object descriptions. IEEE PAM1 14(2):146-156. Bobick, A., and Pinhanez, C. 1995. Using approxi- mate models as source of contextual information for vision processing. In Proc. of the ICCV’95 Workshop on Context-Based Vision, 13-21. Draper, B. A.; Collins, R. T.; Brolio, J.; Griffith, J.; Hanson, A. R.; and Riseman, E. M. 1987. Tools and experiments in the knowledge-directed interpretation of road scenes. In Proc. of the DARPA Image Under- standing Workshop, 178-193. Kalita, J. K. 1991. Natural Language Control of Animation of Task Performance in a Physical Do- main. Ph.D. Dissertation, University of Pennsylvania, Philadelphia, Pennsylvania. Marr, D., and Nishihara, H. K. 1978. Representation and recognition of the spatial organization of three- dimensional shapes. In Proc. R. Sot. Lond. B, volume 200, 269-294. Nagel, H.-H. 1994-5. A vision of ‘vision and language’ comprises action: An example from road traffic. Ar- tificial Intelligence Review 8:189-214. Newtson, D.; Engquist, G.; and Bois, J. 1977. The objective basis of behavior units. Journal 0-f Person- sky and Social Psychology 35( 12):847-862: Rieger III, C. J. 1975. Conceptual memory and infer- ence. In Conceptual Information Processing. North- Holland. chapter 5, 157-288. Schank, R. C. 1975. Conceptual dependency the- ory. In Conceptual Information Processing. North- Holland. chapter 3, 22-82. Siskind, J. M. 1994-5. Grounding language in per- ception. Artificial Intelligence Review 8:371-391. Strat, T. M., and Fischler, M. A. 1991. Context-based vision : Recognizing objects using information from both 2-d and 3-d imagery. IEEE PAMI 13(10):1050- 1065. 0 : action frame obtained from the script (propel (actor chef) (object meat-mallet) jto<location ?in-contact)) (from (location ?not-in-contact)) (result (change (object chicken) (from (flatness ?X)) (to (flatness (greater ?X))))) (instrument (have (object ?in-contact) (attribute (phys-cant chicken)))) (instrument (have (object ?not-in-contact) (attribute (negation (phys-cant chicken))))) (instrument (have (object chicken) (attribute (phys-cant chopping-board))))) 1 : propelling an object (0) requires grasping (grasp (actor chef) (object meat-mallet) (to hands)) 2 : grasping (1) requires physical-contact (have (object hands) (attribute (phys-cant meat-mallet))) 3 : physical-contact (0) implies proximity (have (object ?in-contact) (attribute (proximity chicken))) 4 : physical-contact (0) implies proximity (have (object chicken) (attribute (proximity chopping-board))) 5 : physical-contact (2) Implies proximity (have (object hands) (attribute (proximity meat-mallet))) 6 : the end of propelling (0) implies proximity (have (object meat-mallet) (attribute (proximity ?in-contact))) 7 : transitiveness of proximity, (3) and (6) (have (object chicken) (attribute (proximity meat-mallet))) 8 : transitiveness of proximity, (4) and (7) (have (object chopping-board) (attribute (proximity meat-mallet))) 9 : transitiveness of proximity. (5) and (8) (have (object chopping-board) (attribute (proximity hands))) Tarr, M. J., and Black, M. J. 1994. A computa- tional and evolutionary perspective of the role of rep- resentation in vision. CVGIP: Image Understanding 60( 1):65-73. Thibadeau, R. 1986. Artificial perception of actions. Cognitive Science 10:117-149. Tsotsos, J. K. 1985. Knowledge organization and its role in representation and interpretation of time- varying data: The ALVEN system. Computational Intelligence 1:16-32. Wilks, W. 1975. A preferential, pattern-seeking se- mantics for natural language inference. Artificzal In- telligence 6( 1):53-74. Vision 1123 | 1996 | 166 |
1,806 | Integrating Visual Information Across Camera Movements with a Visual-Motor Calibration Map Peter N. Prokopowicz Department of Computer Science University of Chicago Chicago, IL 60637 peterp@cs.uchicago.edu Paul R. Cooper Department of Computer Science Northwestern University Evanston, IL 60201 cooper@ils.nwu.edu Abstract Facing the competing demands for wider field of view and higher spatial resolution, computer vi- sion will evolve toward greater use of fovea1 sen- sors and frequent camera movements. Integra- tion of visual information across movements be- comes a fundamental problem. We show that in- tegration is possible using a biologically-inspired representation we call the visual-motor calibra- tion map. The map is a memory-based model of the relationship between camera movements and corresponding pixel locations before and af- ter any movement. The map constitutes a self- calibration that can compensate for non-uniform sampling, lens distortion, mechanical misalign- ments, and arbitrary pixel reordering. Integra- tion takes place entirely in a retinotopic frame, using a short-term, predictive visual memory. Introduction The competing demands for wider field of view and higher spatial resolution suggest that computer vision systems will inevitably progress towards the trade- off that evolution selected for animal vision systems: fovea1 (or spatially varying) sampling combined with frequent camera movements. Thus, the integration of visual information across camera movements is a fun- damental prc$&n for lifelike computer vision systems. Figure 1 visually evokes the nature of the task. A variety of possible solutions have been proposed, by researchers from psychology and neurophysiology as well as computer vision. These range from proposals that suggest that integration occurs in “retinotopic” coordinates, through theories that propose that inte- gration occurs in a body- or head-based frame of ref- erence, to theories that suggest integration occurs at a symbolic level of abstraction. Although the problem has received a great deal of attention, no completely convincing model has been developed. We suggest that visual integration can be achieved through a representation we call the visual-motor cal- memory- based “table” represent at ion of the relation- ibration map, and that moreover such a map can be ship between camera movements and corresponding Figure 1: A series of overlapping views taken by a fovea& or spatially-varying, camera. Slight changes in viewpoint emphasize completely different details. Any visual understanding of the scene demands integration across fixations. developmentally acquired. In this paper, we describe what constitutes a visual-motor calibration map, and how it can be used to solve problems of visual integra- tion, including change detection, perceptual stability across eye movements, and the recognition of forms from multiple fovea1 observations. In particular, a developmentally-acquired map is sufficient to replicate psychophysical results on a recognition task involving eye movements. We describe the developmental pro- cess for the map elsewhere (Prokopowicz 1994). In brief, the visual-motor calibration map is a 1124 Perception From: AAAI-96 Proceedings. Copyright © 1996, AAAI (www.aaai.org). All rights reserved. pixel positions before and after a camera movement. Such a representation is neurobiologically plausible and scales-up to a realistically sized visual integration task. The map provides, in effect, a motor coordinate basis for visual information that allows visual tasks of extent larger than a single view to be computed. The map constitutes an adaptive visual-motor calibration that can compensate for effects including spatially-varying fovea1 sampling, random pixel reordering, and arbi- trary (non-linear) lens distortions, all of which would be extraordinarily difficult to model analytically. Visual Integration and Memory-based Calibration It has long been known that our vision is many times more acute in the center of view than outside it, and that, to see, we unconsciously move our eyes two or three times every second, accumulating the details around us (Yarbus 1967). In seeking the mechanisms by which our perception can nevertheless be stable, and relatively uniform, theorists early on appreciated that disrupting the normal relationship between an in- tended eye movement and the viewpoint that would follow throws off these mechanisms: the world seems jumpy (Stratton 1897). Other experiments showed that, over several weeks, this effect can wear off, some- times completely, and later rebound when the disrup- tion is removed. These experiments suggest that in normal seeing, we unconsciously account for the ex- pected visual effects of eye movements, and that this is a learned or adaptive ability. Frames of reference in theories of perceptual integration One theory of perceptual integration holds that the stability and uniformity of perception corresponds to a stable and uniform internal visual representation, formulated in a head or body-centered frame of reference (Feldman 1985). Each succeeding fixation can be viewed as “painting” a different part of a larger inner image, one whose parts persist in mem- ory long enough, perhaps a few seconds, to accumulate a detailed picture of the scene. This representation is invariant with respect to each eye movement, except that the immediate input is directed, or shifted, into the picture according to where the eyes currently point. This stable mental picture, consisting of pixels or some other primitive visual features, constitutes the effective input for the rest of perception, which goes on as if there were no small eye movements at all. Others suggest that eye movements are not ac- counted for at this early a stage of perception, but at a much higher level of abstraction, in terms of the geometric relation between concrete objects or compo- nents (Pollatsek, Rayner, & Collins 1984). When a nose is perceived, for example, a representation of it is tagged with its visual location, taking into account the current direction of gaze. When the mouth is found, the distance between them can be inferred from the remembered position tags, even though they were per- ceived from slightly different viewpoints. Usually, this tagging is thought to go on automat- ically for each item as it is recognized. A recent and more radical proposition holds that visual features are not compulsively memorized and tagged at any level of abstraction simply on the chance that they might later be usefully integrated with other features (O’Regan & Levy-Schoen 1983). Instead, integration across eye movements occurs only for those features that are part of a working model under construction as part of some specific perceptual task. When some unknown partic- ular of the model is needed to continue with the task at hand, eye movements will be made toward where the information is likely to be found. Physiologists have something to say about these hy- potheses. So far, no invariant visual representations have been found; to the contrary, the visual system responds almost entirely to the retinotopic position of features (van Essen et al. 1991). However, it has been observed that in many areas of the visual system, ac- tivity shifts itself during an eye movement, as if to predict what the next retinotopic representation will be (Duhamel, Colby, & Goldberg 1992). This pre- dicted activity can persist even without being rein- forced with an actual input (Sparks & Porter 1983). In other words, features are envisioned retinotopically in a type of very short term visual memory. Also, it has been shown that subjects can often notice when some- thing moves slightly during a saccade, even when they haven’t been looking directly at it. Together, these experiments suggest that there could be a widespread, pre-attentive visual memory for perceptual integration that uses the current retinotopic frame of reference. This is the type of architecture we propose here. Table-based perceptual motor control Regard- less of which frame of reference and level of abstrac- tion underlies integration across eye movements, the process must have access to an accurate quantitative model of the eye/head or camera/mount geometry. Without this, it would be impossible to understand the true geometric relationship between features or objects observed from different viewpoints. Traditional com- puter vision systems, which, to date, typically have not exploited moving cameras, rely on analytically deter- mined models that are calibrated with reference stan- dards. These models are neither easy to develop nor calibrate, especially considering the trend toward ac- tive vision and the potential long-term move towards Vision 1125 Figure 2: Fovea1 images before and after a camera movement are shown in the left and center columns. Predicted image, generated with an acquired visual- motor calibration map, is at right. non-uniform sampling. Recently, there has been grow- ing interest in solving visual problems without requir- ing traditional calibration (Faugeras 1992). Our work takes the approach that the necessary perceptual- mo- tor models can and should be developed by the system itself from natural experience, and tuned continuously as part of every day activity. As the representational basis for visual-motor cal- ibration information, we propose a memory-based or table-based model of the relationship between camera movements and corresponding pixel locations before and after a movement. This representation is similar in spirit to the Mel’s hand-eye kinematic models (Mel 1990) and Atkeson’s arm-joint dynamic models (Atke- son 1990). In table-based control, the relationship be- tween two components of a system are directly stored as tuples for every usable configuration of the system. Mel’s system used the relationship between the joint angles of an articulated arm, and the retinotopic loca- tion of the arm’s end-effector as viewed by a fixed cam- era. Intermediate factors, such as the arm’s component lengths, or the camera’s focal length, are not explicitly represented. Since it concerns only the direct relation- ship between relevant variables, such a model doesn’t change form when intermediate components are added or replaced; it isn’t even necessary to appreciate what these factors are. The Visual Motor Calibration Map: a new representation for perceptual integration across eye movements At its lowest level of abstraction, the problem of per- ceptual integration across eye movements is straight- forward to state: for any particular eye movement, what is the geometric relationship between a point A viewed before the movement and a point B viewed af- ter? This is the basic metrical information needed to combine or integrate visual information, in any form, from successive viewpoints. This relationship can be motor units 00000 00000 000 00000 00000 planned eye previous movement activity fib . -- predicted activity bottom-up visual inputs Figure 3: Connectionist architecture for predicting im- ages across saccades. When a particular relative eye movement is about to be made, a motor unit produces a signal. A visual cell then predicts that its value will be that of the visual cell paired with this motor unit. After the movement, the cell receives bottom-up visual input from the sensor array. described as tuple joining the two lines of sight de- fined by image points A and B, the visual angle V between the lines, and the eye-movement vector M. It is conceivable that this relation R( A, B, A&, V) could be determined and stored. The visual motor calibra- tion map represents a similar relation that defines the image points A and B which correspond to the same line of sight (V = 0), before and after a ‘movement M. To find the corresponding post-movement image lo- cation, given an eye movement and pre-movement lo- cation, the relation can be represented as a two-place look-up table. This has a natural connectionist equiv- alent (fig. 3) that is similar to other reference frame transformation networks (Feldman 1985). To find the third value of the relation given any other two, a slightly more complex network is needed (Feldman & Ballard 1982). This relation, R(A, B, M), can be used to predict an image following a particular camera movement M: for each pixel location A, copy that pixel’s into loca- tion B if and only if R(A, B, 1M) holds. In the same way, the map makes it possible to envision the post- movement location of any feature or object. We will also see that, if you know the pan and tilt angles of each movement M, the visual motor map completely calibrates the visual-motor system. A look at the structure of IRV, our robotic visual motor system, will clarify the specific calibration prob- 1126 Perception lems that the map faces. The camera is a Panasonic GP-KS102, mounted on a computerized pan-tilt head, the Directed Perception model PTU. We restrict the head to 25 possible relative movements of roughly 2.5 to 5 degrees horizontally and/or vertically. The 400 by 400 digital image, spanning a visual angle of about 30 degrees, is resampled with an artificial fovea1 pat- tern (fig. 4) d own to 80 by 80 non-uniform pixels, to produce the images like those in fig. 1. Finally, we randomly reorder the resampled pixels (fig. 7 top). An analytic visual model must account for the optical and sampling properties of the camera and artificial fovea, and the relation between the camera and its mount, which is complicated by the optical and mechanical axes not aligning. Arbitrary pixel ordering constitutes a completely general calibration problem, as well as a worst-case model of a developing organism in which the optic nerve and other fiber bundles scramble the topological ordering originally present on the retin a. As mentioned, a table-based model must fit in a rea- sonable space. We can restrict the visual-motor cali- bration relation so that for each camera movement, ev- ery image location before the movement corresponds to at most one image location after the movement; then, the map can be represented as a table with N movements N pixels entries, 160,000 in IRV’s case. A visual system scaled to roughly human levels would have about 1 million image locations, based on the size of the optic nerve, and 100 X 100 possible relative eye movements (Yarbus 1967), requiring a map with 10 billion entries: 10 billion synapses comprise only 0.001% of the brain’s total. Acquiring a visual motor calibration map The utility of a table-based model as a representation for perceptual-motor coordination depends in large part on whether or not it is actually possible to de- termine the values that constitute the table. Obvi- ously enough, filling such a table in by hand is in- tractable. To learn the visual motor calibration map, IRV makes camera movements, systematically or at random, and fills in the table with observed correspon- dences between pixels before and after a movement. For example, if a particular blue pixel appears at lo- cation A before movement M, and then at location B after the movement, the system would add an entry for W, B, M). However, the visual motor calibration map is not amenable to such simple learning by observation and memorization, simply because, for any two views be- fore and after a camera movement, the question of which pixel corresponds to which is ill-posed. This is especially true when the pixels are arbitrarily or- Figure 4: The approximate size and location of the sampling regions in a 400x400 input, alternately col- ored black and white. Each region is actually square, with some overlap. dered. Regardless of the criteria used to find corre- sponding points in the images, many spurious corre- spondences will occur, and true correspondences will often be missed. The color- matching criteria we used, a 5% difference threshold on the red, green, and blue pixel components, generally produced about 100 false correspondences for every true one, and yet missed true correspondences more than half the time. Despite these difficulties, a successful method for acquiring the visual-motor calibration map has been developed, described in detail elsewhere (Prokopowicz 1994). Very briefly, our solution to this noisy learn- ing problem considers all apparent correspondences as evidence for a possible true correspondence, and accu- mulates evidence by repeating the movements enough times. A straightforward application of this idea ex- plodes the size of the table by a factor of Npixels, which is not feasible on IRV’s or human hardware. A time- space tradeoff is possible that reduces the growth of the table to only a small constant factor while requiring that the movements be repeated a reasonable number of times. Although it must cope with massive ambiguity and uncertainty, our algorithm is able to identify truly cor- responding image locations after only several hundred examples of each movement, in a space only ten times larger than the map size. For IRV, who makes a move- ment every 4 seconds, the whole process typically takes two days. For a human, whose eyes move about 10 times more frequently, and using a parallel implemen- tation of the algorithm, the process could be complete in roughly 80 days, even though a human can make about 400 times as many different eye movements. Vision 1127 Off-line calibration in motor coordinates The visual motor map directly supports tasks that de- mand perceptual integration, by enabling image pre- diction and short-term envisioning of old features in the current retinotopic frame. This will be clarified and demonstrated shortly. But the map also makes it possible to interpret the scrambled, distorted geome- try of a single retinotopic image, which may contain visible and remembered features. Measuring the geo- metric relationships in an image is crucial in computer vision whether the camera moves or not. The interpretation process requires that the system know the mount’s pan and tilt angles for each rela- tive movement in the table. These angles can be used to define a motor-based coordinate system for measur- ing the relative visual angle between any two image locations. If the relation R( A, B, A4) holds for image locations A and B, and platform movement M, then the image locations describe lines of sight separated by an angle proportional to M. Using A and B to find the movement M for which they correspond, it is pos- sible to uncover the true two-dimensional geometric relationship between any pair of points, provided that those points correspond to the same line of sight before and after some known camera movement. We have taken another approach, using only a small subset of all the movements needed for complete cover- age, and an off-line calibration process which yields an- other table that directly supports accurate visual-angle judgments using a motor metric. Instead of consulting the visual motor calibration map to find the motor an- gle between pairs of image locations, we use the map to assign to each scrambled, non-uniform pixel location a canonical coordinate, in a globally consistent way. Each entry in the visual motor calibration map con- strains the coordinates assigned to a pair of points such that they are separated by an angle equal to the move- ment angle for which they correspond (fig. 5). The lo- cal constraints in the map can not normally be satisfied simultaneously, but the total error can be minimized. We use a numerical simulation of a pseudo-physical process that pair-wise forces points into a configura- tion satisfying their constraints. The result is a consistent assignment of a motor- based coordinate to each image location. As the table is filled in and refined with more examples, the con- straints more accurately reflect the underlying imaging geometry (fig. 6). Fig. 7 (top) shows a fovea1 image be- fore and after arbitrary pixel reordering. During map development, the off-line calibration process reorders and redistributes the pixels to their original relative positions. This assignment compensates for any opti- cal distortions, sampling geometry, mechanical align- Figure 5: Three mutually corresponding pairs of pix- els (A,B) (B,C) and (C,A) whose relative locations are not globally consistent with the movement vectors ml, m2, and m3, over which they have been found to cor- respond. ment (fig. 8), and, in the reordering of the pixels. general case, any arbitrary Using the visual motor calibration map for perceptual integration across eye movements The rest of this paper shows how the visual motor cali- bration map and motor-calibrated image locations can be used for two specific tasks that were designed by psychophysical researchers to measure human capac- ity for perceptual integration across eye movements: noticing small movements that take place during the saccade itself, and judging large shapes that sented gradually across a series of fixations. are pre- Stable perception and change detection across saccades A stable world is perceived as stable across saccades under normal conditions, but not when the viewpoint following a movement isn’t as expected, nor when something substantially moves during the sac- cade. These observations motivated the hypothesis that we predict what we will see after a movement and compare it with what we actually get (fig. 9). We tested this feasibility of this hypothesis by using IRV’s acquired visual motor calibration map as an image pre- dicting network (fig. 3). D uring an eye movement, the non-uniform pixels of an image were rearranged, ac- cording to the map, to predict the image that would follow the movement (fig. 2). The actual and predicted images were compared pixel by pixel, and those that differed by less than a 20% threshold were considered to match. For a largely stable scene, most of the image is predicted correctly; the scene is perceived as stable. In fig. 9, two small areas are not predicted correctly. These correspond to the area where a mug was dis- placed slightly on the shelf; IRV noticed this discrep- ancy and registered its surprise. Humans performed similarly in an experiment that measured the sensitiv- ity to small displacements during a saccade (Bridge- man, Hendry, & Stark 1975). Both humans and IRV 1128 Perception Figure 6: Each pixel of the scrambled, foveated visual representation is assigned a motor coordinate through a pair-wise constraint satisfaction process. Here, the pixels are shown migrating to a pattern that reveals the original sampling distribution (fig. 4). This occurs gradually over approximately 25,000 eye movements, which takes about 24 hours for the robot. also notice small global scene shifts during a saccade. Summarizing, through an acquired visual motor cal- ibration map, as embedded in a predictive, retinotopic connectionist architecture, IRV perceives the stabil- ity of the external world across camera movements, despite the radically non-uniform changes that result from the movements. Also, IRV can detect small, local scene displacements that occur during eye movements. Recognition across eye movements Each fixa- tion of a non-uniform sensor gives a highly incomplete view of the world (fig. 1). A crucial component of nor- mal perception under these conditions is recognizing the large-scale relationship among the details acquired from each view. Obviously, this is not the only problem involved in general object recognition, but any theory of recognition that concerns moving, non-uniform sen- sors must support perceptual integration of large-scale visual geometry across frequent movements. We propose that when a form is so large that its defining features are not simultaneously visible, a quick series of movements acquires the features foveally, and the features viewed earlier, although no longer recog- nizable at low resolution, are envisioned in their new retinotopic locations. The basis for visual integra- tion across eye movements is a mental relocation of Figure 7: Top row: An example of how IRV inputs during development are resampled non-uniformly (L) and then reordered (R). Next rows: Each picture shows the same scrambled pixels placed in canonical motor coordinates. As IRV develops an accurate visual-motor calibration map from experience, the assignment of motor-coordinates for distorted and scrambled pixels improves. This sequence represents learning during ap- proximately 25,000 eye movements. all visual features into a new frame of reference with each eye movement. This process can apply to any retinotopic feature map. These maps preserve the non- uniform distribution of spatial acuity, because it is not possible to represent all visual information at the high- est acuity. The visual motor calibration map provides exactly the geometric information needed to envision the retinotopic location of an object or feature after an eye movement, with the same spatial resolution of the sensor. We have already seen that simple color features (pixels) can be envisioned in their new retinotopic lo- cation. The same mechanism is used here to envision the vertices of a triangle as they are presented on suc- cessive fixations. Then, the actual geometry of the triangle is judged using the motor-based coordinates Vision 1129 Figure 9: Top: Consecutive, fovea1 views of a scene. A visual robot will want to know if anything in the scene changed from one view to the next. The com- parisons needed to do this are one form of perceptual integration across eye movements. Bottom: During the camera movement, an expected image was predicted; most of the scene was confirmed (faded). The two ar- eas not predicted correctly correspond to where a mug was moved during the camera movement (shown ana- lytically unresampled at right). Figure 8: In this sequence, the camera was twisted with respect to the pan and tilt axes. The developing visual motor calibration map gradually uncovers the true imaging and sampling geometry. of the envisioned vertices, based on the off-line image calibration process described earlier. the angle between the dots was determined using the motor-based coordinates of each envisioned point. In ten trials, the average absolute error in perceived angle was 3 degrees. The perceptual task is a replication of a psychophys- ical experiment that measures human accuracy of form integration across eye movements (Hayhoe, Lachter, & Feldman 1990). The task is straightforward: judge if the angle formed by three points of light, presented one at a time on successive fixations, is obtuse or acute. An accurate judgment entails knowing the relative po- sitions of the dots with respect to each other, which in turn depend on the size and direction of the in- tervening eye movements. Since the presentation is entirely empty except for the single point on each fixa- tion, the subject has no visual cue to judge how much the viewpoint changed; only non-visual, or so-called extra-retinal (Matin 1986), eye position information can be used. The subjects judged the angles to within a threshold of six degrees. If a single stable visual cue persisted across the saccades, accuracy roughly dou- bled, and was equal to the control case in which all three points were presented simultaneously. Conclusions We have found that the geometric knowledge required for integrating visual information across camera move- ments can be represented conveniently as a visual- motor calibration map. The map defines image points that correspond to the same sight lines before and after a particular camera movement. It has an equivalent connectionist network that can predictively shift re- membered visual information, during a camera move- ment, into the new retinotopic reference frame. The map and its network constitute the representational basis for an alternative to theories relying on sta- ble, head-centered reference frames hypothesized to exist in our visual systems. Such eye-position invari- ant visual responses have yet to found, while pre- dictive activity shifts of the sort proposed here are widespread (Duhamel, Colby, & Goldberg 1992). In our experiments, three dots, defining a right an- gle, were presented one at a time, with small, random camera movement between each presentation. During the camera movement, any previously acquired dots were mentally shifted by IRV into a new, predicted retinotopic position. After the third dot was shown, The experiments described here show that useful processes demanding perceptual integration can be carried out, in a retinocentric frame, by using learned 1130 Perception Figure 10: Schematic form-integration problem. Each frame shows the image of one corner of a quadrilateral, relative to the center of view. In order to determine the overall size and shape of the figure, it is necessary to combine information from the four images using knowl- edge of the intervening eye movements. visual expectations. Unexpected visual changes that occur during eye movements can be detected by notic- ing differences between actual and expected pixels; at the same time, confirmed expectations constitute normal stable perception despite frequently shifting inputs. Visual features that are too small and too far apart to resolve simultaneously can be integrated from successive fixations by continually envisioning the retinotopic positions of remembered features, using the same mechanism of learned visual expectations. If the parameters of the eye/camera movements are known, the map can be used to interpret the relative positions of a pair of image points in terms of the movement for which they most closely correspond. The visual motor calibration map is intrinsically adaptive, since it is acquired from natural, ambiguous visual experience. We have shown how to extrapolate the information in the map to provide a complete cal- ibration of the visual-motor system that accounts for optical parameters and distortions, non-uniform sam- pling, mechanical misalignments, and arbitrary pixel ordering. In short, the visual motor calibration map can serve as the basis for vision with moving, fovea1 cameras, providing both a wider field of view and higher spatial resolution. References Atkeson, C. G. 1990. Using local models to con- trol movement. In Touretzsky, D. S., ed., Advances in Neural Information Processing Systems, 316-323. Morgan Kaufmann. Bridgeman, B. D.; Hendry, D.; and Stark, L. 1975. Failure to detect displacement of the visual world dur- ing saccadic eye movements. Vision Research 15. Duhamel, J. R.; Colby, C. L.; and Goldberg, M. E. 1992. The updating of the representation of visual space in parietal cortex by intended eye movements. Science 255(90):90-92. Faugeras. 1992. What can be seen in three dimes- nions with an uncalibrated stereo rig. In Sandini, G., ed., Proceedings of the 2nd European Conference on Computer Vision. Springer-Verlag. Feldman, J. A., and Ballard, D. H. 1982. Connection- ist models and their properties. Cognitive Science 6. Feldman, J. A. 1985. Four frames suffice: A pro- visional model of vision and space. Behavioral and Brain Sciences 8(2):265-289. Hayhoe, M.; Lachter, J.; and Feldman, J. 1990. Inte- gration of form across saccadic eye movements. Tech- nical report, University of Rochester. Matin, L. 1986. Visual localization and eye move- ments. In Boff, K. R.; Laufman, L.; and Thomas, J. P., eds., Handbook of Perception and Human Per- formance, volume 1. New York: John Wiley and Sons. Mel, B. W. 1990. Connectionist robot motion plan- ning: A neurally-inspired approach to visually-guided reaching, volume 7 of Perspectives In Artificial Intel- ligence. Academic Press. O’Regan, J. K., and Levy-Schoen, A. 1983. Inte- grating visual information from successive fixations: Does trans-saccadic fusion exist? Vision Research 23(8):765-768. Pollatsek, A.; Rayner, K.; and Collins, W. 1984. Integrating pictorial information across eye move- ments. Journal of Experimental Psychology: General 113(3):426-442. Prokopowicz, P. N. 1994. The Development of Per- ceptual Integration Across Eye Movements in Visual Robots. Ph.D. Dissertation, Institute for the Learning Sciences, Northwestern University. Sparks, D. L., and Porter, J. D. 1983. The spatial localization of saccade targets. ii. activity of superior colliculus neurons preceding compensatory saccades. Journal of Neurophysiology 49:64-74. Stratton, G. M. 1897. Vision without inversion of the retinal image. Psychological Review 4:342-360. van Essen, D. C.; Fellernan, D. J.; DeYoe, E. A.; and Knierim, J. J. 1991. Probing the primate visual cor- tex: Pathways and perspectives. In Valberg, A., and Lee, B., eds., Pigments to Perception. Plenum Press, New York. 227-237. Yarbus, A. L. 1967. Eye movements and vision. Plenum Press. Vision 1131 | 1996 | 167 |
1,807 | ias towards izing plans where goal minimization fails Abigail S. Gertner Learning Research & Development Center University of Pittsburgh Pittsburgh PA 15260 gertner+Qpitt.edu Abstract Domains such as multiple trauma management, in which there are multiple interacting goals that change over time, are ones in which plan recog- nition’s standard inductive bias towards a single explanatory goal is inappropriate. In this paper we define and argue for an alternative bias based on identifying contextually “relevant” goals. We support this claim by showing how a comple- mentary planning system in TraumAID 2.0, a decision-support system for the management of multiple trauma, allows us to define a four-level scale of relevance and therefore, of measurable deviations from relevance. This in turn allows definition of a bias towards relevance in the incre- mental recognition of physician plans by Traum- AID’s critiquing interface, TraumaTIQ. Introduction Domains such as multiple trauma management, in which there are multiple interacting goals that change over time, are ones in which plan recognition’s stan- dard inductive bias towards a single explanatory goal is inappropriate. Yet some kind of bias is nevertheless necessary if plan recognition is to identify a best expla- nation for observed actions. In this paper, we describe how plans produced by a complementary planning sys- tem allow us to define an a3ternative bias towards con- textually relevant goals, along with a four-level scale for relevance, which is used in the incremental recogni- tion and evaluation of physician plans. These functions are carried out by TraumAID’s interface, TraumaTIQ, which uses them to produce critiques of physician or- ders in only those cases where it could make a clinically significant difference. The task of TraumaTIQ’s plan recognizer is to build incrementally a model of the physician’s plan based on the actions she has ordered. TraumaTIQ then evalu- ates that plan and compares it to TraumAID’s plan in order to determine potential errors to comment on in I Bonnie L. Webber )epartment of Computer & Information Science University of Pennsylvania Philadelphia PA 19104 bonnie@linc.cis.upenn.edu the critique. The plan evaluation and critique gener- ation components will not be described in this paper. They are discussed in detail in (Gertner 1995). In the next section, we introduce TraumAID 2.0 and describe the representation of planning knowledge and the process by which it generates plans. We then de- scribe the plan recognition algorithm used by Traum- AID’s critiquing module, naumaTIQ, and show how the planning knowledge in TraumAID provides a recog- nition bias based on relevance. We conclude with a discussion of an evaluation performed on TraumaTIQ’s plan recognition algorithm and its implications for fur- ther system development. An Overview of TraumAID 2.0 The TraumAID system is a tool for assisting physicians during the initial definitive management of patients with multiple trauma (Rymon 1993; Webber, Rymon, & Clarke 1992). During this phase of patient care, which often requires urgent action, preliminary diag- noses are pursued and initial treatments are carried out. The current system, TraumAID 2.0, embodies a goal-directed approach to patient management. The system architecture links a rule-based reasoner that derives conclusions and goals from the evidence cur- rently available about the patient, and a planner that constructs a (partially ordered) plan for how best to address the currently relevant goals. TraumAID 2.0’s management plans have been ret- rospectively validated by a panel of three experienced trauma surgeons in a blinded comparison with actual care. Panel members preferred TraumAID’s plans over actual care to a statistically significant extent (Clarke et al. 1993; Gertner, Webber, & Clarke 1996). This result suggests that such plans could provide a valid basis for producing critiques of physician plans which could lead to improvements in patient care. To understand how general knowledge and patient- specific information in TraumAID’s planner allow us to define and use an inductive bias towards contextually Environment 1133 From: AAAI-96 Proceedings. Copyright © 1996, AAAI (www.aaai.org). All rights reserved. relevant goals in TraumaTIQ’s plan recognition, it is important to understand how TraumAID forms goals and clinically appropriate plans for addressing them. When a new piece of evidence is entered, TYaum- AID’s reasoner is triggered, forward chaining through its entire set of rules and generating a list of active goals. When rule activity ceases, the planner is invoked to determine how best to satisfy the current combina- tion of management goals and address the competing diagnostic and therapeutic needs arising from multiple injuries. TraumAID’s plans are constructed out of three types of objects: goals, procedures, and actions (see Fig- ure 1). Part of TraumAID’s general knowledge of goals con- sists of goal-procedure mappings - disjunctive lists of procedures for addressing each goal. Procedures in a mapping are ordered preferentially by their cost, ef- fectiveness, invasiveness, etc. For example, the goal NEED ACCESS CHEST CAVITY can be addressed by either PERFORM THORACOTOMY or PERFORM BI- LATERAL THORACOTOMY WITH TRANSVERSE STER- NOTOMY, but the former is preferred. Given a set of goals, TraumAID’s planner selects one procedure for each goal from its goal-procedure map- ping. Selection depends on both the a priori preference ordering and a more global need to address multiple goals efficiently, since one procedure can sometimes be used to address more than one goal. A procedure comprises an ordered sequence of ac- tions and/or sub-goals, stored in a procedure-action mapping. The use of sub-goals allows TraumAID’s planner to delay certain decisions about how to ad- dress top-level goals. For example, if TraumAID is planning to address the goal TREAT UPPER THO- RACIC ESOPHAGEAL INJURY by performing PERFORM UPPER ESOPHAGUS REPAIR, it can commit early on to its specific component actions, GIVE ANTIBIOTICS and ESOPHAGUS REPAIR AND DRAIN, while basing its choice of how to address NEED ACCESS CHEST CAV- ITY on the other currently relevant goals. Another feature of TraumAID’s goal posting and planning is that its reasoner embeds a conservative, staged strategy for selecting diagnosis and treatment goals (Rymon 1993) : goals whose satisfaction requires expensive and/or risky procedures are not included in a plan until they are justified by less costly tests or ob- servations. These strategies appear in the knowledge base as implicitly related management goals, such as a DIAGNOSE HEMATURIA (blood in the urine), which if present, triggers DIAGNOSE BLADDER INJURY, which in turn canleadto agoal TREAT BLADDER INJURY. 1134 Planning Using context to bias search in plan recognition Intelligent interaction with another agent often de- pends on understanding the agent’s underlying mentak states that lead her to act as she does. These mental states include beliefs about the world, desires for the future state of the world, and intentions to act in cer- tain ways. The process of inferring these mental states is generally referred to as plan recognition. The importance of plan recognition for automated decision support has been recognized by both our- selves and Shahar and Musen (Shahar & Musen 1995). In connection with automated decision support, plan recognition can support several elements of critiquing, including flexible plan evaluation, explanation of cri- tiques, and proposing alternative actions and goals. Since there are theoretically many possible expla- nations for any set or sequence of observations, plan recognition requires an inductive bias. Previous plan recognition algorithms, most notably (Kautz 1990)) in- corporated a bias towards minimizing the number of goals used to explain the observed actions. Such a bias is inappropriate in a domain such as multiple trauma management where, as discussed in the preceding sec- tion, a range of independent diagnostic and therapeutic goals may be active simultaneously. Other factors also constrain the kind of bias that can be used: physician orders (which serve the role of ob- served actions) are not necessarily given and entered in the order in which they are intended to be performed. TraumaTIQ therefore cannot assume that consecutive orders address the same or similar goals. In addition, physicians’ plans are not always correct. Since the set of incorrect plans is too large to encode a priori, a bias is needed that will still allow interpretation of orders that do not correspond exactly with its knowledge of clinically appropriate plans. Given these constraints, TraumaTIQ’s plan recog- nizer employs a bias towards relevance, attempting to explain physician orders as closely as possible in con- formance with the principles of trauma care encoded in TraumAID. TraumAID’s current goals and plan then provide a standard of relevance, with ways of interpret- ing deviations from relevance following from Traum- AID’s extensive general knowledge base of conclusions, goals, and actions in the domain. Several researchers have pointed out the advan- tages of using contextual knowledge and basic do- main principles to bias the search for an explana- tory plan (Huff & Lesser 1993; Hill & Johnson 1995’; London & Clancey 1982). The basic idea is that the plan recognizer can use its knowledge of what actions are appropriate in the current situation to reduce am- Goal: Treat Lower Thoracic Esophag=l hp 4 PmG Perform Lower Esophagus Repair Action: Gi Antibiotics : Esophagus and Drain Goal: N&d Access Goal: Need Access Thoracotomy~ightl I JI Action: Thoracotomy~ight] with Transverse Sternotomy I v Action: Bilateral Thoracotomy Transverse Sternotomy with ThoracotomylZeftl I 0 Action: ThoracotomylLeft] Figure 1: An example plan graph. Dotted arrows indicate disjunctive goal-procedure mappings, while solid arrows indicate conjunctive procedure-action mappings. biguities in interpreting observed actions. We believe this is an appropriate bias to use in TraumaTIQ be- cause we can assume: The head of the trauma team will have training and experience, and will usually develop plans that are similar to TraumAID’s. The head of the trauma team is more likely to have appropriate goals but be addressing them in a sub- optimal way, than to be pursuing the wrong goals altogether. While TraumAID follows a conservative strategy for pursuing diagnosis and treatment from observations, the head of the trauma team may proceed more rapidly, pursuing a goal for which TraumAID does not yet have enough evidence to conclude its rele- vance. The first two assumptions motivate a policy of giv- ing the physician “the benefit of the doubt”: if an or- dered action can be explained in terms of TraumAID’s current goal set, the physician will be assumed to be pursuing the explanatory goal(s). An ordered action can be explained if it appears in TraumAID’s plan for addressing a goal in the goal set, or if TraumAID has chosen a different action to address this goal. Informally, our plan recognition algorithm works by first enumerating the set of possible explanations for all actions that have been ordered. Each explanation consists of a path in the plan graph from the ordered action to a procedure in which the action plays a part, back to a top level goal. The path may pass through a series of sub-goals and procedures before reaching a top level goal. Since the same goal may be addressed by more than one procedure, an action may be explained by one goal in the context of two different procedures. The possible explanations are evaluated in two phases. The first phase considers the top level goals. These are sorted according to their relevance in the cur- rent situation, and the most relevant ones are selected as candidate explanations. The plan recognizer cate- gorizes potential explanatory goals on a 4-level scale: 1. Relevant goals: goals in TraumAID’s set of goals to be pursued. The third assumption allows the plan recognizer to 2. Potentially relevant goals: goals that are part of a interpret actions that could be justified by more evi- currently active diagnostic strategy, as described ear- dence. Using knowledge about the strategic relation- ships between goals, TraumaTIQ can identify when the physician’s orders may be motivated by a goal that is partially but not yet completely supported by the evi- dence. The Plan Recognition algorithm Environment 1135 3 lier. For example, if the goal of diagnosing a frac- tured rib is currently relevant, then the goal of treat- ing a fractured rib is potentially relevant, depending on the result of the diagnostic test. Previously relevant goals: goals that were once rel- evant but are no longer so, because either already addressed or ruled out by new evidence. . 4. Irrelevant goals: all other goals. The bias embodied in this phase of plan recognition is that the higher a goal is on this scale, the more likely the physician is considered to be pursuing it. Formally, phase one of the algorithm can be specified as follows: For each action a ordered, TraumaTIQ’s plan rec- ognizer extracts from TraumAID’s knowledge base a set of explanatory procedure-goal chains, PG,, that could explain the presence of that action: PG a = {(P.. . G)l,. . . , (P.. . G)n} where P is a procedure containing Q in its decom- position, and (P . . . G)i is a backward path through the plan graph ending with the goal G. Now consider the set l? = {Gi} where Gi is the top level goal ending (P . , . G)i. In rank order, I consists of: l?r the relevant goals, I’2 the potentially relevant goals, Is the previously relevant goals, and I4 all other goals. Let I’ = {Gj} be the highest ranking non-empty subset of I’. If I? is the set of irrelevant goals, halt here and add o to the plan with no ex- planatory procedure-goal chains. The second phase considers the procedures in the re- maining explanations. These are evaluated according to how strongly the physician’s other actions/orders provide additional evidence for them. All procedures in the highest non-empty category are accepted as expla- nations for the action. For simplicity, the procedures are actually sorted according to a four-level scale of evidence: 1. Completed procedures: procedures for which actions have been ordered by the physician. all the 2. Partially completed procedures: procedures for which some of the actions have been ordered. 3. Relevant procedures: procedures that are currently in TraumAID’s plan. This means that if an action could address a goal by playing a role in two different procedures, the one in TraumAID’s plan is preferred as the explanation for the physician’s action. 4. All other procedures. Formally, phase two of the algorithm can be specified as follows: 3. Let P = {Pj} where Pj is the procedure that is the child of Gj in PG,. In rank order, P consists of: PI, procedures for which all the actions have been ordered, P2, procedures for which some actions have been ordered, P3, procedures currently in T’raum- AID’s plan, and PJ, all other procedures. Let P’ be the highest ranking non-empty subset of P. 4. Select the paths PG’ E PG such that PG’ contains all paths ending with goals in I” with children in P’. Finally, the explanations with the most relevant top- level goals and the highest level of evidence (i.e., the paths in PG’) are ascribed to the physician and in- corporated into TraumaTIQ’s model of the physician’s plan. Incorporating a new explanation into the plan involves adding new procedures and goals if they are not already present, and adding links between items that are not already connected. Note that there may be more than one explanation for a given action, if the explanatory goals are equally relevant and the procedures equally manifested. For example, TREAT UPPER THORACICESOPHAGEAL IN- JURY and TREAT LOWER THORACIC ESOPHAGEAL INJURY might be accepted as explanatory goals for the action ESOPHAGUS REPAIR AND DRAIN, provided that both goals are in the same category of relevance. An example of TraumaTIQ’s plan recognition process The use of context to bias the search for explanatory goals means that ‘IraumaTIQ’s plan recognizer can dis- tinguish between goals that are otherwise equally good explanations of the observed actions. Continuing the example from Figure 1, suppose that TREAT UPPER THORACIC ESOPHAGEAL INJURY is currently the only goal in TraumAID’s relevant goal set, but the physi- cian is erroneously pursuing the goal of treating an lower thoracic esophageal injury. If the physician first orders ANTIBIOTICS, TraumaTIQ will infer that they are being given as part of the procedure to treat the upper esophageal injury, even though antibiotics may play a role in a number of other procedures, including treating a lower thoracic esophageal injury. Next, if the physician orders a BILATERAL THORA- COTOMY in order to get access to the left chest, this action will also be inferred as part of TREAT UPPER THORACIC ESOPHAGEAL INJURY. However,sincethis is the less preferred procedure for addressing that goal, a comment will be produced to the effect that “Doing 1136 Planning a right thoracotomy is preferred over doing a bilat- eral thoracotomy with a transverse sternotomy to get access to the right chest cavity.” Note that such a com- ment leaves it to the physician to determine that the correct sub-goal is getting access to the right half of the chest in order to treat the upper esophagus. If, on the other hand, the physician orders a LEFT THORACOTOMY, this action is inconsistent with the goal of treating an upper esophageal injury, and so ‘I’raumaTIQ infers that it is being done for some reason that TraumAID does not currently consider relevant. This will result in the comment, “Doing a left thora- cotomy is not justified at this time. Please reconsider this order or provide justification.” Furthermore, since the physician has failed to order a procedure to get access to the right chest cavity, TraumaTIQ will also produce the comment, “Please consider doing a right thoracotomy and repairing and draining the esopha- gus in order to treat the upper thoracic esophageal injury.” Such a comment would make explicit a pos- sible discrepancy in goals between the physician and TraumAID. Analysis of the plan recognition algorithm A serious criticism of previous approaches to plan recognition is that they are computationally in- tractable (Charniak & Goldman 1993; Goodman & Litman 1992). In a time-critical domain like trauma management, it is essential for TraumaTIQ to respond quickly. The complexity of the algorithm is not really a problem in the current implementation of TraumaTIQ because of the limited size and complexity of the plans generated by TraumAID 2.0. To demonstrate how fast the implementation actually is in practice: Trauma- TIQ’s plan recognizer, implemented in Lucid Common Lisp and compiled on a Sun 4 processed 584 actions in an average of 0.023 cpu seconds per action. The problem arises when we consider extending the system to cover other areas of the body and/or blunt injury, increasing the number of procedures and goals that might explain an action in the knowledge base. To allow for the growth of the system, it is important that the plan recognition algorithm scale up efficiently. As Rymon (1993) points out, plan recognition can be formalized as a set-covering problem in which two sets of observations, symptoms and actions, are mapped onto a set of goals which covers both of them: every symptom motivates some goal and every action is moti- vated by some goal in the covering set (Figure 2). The covering set is optimized according to some cost func- tion, such as set minimization. Since the set covering problem in general is NP-hard, so is this formalization of plan recognition. Diseases/Goals: Figure 2: Plan Recognition as a set covering problem In general, any plan recognition algorithm that con- siders all possible combinations of explanatory goals for the observed actions is going to grow exponen- tially with the number of actions. The algorithm we present here avoids the need for an exponential search by grouping the potential explanations according to relevance and then greedily accepting all the explana- tions in the most relevant group. One way to look at this is that rather than trying to optimize the covering goal set according to a cost function, we simply choose to maximize the number of relevant goals in the cov- ering set. In doing this, for each ordered action a, the algo- rithm only has to consider II’1 goals, where l? is the set of possible explanatory goals for (u, and xlr,, lpr, 1 procedures, where I” .is the most relevant non-empty subset of I, and Prj are the procedures linked to each goal rj in I”. For each procedure, it has to look at ldpl actions in the procedure, and compare them with at most all of the actions that have been ordered. So the total cost of inferring a plan from a set of orders, d, is at most I4 * (lrl + CC IPrj I * ISZP, I * 14)) P-1 Thus, this algorithm is polynomial in the number of ordered actions, and linear in the number of possible goals per action, the number of goals in the most rele- vant goals set, and the number of possible procedures per action. Evaluation and Discussion We evaluated the performance of the plan recognition algorithm by applying it to the management plans from the 97 actual trauma cases from the Medical College of Pennsylvania used in the retrospective validation of TraumAID (Clarke et al. 1993; Gertner, Webber, & Clarke 1996). Out of 584 actions, 234 of them were not also part of TraumAID’s plan at the time that they were ordered. Environment 1137 Of these 234, 15 of them could be explained by a goal that was currently in TraumAID’s relevant goal set. Of the remaining 219,69 could be explained by a goal that was considered to be potentially relevant, given TraumAID’s current knowledge about the state of the patient. The plan recognizer failed to explain the re- maining 148 actions in terms of relevant or potentially relevant goals. Many of the actions that naumaTIQ fails to infer a goal for are broad diagnostic tests that can be used to look for a number of conditions, and the physician may not actually have a specific goal in mind when or- dering them. To understand physicians’ plans in such cases it is necessary to have a more complete abstrac- tion hierarchy for goals than is currently available in TraumAID 2.0. Since the knowledge base was imple- mented in support of plan generation rather than plan recognition, only goals that could be directly opera- tionalized as actions were included. Second, some goals that physicians may pursue in these cases are not included in TraumAID’s knowledge base because its designers opted not to pursue these goals under any circumstances relevant to the current domain of the system. To have a complete plan recog- nition system, it will be necessary to include such goals in the knowledge base. Summary and Conclusion In this paper we have pointed standard inductive biases, such out the weakness of as goal minimization in domains where agents can have multiple indepen- dent goals. We have further argued that the goals and plans that a decision support system would adopt un- der the circumstances can provide a workable inductive bias. To show this, we have described how TraumAID’s planner provides a standard of relevance and of mea- surable deviations from relevance, providing in turn a context for the incremental recognition of physician plans by TraumaTIQ. The approach to plan recogni- tion presented here is computationally efficient and can be applied in any domain where the user’s behavior can be predicted on the basis of contextual information. Acknowledgments This work has been supported in part by the National Library of Medicine under grants R01 LM05217-03 and ROl LM05764-01 and the Agency for Health Care Pol- icy and Research under grant ROl HS06740. Clarke, J. R.; Rymon, R.; Webber, B.; Hayward, C.; Santora, T.; Wagner, D.; and Ruffin, A. 1993. The importance of planning in the provision of medical care. Medical Decision Making 13(4):383. abstract. Gertner, A.; Webber, B.; and Clarke, J. 1996. On-line quality assurance in the initial definitive management of multiple trauma. submitted to Artificial Intelligence in Medicine. Gertner, A. S. 1995. Critiquing: Efective Decision Support in Time-Critical Domains. Ph.D. Disserta- tion, University of Pennsylvania, Philadelphia, Penn- sylvania. Goodman, B. A., and Litman, D. J. 1992. On the interaction between plan recognition and intelligent interfaces. User Modeling and User-Adapted Interac- tion 2( l-2) :55-82. Hill, R. W., and Johnson, W. L. 1995. Situated plan attribution. Joumak of Artificial Intelligence in Edu- cation. to appear. Huff, K. E., and Lesser, V. R. 1993. Integrating plausible reasoning in an incremental plan recognizer. Technical Report 93-72, University of Massachusetts, Amherst. Kautz, H. 1990. A circumscriptive theory of plan recognition. In Philip R. Cohen, J. M., and Pollack, M. E., eds., Intentions in Communication. Bradford Books. London, R., and Clancey, W. J. 1982. Plan recogni- tion strategies in student modelling: prediction and description. In Proceedings of the American Associa- tion for Artificial Intelligence, 335-338. Rymon, R. 1993. Diagnostic Reasoning and PZan- ning in ExpZoratory-Corrective Domains. Ph.D. Dis- sertation, Department of Computer & Information Science, University of Pennsylvania. Appears as Tech- nical Report MS-CIS-93-84. Shahar, Y., and Musen, M. A. 1995. Plan recogni- tion and revision in support of guideline-based care. In Working notes of the AAAI Spring Symposium on Representing Mental States and Mechanisms, 118- 126. Webber, B. L.; Rymon, R.; and Clarke, J. R. 1992. Flexible support for trauma management through goal-directed reasoning and planning. Artificial In- telligence in Medicine 4(2):145-163. References Charniak, E., and Goldman, R. P. 1993. A Bayesian model of plan recognition. Artificial Intelligence 64:53-79. 1138 Planning | 1996 | 168 |
1,808 | What is planning in the resence of sensing?* Hector J. Levesque Department of Computer Science University of Toronto Toronto, ON, M5S 3H5 Canada hector@cs.toronto.edu Abstract Despite the existence of programs that are able to generate so-called conditional plans, there has yet to emerge a clear and general specification of what it is these programs are looking for: what exactly is a plan in this setting, and when is it correct? In this paper, we develop and motivate a speci- fication within the situation calculus of conditional and iter- ative plans over domains that include binary sensing actions. The account is built on an existing theory of action which includes a solution to the frame problem, and an extension to it that handles sensing actions and the effect they have on the knowledge of a robot. Plans are taken to be programs in a new simple robot program language, and the planning task is to find a program that would be known by the robot at the outset to lead to a final situation where the goal is sat- isfied. This specification is used to analyze the correctness of a small example plan, as well as variants that have redun- dant or missing sensing actions. We also investigate whether the proposed robot program language is powerful enough to serve for any intuitively achievable goal. Much of high-level symbolic AI research has been con- cerned with planning: specifying the behaviour of intelli- gent agents by providing goals to be achieved or maintained. In the simplest case, the output of a planner is a sequence of actions to be performed by the agent. However, a number of researchers are investigating the topic of conditional plan- ning (see for example, [3, 9, 14, 171) where the output, for one reason or another, is not expected to be a fixed sequence of actions, but a more general specification involving con- ditionals and iteration. In this paper, we will be concerned with conditional planning problems where what action to perform next in a plan may depend on the result of an ear- lier sensing action. Consider the following motivating example: *Thanks to the members of the University of Toronto Cognitive Robotics group (Yves Lesperance, Fangzhen Lin, Daniel Marcu, Ray Reiter, and Richard Scherl) and to Fahiem Bacchus, for dis- cussion, comments, and suggestions. A special thanks to Yves for helping with the definition in Section 3, and to Fangzhen for asking and helping to answer the question of Section 5. This research was made possible by financial support from the Information Technol- ogy Research Center, the Institute for Robotics and Intelligent Sys- tems, and the Natural Science and Engineering Research Council. They also had to pick up the asinine AAAI fee for extra pages. The Airport Example The local airport has only two boarding gates, Gate A and Gate B. Every plane will be parked at one of the two gates. In the initial state, you are at home. From home, it is possible to go to the airport, and from there you can go directly to either gate. At the airport, it is also possible to check the departures screen, to find out what gate a flight will be using. Once at a gate, the only thing to do is to board the plane that is parked there. The goal is to be on the plane for Flight 123. There clearly is no sequence of actions that can be shown to achieve the desired goal: which gate to go to depends on the (runtime) result of checking the departure screen. Surprisingly, despite the existence of planners that are able to solve simple problems like this, there has yet to emerge a clear specification of what it is that these planners are looking for: what is a plan in this setting, and when is it correct? In this paper, we will propose a new definition, show some examples of plans that meet (and fail to meet) the specification, and argue for the utility of this specifica- tion independent of plan generation. What we will not do in this paper is propose a new plan- ning procedure. In many cases, existing procedures like the one presented in [3] will be adequate, given various repre- sentational restrictions. Moreover, our specification goes beyond what can be handled by existing planning proce- dures, including problems like the following: The Omelette Example We begin with a supply of eggs, some of which may be bad, but at least 3 of which are good. We have a bowl and a saucer, which can be emptied at any time. It is possible to break a new egg into the saucer, if it is empty, or into the bowl. By smelling a container, it is possible to tell if it contains a bad egg. Also, the contents of the saucer can be trans- ferred to the bowl. The goal is to get 3 good eggs and no bad ones into the bowl. While it is far from clear how to automatically generate a plan to solve a problem like this,’ our account, at least, will make clear what a solution ought to be. ‘However, see [IO] for some ideas on how to generate plans containing loops (when there is no sensing). Environment 1139 From: AAAI-96 Proceedings. Copyright © 1996, AAAI (www.aaai.org). All rights reserved. Classical planning There are a number of ways of making the planning task precise, but perhaps the most appealing is to put aside all algorithmic concerns, and come up with a specification in terms of a general theory of action. In the absence of sens- ing actions, one candidate language for formulating such a theory is the situation calculus [ 121. We will not go over the language here except to note the following components: there is a special constant Sa used to denote the initial sit- uation, namely that situation in which no actions have yet occurred; there is a distinguished binary function symbol do where do(a, s) denotes the successor situation to s resulting from performing the action a; relations whose truth values vary from situation to situation, are called (relational)JEu- ents, and are denoted by predicate symbols taking a situa- tion term as their last argument; finally, there is a special predicate Poss(a, s) used to state that action a is executable in situation s. Within this language, we can formulate domain theories which describe how the world changes as the result of the available actions. One possibility is a theory of the follow- ing form [15]: Axioms describing the initial situation, So. Action precondition axioms, one for each primitive ac- tion a, characterizing Poss(u, s). Successor state axioms, one for each fluent F, stating un- der what conditions F(Z, do(u, s)) holds as function of what holds in situation s. These take the place of the so- called effect axioms, but also provide a solution to the frame problem [ 151. Unique names axioms for the primitive actions. Some foundational, domain independent axioms. For any domain theory of this sort, we have a very clean specification of the planning task (in the absence of sensing actions), which dates back to the work of Green [5]: Classical Planning: Given a domain theory Axioms as above, and a goal formula 4(s) with a single free- variable s, the planning task is to find a sequence of actions2 a’ such that Axioms /= Lega@i, SO) A &do(a’, SO)) where do([ul , . . . , a,], s) is an abbreviation for do(un, do(u,-l, . . . , do(q) s) . . .)), and where LegaZ([ul , . . . , a,], s) stands for Poss(u1 ) s) A . . . A Poss(u,, do([q . . . , a,-~], s)). In other words, the task is to find a sequence of actions that is executable (each action is executed in a context where its precondition is satisfied) and that achieves the goal (the ‘To be precise, what we need here (and similarly below for robot programs) are not actions, but ground terms of the action sort that contain no terms of the situation sort. goal formula 4 holds in the final state that results from per- forming the actions in sequence).” A planner is sound if any sequence of actions it returns satisfies the entailment; it is complete if it is able to find such a sequence when one exists. Of course in real applications, for efficiency reasons, we may need to move away from the full generality of this spec- ification. In some circumstances, we may settle for a sound but incomplete planner. We may also impose constraints on what what sorts of domain theories or goals are allowed. For example, we might insist that SO be described by just a finite set of atomic formulas and a closed world assump- tion, or that the effect of executable actions not depend on the context of execution, as in most STRIPS-like systems. However, it is clearly useful to understand these moves in terms of a specification that is unrelated to the limitations of any algorithm or data structure. Note, in particular, that the above account assumes nothing about the form of the preconditions or effects of actions, uses none of the termi- nology of STRIPS (add or delete lists etc.), and none of the terminology of “partial order planning” (threats, protecting links etc.). It is neutral but perfectly compatible with a wide range of planners. Indeed the STRIPS representation can be viewed as an implementation strategy for a class of planning tasks of this form [6]. Incorporating sensing actions In classical planning, it is assumed that what conditions hold or do not hold at any point in the plan is logically determined by the background domain theory. However, agents acting in the world may require sensing for a number of reasons:4 There may be incomplete knowledge of the initial state. In the Airport example above, nothing in the background theory specifies where Flight 123 is parked, and the agent needs to check the departure screen at the airport to find out. The Omelette example is similar. There may be exogenous actions. The agent may know everything about the initial state of the world, but the world may change as the result of actions performed by other agents or nature. For example, a robot may need to check whether or not a door is open, in case someone has closed it since the last time it checked. The effects of actions may be uncertain. For example, a tree-chopping robot may have to check if the tree went down the last time it hit it with the axe. This then raises an interesting question: is there a specifica- tion of the planning task in a domain that includes sensing actions like these which, once again, is neutral with respect to the choice of algorithm and data structure? Informally, what we expect of a plan in this setting is that it be some sort of program that leads to a goal state no mat- ter how the sensing turns out. For the Airport example, an expected solution might be something like “This definition is easily augmented to accommodate mainte- nance goals, conditions that must remain true throughout the exe- cution. For space reasons, we ignore them here. 41n this paper, we limit ourselves to the first of these. 1140 Planning go to the airport; check the departures screen; if Flight 123 is boarding at Gate A then go to Gate A else go to Gate B; board the plane. Similarly, in the case of the Omelette, we might expect a plan like /* Assuming the bowl and saucer are empty initially */ until there are 3 eggs in the bowl do until there is an egg in the saucer do break an egg into the saucer; smell the saucer; if the saucer has a bad egg then discard its contents end until; transfer the contents of the saucer to the bowl end until Note that in either case, the plan would not be correct with- out the appropriate sensing action. The closest candidate that I could find to a formal speci- fication of a plan in this setting is that of Etzioni et al in [3]. In addition to a partial-order plan-generation procedure in the style of SNLP [ 111, they propose a definition of a plan, and what it means for a plan containing sensing actions to be valid (achieve a desired goal for a given initial state). Unfortunately, as a specification, their account has a num- ber of drawbacks. For one thing, it is formulated as a rather complex refinement of the STRIPS account. It deals only with atomic conditions or their negations, assumes that we I will be able to “match” the effects of actions with goals to be achieved, and so on. There are also other representa- tional limitations: it does not allow preconditions on sens- ing actions, and does not handle iteration (and so could not deal with the Omelette example). While limitations like these may be perfectly reasonable and even necessary when it comes to formulating efficient planning procedures, they tend to obscure the logic behind the procedure. There are other problems as we11.5 In describing plan validity, they insist that every branch of a plan must be valid, where a branch is one of the possible executions paths through any if-then-else in the plan. But this is overly strict in one sense, and not strict enough in another. Imagine a plan like the Airport one above except that it says that if Flight 123 is at Gate A, the agent should jump off the roof of the airport. Suppose however, that the sensing happens to be redundant because the agent already knows that the gate for Flight 123 is Gate B. In this context, the plan should be considered to be correct, despite what appears to be a bad branch. Next, imagine a plan like the Airport one above, but without the sensing action. Even though both branches of the if-then-else are handled properly, the plan is now in- correct since an agent executing it would not know the truth value of the condition. This is not to suggest that the plan- ner developed by Etzioni et al is buggy; they may never end up generating plans like the above. However, as a procedure independent specification, we should be able to evaluate the appropriateness of plans with extra or missing sensing ac- tions. Instead of building on a STRIPS-like definition of plan- ning, we might again try to formulate a specification of the planning task in terms of a general theory of action, but this time including sensing actions and the effect they have on the knowledge of the agent or robot executing them. As it turns out a theory of this sort already exists. Build- ing on the work of Moore [ 131, Scherl and Levesque have provided a theory of sensing actions in the situation calculus [16]. Briefly, what they propose is a new fluent li’, whose first argument is also a situation: informally, K(s’, s) holds when the agent in situation s, unsure of what situation it is in, thinks it could very well be in situation s’. Since different fluents hold in different situations, the agent is also implic- itly thinking about what could be true. Knowledge for the agent, then, is what must be true because it holds in all of these so-called accessible situations: Know(+, s) is an ab- breviation for the formula Vs’[K(s’, s) > $(s’)]. Beyond this encoding of traditional modal logic into the situation calculus, Scherl and Levesque provide a successor state ax- iom for Ii, that is an axiom which describes for any action (ordinary or sensing) the knowledge of the agent after the action as a function of its knowledge and other conditions before the action. Assume for simplicity that we have two types of primi- tive actions: ordinary ones that change the world, and bi- nary sensing actions, that is, sensing actions that tell the agent whether or not some condition +a holds in the cur- rent situation.6 For each sensing action a, we assume the domain theory entails a sensedfluent axiom of the form where SFis a distinguished predicate like Poss, relating the action to the fluent. For the Airport example, we might have SF(check-departures, s) 3 Parked(FZightl23, gateA, s) SF(smeZZ(c), s) G 3e.Bad-egg(e, s) A Contains(c, e, s) says that smelling a container c tells the agent whether or not c contains a bad egg e. We also assume that the domain theory entails [SF(u, s) E True] for every ordinary non- sensing action a. Under these assumptions, we have the following successor state axiom for Ii’: Poss(u, s) > { K(s”, do(u, s)) E 3s’. s” = do(u, s’) A K(s’, s) A Poss(u, s’) A [SF(u, s’) E SF(u, s)]} Roughly speaking, this says that after doing any action a in situation s, the agent thinks it could be in a situation s” iff s” is the result of performing a in some previously accessible “Later we discuss other types of sensing, involves a sensor reading. especially sensing that % may be possible to fix their definition to handle these [4]. Environment 1141 s’, provided th t a action a is possible in s’ and s’ is identical to s in terms of what is being sensed, if anything. For ex- ample, if a is check-departures, this would have the effect of ensuring that any such s’ would have Flight 123 parked at the same gate as in s. Assuming the successor state ax- iom for Parked is such that where a plane is parked is unaf- fected by sensing, any accessible s” would also have Flight 123 parked at the same gate as in s. Thus, the result is that after check-departures, the agent will know whether or not Flight 123 is parked at Gate A. More generally, the set of ac- cessible situations after performing any action is completely determined by the action, the state of the world, and the ac- cessible situations before the action. This therefore extends Reiter’s solution to the frame problem to the K fluent. Robot programs While the above theory provides an account of the relation- ship between knowledge and action, it does not allow us to use the classical definition of a plan. This is because, in general, there is no sequence of actions that can be shown to achieve a desired goal; typically, what actions are required depends on the runtime results of earlier sensing actions. It is tempting to amend the classical definition of plan- ning to say that the task is now to find a program (which may contain conditionals or loops) that achieves the goal, a sequence of actions being merely a special case. But saying we need a program is not enough. We need a program that does not contain conditions whose truth value (nor terms whose denotations) would be unknown to the agent at the required time: that is, the agent needs to know how to execute the program. One possibility is to develop an account of what it means to know how to execute an ar- bitrary program, for example, as was done by Davis in 121. While this approach is certainly workable, it does lead to some complications. There may be programs that the agent “knows how” to execute in this sense but that we do not want to consider as plans. 7 Here, we make a much simpler pro- posal: invent a programming language R whose programs include both ordinary and sensing actions, and which are all so clearly executable that an agent will trivially know how to do so. Consider the following simple programming language, defined as the least set of terms satisfying the following: 1. & and exit are programs; 2. If a is an ordinary action and r is a program, then seq(u, r) is a program; 3. If a is a binary sensing action and ri and ~2 are programs, then branch(u, rr,r2) is a program; 4. If TI and r2 are programs, then loop(ri, ~2) is a program. We will call such terms robot programs and the resulting set of terms R, the robot programming language. Informally, these programs are executed by an agent as follows: to execute nil the agent does nothing; to execute - 7Consider, for example, the program that says (the equivalent of) “find a plan and then execute it.” While this program is easy enough to generate, figuring out how to execute it sensibly is as hard as the original planning problem. exit it must be executing a loop, in which case see below; to execute &a, r), it executes primitive action a, and then r; to execute branch(u, r1, ~2) it executes a which is sup- posed to tell it whether or not some condition 4a holds, and so it executes ~1 if it does, and ~2 otherwise; to exe- cute loop(ri ,77), it executes the body ~1, and if it ends with rz& it repeats rl again, and continues doing so until it ends with e&, in which case it finishes by executing Q. The reason a loop-exit construct is used instead of the more “structured” while-loop, is that to ensure that an agent would always know how to execute a robot program, R does not include any conditions involving fluents. Thus, although we want robot programs to contain branches and loops, we cannot use the traditional if-then-else or while- loop constructs. Ris a minimal language satisfying our cri- teria, but other designs are certainly possible. Note that it will not be our intent to ever write programs in this lan- guage; it should be thought of as an “assembly language” into which planning goals will compile. Here are two example robot programs. The first, Rair, is from the Airport domain: seq(go(airport), branch(check-departures, seq(go(gateA),seq(board_pZane(FZight123),&Z)) seq(go(gateB),seq(board_pZane(FZightl23),&Z)))); the second, Regg, is from the Omelette domain: loap(body, seq(transfer(sauce< bowl), -(body, seq(transfer(sauce< bowl), loop( body, seq(transfer(saucec bowZ),niZ)))))), where body stands for the program seq(break-new-egg(saucer), branch(smeZZ(saucer), &dump(saucer),nil), exit)). There is an equivalent formulation of robot programs as finite directed graphs. See the regrettably tiny figures squeezed in after the references. Intuitively at least, the following should be clear: a An agent can always be assumed to know how to execute a robot program. These programs are completely deter- ministic, and do not mention any fluents. Assuming the binary sensing actions return a single bit of information to the agent, there is nothing else it should need to know. o The example robot programs above, when executed, re- sult in final situations where the goals of the above plan- ning problems are satisfied: Rair gets the agent on Flight 123, and Regg g ets 3 good eggs into the bowl. In this sense, the programs above constitute a solution to the earlier planning problems. To be precise about this, we need to first define what sit- uation is the final one resulting from executing a robot pro- gram Y in an initial situation s. Because a robot program 1142 Planning could conceivably loop forever (e.g. Zoop(niZ,niZ)), we will use a formula Rdo(r, s, s’) to mean that T terminates legally when started in s, and s’ is the final situation. Formally, Rdo is an abbreviation for the following second-order formula: Rdo(r, Sl, 232) d2 VP[. . . 3 w, Sl, s2,1)1 where the ellipsis is (the conjunction of the universal closure of) the following: 1. Termination, normal case: P(niz, s, s, 1); 2. Termination, loop body: P(&, s, s, 0); 3. Ordinary actions: 4. 5. 6. 7. Poss(a, s) A P(r’, do@, s), s’, 2) 3 P(seq(a, r’), s, s’, 2); Sensing actions, true case: Poss(a, s) A SF(u, s) A P(r’, do(u, s), s’, 2) > P(branch(u, r’, r”), s, s’, 2); Sensing actions, false case: Poss(u, s) A +F(u, s) A P(r”, do(u, s), s’, 2) > P(branch(u, r’, r”), s, s’, 2); Loops, exit case: P(r’, s, s”, 0) A P(r”, s”, s’, z) > P(loop(r’, r”), s, s’, 2); Loops, repeat case: P(r’, s, s”, 1) A P(&(r’, r”), s”, s’, z) > P(laop(r’, r”), s, s’, z). By using second-order quantification in this way, we are defining Rdo recursively as the least predicate P satisfying the constraints in the ellipsis. Second-order logic is neces- sary here since there is no way to characterize the transitive closure implicit in unbounded iteration in first-order terms. Within this definition, the relation P(r, s, s’, 0) is in- tended to hold when executing r starting in s terminates at s’ with e&; P(r, s, s’, 1) is the same but terminating with nil. The difference shows up when executing Zoop(r, r’): in the former case, we exit the loop and continue with r’; in the latter, we continue the iteration by repeating loop(r, r’) once more. It is not hard to show that these robot programs are de- terministic, in that there is at most a single s’ such that Rdo(r, s, s’) holds. Less trivially, we also get: Theorem 1: The following formulas are logically valid: 1. Rdo(nil, s, s’) E (s = s’). 2. Rdo(seq(u , r), s, s’) E Poss(u, s) A Rdo(r, do(u, s), s’). 3. Rdo(branch(u, r, r’), s, s’) E Poss(u, s) A [SF(u, s) > Rdo(r, do(u, s), s’)] A [lSF(u, s) > Rdo(r’, do(u, s), s’)]. 4. Rdo(loop(r, r’), s, s’) E Rdo(unwind(r, r’, loop(r, r’)), s, s’) where unwind(r, r’, r”) is defined recursively by (a) unwind(&, r’, r”) = r’ (b) unwind@, r’, r”) = r” (c) unwind&(u, r), r’, r”) = seq(u, unwind(r, r’, r”)) (d) unwind(branch(u, rl, r2), r’, r”) = branch@, unwind(ri, r’, r”), unwind(r2, r’, r”)) (e) unwind(loop(ri , rz), r’, r”) = loop(q) unwind(r;!, r’, r”)) This theorem tells us how to build an interpreter for robot programs. For example, to execute Zoop(branch(u, exit, seq(b, niZn), r), we can unwind the loop and execute branch(u, r, seq(b, Zoop(branch(u, exit, seq(b, &ZZ), r))). Note that we should not try to define Rdo “axiomatically” using axioms like these (as in [ 131, for example) since they are first-order, and not strong enough to characterize loop termination. The revised planning task With the definition of a plan as a robot program, we are now ready to generalize the classical planning task: Revised Planning: Given a domain theory Axioms and a goal formula &s’) with a single free-variable s’, the planning task is to find a robot program r in the language 72 such that: Axioms k Vs.K(s, So) > 3s’[Rdo(r, s, s’) A #(s’)] where Axioms can be similar to what it was, but now covering sensing actions and the 1-C fluent. To paraphrase: we are looking for a robot program r such that it is known in the initial situation that the program will terminate in a goal state. 8 This reduces to the classical defi- nition when there are no sensing actions, and K(s, Se) holds iff (s = So). In this case, it is sufficient to find an r of the form seq(ui, seq(u2, . . . r&n). Note that we are requiring that the program lead to a goal state s’ starting in any s such that K(s, So); in different s, r may produce very different sequences of actions. To show this definition in action, we will formalize a ver- sion of the Airport problem and establish the correctness of the above robot program and a few variants. For our purposes, there are two ordinary actions go(z) and board-plane(p), one sensing action check-departures, and three relational fluents At(a:, s), On-pZane(p, s) and Parked(p, 2, s), where x is a location, either home, airport, gateA, or gateB, and p is a plane. We have the following domain theory:’ ‘We are requiring the agent to know how to achieve the goal, in that the desired T must be known initially to achieve 4. A variant would require an T that achieved 4 starting in SO, but perhaps un- beknownst to the agent. A third variant might require not merely 4, but that the agent know that 4 at the end. So many variants; so little space. ‘We omit here unique name axioms for constants, as well as domain closure axioms, including one saying that Gate A and Gate B are the only gates. Environment 1143 Precondition axioms: Poss(board-plane(p), s) z Elx.Parked(p, x, s) A At(x, s) Poss(check-departures) z lAt(home, s) Poss(go(x), s) f 2 = airport V At(airport, s); Successor state axioms: the one above for I< and Poss(u, s) > {At(x, do(u, s)) E a = go(x) V (At(x, s) A 13y.u = go(y))} Poss(u, s) > (On-pZane(p, do(u, s) E a = board-plane(p) V On-pZane(p, s)) Poss(u, s) > (Parked@, x, do(u, s)) E Parked(p, x, s)); Sensed fluent axiom: SF(go(x), s) A SF(board-plane(p), s) A [SF(check-departures, s) E Parked(FZightl23, gateA, s)] . The goal +(s’) to be satisfied is OnpZane(FZightl23, s’). We claim that a solution to this planning problem is the earlier robot program Rair. Using the above theorem, the proof is straightforward: lo We need to show lqs, So) 3 3s’ [Rdo(Rairy s, s’) A OnpZane(FZightl23, s’)]. So let us imagine that K(s, SO) and show that there is an appropriate s’ . There are two cases: first suppose that Parked(FZightl23, gateA, s). 1. 2. 3. 4. Let al = go(airport) and si = s. The program Rair is of the form seq(ut , RI). By a precondition axiom, we have Poss(u1, ~1). So by the Theorem above, Rdo(Rairy s, s’) if Rdo(R1, do(u1, sl), s’). Let u2 = check-departures and s2 = do(ul, ~1). RI is of the form branch(u2, Rza, Rsb). By the successor state ax- iom for At, we have At(airport, s2), and so by a precondi- tion axiom, we have Poss(u2, ~2). By the successor state axiom for Parked, we have Parked(FZightl23, gateA, s2), and so SF(check-departures, ~2). So by the Theorem, Rdo(R1, ~2, s’) if Rdo(Rz,, do(u2, s2), s’). Let u3 = go(gateA) and sg = do(u2, ~2). Rza is of the form m(us, R3). By the successor state axiom for At, we have At(airport, ss), and so by a precondition axiom, we have Poss(us, ~3). By the successor state axiom for Parked, we have Parked(FZightl23, gateA, ~3). So by the Theorem, Rdo(Rz,, ~3, s’) if Rdo(R3, do(u3, Q), s’). Let u4 = board-pZane(FZightl23) and s4 = do(u3, ~3). R3 is the robot program seq(uh,niZl). By the succes- sor state axiom for At, we have At(gateA, sq), and by the successor state axiom for Parked, we have Parked(FZightl23, gateA, ~4). Thus, by a precondition axiom, we have Poss(u4, ~4). So by the Theorem, Rdo( R3, ~4, s’) if s’ = do(u4, ~4). Moreover, for this s’ we have by the successor state axiom for On-plane that On-pZane(FZightl23, s’). “‘In the following, for convenience, we will systematically be confusing use with mention: we will be saying that p where p is a logical sentence, meaning that it is true in any interpretation satis- fying the above axioms. Putting all the pieces together, we can see that for any s such that Parked(FZightl23, gate/-i, s), there is an s’ such that Rdo( Rair, s, s’) and On-pZane(FZightl23, s’), namely s’ = do([go(airport), check-departures, go(gateA), board-pZane( Flight1 23)], s). The case where Parked(FZightl23, gateB, s) is completely analogous, but leads to s’ = do([go(airport), check-departures, go(gateB), board-pZane(FZightl23)], s). Note that in each case there also exists a sequence of ac- tions not containing check-departures that puts the agent on Flight 123. However, no robot program without sens- ing would be able to generate both cases. We can also consider what happens if the agent knows initially where the plane is parked: 0 Initial State: Vs.K(s, So) > Parked(FZightl23, gateB, s). The argument above shows that Rair continues to work in this context even with the redundant sensing (there is only one case to consider now). The same argument also shows that if we replace the &go(gateA), . . .) part in Rair by any- thing at all, the program still works. Of course, the program with no sensing would work here too. Observe that the derivation above does not make use of the successor state axiom for Ii. This is because the agent was not required to know anything in the fi- nal state. It is not hard to prove that not only does Rair achieve the goal On-pZane(FZightl23, s’), it also achieves Know(OnpZane(FZightl23), s’). We can also imagine new primitive actions that depend on knowledge preconditions, such as “going to the gate of a flight,” which can only be executed if the agent knows where the plane is parked: Poss(go-gate(p), s) G A t(airport, s) A 3x. Know(Parked(p, x), s). With a suitable modification to the successor state axiom for At to accommodate this new action, an argument like the one above shows that the robot program seq(go(airport),seq(check-departures, &go-gate( Flight1 23), seq(board_pZane(FZight123),niZ)))), with no conditional branching, achieves the goal. This shows that whether a plan containing sensing needs to be conditional depends on the primitive actions available. One clear advantage of a specification like ours is that in being independent of any planner, it gives us the freedom to look at plans like these that might never be generated by a planner. This is especially useful if we are using a plan critic of some sort to modify an existing plan to reduce cost, or risk, or perhaps just to make sensing actions happen as early as possible. Plan correctness is not tied to any assumptions about how the plan was produced. 1144 Planning Are robot programs enough? Given the extreme simplicity of the robot program language R, and given that the planning task is defined in terms of the existence of robot programs, one might reasonably won- der if the restriction to R rules out goals that are intuitively achievable. Consider the following two examples: The Odd Good Eggs Example The setup is exactly like the Omelette example, except that there is an ad- ditional sensing action, which tells you when the sup- ply of eggs is exhausted. The goal is to have a single good egg in the bowl, but only if the supply contains an odd number of good eggs; otherwise, the bowl should remain empty. The More Good Eggs Example The setup is as above. The goal now is to have a single good egg in the bowl, but only if the supply contains more good eggs than bad; otherwise, the bowl should be empty. These are unusual goals, admittedly. But they do show that it is possible to encode language-recognition problems (over strings of eggs!) in a robotic setting. Informally, both goals are achievable in that we can imagine physical devices that are able to do so. The formal claim here is this: there is a robot program that achieves the first goal (which we omit for space reasons), but there is provably none that achieves the second. The proof is essentially the proof that a finite automaton cannot recognize the language consisting of bi- nary strings with more l’s than 0’s. To do so, you need the equivalent of a counter. To preserve the simple structure of R, we augment our set of primitive actions to give the robot a memory. Thus, we assume that apart from those of the background theory, we have 5 special actions, left, right, mark, erase, read-mark, and two special fluents Marked, pos, characterized by the following axioms: 1. Precondition: the 5 actions are always possible Poss(Zeft, s) A Poss(right, s) A Poss(mark, s) A Poss(erase, s) A Poss(read-mark, s); 2. Successor state: only erase and mark change Marked Poss(u, s) 3 {Marked(n, do(u, s)) = a = mark A pas(s) = n V Marked(n, s) A ~[a = erase Apes(s) = n]}; 3. Successor state: only left and right change the pos fluent Poss(u, s) 3 {pos(do(u, s)) = n E u=ZeftApos(s)=n+l V a = right A pas(s) = n - 1 V pas(s) = n A a $ left A a $ right}; 4. Sensed fluent: read-mark tells the agent whether the cur- rent position is marked SF(Zeft, s) A SF(right, s) A SF(erase, s) A SF(mark, s) A [SF(read-mark, s) E Marked(pos(s), s)]. These axioms ensure that the 5 special actions provide the robot with what amounts to a Turing machine tape. The idea is that when solving a planning task wrt a background theory I;, we look for a robot program that works wrt (X U TM), where TM is the set of axioms above. We can then prove that the More Good Eggs example is now solvable (again, we omit the program). We believe that no further extensions to R will be needed. However, to prove this, we would want to show that any “ef- fectively achievable” goal can be achieved by some robot program. But this requires an independent account of effec- tive achievability, that is, an analogue of computability for robots over a domain-dependent set of actions whose effects are characterized by a set of axioms. To our knowledge, no such account yet exists, so we are developing one. Conclusion One limitation of the work presented here is that it offers no suggestions about how to automatically generate plans like those above in a reasonable way. Of course, our specifica- tion does provide us with a planning procedure (of sorts): Planning Procedure (4) { repeat with r E R { if Axioms b Vs.K(s, SO) > 3s’ [Rdo(r, s, s’) A q5(s’)] then return r }} We can also think of the r as being returned by ans traction [5] from an attempt to prove the following: wer ex- Axioms b 3rVs..K(s, So) > 3s’ [Rdo(r, s, s’) A q5(s’)] Either way, the procedure would be problematic: we are searching blindly through the space of all possible robot pro- grams, and for each one, the constraint to check involves using the Ii fluent explicitly as well as the (second-order!) Rdo formula. However, we do not want to suggest that a specification of the planning task ought to be used this way to generate plans. Indeed, our criticism of earlier accounts was precisely that they were overly tied up with specific planning procedures. In our own work in Cognitive Robotics, we take a slightly different approach. Instead of planning tasks, we focus on the execution of high-level programs written in the GOLOG programming language [7]. GOLOG programs look like ordinary block-structured imperative programs except that they are nondeterministic, and they use the primitive actions and fluents of a user-supplied domain theory. There is a formula of the situation calculus Do(S, s, s’), analogous to Rdo, which says that s’ is one of potentially many terminat- ing situations of GOLOG program S when started in initial situation s. To execute 6 (when there are no sensing ac- tions), a GOLOG processor must first find a legal sequence of primitive actions a’ such that Axioms 1 Do(S, SO, cZo(a’, SO)), which it can then pass to a robot for actual execution. This is obviously a special case of planning. Furthermore, when S contains sensing actions, an argument analogous to the one presented here suggests that instead of a’, the GOLOG processor would need to find a robot program r [8]. With or without sensing, considerable searching may be required to do this type of processing. To illustrate an ex- treme case, the GOLOG program while ~Goul do (n u)[Appropriute(u)?; a] end, Environment 1145 repeatedly selects an appropriate action and performs it un- til some goal is achieved. Finding a sequence of actions in this case is simply a reformulation of the planning problem. However, the key point here is that at the other extreme, when the GOLOG program is fully deterministic, execu- tion can be extremely efficient since little or no searching is required. The hope is that many useful cases of high- level agent control will lie somewhere between these two extremes. A major representational limitation of the approach pre- sented here concerns the binary sensing actions and the de- sire to avoid mentioning fluents in a robot program. Sens- ing actions that return one of a small set of values (such as reading a digit on a piece of paper, or detecting the colour of an object) can be handled readily by a case-like construct. Even a large or infinite set might be handled, if the values can be ordered in a natural way. But suppose that sensing involves reading from a noisy sensor, so that instead of returning (say) the distance to the nearest wall, we get a number from a sensor that is only correlated with that distance. An account already exists of how to characterize in the situation calculus such sensing actions, and the effect they have not on knowledge now, but on degrees of belief [ 11. However, how robot programs or planning could be defined in terms of this account still re- mains to be seen. References [l] F. Bacchus, J. Halpern, and H. Levesque. Reasoning about noisy sensors in the situation calculus. In Proc. of IJCAI-95, pp. 1933-1940, Montreal, August 1995. Morgan Kaufmann Publishing. [2] E. Davis. Knowledge preconditions for plans. Techni- cal Report 637, Computer Science Department, New York University, 1993. [3] 0. Etzioni, S. Hanks, D. Weld, D. Draper, N. Lesh, and M. Williamson. An approach to planning with incom- plete information. In Principles of Knowledge Rep- resentation and Reasoning: Proceedings of the Third International Conference, pp. 115-125, Cambridge, MA, 1992. Morgan Kaufmann Publishing. [4] 0. Etzioni and S. Hanks. Personal comm., 1995. [5] Cordell C. Green. Theorem proving by resolution as a basis for question-answering systems. In Machine In- telligence 4, pp. 183-205. Edinburgh University Press, 1969. [6] F. Lin and R. Reiter. How to progress a database II: The STRIPS connection. In Proceedings of ZJCAI-9.5, pp. 2001-2007, Montreal, Aug. 20-25, 1995. [7] H. J. Levesque, R. Reiter, Y. Lesperance, F. Lin, and R. Scherl. GOLOG: A logic programming language for dynamic domains. To appear in the Journal of Logic Programming, 1996. [8] Hector J. Levesque. The execution of high-level robot programs with sensing actions: theory and implemen- tation. In preparation, 1996. 1146 Phnning [9] K. Krebsbach, D. Olawsky, and M. Gini. An empirical study of sensing and defaulting in planning. In Proc. of 1st Conference on AI Planning Systems, pp. 136-144, San Mateo CA, 1992. [lo] Z. Manna and R. Waldinger. How to clear a block: A theory of plans. Journal of Automated Reasoning, 31343-377, 1987. [ll] D. McAllester and D. Rosenblitt. Systematic non- linear planning. In Proc. of AAAI-9I, pp. 634-639, Menlo Park, CA, July 199 1. [ 121 J. McCarthy and P. J. Hayes. Some philosophical prob- lems from the standpoint of artificial intelligence. In Machine Intelligence 4, pp. 463-502. Edinburgh Uni- versity Press, 1969. [ 131 R. C. Moore. A formal theory of knowledge and ac- tion. In J. R. Hobbs and R. C. Moore, editors, For- mal theories of the common sense world, pp. 3 19-358. Ablex Publishing, Norwood, NJ, 1985. [ 141 M. Peot and D. Smith. Conditional nonlinear planning. In Proc. of 1st Conference on AI Planning Systems, pp. 189-l 97, San Mateo CA, 1992. [ 151 R. Reiter. The frame problem in the situation calculus: A simple solution (sometimes) and a completeness re- sult for goal regression. In Vladimir Lifschitz, edi- tor, Artificial Intelligence and Mathematical Theory of Computation: Papers in Honor of John McCarthy, pp. 359-380. Academic Press, San Diego, CA, 199 1. [16] R. Scherl and H. Levesque. The frame problem and knowledge-producing actions. In Proc. of AAAI- 93, pp. 689-695, Washington, DC, July 1993. AAAI Press/The MIT Press. [ 171 M. Schoppers. Building plans to monitor and exploit open-loop and closed-loop dynamics. In Proc. of 1st Conference on AI Planning Systems, pp. 204-2 13, San Mateo CA, 1992. break-new-egg(saucer) 1 ,1 smell(saucer) dump(saucer) Getting on Flight 123 0 I transfer(saucer.bowt) go(airpoN 4 1 break-new-egg(saucer) check-departures I 0 J \ w(gateA) w(gaW 1 ,1 smell(saucer) dump(saucer) 0 I transfer(saucer.bowl) 1 Getting 3 good eggs board(flightl23) into the bowl break-new-egg(saucer) 1 ,1 smell(saucer) dump(saucer) 0 c transfer(saucer.bowl) | 1996 | 169 |
1,809 | Learning other agents7 preferences in multiagent negotiation H. H. Bui, D. Kieronska and S. Venkatesh Department of Computer Science Curtin University of Technology Perth, WA 6001, Australia buihh, dorota, svethaQcs.curtin.edu.au Abstract In multiagent systems, an agent does not usu- ally have complete information about the pref- erences and decision making processes of other agents. This might prevent the agents from mak- ing coordinated choices, purely due to their ig- norance of what others want. This paper de- scribes the integration of a learning module into a communication-intensive negotiating agent ar- chitecture. The learning module gives the agents the ability to learn about other agents’ prefer- ences via past interactions. Over time, the agents can incrementally update their models of other agents’ preferences and use them to make better coordinated decisions. Combining both commu- nication and learning, as two complement knowl- edge acquisition methods, helps to reduce the amount of communication needed on average, and is justified in situations where communica- tion is computationally costly or simply not de- sirable (e.g. to preserve the individual privacy). Introduction Multiagent systems are networks of loosely-coupled computational agents that can interact with one an- other in solving problems. In such systems, it is often not feasible for any agent to have complete and up-to- date knowledge about the state of the entire system. Rather, the agents must be able to work together, with- out prior knowledge about other agents’ mental (inter- nal) states. Traditional work in distributed problem solving re- lies heavily on the communication between problem solving nodes in order to provide the kind of coordina- tion necessary in a distributed system (Bond & Gasser 1988). Research in theories of agency based on the formulation of agents’ mental states also uses commu- nication as the only method for acquiring knowledge about other agents’ mental states (Woodridge & Jen- nings 1995). Driven by the costs and problems as- sociated with communication, recent work in multia- gent learning has suggested learning as an alternative knowledge acquisition method. It has been shown that 114 Agents even without communication, agents can learn to co- ordinate their tasks in simple multiagent settings (Sen & Sekaran 1995),(S en, Sekaran, & Hale 1994). Our on-going research goal is to design a generic ar- chitecture for negotiating agents. In this domain, the importance of learning from previous experiences was documented in (Sycara 1989) where case-based reason- ing techniques were used to reduce the communication overhead in the PERSUADER system. Since nego- tiation is a communication-intensive task, rather than using learning as a complete replacement for communi- cation (Sen & Sekaran 1995), we view both communi- cation and learning as two complementary knowledge acquisition techniques, each with its own strengths and weaknesses. Communication, typically, is more expen- sive (in terms of time and resource) than computa- tion and can become a bottleneck of the negotiation process. However when one asks the right question and gets back the correct response, the information one gathers is certain. On the other hand, learning is performed locally by each individual agent and is thus less costly, however, the information acquired is mostly uncertain. The contrasting characteristics of the two knowledge acquisition methods make a hybrid approach an attractive alternative. This paper describes how to integrate a learn- ing component into a reactive agent architecture in which the agents negotiate by refining a joint intention gradually until a common consensus is reached (Bui, Venkatesh, & Kieronska 1995). Here, we assume that the agents are cooperative and sincere. We use a sim- ple learning mechanism that allows an agent to make predictions about other agents’ preferences by building statistical models of others’ preference functions from its past interactions with them. The learning mech- anism helps to reduce the amount of communication needed, and thus improves the overall efficiency of the negotiation process. The approach is illustrated with an example from the distributed meeting scheduling domain. The paper is organised as follows: the following sec- From: AAAI-96 Proceedings. Copyright © 1996, AAAI (www.aaai.org). All rights reserved. tion introduces the negotiation context under which our agents interact; next, we describe the learning mechanism and how it is integrated into the agent’s architecture; finally we show some initial experimental results in the distributed meeting scheduling domain and provide a preliminary evaluation of the approach. The negotiation context Definitions We use the term negotiation contezt to refer to situ- ations where a group of agents with different prefer- ences are trying to achieve a common agreement. Due to the distributed nature of the problem, the agents, at best, possess only partial knowledge about other agents’ preferences. Such problems turn out to be ubiquitous in Distributed Artificial Intelligence (Bond & Gasser 1988). Although a number of negotiation protocols (Smith 1980),(Conry, Meyer, & Lesser 1988) and agent architectures (Laasri et al. 1992) have been proposed, attempts to formalise and construct a generic agent architecture have proved to be quite complex (Woodridge & Jennings 1994). In order to aid the clarity of further discussions, we define here a formal notion of a simple negotiation con- text N as follows: A group of agents A involved in the negotia- tion. Subsequently, we will use the capital letters A, B, C.. . to denote members of A. o A domain V represents the set of all possible agree- ments. Let 6 C V be a subset containing some agree- ments. We use the notion of intention Int(A, 6) to denote agent A “intends to look for the final agree- ment” within 6. Similarly, JInt(A, 6) denotes the joint intention of all the agents in A to look for the final agreement within 6. If Jlnt(A, 5) holds, 6 is termed the current agreement set of the agents in A. For each agent A E A, a function fA : V + R (the set of real numbers) represents the preferences of agent A over the set of possible agreements V. For 6 C V, TA(6) denotes the mean Of fA(d), d E 6. One can think of the preference fA (d) as the amount of money A would earn if d were accepted as the final agreement. The preferences are thus additive, and we define the preferences of the group A by the sum of its members’ preferences F4(d) = xAEA fA(d) ‘. The negotiation process Throughout the negotiation process, the agents at- tempt to find a common agreement by refining their l Since the pr oduct of positive numbers corresponds to a sum via a logarithmic transformation, the results of this paper still applies when the group’s preferences are defined as the product of the (positive) individual’s preferences. joint intentions incrementally. At the start of the ne- gotiation process, 5 = V. The incremental behaviour of the negotiation process is guided by an agreement tree defined as a tree structure whose nodes are agree- ment sets with the following properties: (1) the root node is V9 (2) all the leaf nodes are singleton sets, and (3) the set of all children of a node is a partition of that node. At the k-th iteration of the negotiation process, each agent A would attempt to refine the current joint agreement set 61, (at level k in the tree structure) to some new tentative agreement set S$l+1 C Sk (at level k+l). The choice of Sf+l depends on A’s perception of the expected utility of those possible agreements within the agreement set 6f+1. The choice of refinement be- comes the agent’s individual intention and is broadcast to other agents in the group. If all individual refinement choices agree, the group’s refinement choice becomes the new joint agreement set of the agents. Otherwise, the differences in the indi- vidual refinement choices are resolved through further communication between the agents in three steps: (1) each agent collects other agents’ preferences of its own refinement choice; (2) each agent calculates the group’s preference for its refinement choice and uses this pref- erence as a new ranking value for its own choice; and (3) the agents choose a winner among themselves on the basis of maximal ranking value (to assure a clear winner, a small random perturbation can be added to the ranking value in step 2). Subsequently, the win- ner’s refinement choice is adopted by the whole group of agents. At the end of the k-th iteration, all the agents in the group should form a new agreement set &+I c 61, or decide that the agreement set 61, is over-constrained and backtrack to 6k-1. The iterative negotiation pro- cess ends when either an agreement set 6, = (d} at the leaf level is reached, or 50 = V is over-constrained itself. In the former case, a solution is found whereas in the latter case, the negotiation is regarded as failing. roblems with incomplete knowledge Crucial to the performance of the above negotiation protocol is the decision involved in choosing the refine- ment of an agreement set. Ideally, the agents should choose a refinement &+I for 51, so as to maximise the grOUp'S preferenCeS Fd. Given that an agent A’s preference value for a refine- ment choice 5 is TA(S), th e sum of all group members’ preferences for 6 is-@d(b) = x&A ?A(@. In the ideal case where every agent uses Fd (6) to select a refine- ment, all individual refinements and intentions will be the same, hence a new agreement set can be formed immediately without further complication. Negotiation & Coalition 115 To see why the ideal case might not happen in prac- tice, let’s reWrite pd(6) as: Unfortunately, the component Fother,A (6) is usually not readily available to A since it requires knowledge about other agents’ preference functions. In situations where other agents are eager to reveal their preference functions, A can directly ask other agents about their preferences (ask-first selection method). Such an ap- proach requires additional communication and may not be feasible in circumstances where exposure of individ- ual preference is not desirable. When asking others is costly, the agents can choose the refinement by maximizing only their own prefer- ences (don’t-ask selection method). However, this ap- proach usually leads to diverging and conflicting indi- vidual intentions and requires a lengthy conflict reso- lution stage. We propose the use of learning as an alternative knowledge acquisition method to counter the prob- lem of incomplete knowledge. If it is not desirable to acquire the knowledge from asking questions di- rectly, why not learn to predict what the answers would be? Furthermore, in our negotiation context, making a false prediction will not result in a catastrophe (the worst situation is when extra exchange of messages is needed). With a mechanism to make reasonably good predictions about other agents’ preferences, we are likely to improve the efficiency of the whole nego- tiation process. Learning other agents’ preferences Learning data A negotiating agent throughout its lifetime will partic- ipate in a potentially large number of different negoti- ation contexts. Although each negotiation context has a different set of participating members, closely affili- ated agents are likely to engage in the same negotiation context more often. Furthermore, the domains of these negotiation contexts are usually subsets of a common domain. For example, in resource allocation, the set of resources to be allocated might be different from one negotiation to another, however, they are usually drawn out of one common set of resources frequently shared by the agents. In meeting scheduling, the time windows for the meetings to be scheduled are different, however, again, they are subsets of one common time line. We denote this common domain of all negotiation contexts by V*. Formally, V* is the union of the do- mains of all negotiation contexts: V* = Un/Vti where VN denotes the domain of the negotiation context N. An agent has the opportunities to acquire sample data about others’ preference functions via the number of exchanges of preferences taking place in previous negotiation contexts. For example, from the agent A’s viewpoint, the accumulated samples of f~ are the set of values f~ (o!) for some random d’s drawn out of V* . This sample data in turn can help the agent in making predictions about others’ preferences should they be in the same negotiation context in the future. earning mechanism This subsection describes how an agent can use a sim- ple learning mechanism to accumulate samples of other agents’ preference functions and make statistically- based predictions of their future values. To see how the mechanism works, consider a negoti- ation context with A = (A, B, C}. Facing the problem of choosing a refinement for the current agreement set, agent A is trying to guess the values of agents B and C’s preference functions fB and fc. Like most learning methods, the first stage is feature selection. In this stage, the domain V* is partitioned into a number of subsets (Ei}, where each subset cor- responds to a region in the feature space. The values of fB(d) with d chosen randomly from Ei then define a random variable XB,E; on the sample space Ei. Given a point d E Ei’ the estimation of fB(d) is characterised by p(fBtd> = z/d E Ei) which is the probability den- sity function of XB,E;. If we know that a refinement choice b is a subset of Ei, we can proceed to approximate the function F &her,A(6) = TB(6)+Fc(6) by a random variable XE;, the sum of two random variables XB,E; and XC,E;, with the mean y,$ = XB E; + rc,E; and the stan- dard deviation o2 ( XE; ) = ~“(XB,E~)+O”(XC,E~). For the purpose of predicting Fother,A(fs), agent A can use its expected value z,?& with ~(XE,) as the prediction expected error 2. To minimise the prediction expected error, the par- tition (Ei} should be formed so that given an agent B, its preference values for the agreements in Ei is uniform (e.g. 0(X&) is small). This partition is sim- ilar to the organizational structure for storing cases in the case-based reasoning approach (Sycara 1989). The formation of such a partition largely depends on the domain structure and knowledge. In the meeting scheduling domain, we choose to partition V* (which is the time line) into periodic intervals such as all Mon- day mornings, Monday afternoons, Tuesday mornings, etc. Since the users tend to have appointments that 2A further conjecture based on the central limit theo- rem is that, when there are many agents participating in the negotiation context and when their preferences are sta- tistically independent, the probability density function of XE; would be approximately gaussian. 116 Agents happen on a regular basis, these periodic intervals can The distributed meeting scheduling yield good predictive value. domain If A has to choose among two refinement choices S1 and 62, the agent will use the following steps to decide which refinement choice to take: We chose the distributed meeting scheduling domain as a testbed for the performance of the learning agents. In distributed meeting scheduling, the agents are the managers of the users’ personal schedules. In a typical meeting scheduling situation, given an allowed time- window, a group of agents have to decide on a com- mon slot within the given time-window as their agreed meeting time. Meanwhile, each member of the group has different preferences of what the desired slot should be and no agent has complete information about other agents’ preferences. The problem is further compli- cated by the desired property to preserve the privacy of the personal schedules. This places an upper bound on the amount of information that can be exchanged among the group of agents. a Identify the sample spaces that Si belongs to. As- sume that Si C Ei. @ From the samples of fB and fc accumulated from A’s previous exchanges of preferences with B and C, calculate the average of fB(d) and fc(d) with d E Ei. The results give the estimates of XB,E; and ZCY,E; respectively. If the distribution of XB,E; (or XC,Ei) h g c an es over time, a better estimation can be obtained if A only remembers and averages the most recent m values of fB (d), (d E Ei) for some positive integer m. We have: 0 Choose 6’ such that FF”(@) is maximum. Generally, for an arbitrary number of agents in the group, the function used by A in evaluating its refine- ment choices Fyt is given by: FTt(6) = ?A@) + x %,Es Ocd-A where EJ 3 6. From A’s point of view, FTt is the expected value of the group’s preference function Fd. The learning mechanism involves incrementally updating the func- tion FTt when new data is available. To incorporate learning into the neptiation scheme, instead of using the usual function f A (6) to evaluate A’s refinement choices, the new function FTt is used. Since 3’Ft in- cludes the pattern of other agents’ preferences, it can facilitate A and other learning agents in making better coordinated decisions. Benefits of learning Evaluation of the hybrid method requires the consid- eration of many factors, such as how often the agents need to conduct new negotiations and if there are any patterns to individual agent’s preferences. In this sec- tion, we present preliminary results of applying the proposed hybrid method to solve the meeting schedul- ing problem. A single meeting scheduling scenario involves a set of participating agents, a time window (W), the duration for the meeting being scheduled (6), and for each agent A a set of existing appointments (App~ = (api}) such that V’i # j, ami namj = 0. For each appointment app we use cost(app) to denote its cancellation cost. The continuous timeline is discretised and modelled by the set Time = {to + i 1 i = 0, 1,. . .}. Such a meeting scheduling scenario constitutes a negotiation context in which: a~ The set of all agents A are the set pating in the meeting scheduling. of agents partici- e The set of all possible agreements V is derived from W and I as 2, = (t E Time 1 [t, t + Z] C_ W}. o For each agent A and a possible agreement t E V, the preference of A for t is fA(t) = c -c0.9t(upp) aPP E &PA ~PPm,t+q #0 The domain of all negotiation contexts I)* becomes the timeline itself V* = Time. For the learning mecha- nism, we partition Time into periodic intervals such as morning, afiernoon (daily intervals) or monday morn- ing, monday afternoon (weekly intervals). We choose this partition since the preferences of the agents also tend to have daily and weekly periods. Preliminary analysis Table 1 compares the expected performance of three refinement selection methods: don’t- ask: (choose the re- finement to maximise own preference), ask-first (query other agents’ preferences first before choosing a refine- ment), and learning (as described above). We assume Negotiation & Coalition 117 Refinement Function Prior-decision messages Post-decision messages selection method maximized Querying others Resolving conflict Ask-first Fd hl(L)n2 Don’t-ask fA 0 o(lo!&)n2) Learning Expected value of Fd 0 Wos(L)n2> Table 1: Comparison between refinement selection methods that the agents are using a binary tree as their agree- ment tree. The performance of each method is mea- sured in terms of the total number of messages ex- changed among the set of agents in one negotiation context. Here n denotes the number of participating agents and L is the number of possible agreements. The numbers show that the ask-first selection method always incurs a number of messages of (Zog( L)n2) or- der of magnitude, which is the expected performance of the don’t-ask: and learning selection method in the worst case. The trade-off exists, however, since the for- mer method always guarantees to find the best optimal solution while the latter two do not. Further, it is interesting to evaluate the relative per- formance of the agents using ddt-ask: selection and those augmented with a learning component. Since learning agents are more aware of other agents’ pref- erence functions, they can be better coordinated in selecting a refinement choice even without any prior- decision communication. Experiments with these two types of selection methods are presented in the next subsection. Experiments Our preliminary set of experiments involve two agents implemented in Dynaclips 3.1/Clips 6.0 running under SunOS operating system. The agents can run with or without the learning module. The aim of the experi- ment is to collect initial data confirming the benefits of the agents running with the learning module as op- posed to those running without learning. We model the timeline as a set of discrete points 30 minutes apart. Each day, and for a period of 20 days, the agents have to schedule a meeting with the dura- tion of 2 units and within the time-window [8,16.30]. The possible agreement set ZJ with its tree structure is shown in figure 1. The agents’ preferences are pe- riodic, with the period of 1 day. Noise can be added to the preference functions and this is interpreted as new non-periodic appointments, or the cancellation of existing periodic appointments. The results of the experiment shown in figure 2 demonstrate that learning agents perform relatively su- perior when compared to the agents running without the learning module. The difference in performance, however, is reduced as the level of noise is increased. 118 Agents Day * * I , I I Morning +‘ Afternoon [ I I I : : I I I I D = {8,8.30,9,.., 15.30) W = [S, 16.301 I=2 Figure 1: The domain V and its tree structure This agrees with common sense as learning method would only show its benefits if the agents’ preferences are periodic and can be learned. Also, the more often the non-learning agents are in conflict, the relatively better are the learning agents. This is because the learning mechanism works by learning from previous conflicts to prevent the same type of conflict from oc- curring in the future. iscussion and Conclusions In this paper, we have presented a method to incorpo- rate a learning component into the negotiating agent architecture. This gives the agents the ability to learn about other agents’ preferences from the interactions during their past negotiations. With the knowledge learned, the experienced agents are able to make better coordinated decisions in the future negotiations with their peers, thus improving the performance of the sys- tem over time. This illustrates that learning techniques can be used as an alternative knowledge acquisition to complement direct querying in negotiation. Although not designed to replace direct queries, the ability to complement direct queries with learning can be useful when communication costs are high, or when high level of inter-agent communication is not desirable (e.g. to preserve individual privacy). Such a technique proves to be quite useful in the distributed meeting scheduling domain. In this do- No noise Noise percentage: 6% 11 yy++-+++-;~[ 0 2 4 6 6 10 12 14 16 18 20 Days Noise percentage: 12% without learning + with learning -+ - - ” ” ” ” ” 0 2 4 6 8 10 12 14 16 18 20 Days 0 2 4 6 8 10 12 14 16 18 20 Days Noise percentage: 25% EEi - I I I 1 I I I I I I- without learning +- - 15 - with learning -+-- - 10 ’ ’ ’ ’ ’ ’ ’ ’ ’ ’ 0 2 4 6 8 10 12 14 16 18 20 Days Figure 2: Relative performance of ordinary agents and learning agents main, the agents’ preferences tend to be repetitive with a fixed period; thus the learning mechanism can be simple and yet gives positive results. When there are a large number of agents involved, a saving in the amount of communication can save useful system re- sources required by the schedulers. Furthermore, it also preserves the privacy of the individual schedules. The work presented here can be extended in a num- ber of different directions. Firstly, based on the results of our initial experiments, we are planning to carry out more experiments to investigate the behaviour of the learning agents when there are a large number of agents and when the groups of agents are formed dynamically. Secondly, to evaluate the benefits of learning to its full extent, it is necessary to develop a common framework in which the accuracy of learned knowledge and the cost of communication can be examined together. References Bond, A. H., and Gasser, L., eds. 1988. Distributed Artificial Intelligence. Morgan Kaufman. Bui, H. H.; Venkatesh, S.; and Kieronska, D. 1995. A multi-agent incremental negotiation scheme for meet- ings scheduling. In proceedings A NZIIS-95, the Third Australian and New Zealand Conference on Intelli- gent Information Systems. Conry, S. E.; Meyer, R. A.; and Lesser, V. R. 1988. Multistage negotiation in distributed planning. In Bond and Gasser (1988), 367-384. Laasri, B.; Laasri, H.; Lander, S.; and Lesser, V. 1992. A generic model for intelligent negotiating agents. In- ternational Journal on Intelligent Cooperative Infor- mation Systems 1(2):219-317. Sen, S., and Sekaran, M. 1995. Multiagent coordina- tion with learning classifier systems. In Proceedings of the IJCAI-95 workshop on Adaptation and Learning in Multiagent Systems. Sen, S.; Sekaran, M.; and Hale, J. 1994. Learning to coordinate without sharing information. In Proceed- ings of the National Conference on Artificial Intelli- gence (AAAI-94), 426-431. Smith, R. 6. 1980. The contract net protocol: High-level communication and control in a distributed problem solver. IEEE Transactions on Computers 29(12):1104-1113. Sycara, K. 1989. Multiagent compromise via ne- gotiation. In Gasser, L., and Huhns, M. N., eds., Distributed Artificial Intelligence, volume 2, 119-137. Morgan Kaufman. Woodridge, M. J., and Jennings, N. R. 1994. For- malizing the cooperative problem solving process. In Proceedings of the 13th International Workshop on Distributed Artificial Intelligence (IWDA I- 94), 403- 417. Woodridge, M. J., and Jennings, N. R. 1995. Agent theories, architectures and languages: a survey. In Woodldridge, M. J., and Jennings, N. R., eds., Inteb- ligent Agents, ECAI-94 Workshop on Agent Theories, Architectures, and Languages, number 890 in Lecture Notes in Artificial Intelligence, l-39. Negotiation & Coalition 119 | 1996 | 17 |
1,810 | Opportunity recognition in complex environments Louise Pryor Department of Artificial Intelligence University of Edinburgh 80 South Bridge Edinburgh EHl 1HN Scotland louisep@aisb.ed.ac.uk Abstract An agent operating in an unpredictable world must be able to take advantage of opportunities but cannot afford to perform a detailed analysis of the effects of every nuance of the current sit- uation on its goals if it is to respond in a timely manner. This paper describes a filtering mecha- nism that enables the effective recognition of op- portunities. The mechanism is based on a char- acterization of the world in terms of reference fea- tzrrees, features that are both cheap and functional and that appear to be prevalent in everyday life. Its use enables the plan execution system PARETO to recognize types of opportunities that other sys- tems cannot. Reference features can also play a r6le in the detection of threats, and may be in- volved in the development of expertise. Introduction The world is unpredictable-it is impossible to tell in advance exactly what its state will be at any future time-so plans on their own are not sufficient to gov- ern the behavior of a goal-driven agent. For example, a robot roaming the surface of a strange planet will have to be able to decide for itself where it should take soil samples, what areas to explore and what routes to take. The samples that should be taken may depend inter alia on the terrain that is encountered, atmospheric conditions, and the results of tests on earlier samples, none of which can be predicted in detail. Moreover, the number of possibilities is huge: it would be impossi- ble to construct contingency plans for all combinations of circumstances that could be encountered, even if it were possible to predict what they might be. The traditional approach to AI planning assumes that plans are guaranteed to work and that noth- ing unexpected can happen during their execution. However, in many real-world domains the inevitabil- ity of the unexpected means that any plans that are made in advance will have to be changed during ex- ecution (Alterman 1986; Firby 1987; Hammond 1989; Pryor & Collins 1994). Expending a great deal of ef- fort on constructing elaborate and detailed plans by trying to predict exactly what will happen is therefore often unproductive. A more effective approach is to choose simple plans and adapt them when unforeseen circumstances are encountered An agent following this approach must be able to determine when it should change its current plan. This determination is not triv- ial. Any aspect of the world may in principle affect any of the agent’s goals: any change in the world may thus make it desirable to change plans. The agent cannot afford to analyze in detail every circumstance that it encounters if it is to respond appropriately. This paper presents a filtering mechanism that can be used to decide when a detailed analysis of the agent’s circumstances should be performed. The mech- anism has been implemented in the plan execution system PARETO. 1 The mechanism used in PARETO is based on the observation that the world, although un- predictable in detail, is in essence very regular. It uses the fact that particular aspects of the world are of- ten associated with the achievement or frustration of certain types of goals. For example, the presence of a sharp object is often implicated in the ability to achieve a goal of cutting something. The sharpness of an ob- ject indicates its effect on goals whose achievement is affected if the object cuts something else. There are many concepts such as sharp that describe effects on goals: they form the basis of an effective method of opportunity recognition.2 The problem An agent should change its plans when an unpre- dicted environmental factor affects the achievement of its goals. Goals can be affected in two ways: either op- portunities are presented, or threats are posed. This paper concentrates on the recognition of opportunities; the principles involved are much the same for threats. Opportunity recognition is complex in two ways. First, there is an enormous number of elements in every ‘Planning and Acting in Realistic Environments by Thinking about Opportunities. The economist, sociologist and philosopher Vilfredo Pareto (1848-1923) is best known for the notion of Pareto optimality and for the Pareto dis- tribution, neither of which is used in this work. 2This work is described in more detail in (Pryor 1994). Environment 1147 From: AAAI-96 Proceedings. Copyright © 1996, AAAI (www.aaai.org). All rights reserved. situation that no agent can possibly predict (Brand & Birnbaum 1990). None of these elements can be ruled out a priori as never being relevant to any goal. Sup- pose, for example, that you have a goal to open a can of paint. The objects in your garage include your car, the shelves on the wall, engine oil, etc. Few of these are relevant to your current goal, but they may be relevant to other goals at other times. Second, there are many subgoals involved in achieving a goal. For instance, to pry the lid off the paint can you must find something that is the right size and shape to act as a lever, a strong rigid surface on which you can rest the can at a convenient height, and so on. The analysis of each situation element of a com- plex environment in the light of each of the agent’s many goals would involve huge numbers of subgoals and situation elements and would preclude a timely response to unforeseen situations. Moreover, such an analysis would require the determination of each goal’s subgoals, thus demanding the existence of a plan to achieve that goal. However, the recognition of an op- portunity may trigger a radical change of plan or the construction of a plan for a hitherto unplanned-for goal. PARETO can recognize and take advantage of opportunities in these circumstances. Related work Most research on the issues of replanning during plan execution fails to address the issue of opportunism. The need to replan is usually determined by project- ing the agent’s current plans explicitly (Ferguson 1992; McDermott 1992); as projection may involve arbitrar- ily complex reasoning, the problem of the timely recog- nition of opportunities is not addressed. Hayes-Roth and Hayes-Roth (1979) look at oppor- tunism in plan construction but do not extend the con- cept to plan execution. Birnbaum and Collins (1984) present a theory of opportunism in plan execution based on the idea that opportunity recognition should be goal-driven rather than environment-driven. They suggest that each goal should be an active mental agent, performing the necessary reasoning to recognize opportunities to achieve itself. They do not address the issue of how agents can recognize opportunities with little reasoning: sharing the reasoning among the goals does not necessarily reduce the amount required. Hammond et al. (1993) suggest that each goal should have associated with it the features in the environment that will be involved in achieving it. Such features might include tools and resources, locations at which the goal can be achieved, etc. The agent then recog- nizes an opportunity for the goal when the relevant feature appears in the environment. This approach relies on having specified the plan that will be used in enough detail that the relevant features are already known. It does not allow an agent to recognize oppor- tunities that require a method of achievement other than that in the current plan or for goals that it has Figure 1: A filter for potential opportunities not yet decided how to achieve. The RAPS plan execution system recognizes oppor- tunities only when their possibility has been rep- resented in the plans being executed (Firby 1987; 1989). Reactive systems such as PENGI require the sys- tem designer to have encoded potential opportunities into the reaction rules (Agre & Chapman 1987). PARETO’s approach extends the range of oppor- tunism by enabling the speedy recognition of oppor- tunities for goals for which a plan has not yet been chosen and of opportunities whose possibility has not been explicitly foreseen. A filter for opportunity recognition PARETO uses a filtering mechanism that indicates those situations in which there are likely to be oppor- tunities and that will therefore repay further analysis (Figure 1). It is based on two key observations: first that there is a large class of opportunities whose pres- ence is indicated by a single critical factor; and second that the causal properties of situation elements, and thus their effects on goals, are in general predictable. The critical factor hypothesis Although an opportunity usually arises as the result of a number of factors, in many cases few of these factors by themselves indicate the presence of an op- portunity. For example, the presence of a table does not usually constitute an opportunity to open a can of paint. There are many different ways in which the requirement of having somewhere to rest the can could be met, and it happens that the presence of the table is one of them. However, the presence of a screwdriver is more significant; there are comparatively few objects that are suitable for use as a lever. The critical factor hypothesis states that the presence of a single factor is often crucial for the existence of an opportunity. The critical factor hypothesis relies on the obser- vation that many situation elements are stable across many different situations. There is more stability at the functional level than at the purely physical: while there may not always be a table, there is usually some- thing that you can use to support an object. It is thus usually easy to meet most of the preconditions for a given goal and the presence of situation elements that allow you to meet these preconditions does not greatly affect goal achievement. However, there is often one precondition that is more difficult to achieve, such as the availability of a lever that will fit under the lid 1148 Planning of the can. The presence of a situation element that enables the achievement of this precondition is then sufficient to indicate the presence of an opportunity. The critical factor hypothesis appears to hold for a large number of opportunities and forms the basis of an effective filtering mechanism for recognizing oppor- tunities. An agent can take the presence of a factor that is critical for one of its goals as an indication that the goal is worth analyzing in more detail. Reference features A filtering process is only effective if it is cheap and adequately predictive. 3 The critical factor hypothesis means that an agent need only recognize the presence of the critical factor for a goal to realize that there may be an opportunity for it. However, the filtering , process cannot involve a detailed functional analysis of all the elements of the current situation to determine whether they are critical factors for any of the agent’s goals. Instead, we can use the fact that the causal properties of objects tend to be stable across situations: e.g., knives and scissors tend to cut things, tables to support things, and cups to hold liquids. These functional tendencies can be labeled; for in- stance, objects that cut other things are labeled as be- ing sharp. There are many similar descriptive terms that label functional properties of objects. Words such as absorbent, sturdy and fragile indicate that the object with the property either affects other objects in some way or is itself affected. These properties can be used when planning to achieve goals: something sharp can be used to cut another object, something absorbent to mop up a spill, and something sturdy to support a heavy object. We call these labels reference features. Reference features are cheap to compute To be effective the filtering process must involve minimal rea- soning . An important characteristic of reference fea- tures is thus the ease of infering them in most situa- tions in which the associated causal effects are present. Reference features are often associated with percep- tual cues: e.g., a sharp object has a well-defined edge between two nearly parallel planes. In such cases, the sight of an object is enough to bring the reference fea- ture to mind. There are obvious links to the theory of aflordances proposed by Gibson (1979): an environ- ment’s affordances are the functionalities it offers to an animal in it. There are two main differences between affordances and reference features. First, reference fea- tures are not limited to interactions between the agent and its environment: e.g., a surface can have the ref- erence feature support even if it could not bear the agent’s own weight, as long as its ability to support other objects is useful to the agent. Second, reference features need not be directly linked to perceptual fea- tures: e.g., we do not have to feel or even see a rubber 3 It is in the natur e of heuristic filtering processes such as this that some mistakes will be made. slippery, rough, gritty, sturdy, tough, heavy, permeable, absorbent, Table 1: Some reference features in everyday domains band in order to apply the term elastic to it. Sloman (1989) proposed compact functional descrip- tions derived from the possibilities for motion provided by physical descriptions of objects. Reference features are more general, as they are not concerned with mo- tion alone but with any type of relevant functional- ity. Some reference features from everyday domains are shown in Table 1. Reference features are highly predictive The filtering process used in PARETO is predictive because it is based on the critical factor hypothesis. A critical factor in the achievement of a goal is characterized in functional terms, i.e., in terms of its causal effects on goals. For example, objects that can cut things are critical factors for the goal of cutting string. Refer- ence features, which label the functional tendencies of objects, can thus be used to recognize critical factors. It is important to note that reference features do not form a complete functional classification of situa- tion elements. For example, not all objects with the reference feature sharp help with the goal of finding something to cut a cake. Reference features are useful only because they form the basis of a heuristic filter that indicates some goals as being potentially easy to achieve. Goals that pass the filter can be analyzed further: although the filtering mechanism must reduce the amount of detailed reasoning required, it need not obviate the need for all reasoning. Reference features form a simplified classification of objects by their func- tionally interesting properties. Reference features form the basis of an effective filter for opportunities because they constitute an interme- diate level of conceptualizing the world between the physical vocabulary provided by perception and the functional vocabulary required to reason about goals. They are thus both cheap and highly predictive. Recognizing opportunities Reference features are attached to the situation ele- ments that tend to cause the effects that they label. They can also be attached to goals by labeling each goal with the reference feature of its critical factor. If Environment 1149 there is a situation element that shares a reference fea- ture with one of the agent’s goals, that goal is likely to be easily achievable. An agent can then use the follow- ing filtering process to spot potential opportunities: 1. Find the reference features in the current situation, indicating the easily achievable effects. 2. Find the reference features of its current goals, indi- cating the effects needed for goal achievement. 3. Compare the two sets of reference features. The goals whose reference features are found in the cur- rent situation are likely to be easily achievable. The agent then analyzes the indicated goals whether they are genuinely easy to achieve. to see PARETO PARETO'S world was built using the TRUCKWORLD simulator (Firby & Hanks 1987; Hanks, Pollack, & Cohen 1993). A delivery truck travels between lo- cations, encountering and manipulating objects as it goes. There are three building sites whose workers use the truck to run delivery errands such as “fetch something to carry my tools in.” There are over 30 types of object in PARETO'S world, of which 20 are used for deliveries. At any moment, there are typically well over 100 different objects at the various locations. PARETO receives delivery orders at random intervals, with a typical run involving between seven and twelve separate deliveries. The world is unpredictable: the truck can sense only those objects at its current loca- tion; objects may spontaneously change location; and actions may fail to have the desired results. How PARETO works PARETO is based on the RAPS plan execution system (Firby 1987; 1989). When PARETO acquires a new goal, it looks in its library of RAPS (sketchy plans) for one that will achieve the goal. A RAP (Reactive Action Package) specifies all the different methods that might be used to achieve a goal. Each method consists of subgoals that must be achieved to execute the plan. Having chosen a RAP for the goal, PARETO adds a new task to its task agenda. A task consists of a goal and a RAP that will achieve it. PARETO'S execution cycle consists of choosing a task and processing it: Either If it has succeeded remove it from the agenda; Or If it is described by a RAP choose the appropriate method, based on the state of the world at the time that the processing takes place, and add a new task to the task agenda for each new goal; Or Perform the primitive action specified by the task. The original task is reprocessed after all its subtasks have succeeded. Their success does not guarantee the success of the original task as some time may pass be- tween their execution and its repeat processing. At any time during execution the agenda may hold tasks at varying levels of abstraction, ranging from tasks that have not been expanded at all to those con- sisting of single primitive actions. PARETO can choose any task on the agenda for processing: in general, how- ever, constraints are observed that ensure subgoals are addressed in the correct order and that a parent task is not reprocessed until its subtasks have all succeeded. PARETO'S task selection algorithm, unlike that of the RAPS system, incorporates an opportunity recog- nition mechanism. Reference features are used to in- dicate tasks for which there are potential opportuni- ties; they are then analyzed in more detail to deter- mine whether the opportunities are genuine. Finally, PARETO chooses a task from among the set of those for which there are genuine opportunities together with those that are ready for expansion according to the constraints on the task agenda. A set of heuristics is used to ensure that opportunities are pursued when desirable without abandoning the original goal. Reference features in PARETO The reference features of situation elements are directly related to the goals they help achieve. PARETO is asked to deliver objects to building sites. Requests are often specified in terms of the type of object required, so ev- ery object has its type as a reference feature. Other goals are more loosely specified: e.g., PARETO may be asked to deliver something that can be used to carry tools. Several types of object can be used to carry things; all have the reference feature carrier. The functional aspects of objects are relevant only insofar a5 they affect PARETO's goals; PARETO's reference fea- tures may not be those we would attach to the objects’ real-world counterparts. The critical factors for the goals that involve deliv- ering objects are the objects to be delivered; the goals’ reference features reflect the objects’ properties. A de- livery goal that specifies only the class of object to be delivered has the reference features of that class of ob- ject. More complex criteria have their own specific ref- erence features: e.g., a task to deliver something with which to carry tools has the reference feature carrier. Other tasks include those to travel to the location of a building site or unload an object at a building site, both of which have the reference feature site, and the task to refill the fuel tank, which has the reference fea- tures fuel and fuel-drum. There is an inheritance mechanism that ensures that subtasks inherit relevant reference features from their parent tasks. Opportunity n?COgnitiOn in PARETO This section gives a detailed example of PARETO'S opportunity recognition. 4 PARETO has just received two requests: a carry-tools goal for maple-ave and a cut-twine goal for Sheridan-rd. Each de- livery goal has a deliver-object task on the task agenda. The first task PARETO chooses to process 4Program output is edited for brevity. 1150 Planning is <deliver-object carry-tools> which it expands into several subtasks, one of which is a find-object task, which is processed next. Boxes (which are suit- able for carry ing tools) are generally to‘be found at the lumberyard, so a <truck-travel -to lumberyard> subtask is added to the agenda and is then processed. On the way to the lumberyard PARETO arrives at warehouse-2 and scans its neighborhood as it always does when it arrives at a new place. Most of the objects present are irrelevant to its goals but two of them, a pair of scissors and a bag, are directly relevant to its cut-twine and carry-tools goals. The first step of PARETO's filtering process is to find the reference features of the current situation, which include: Reference features for ITEM-20: (BAG CARRIER) Reference features for ITEM-13: (SCISSORS SHARP) The second step is to find the reference features of its current goals. PARETO attaches reference features directly to tasks on the task agend .a: the carry-tools tasks have the reference feature carrier, while the cut-twine task has sharp. The third step is to match the two sets of reference features. The bag indicates potential opportunities for several of the subtasks of the carry-tools goal: Potent ial opportunity for <DELIVER-OBJECT CARRY-TOOLS MAPLE-AVE> ITEM-20 (BAG) has reference feature CARRIER Potential opportunity for <FIND-OBJECT CARRY-TOOLS> ITEM-20 (BAG) has reference feature CARRIER Potential opportunity for <LOAD-PAYLOAD-OBJECT CARRY-TOOLS> ITEM-20 (BAG) has reference feature CARRIER The top level <deliver-object cut-twine> task has not yet been processed; it is therefore the only task for its goal. It has the reference feature sharp, as do the scissors that are present at the current location: Potential opportunity for <DELIVER-OBJECT CUT-TWINE SHERIDAN-RD> ITEM-13 (SCISSORS) has reference feature SHARP Finally, PARETO analyzes the potential opportuni- ties to determine which are genuine. There are two ways in which one of PARETO'S task may be easily achievable: if it has already succeeded or if it is ready for processing, i.e., not waiting for any others to be completed (Pryor & Collins 1994 . k In this case the <find-object carry-tools> tas has already suc- ceeded. It therefore constitutes a valid opportunity, and PARETO loads the bag into its cargo bay. Has already succeeded <FIND-OBJECT CARRY-TOOLS> Taking unexpected opportunity: <FIND-OBJECT CARRY-TOOLS> PARETO also decides that there is an opportunity for the <deliver-object cut-twine> task: the presence of the scissors prompts it to expand the task. Taking expected opportunity: <DELIVER-OBJECT CUT-TWINE SHERIDAN-RD> Processing task: <DELIVER-OBJECT CUT-TWINE SHERIDAN-RD> It then loads the scissors. need to go to the lumberyard, Since there PARETO now is now no makes its way directly to maple-ave where it delivers the bag and then to Sheridan-avs to deliver the scissors. In this example we have seen how PARETO uses refer- ence features to recognize potential opportunities, re- gardless of whether the goals are being actively pur- sued. The opportunities that are recognized might in- volve abandoning an existing plan or choosing a plan for a goal that has not hitherto been addressed. Diseussion This paper has described a filtering mechanism for op- portunity recognition, and its implementation in the plan execution System PARETO. Reference features and threats This paper has described how reference features can be used in the recognition of opportunites: they can also be used to recognize threats. Many reference fea- tures indicate possible threats to goals. For example, in PARETO'S world a sharp object, such as a knife, may cut a soft object, such as a ball of twine thus rendering it unsuitable for its purpose and making it unacceptable to the worker who has requested it. A limited form of threat detection through the use of reference features has been implemented in PARETO (Pryor & Collins 1992). PARETO's analysis of potential opportunities is performed in two steps: the goal is first checked to see whether it is easily achiev- able and then its potential interactions with other goals are examined. A goal with problematic interactions does not constitute an opportunity. Detecting prob- lematic interactions involves the same computational problems as recognizing opportunities: the agent may have many goals, each with a large number of subgoals, and in any case might not have chosen plans for all of them. PARETO uses a filtering process that is very sim- ilar to the one used for opportunity recognition. Ref- erence features are used to indicate potentially prob- lematic interactions. The potential interaction is then analyzed in more detail to determine whether it is gen- uine. The reference features not only indicate that a problematic interaction is possible, they also indicate its probable form thus assisting its avoidance. Reference features and expertise The concept sharp is useful just because many com- monly arising human goals involve structural integrity. If, however, we lived in a world in which structural in- tegrity was unimportant, we might well not even have such a concept, let alone find it useful. The reference features that an agent finds useful depend on the tasks that it habitually performs. Agents performing dif- ferent tasks in the same world may attach completely different reference features to objects in their environ- ments. Properties that are significant to one agent may be completely meaningless to another. An important corollary to the task-related function- ality of reference features is their role in the develop- Environment 1151 ment of expertise. A domain expert is certainly ex- References petted to be able to recognize opportunities, both rou- tine and novel. A domain novice, on the other hand, Agre, P. E., and Chapman, D. 1987. Pengi: An imple- mentation of a theory of activity. In Proc. 6th Nat. Conf. may not recognize even routine opportunities. The dif- ference between the two is due to their relative famil- iarities with the domain. An interesting hypothesis is that an expert is one who has a comprehensive set of reference features for the domain: a novice has a lim- ited or nonexistent set. Testing this hypothesis is an important area of future work. A related issue is the question of how reference features are acquired. The development of expertise involves learning about the functional aspects of the domain and learning to asso- ciate perceptual cues or other easily inferable features with them. This learning may consist of associating functional tendencies with features that are already available, or of learning to recognize new features. Reference features appear to provide a bridge be- tween the perceptual cues an agent receives and the functional judgments it must make. However, the question remains as to whether they are a genuine psy- chological phenomenon or simply the basis of an effec- tive mechanism totally unlike anything actually used by people. Further work is needed to investigate their reality, both by examining their possible use by domain experts and by investigating their use in opportunity recognition, possibly along the lines of the experiments performed by Patalano et ul. (1993). on Artificial Intelligence. AAAI. Alterman, R. 1986. An adaptive planner. In Proc. 5th Nat. Conf. on Artificial Intelligence, 65-69. AAAI. Birnbaum, L., and Collins, G. 1984. Opportunistic plan- ning and freudian slips. In Proc. 6th Ann. Conf, Cognitive Science Society. Boulder, CO: LEA. Brand, M., and Birnbaum, L. 1990. Noticing opportu- nities in a rich environment. In Proc. 12th Ann. Conf. Cognitive Science Society. Cambridge, MA: LEA. Ferguson, I. A. 1992. TouringMachines: An architecture for dynamic, rational, mobile agents. Technical Report No. 273, Computer Laboratory, University of Cambridge. Firby, R. J., and Hanks, S. 1987. The simulator manual. Technical Report YALEU/CSD/RR 563, Department of Computer Science, Yale University. Firby, R. J. 1987. An investigation into reactive planning in complex domains. In Proc. 6th Nat. Conf. on Artificial Intelligence, 202-206. Seattle, WA: AAAI. Firby, R. J. 1989. Adaptive execution in complex dy- namic worlds. Technical Report YALEU/CSD/RR 672, Department of Computer Science, Yale University. Gibson, J. J. 1979. The ecological approach to visual perception. Boston, MA: Houghton Mifflin. Hammond, K.; Converse, T.; Marks, M.; and Seifert, C. 1993. Opportunism and learning. Machine Learning 10:279-309. Hammond, K. 1989. Opportunistic memory. In Proc. 11th Conclusion Int. Joint Conf. on Artificial Intelligence, 504-510. Hanks, S.; PoIIack, M. E.; and Cohen, P. R. 1993. Bench- The implementation of PARETO has demonstrated that opportunity recognition based on reference features is feasible. To demonstrate its effectiveness the mecha- nism must be scaled up to a real world domain. This will involve the construction of a set of reference fea- tures for that domain. Scaling up will also involve the use of computationally efficient matching algorithms in the filtering stage of the process. The mechanism described in this paper can recog- nize only those opportunities indicated by critical fac- tors. PARETO cannot, for example, make a detour while traveling to a specific location. This type of opportunity can be recognized only through detailed analysis and projection of a type that PARETO is not designed to perform. Instead of deciding in advance how the plans for its various goals will be combined, PARETO combines them on the fly through opportunity recognition. This enables robust and reactive behavior because its method of opportunity recognition does not rely on a plan already having been chosen, and allows the recognition of opportunities whose pursuit involves abandoning existing plans. Acknowledgments. This work was principally per- formed at the Institute for the Learning Sciences, North- western University. Thanks to Gregg Collins for criticism and encouragement, and to Matt Brand, WiII Fitzgerald, Eric Jones and Bruce KruIwich for many useful discussions. marks, testbeds, controlled experimentation, and the de- sign of agent architectures. AI Magazine 14(4):17-42. Hayes-Roth, B., and Hayes-Roth, F. 1979. A cognitive model of planning. Cognitive Science 3(4):275-310. McDermott, D. 1992. Transformational planning of reac- tive behavior. Technical Report YALEU/CSD/RR 941, Yale University, Department of Computer Science. PataIano, A. L.; Seifert, C. M.; and Hammond, K. J. 1993. Predictive encoding: Planning for opportunities. In Proc. 15th Ann. Conf. Cognitive Science Society, 800-805. Boul- der, CO: LEA. Pryor, L., and Collins, G. 1992. Reference features as guides to reasoning about opportunities. In Proc. 14th Ann. Conf. Cognitive Science Society, 230-235. Bloom- ington, IN: LEA. Pryor, L., and Collins, G. 1994. Opportunities: A unifying framework for planning and execution. In Proc. 2nd Int. Conf. on Artificial Intelligence Planning Systems, 329- 334. Chicago, IL: AAAI Press. Pryor, L. 1994. Opportunities and planning in an un- predictable world. Technical Report 53, Institute for the Learning Sciences, Northwestern University. Sloman, A. 1989. On designing a visual system: To- wards a Gibsonian computational theory of vision. Jour- nal of Experimental and Theoretical Artificial Intelligence 1(4):189-337. 1152 Planning | 1996 | 170 |
1,811 | Generalizing Indexical-Functional Reference Marcel Schoppers and ichard Shu Robotics Research Harvesting, PO Box 2111, Redwood City, CA 94063 Abstract The goals of situated agents generally do not spec- ify particular objects: they require only that some suitable object should be chosen and manipulated (e.g. any red block). Situated agents engaged in de- ictic reference grounding, however, may well track a chosen referent object with such fixity of purpose that an unchosen object may be regarded as an obstacle even though it satisfies the agent’s goals. In earlier work this problem was bridged by hand-coding. This paper lifts the problem to the symbol level, endow- ing agents with perceptual referent selection actions and performing those actions as required to allow or disallow opportunistic re-selection of referents. Our work preserves the ability of situated agents to find and track specific objects, adds an ability to automat- ically exploit the opportunities allowed by nonspecific references, and provides a starting point for studying how much opportunistic perception is appropriate. Introduction If an artificial agent is to interact with real objects, associations between references inside the agent and objects outside the agent must be maintained by the agent itself. To solve this basic problem, (Agre 1988) devised PENGI, which showed how to ground and ma- nipulate indexical-functional references (IFR) . Subsequently, (Schoppers&Shu 1990) built the ba- sic IFR capabilities into an execution engine for a symbolic plan representation, and reported that the plan’s execution-time behavior was not what they had wished. They gave the agent a goal to stack any red block on any blue block. After waiting for the agent to find red and blue blocks, they then put a different red block on the blue block chosen by the agent. Instead of recognizing this as a serendipitous achievement of its goal, the agent put down its chosen red block, re- moved the unwanted red block, and then resumed the activity of placing its chosen red block atop the blue block. This behavior was of course appropriate for the agent’s construction: the agent was designed to use all variables as vehicles for indexical-functional references, and to ground each such reference in any one suitable object, permanently. Such grounding corresponds to what might be expected if the agent were given an instruction using a definite noun phrase: “Put the (whichever) red block you see first on the (whichever) blue block you see first”. But because the references associated with the plan’s variables were initially un- grounded, it was easy to want behavior appropriate to a nonspecific indefinite noun phrase (“Put a red block on a blue block”). The foregoing analysis moves the problem into a lin- guistic realm where there are many varieties of refer- ence, with IFR being only one special case. IFR hap- pens to be basic to the tracking of physical objects, but is relatively rare in statements of things to be ac- complished. Usually, any suitable object will do. Even when the language of an instruction identifies a specific object, this often occurs not because that one object must be used, but because the speaker believes one such object to be especially convenient. (Consider the written assembly instruction “With the Phillips screw- driver, . ..” when the reader has several but was asked to fetch one in previous instructions.) Thus, the prob- lem is that there remains a large gap to be bridged between implemented reference grounding capabilities (currently for IFR only) and the varieties of object reference available to humans when expressing desired behavior. If we are to produce agents capable of effi- ciently carrying out human instructions, whether ver- bal or programmed, we must find ways to make agents more responsive to the variety of object references used by humans. Such responsiveness must be provided both in instruction understanding and in instruction enactment. It might be argued that PENGI, by using one marker to track the bee believed to be closest to the penguin and a second marker to compare other candidate bees, went beyond enactment of references of the form “that bee” to enact identification of “the nearest bee”. While Environment 1153 From: AAAI-96 Proceedings. Copyright © 1996, AAAI (www.aaai.org). All rights reserved. true, this was accomplished by hand-coding a specific defensive behavior, where we wish to devise a mecha- nism that can be parameterized to allow an automatic choice from a variety of reference types. The present paper is concerned with augmenting the varieties of reference that can be enucted by situated agents; it is not a Natural Language paper. We as- sume the existence of a Natural Language component capable of maintaining a discourse model, of resolving co-references, and of deciding what kind of reference grounding behavior is implied by received instructions. In the next section, where we are obliged to define a few terms for the sake of describing certain implemented capabilities, it is merely an accident that the terms we find most useful come from the domain of NL research. In the next section we show that the kind of ref- erence used in a goal restricts the range of situations that satisfy the goal. Then, since nonspecific indefinite reference is common in goals, we describe how we co- opted mechanisms for grounding IFR, to make them produce behavior appropriate to nonspecific indefinite references. As a result, our agent can behave sensibly when given goals combining nonspecific indefinite ref- erences with deictic references (“See to it that there’s a red block on that blue block.“) Defining the Problem Varieties of Reference In English, noun phrases and their reference mean- ings can be distinguished in many ways, and we list a few distinctions below. Our objective in this and the next subsections is to exhibit some of the most obvious ways in which an artificial agent’s behavior must be influenced by some basic varieties of reference that can occur in instructions. Our definitions are pur- posely ad hoc, i.e. for the sake of succintly describing what we have implemented we are making some dis- tinctions in slighly different ways than linguists would. See e.g. (Quirk et al 1985) for a more complete classi- fication. e Referential noun phrases (NPs) expect the agent to identify the referent using either contextual or gen- eral knowledge. Since we are working on grounding references in physical objects, and since attributions occur at the linguistic level, we consider only refer- ential NPs here. e In a specific reference - and under the restriction to bounded physical objects - the instructor has a particular object “in mind”. Only in the specific case can the speaker be expected to answer the question, “Who/what is it?” o A deictic reference depends for its grounding on the instructor’s place in space and time, and is used only 1154 Planning when the instructor expects the referent to be imme- diately identifiable in context. Indexicals are deictic ( “I”, “tomorrow”). The distinction between definite and indefinite NPs is syntactic, based on the use of such determiners as “the” and “a”. However, a speaker using a defi- nite NP expects one referent to be uniquely relevant, though perhaps only in some later context, e.g. “Go into the house and pick up the amulet [you will find there]” (Webber 1991). Demonstrative NPs are indicated syntactically by the presence of the determiners “this”, “that”, “these” , or “those” . The associated references are usually specific, definite, and deictic. Specific reference is especially hard to isolate. For example, the NP “Olaf Palme’s assassin” refers (per- haps) to a common knowledge entity, but no-one knows who that person is. Similarly, there is room to debate the specificity of superlatives such as “the biggest ap- ple you can find,” of ordinals such as “the first apple you touch,” and of collectives such as “the 10,000 men around the castle.” Since this paper is concerned with encoding and enacting plans that can identify and ma- nipulate only singular physical objects, we gladly side- step such complexities. To simplify the rest of this paper we propose that superlative, ordinal, and sim- ilar references which are not immediately groundable but may be made so in the future (including some func- tionals) be considered nonspecific references. The distinctions just elaborated give rise to the fol- lowing classification of referential, non-generic, non- coreferring, references to singular, physical, bounded, non-collective, objects. We include demonstrative NPs under deictic references. nonspecific, indefinite: “Pick up a red block.” nonspecific, definite, nondeictic: “Identify the heav- iest Mock [on the table]” (when the blocks are oth- erwise indistinguishable). “Olaf Palme’s assassin.” nonspecific, definite, deictic: “Pick up the block clos- est to your hand” (when the instructor can’t see it). specific, indefinite: “A red block just fell off the ta- ble” (while the instructor is looking at it). specific, definite, nondeictic: “Pick up the red b2ock on the floor.” specific, definite, deictic: “Pick up the block I’m pointing at. ” “Pick up that block.” Varieties of Serendipity Now let us consider what behavior we expect to see from an intelligent embedded agent when it is given instructions specifying goals (desired states) involving the kinds of reference listed above. First, let us give the agent a goal to “Obtain a state in which a red block is resting on a blue block” (non- specific indefinite reference). This goal is satisfied by any world state in which any red block is on any blue block; the agent need do no more than scan the table. Next, let us consider the use of what we have decided to call nonspecific definite reference. Strictly speak- ing, only the object being “identified” will satisfy the request, but by definition, the speaker does not know which object that is and cannot verify that exactly the right one has been found (without executing her own instruction and assuming that the world has not changed)! That being the case, nonspecific definite in- structions must be meant either as 1. requests to identify objects (e.g. find Olaf Palme’s assassin) to save the instructor some time; or as 2. abstractions (e.g. retrieve the astronaut that just fell off the Shuttle, who it is doesn’t matter); or as 3. simplifications that only approximate the instruc- tor’s real wishes (e.g. bring the next person in line, assuming no unruly behavior in the queue; or, use the first X you find, instead of “be quick”). In all cases, the definiteness of the reference serves only to communicate the instructor’s (possibly false) expec- tation that one identifiable object is “the right one”. Beyond that, it makes no difference to the desired be- havior - the agent should find any person who is Olaf Palme’s assassin, retrieve any astronaut who fell off the station, and bring whichever person is next in line. Consequently, the distinction between nonspecific in- definite and nonspecific definite instructions is not im- portant to agent implementation. Next, we find that specific indefinite reference can- not be used for posing goals. Any instruction forces the agent to choose objects to manipulate, but in a specific indefinite instruction the speaker already has specific objects in mind and refuses to say which objects! Next, let us use specific definite reference (other than a coreference) and give the agent a goal to “Obtain a situation in which that (pointing) red block is resting on that (pointing) blue block.” This goal is narrow: exactly the indicated blocks must be made to satisfy the goal condition, no other blocks will do. However, if the two indicated blocks are already in the desired configuration, the agent needs to do no more than look. Finally, instead of describing a desired situation we can ask the agent to perform an action, such as “Put a red block on a blue block”. In this case, no matter how many red blocks are already atop blue blocks, the agent is nevertheless obliged to build another such tower. The two major kinds of reference (nonspecific and specific) usable in goals, and the distinction between requests for conditions and requests for action, to- gether produce three kinds of behavior that may be distinguished from each other by the kinds of serendip- itous circumstances an embedded agent may exploit while performing the requested task. We distinguish the following cases of admissible serendipity: CB Nonspecific serendipity. The agent may satisfy the goal using any suitable objects, and is not required to act if suitable objects already satisfy the goal. e Specific serendipity. The agent may satisfy the goal using only the objects specified by the references in- cluded as part of the goal, but is not required to act if those specific objects already satisfy the goal. a No serendipity. The agent must not exploit the pres- ence of suitable objects that already satisfy the goal. Problem Statement The three kinds of serendipity also allow us to suc- cinctly describe the problem with implementations of indexical-functional reference (IFR), as follows: e IFR is a form of specific definite reference which nat- urally delivers only specific serendipity, and which can be made to deliver nonspecific serendipity only with considerable effort. Naive attempts to make implementations of IFR deliver nonspecific serendipity result in referential thrashing, i.e. an agent too confused about object selection to get anything done. Our objective is to use explicit representations, not hand-coding, to specify the behavior of embedded agents, including behavior involving interaction with several indistinguishable objects at the same time. Two problems must be solved to attain our objective. 1. Symbolic plans using IFR must become capable of noticing and exploiting nonspecific serendipity. 2. Symbolic plan representations must become capable of expressing a demand for specific serendipity. The second problem may be solvable by building on the distinction between rigid and nonrigid designators, as has been done in e.g. (Appelt 1982). In this pa- per we address the first problem. In particular, we wish to define general, domain-independent notations and plan execution machinery, such that the execution- time specificity of each of the agent’s references can be controlled by domain-specific knowledge (which knowl- edge may therefore be regarded as a parameter to the agent’s re-usable algorithms). The Experimental Setup Before we describe our work, the reader must under- stand some details of the plan encoding we worked with. Our agent consisted of a simulated arm, some simulated sensors, and a repetitively executed decision tree, all operating in a world of simulated blocks. The Environment 1155 on(A,B) ? T) NO-OP F) box(B) 2 T) clear(B) ? . T) holding(A) ? . . T) over(B) ? . . . T) LOWER . . . F) at(top) ? . . . T) LATERAL . . . F) RAISE . . F) [subplan to GRASP A] . F) [subplan to CLEAROFF B] F) FAIL Figure 1: Decision Tree Schema for Block Stacking. simulated arm and camera moved horizontally and ver- tically, in increments that were much smaller than the size of a block. An example decision tree is shown in Figure 1. We refer to each traversal of the decision tree a~ a execution cycle. Blocks had names and colors, and could be indistinguishable. The human observer could move blocks about in the simulated world, thus bringing good or bad luck at will. Notice that the given decision tree is a schema con- taining unbound variables (in the Prolog convention logical variables begin with an upper case letter). This was important in allowing us to invoke the plan (tree) with whatever parameters we wanted. (The parame- ters finally became records that contained two position vectors, one for predicted/believed position and one for perceived/known position; lack of information was in- dicated as a zero vector.) Following (Schoppers&Shu 1990) we regarded object descriptions as goals. When we wanted a plan to put any red sphere on any blue box, we implemented a plan that achieved color(X,red) A shape(X,sphere) A shape(Y,box) A on(X,Y) and we expected the agent to find suitable objects for X and Y to refer to. This view of descriptions is not as strange as it may seem at first. If we were to augment the agent’s ca- pabilities by introducing a painting action, the above goal might induce the agent to paint things, a poten- tially appropriate behavior. At the same time, one way of satisfying a color goal for a nonspecific indef- inite object is to (physically) look for an object that already has the desired color; indeed, that is the only way to achieve a color goal when there is no painting action. Thus we came to regard painting and scanning as alternative ways of achieving color goals. At the start of the agent’s activities, the agent knew nothing at all about the state of the (simulated) world. In particular, when the plan specified that some block should be moved, the agent had first to find a block to move. To solve this problem we implemented a camera movement procedure that systematically scanned the table until the camera viewed a block. This whole scanning procedure was controlled by means of camera positioning coordinates. As a result, the executing plan could refer to objects by knowing in what direction the camera should be pointed in order to make the object appear in the cam- era’s field of view. That directional knowledge allowed the agent to verify visible properties of objects when- ever the property verification could be implemented as a test on the camera image. To test relative posi- tions of blocks, however, or to test the position of the robot arm relative to a block, it was necessary to know both the direction and the range to a block, i.e. the block’s position in three dimensions. Thus, “referring to an object” came to mean “truly knowing the ob- ject’s current 3D position” (versus having a belief or expectation). The required position information was stored (and updated) in records that were passed as parameters to the decision tree. Thus we could define a predicate known located(X) to test that variable X was bound to a record containing visually verified in- formation about current position. We also defined be- lieved located(X) to test that X’s record contained an ezpectution of current position. From there we could define actions that achieved known located(X) (see next section). Referent Selection Actions (Cohen 1981) examined dialogues in which speakers at- tempted to get hearers to identify specific objects, and argued for extending a plan-based theory of commu- nication with explicit actions representing the hearer’s identification of objects. The speaker would adopt a goal of getting the hearer to identify something, and would communicate that goal to the hearer, who would then try to achieve the goal by means of an IDENTIFY action. Our block-stacking agent’s decision tree both tests object identification goals and achieves them with IDENTIFY-like actions. However, we have found it use- ful to endow our agent with many such actions, most of which come down to a visual search that is special- ized to exploit known visual features of the object to be identified. Additionally, many of our “referent se- lection actions” exploit positional expectations. Candidate referents must be found for each plan (ob- ject) variable before any of the conditions mentioning that variable can be tested. The very first thing most plans must do is cause the performance of perceptual searching activities to locate candidate objects. Once 1156 Planning candidate objects are found, plan execution can track them and apply perceptual tests to them. But clearly, if objects are selected only once, at the beginning of plan execution, and are tracked thereafter, the result- ing behavior can exploit specific serendipity at best. To achieve nonspecific serendipity it must be possible to select new objects for references that already have ref- erents. The cyclic execution paradigm makes this very easy by providing the opportunity to revise all referent selections once per execution cycle, and our definition of known located(X) plays the role of having to be constantly reachieved (because a remembered or pre- dicted location is not a known location). The main challenge is to determine a suitable set of referent se- lection and reselection actions, and to integrate those actions with the existing tracking machinery. Identifying Efficiently There are many efficient ways to locate and identify objects. For example, given a goal to put a red block on a blue block, you will first look for either a red or blue block. If the perceptual apparatus can be primed to look for colors, this is already much more efficient than looking for any object at all. Now suppose you have found a blue block. Given that you have been asked to put a red block on top of it, the natural thing to do is: look for a red block on top of the blue block you just found. This behavior is efficient because it tells the perceptual system exactly where to look, and also because finding a red block there would allow you to consider yourself finished with the task. The preceding paragraph implicitly states the heuristic that, to find referents for nonspecific indef- inite references, it is efficient to limit perception by using knowledge of what’s wanted, as indicated by current plan goals. This heuristic is the basis of a large body of work on task-directed perception. We are less concerned with the details of perception than with managing perception to support opportunistic task performance. In our simulated domain we found that there were numerous goals that suggested perceptually efficient referent selection actions. For example: on(X,Y) To find a referent for X when Y has a refer- ent, look just above the location of Y. on(X,Y) To find a referent for Y when X has a refer- ent, look just below the location of X. over(X) To find a referent for X (such that the agent’s hand is over it), look at the agent’s hand and move the camera in a vertical line downwards. clear(X) To find a referent for X, start the camera at the left end of the table and follow the “skyline”. holding(X) To find a referent for X, look at the lo- cation of the agent’s hand, which is always known. Similarly for around(X) . shape(X,box) To find a referent for X, look for any rectangular thing (similarly for other shapes). color(X,red) To find a referent for X, look for any red thing (similarly for other colors). For the first few goals above, referents can be found ef- ficiently because the goals are prepositions that iden- tify a direction vector. The more general source of efficiency, as evidenced by the last few goals, is ex- ploitation of capabilities of the perceptual system it- self, e.g. the ability to locate only certain colors or shapes. The capability to rapidly look at specified lines or locations is responsible for the usefulness of spatial prepositions in referent selection. Some Consequences Because of the association between particular goals and particular actions for finding referents efficiently, we now give referent selection actions postconditions in- dicating their special suitability. For example, under a standard approach to action representation the ac- tion for visually locating red things should have the postconditions known color(X,red) A known lo- cated(X), where known located(X) is as defined in the previous section: perceptually finding a red object also finds the object’s position. But these postcon- ditions would represent that the identify-a-red-thing action is also useful for locating nondescript objects, leading to inefficient behavior when the plan wants any nondescript object and all the available objects are blue. Indeed, many color-seeking, shape-seeking, and texture-seeking visual searches might be tried in vain. On the other hand, there are efficient ways to find nondescript objects. Consequently our representation of the identify-a-red-thing action omits the known located(X) postcondition. Conversely, even though an identify-anything action can eventually find a red object if there is one, our representation of identify- anything actions has no known color(X,red) post- condition. Thus, postconditions now have less to do with actual effects than with the utility of the described action. (We recognize that this is a poor substitute for explicit knowledge about the utility of actions, see the “Future Work” section.) When a plan variable occurs in several goals, the referent selection actions associated with any of those goals might be used to find a referent. For exam- ple, when the plan’s goal is a conjunction such as color(X,red) A shape(X,sphere) A shape(Y,box) A on(X,Y) there are two referent selection actions that can be used immediately to find a referent for X (using color and shape as guides) after which there are Environment 1157 also two referent selection actions for Y (using shape and the location under X). To ensure that every object variable indexes at least one referent selection action, all goals are enlarged with additional conjuncts known located(Xi), one per ob- ject variable Xi appearing in the goal. As a result, most referents can be selected with both goal-specific referent selection actions and a general visual search.l Since the known located(Xi) conjuncts need con- stantly to be reachieved, every execution cycle offers an opportunity to change the object being referred to by Xi. To make the reselection, all of the goals and subgoals in which Xi appeared during the previous ex- ecution cycle may provide useful information. For ex- ample, on(X,Y) is achieved by LOWERING the hand when holding(X), which is therefore a subgoal; this subgoal is achieved by GRASPING when around(X), which is therefore a subsubgoal; this subsubgoal is achieved by lowering when clear(X), and so on. To achieve known located(X) by possibly choosing a different X, it is useful to look at what’s already on Y, at what’s already in the agent’s hand, and at the “sky-line” of the current block configuration. This uniform relevance of everything in the goal stack was especially appreciated in the case of the goal holding(X), whose associated referent selection ac- tion caused X to refer to “the-thing-I’m-holding” - a reference that is both indexical and functional. Indeed, when the referential guidance provided by goals is taken seriously, the meaning of a reference comes to depend on which goals apply to the referent. Hence the meanings of our references change dynami- cally as the agent’s goals change, and the semantics of our references is compositional.2 Recovering Opportunistic Behavior Finally we come to the issue of integrating referent (re)selection with deictic tracking. The normal reac- tive execution cycle is constantly re-achieving known located(Xi) for each Xi. The first referent selection action that is relevant and that succeeds in finding a referent for the given Xi will have satisfied the goal, ‘It does not matter that the same variable can ap- pear in many known located(Xi) conjuncts. For each Xi, the first known located(Xi) conjunct encountered in any given execution cycle initiates perceptual activ- ity. This achieves known located(Xi) and forestalls re- achievement for the rest of that execution cycle. An equiv- alent approach could attach a known located(Xi) con- junct only to the goal at the root of the subtree containing Xi. 2The semantics of (Agre 1988)‘s deictic representation was not compositional, i.e. there was no semantic relation- ship between the notations “the-thing-I’m-holding” and “‘the-red-thing-I’m-holding”. thus preempting the use of other referent selection ac- tions (for the same reference and execution cycle). This means that we have to be careful with the order in which referent selection actions are tried: e As soon as Xi becomes grounded, deictic tracking can maintain the grounding, but that produces ex- actly the single-minded tracking we are mitigating. To have any effect, goal-related referent selection ac- tions must be tried before tracking. They can then look for objects in places that will cause the agent to notice nonspecific serendipities. A serendipitous object will ground the reference and preempt deictic tracking; otherwise deictic tracking can continue. e Similarly, general visual scanning can always find a candidate object for any reference. If it were tried before tracking, then tracking would never be used. Hence, such general backup methods must be tried only after tracking has failed. Thus we were obliged to try referent selection actions in the specific order 1) goal-related referent selection actions, 2) tracking (or, re-perceiving the referent of the preceding execution cycle), and 3) general, goal- independent visual searching. Nonspecific indefinite reference, and the correspond- ing ability to exploit nonspecific serendipity, resulted from trying all the relevant actions in the order just stated, until some action succeeded in finding a refer- ent. Specific definite (deictic) reference could be recov- ered by disabling the use of (category 1) goal-related referent selection actions. Disabling the use of (2) de- ictic tracking produced referential thrashing; disabling the use of (3) goal-independent visual searching left the agent unable to find anything. Summary and Future Work In this work we have shown that the type of refer- ence used in specifying agent behavior affects the kinds of serendipities the agent may exploit. We have har- nassed IFR to provide two additional kinds of situ- ated object reference, namely specific definite refer- ence and nonspecific indefinite reference, as different points on a continuum defined by the presence or ab- sence of three kinds of referent selection actions. The three kinds of referent selection actions are 1) goal- related, which can detect that goals have been satis- fied by objects other than those chosen by the agent; 2) tracking, which provides referential stability; and 3) general visual searches, which locate objects when none are known. Our work raises many performance issues. An agent cannot always afford to re-perceive, in each execution cycle, every object it is already interacting with, let alone looking for serendipitous objects. There is a 1158 Planning small body of work discussing when it is worth-while for agents to engage in sensing (Abramson 1993; Chris- man&Simmons 1991; Dean et al 1990; Doyle et al 1989; Draper et al 1994; Hansen 1994). Our own previous work (&hoppers 1995) made sensing intermittent and dependent on environmental dynamics. Our present implementation side-steps these issues by a) treating action descriptions as heuristics, and making postcon- ditions encode our subjective judgments about the util- ity of actions for specific goals3; and b) restricting our goal-related referent selection actions to look only for serendipities involving previously selected objects. Thus, our agent will notice if another block is put down on one of the blocks the agent was using, but will not notice if someone builds a tower that satis- fies the agent’s goals by using only blocks the agent doesn’t care about. In future work we intend to make the agent more conscious of the likelihoods that partic- ular goals will be achieved serendipitously, of the time costs of perceiving those particular serendipities, and of the time costs of achieving the goals deliberately. Performing, in each execution cycle, the perception or action process having the highest probability of goal achievement per unit time cost, will then yield opti- mally efficient behavior (Simon&Kadane 1975). Since looking above a block already in the field of view is almost free, while looking in a random place for a par- ticular block arrangement is both expensive and prob- ably futile, we expect the resulting behavior to be very similar to what we have now. Other relevant issues include: When the agent must find objects to ground sev- eral references, the order in which references are grounded may affect the agent’s efficiency. There may be useful ways to order the referent se- lection actions applicable at a given time. There are choices to be made between: physically moving lots of things around to find an object that perfectly satisfies a description, versus purely per- ceptual searching for a perfect object, versus finding an object that is nearly right and then modifying it, versus just building a suitable object. Agents could be made very sophisticated about what serendipities they deem likely at any given time, and how much effort to spend on looking for them. More long-range possibilities include, finding ways to mentally distinguish a selected referent (e.g. you can track one pigeon among a flock by using a distinctive feature); finding ways to m&e an indistinguishable ref- 3R,eaders who feel that an effect is an effect no matter how expensive the action is, might yet hesitate to represent an effect having very low probability, and probability is merely the other factor in low utility. erent distinguishable; how to handle plural references; and how to efficiently spot perceptually complex ref- erents (since the predicate block(X) should automat- ically lead to a search for appropriate features such as lines and angles). References Abramson, B. 1993. A decision-theoretic framework for integrating sensors into plans. IEEE Trans SMC 23(2):366-373. Agre, P. 1988. The Dynamic Structure of Everyday Life. Tech rept 1085, AI Lab, MIT. Appelt, D. 1982. Planning natural language utter- ances to satisfy multiple goals. Tech Note 259, SRI International, Menlo Park, California. Chrisman, L. & Simmons, R. 1991. Sensible planning: focusing perceptual attention. Proc AAAI Nat’1 Conf on AI:756-761. Cohen, Phil. 1981. The need for identification as a planned action. Proc 7th IJCAI:31-36. Dean, T. Basye, K. & Lejter, M. 1990. Planning and active perception. Proc DARPA Workshop on Innovative Approaches to Planning, Scheduling and Control:271-276. Doyle, R. Sellers, S. & Atkinson, D. 1989. A focused, context-sensitive approach to monitoring. Proc 1 lth IJCAI:1231-1237. Draper, D. Hanks, S. & Weld, D. 1994. Probabilistic planning with information gathering and contingent execution. Proc Int’l Conf on AI Planning Systems AIPS:31-36. Hansen, E. 1994. Cost-effective sensing during plan execution. Proc AAAI Nat’1 Conf on AI:1029-1035. Quirk, R. et al. 1985. A Comprehensive Grammar of the EngZish Language. Longman Inc., New York. Schoppers, M. & Shu, R. 1990. An implementation of indexical-functional reference for the embedded exe- cution of symbolic plans. Proc DARPA Workshop on Innovative Approaches to Planning, Scheduling and Control:490496. Schoppers, M. 1995. The use of dynamics in an intel- ligent controller for a space-faring rescue robot. Arti- ficial Intelligence 73( 1):175-230. Simon, H. & Kadane, J. 1975. Optimal problem- solving search: all-or-none solutions. Artificial In- telligence 6(3):235-247. Webber, B. 1991. Indexical-functional reference: a natural language perspective. Unpublished extended abstract. Environment 1159 | 1996 | 171 |
1,812 | Fahiem Bacchus Craig Boutilier Dept. Computer Science Dept. Computer Science University of Waterloo University of British Columbia Waterloo, Ontario Vancouver, B.C. Canada, N2L 3Gl Canada, V6T 124 fbacchus @logos.uwaterloo.ca cebly@cs.ubc.cs Adam Grove NEC Research Institute 4 Independence Way Princeton NJ 08540, USA grove@research.nj.nec.com Abstract Markov decision processes (MDPs) are a very popular tool for decision theoretic planning (DTP), partly because of the well- developed, expressive theory that includes effective solution techniques. But the Markov assumption-that dynamics and rewards depend on the current state only, and not on history- is often inappropriate. This is especially true of rewards: we frequently wish to associate rewards with behaviors that extend over time. Of course, such reward processes can be encoded in an MDP should we have a rich enough state space (where states encode enough history). However it is often difficult to “hand craft” suitable state spaces that encode an appropriate amount of history. We consider this problem in the case where non-Markovian re- wards are encoded by assigning values to formulas of a tempo- ral logic. These formulas characterize the value of temporally extended behaviors. We argue that this allows a natural rep- resentation of many commonly encountered non-Markovian rewards. The main result is an algorithm which, given a de- cision process with non-Markovian rewards expressed in this manner, automatically constructs an equivalent MDP (with Markovian reward structure), allowing optimal policy con- struction using standard techniques. 1 Introduction Recent years have seen a tremendous interest in extending the classical planning paradigm to deal with domains involv- ing uncertain information, actions with uncertain effects, and problems with competing objectives. Much work in deci- sion theoretic planning (DTP), generally aimed at address- ing these issues, has adopted the theory of Markov decision processes (MDPs) as the underlying conceptual and compu- tational model [DKKN93, TR94, BD94, BDG95]. MDPs allow one to formulate problems in which an agent is in- volved in an on-going, process-oriented interaction with the environment and receives rewards at various system states. This generalizes the classical goal-oriented view of plan- ning [BP95]. Instead of classical plans, one considers the more flexible concept of a policy, namely a mapping from each state to the action that should be executed in that state. Effective optimization methods exist for computing policies such that an agent executing the policy will maximize its accumulated reward over time [Put94], 1160 Planning The fundamental assumption underling the formulation of a planning problem as an MDP is that the system dynamics and rewards are Markovian. That is, the manner in which the system behaves when an action is executed, and the rewards received, depend only on the system’s current state, not on states previously visited. For example, if we wish to control a robot it is usually not difficult to find a state space in which the robot’s actions can be described as Markovian (stochastic) state transitions. In fact, this is often the most natural way to represent the effects of actions. Assigning natural Markovian rewards can be more problematic. Although it is sometimes easy to associate rewards with individual states (e.g., in a navigation problem where rewards are associated with locations), often a reward is most natu- rally assigned to some behavior that occurs over an extended period. In such cases, it can be difficult to encode the reward as a function of state. For instance, we may reward an agent in states where coffee has just been delivered, but only if this state was preceded by a state (perhaps within Ic steps) where a coffee request was issued, withholding reward for spurious delivery. This reward is properly a function of the system trajectory or history, and not of the state alone. Typical forms of desirable temporally extended behaviors include response to requests, bounded response, lack of response, maintaining safety constraints, and so on. Temporally extended goals of this nature have been examined to some extent in the litera- ture [HH92, Dru89, Kab90, GK91], but not in the context of generating effective policies. The key difficulty with non-Markovian rewards is that standard optimization techniques, most based on Bellman’s [Be1571 dynamic programming principle, cannot be used. One way of dealing with this predicament is to formulate an equivalent decision problem in which the rewards are Markovian. In particular, one can augment the state space of the underlying system by adding variables that keep track of the history relevant to the reward function. For instance, Boutilier and Puterman [BP951 suggest straightforward ways of encoding reward functions that involve simple requests. This approach has the advantage that existing optimization methods for MDPs can be used. Unfortunately, in general, finding a good way to augment the state space requires considerable cleverness-especially if we are concerned with minimizing the size of the resulting From: AAAI-96 Proceedings. Copyright © 1996, AAAI (www.aaai.org). All rights reserved. augmented space for computational reasons. In this paper, using probabilistic STRIPS rules [KHW94, BD94], Bayes we examine the problem of rewarding temporally extended nets [DK89, BDG95] or other action representations. behaviors. We provide a natural, and quite expressive, means for specifying rewards attached to behaviors extended over time. Furthermore, we solve the problem of computing poli- cies in the face of these non-Markovian rewards by develop- ing an algorithm that automatically constructs a Markovian reward process and associated MDP. Our algorithm auto- mates the process of generating an appropriate augmentation of the state space, and, when coupled with traditional pol- icy construction techniques, provides a way of computing policies for a much richer range of reward functions. In Section 2 we introduce NMRDPs, essentially MDPs with non-Markovian reward. System dynamics are specified as with MDPs, but rewards are associated with formulas in a suitable temporal logic. We define temporally-extended reward functions (TERFs) by requiring that the reward as- sociated with a formula be given at any state in which the formula is satisfied. We note that the decision to reward an agent in a given state should depend only on past states, not on future states. For this reason, it will be more natural to en- code our reward formulas using a past or backward-looking temporal logic rather than the usual future or forward logics like LTL, CTL [EmegO] or MTL [AHgO]. In Section 3, we describe a number of interesting and useful classes of target behaviors and show how they can be encoded by TERFs. In Section 4, we consider the problem of constructing optimal policies for NMRDPs. As mentioned, dynamic pro- gramming cannot be used to construct policies in this setting. Nominally, this requires one to resort to optimization over a policy space that maps histories (rather than states) into actions, a process that would incur great computational ex- pense. We present a procedure that, instead, expands the original state space by attaching a temporal formula to each state. This formula keeps track of an appropriate amount of relevant history. By constructing a state-based (Markovian) reward function for the extended state space, we convert the NMRDP into an equivalent MDP; in particular, optimal poli- cies for this MDP determine optimal policies for the original NMRDP in a natural way. In this way, we obtain a com- pact representation of the required history-dependent policy by considering only relevant history, and can produce this policy using computationally-effective MDP algorithms. 2 Non-Markovian Rewards 2.1 Markov Decision Processes Much recent work in DTP considers planning problems that can be modeled by completely observable Markov Decision Processes [How60, Put94]. In this model, we assume that there is a finite set of system states S, a set of actions A, and a reward function R. The effects of actions cannot be predicted with certainty; hence we write Pr(si, a, ~2) = p (or sr qs2) to denote that s2 is reached with probability p when action a is performed in state s1 . Complete observability entails that the agent always knows what state it is in. We assume that the state space is characterized by a set of features, or logical propositions. This allows actions to be described compactly A real-valued reward junction R reflects the objectives, tasks and goals to be accomplished by the agent, with R(s) denoting the (immediate) utility of being in state s. For our purposes, then, an MDP consists of S, A, R and the set of transition distributions {Pr(., a, -) : a E A). A stationary Markovian policy is a mapping x : S -+ A, where r(s) denotes the action an agent should perform whenever it is in state s. One might think of such poli- cies as reactive or universal plans [Sch87]. Given an MDP, an agent ought to adopt a policy that maximizes the ex- pected value over its (potentially infinite) trajectory through the state space. The most common value criterion in DTP for infinite-horizon problems is discounted total reward: the current value of future rewards is discounted by some factor p (0 < ,0 < l), and we maximize the expected accumu- lated discounted rewards over an infinite time period. The expected value of a fixed policy 7r at any given state s can be shown to satisfy [How60]: G(S) = R(s) + P x Pr(s, r(s), t> - G(t) tes The value of x at any initial state s can be computed by solving this system of linear equations. A policy 7~ is optimal if V, (s) 2 V,! (s) for all s E S and policies x’. Techniques for constructing optimal policies in the case of discounted rewards have been well-studied, and include algorithms such as value iteration [Be1571 and policy itera- tion [How60]. It should be noted that each of these algo- rithms exploits the Markovian nature of the reward process. We refer to [Put941 for an excellent treatment of MDPs and associated computational methods. 2.2 A Temporal Logic of the Past To reward agents for (temporally extended) behaviors, as opposed to simply reaching certain states, we need a means to specify rewards for specific trajectories through the state space. Generally, we want to associate rewards with prop- erties of trajectories rather than rewarding individual trajec- tories. For example, we might reward an agent whenever condition Q is achieved within k steps of condition P, with- out regard for the particular trajectory the agent is traversing. Therefore, we associate rewards (or penalties) with desirable (or undesirable) formulas in a suitable temporal logic that describes such trajectory properties. The logic we consider is “backward”, or past looking. That is, the truth of a temporal formula depends on prior states only, not on what will happen in the future. This accords well with our view of reward processes because, in most contexts, rewards should be earned based on what has actually happened. We present a past version of LTL [EmegO] called PLTL. We assume an underlying finite set of propositional constants P, the usual truth functional connectives, and the following temporal operators: S (since), q (always in the past), 0 (once, or sometime in the past) and 0 (previously).’ The ‘These are the backward analogs of the LTL operators until, Handling Uncertainty 1161 formulas 41 S 42, O&,0& and 04, are well-formed when 41 and 42 are. 2 We use T and I to denote truth and falsity, respectively. The semantics of PLTL is described with re- spect to models of the form T = (se, . . . , s,) , n 2 0, where each si is a state or valuation over P (i.e., si E 2’). Such a T is called a (‘nite) trajectory, or partial history. For any trajectory T = (so, - . . , s,), and any 0 < i 5 n, let T(i) denote the initial segment T(i) = (so, - . . , si). Intuitively, a temporal formula is true of T = (so, - . . , s,) if it is true at the last (or current state) with respect to the history reflected in the trajectory. We define the truth of formulas inductively as follows: T + $1 S 42 iff there is some i 5 n s.t. T(i) + 42 and for all i < j 5 n, T(j) b 41 (intuitively, $1 has been true since the last time 42 held) 2’ b I34 iff for all 0 5 i 5 n, T(i) b 4 (4 has been true at each point in the past) T b 04 iff for some 0 5 i 5 12, T(i) k q3 (4 was true at some point in the past) Tb04iffn>OandT(n-l)/=+ (4wastrueatthe previous state) One notable consequence of this semantics is the fact that while { 04, O--@} is unsatisfiable, { +3$,lO+} is satisfi- able: any model of the form (s) satisfies the latter. It is well-known that the modalities in LTL can be decom- posed into present and future components [EmegO]. Simi- larly, modalities of PLTL can be decomposed into present and past components. For example, 13 4 is equivalent to 00 4A4. That is, O+ is true iff 4 is true of the current state and 04 is true of the previous state. Using these equivalences we can determine, for any formula 4, what must have been true in the previous state in order that 4 be true now. We call this the regression of 4 through the current state. Note that if the current component of 4 is falsified by the current state, then nothing about the previous state can make 4 true now. In this case the regression of 4 is 1. Definition 2.1 The regression of 4 through s, denoted Regr(4, s), is a formula in PLTL such that, for all trajec- tories T of length n > 1 withJina1 state s, we have T j= C#I iff T(n - 1) + Regr(4, s) Regr(+, s) can be computed recursively: l If 4 E P, Regr(+, s) = T ifs b 4, and I otherwise 0 Regr(41 A 42, s> = Regr(41, s) A M&h, s) 0 Regr(l#i, s) = +Wg(h, s> e Regr(O+, s) = 4 always, eventually and next, respectively. 2We use the abbreviation 0” for k iterations of the 0 modality (e.g., 03@ E OOOq5), and OS’ to stand for the disjunction of Oi for 1 5 i 5 k, (e.g., OS24 S 04 V OO+). e W&h W2, s> = Regr(42, s>v&w-(4l, s)A(h %b)) 0 Regr( 041, s) = Red&, s> V WI e Regr(Ph , s> = Regr(h , s) A 041 Finally, we define some useful notation. For an MDP (or NMRDP) with actions A and transition probabilities Pr, a trajectory (se,. . . , spz) is feasible iff there are actions al,-*,% E A such that Pr(si,ai, si+r) > 0. If 41 and 42 are PLTL formulas, ~$1 determines 4)~ iff either $1 k $2 or 41 b 142 hold. Given any PLTL formula 4, we define Subformulas(4) to be the set of all subformulas of 4 (includ- ing d, itself). Note that ISubformulas(4)l 5 length(q5). 2.3 Rewarding Temporally-Extended Behaviors To reward behaviors, we must adopt a generalization of MDPs that allows the reward given at any stage of the pro- cess to depend on past history. A decision process with non-Markovian reward, or NMRDP, is similar to an MDP with the exception that the reward function R takes as its do- main histories of the form (so, - - - , s,) for all n. Intuitively, the agent receives reward R( (so, . . e , s,)) at stage n if the process has passed through state si at stage i for all i 5 n. Clearly, the explicit specification of such a reward function is impossible since there are an infinite number of different histories. Instead, we assume that the reward function of an NMRDP can be specified more compactly. In particular, we assume that the reward function is defined by a finite set @ of rewardformulas expressed in PLTL, together with a real- valued reward ri associated with each & E @ (we sometimes write this & : pi). The temporally extended reward function (TERF) R is then defined as follows: R( (so, - l - ,s n>> = ) ☺ri : (S O ? AZ) c= A> This formulation gives a reward of ki at each state that sat- isfies formula &; if & has a nontrivial temporal component then the reward is history-dependent. Because reward for- mulas are expressed in PLTL, rewards depend only on past states, and the TERF can be unambiguously evaluated at each stage of the process.” Consideration should not be restricted to Markovian poli- cies when dealing with NMRDPs. The value, and hence the choice, of action at any stage may depend on history. We thus take policies to be mappings from histories to actions. As usual, the value of a given policy x is taken to be the expectation of the discounted accumulated reward: v&o) = ~{~P”R((so,s~,...,sn))llr). n=O Since TERFs are finitely specified, we can find good ways of encoding and computing optimal policies (see Section 4). But first we examine the expressive power of TERFs. 3The ri are assumed to be additive and independent (this is not restrictive). Any (history independent) MDP can be expressed this way by restricting Q, to contain no temporal operators. 1162 Phnning 3 Encoding ‘I&pica1 Forms of Behavior To demonstrate that TERFs provide an appropriate and use- ful language in which to specify rewards for NMRDPs, we examine several common examples to see how they can be encoded in PLTL. We make no claim that all interesting re- wards can be encoded in this way, but the evidence suggests that PLTL and TERFs can capture a very large and useful class of reward functions. Among the common types of behaviors, simple goal achievement has retained a special place in classical plan- ning. However, in a process-oriented model, like an MDP or NMRDP, a number of subtleties arise in giving “goal achieve- ment” a precise interpretation. We describe several possibili- ties. Assume one such goal is the proposition G: we wish the agent to reach a state in which G holds and will reward it with r if it does so. The simplest reward formula for this goal is G. As a TERF, this rewards the agent at every state satisfying 6, and hence the agent is more highly rewarded (roughly) the larger fraction of its time it spends in G-states. This provides incentive for the agent to constantly maintain G if r is greater than rewards it may receive for other behaviors. In many cases, this is not the intended effect of specifying a goal G. If we only care that G is achieved once, there are several different interpretations that can be provided. The strictest offers reward r only to the first state at which G holds; that is, (G A l@@G) : r. A more generous formula, OG : r, rewards every state that follows the achievement of G. Finally, we may reward G periodically, but not encourage constant maintenance of G, by rewarding G at most once ev- ery k stages: formula GA l(6’G) : r will reward G-states that have not occurred within k-stages of a previous G-state. Yet another option rewards any G-state that occurs within k-stages of some 1G-state (allowing up to k consecutive G-rewards), using G A @s’lG : r. In addition, PLTL allows one to formulate temporally ex- tended goal sequences. For instance, if the agent is to be rewarded for achieving G, followed immediately by H and then by I, the reward formula 02G A OH A I can be used. Periodic reward of such behavior, or the similar behavior in which other steps are allowed to intervene between G, H, and I, can also be prescribed in a straightforward fashion. The formulations above assume that there is some goal G that is constantly desirable, a vestige of the classical in- terpretation of goals. Such behaviors are more suited to background, maintenance goals. In a process-oriented set- ting, we are likely to want the agent to respond to requests or commands to bring about some goal. In these settings, goals are not constant: they arise periodically, can be fulfilled, for- gotten, preempted, and might even expire. We model these in PLTL using response formulas which specify a relation between a command C and rewarded goal achievement G. The most basic response formula is that of eventual re- sponse, G A OC-the agent is rewarded at any G-state that follows a C-state in which the command is given (or is out- standing). As usual, we may only wish to reward the first state at which G holds following the command, in which case GA @(lG S C) suffices. Many requests must be achieved in a timely fashion. Immediate response formulas have the form G A OC, re- warding a goal achieved at the state following a command. More generally, we have bounded response formulas of the type G A OlrcC which reward goal achievement within k steps of a request. This formula does not preclude multi- ple rewards for a single request, so we might instead pre- fer G A O<‘EC A @(lG S C), which rewards only the first goal state. Finally, a graded reward can be given for faster achievement of G (within limits). For instance, the set (GAOC:rl, G A Os2C : r2, G A &C : r3) rewards goal achievement in one step with reward r I+ r2 + r3, in two steps with r2 + r3, and in three steps with r3. In a longer version of this paper, we describe additional types of behaviors, as well as the possibility of using other logics to express different kinds of reward. 4 Modeling NMRDPs with MDPs As has been pointed out, constructing optimal policies in settings of non-Markovian reward can be computationally prohibitive. In this section, we describe a method of state- space expansion that determines the aspects of history that are relevant to an NMRDP (i.e., which must be recorded so that we can verify the truth of the temporal reward formulas), and encodes this history within the state. A straightforward trans- formation of the reward function, so that rewards are attached to such extended states rather than trajectories, restores the Markovian reward property. Together with an adjustment in action descriptions to deal with the new state space, we then have a (fully-observable) MDP that accurately reflects the NMRDP, that can be solved by standard (relatively efficient) methods. We begin by discussing the basic properties that such a transformation should satisfy, and then specialize to the case of rewards that are given by TERFs. 4.1 Markovian Transformations To transform an NMRDP into an equivalent MDP requires that we expand the state space S of the NMRDP so that each new state in the expanded state space ES carries not just the original state information, but also any additional information required to render reward ascription independent of history.4 As we shall see, we can think of expanded states as consisting of a base state annotated with a label that summarizes rele- vant history. If Gs = (S, A, R) is the NMRDP in question, then we wish to produce an MDP GES = (ES, A, &s) with expanded space ES. The actions A available to the agent remain unchanged (since the aim is to produce a policy suit- able for the original NMRDP), but the reward function REP is now Markovian: it assigns rewards to (expanded) states. For the new MDP to be useful, we would expect it to bear a strong relationship to the NMRDP from which it was constructed. In particular, we define a strong correspondence between the two as follows: 4Here we are concerned only with dynamics are already Markovian. reward ascription; the system Handling Uncertainty 1163 Definition 4.1 An A4DP GES = (ES, A, RES) is an expansion Proposition 4.3 For any policy r’ for A4DP GEs, corre- of an NMRDP Gs = (S, A, R) iftherearefunctionsr : ES I--+ sponding policy x for Gs, and s E S, we have V,(s) = S and D : S I+ ES such that: Kr, (44). 1. For all s E S, 7(0(s)) = s, 2. For all s, s’ E S and es E ES, ifPr(s, a, s’) = p > 0 and +4 = s, then there is a unique es’, r(es’) = s’, such that Pr(es, a, es’) = p. 3. For any feasible trajectories (so,. . v I s,) in Gs and @so, 9 -a, es,) in GES, where r(esi) = si and a( SO) = eso, we have R((so, . . . , s,)) = R&es,). Intuitively, T(es) is the base state for es, the state in S extended by es. For this reason, we will often speak of extended states being labeled or annotated: each extended state can be written sol, where s E S is the base state, and 1 is a label that distinguishes es from other extensions of s. However, among the extensions of s, we must pick out a unique a(s) E ES as the “start state” corresponding to s. In other words, a(s) should be thought of as that annotation of s with an “empty” history; i.e., corresponding to an occurrence of s at the very start of a trajectory. We will see below why it is important to distinguish this extension of s from other extensions. The important parts of this definition are clauses (2) and (3), which assert that GES and Gs are equivalent (with respect to base states) in both their dynamics and reward structure. In particular, clause (2) ensures, for any trajectory in Gs soa*s, ’ * ’ s,-,atlz;ns, and extended state es0 with base state SO, that there is a trajectory in GES of similar structure esga3esl - - - es,- 1 a* esn where T(esi) = si for all i. We call (eso, ‘. . , es,) and (so, * - - 7 s,) weakly corresponding trajectories in this case. Clause (3) imposes strong requirements on the reward as- signed to the individual states in GES. In particular, if (es0, - - -, es,) and (SO, . . . , s,) are weakly corresponding, and a(so) = es0 (i.e., es0 is a start state), we say these tra- jectories are strongly corresponding. It is not hard to see that this relationship is one-to-one: each (SO,. . . , s,) has a unique strongly corresponding trajectory, and (eso, . . . , es,) has a unique strongly corresponding trajectory iff es0 is a start state. Clause (3) requires that RES assign rewards to extended states in such a manner that strongly correspond- ing trajectories receive the same reward. This need not be the case for weakly corresponding trajectories since, intu- itively, different annotations (extensions) of SO correspond to different possible histories. If we can produce an MDP GES that is an expansion of an NMRDP Gs as specified by Defn. 4.1, then we can find optimal policies for Gs by solving GES instead. Corollary 4.4 Let r’ be an optimal policy for MDP GEM. Then the corresponding policy r is optimal for NMRDP Gs. Thus, given a suitable expanded MDP and an optimal policy 7c’, one can produce an optimal policy x for the NMRDP quite easily. In practice, the agent need not construct 7r ex- plicitly. Instead, it can run X’ over the expanded MDP. Once the agent knows what base state it starts in, it determines the corresponding extended state using the function 0. Further- more, the dynamics of the expanded MDP ensures that it can keep track of the current extended state simply by observing the base state to which each transition is made. Finally, we should consider the size of the expanded MDP. Often, we can fulfill the requirements of Defn. 4.1 with a trivial MDP, that has states encoding complete trajectory in- formation over some finite horizon. But such an expanded space grows exponentially with the horizon. Furthermore, even simple rewards -like OG, which only require one item of history (a bit indicating if a G state has been passed through)-can require in infinite amount of complete trajec- tory history using this naive approach. If possible, we want to encode only the relevant history, and find an MDP which has a few states as possible (subject to Defn. 4.1). Note that state- space size tends to be the dominant complexity-determining factor in standard MDP solution techniques, especially as applied to planning problems.” 4.2 Transformations using TERFs The problem of finding a small MDP that expands a given NMRDP is made easier if the latter’s rewards are given by a TERF. In this case, it is natural to label states with PLTL for- mulas that summarize history. More precisely, the new state space ES consists of annotated states, of the form s o f where s E S and f is a formula in PLTL. These annotations will be meaningful and correct assertions about history, in a sense to be made precise below. We give an algorithm that constructs an expansion of the state space by producing labelings of states that are sufficient to determine future reward. We begin with a simple example to illustrate the essential ideas. Consider a single reward formula $R = Q A @OP. Recall that our goal is to encode all relevant history in a state’s annotation. Thus, for each state s in which 4~ might possibly be true, we need at least two distinct labels, one implying the truth of 4~ and one its falsity. Next, imagine that we have an extended state es = so+, where Q is true in s and ti = @OP. (Thus es implies that 43, is true.) Next, suppose that s is reachable from some other state s- (i.e., there is some transition in the NMRDP from s- to s). Since we must ensure that es’s label $ is a correct as- sertion about its history, in the expanded MDP any transition from an extended version of s- (es-, say) to es must satisfy Definition 4.2 Let r’ be a policy for MDP GEM. The corresponding policy x for the NMRDP Gs is defined as rr((so,---,s,)) = d(es,), where (eso;..,es,) is the strongly corresponding trajectory for (SO, - - - , s,). ‘See [LDK95] on the complexity of solving MDPs. Generally, the state space is problematic in planning problems because it grows exponentially with the number of atomic propositions. Adding his- tory “naively” to the domain exacerbates this problem considerably. 1164 Planning the “historical” constraints imposed by $. In this example, if there is a transition from es- to es it must be the case that es- satisfies OP (otherwise, es might not satisfy OOP). In general, we can use the regression operator to determine what must have been true at earlier states. A reward for- mula C$ is true of a trajectory terminating in es iff Regr(+, s) holds at es’s predecessor. Thus, the formula Regr(+, s)- or a stronger formula implying Regr(+, s)-must be part of any label attached to states that reach es = so4 in one step. This process is, naturally, repeated (states reaching es- must satisfy P, etc.). To quickly summarize this example, suppose that every state is reachable from any other, and that P and Q are the only propositions (hence, there are exactly 4 base states). Then 12 extended states are necessary. For each base state where Q is false (i.e., P A 1Q and 1P A 1Q) we need one extension labeled with TOP and another with OP. For each of the two base states in which Q is true, we need 4 extended states, with the labels OP A OOP, TOP A OOP, OP A -@OP, and TOP A 1OOP. Note that every extended state has the property that we can easily tell whether the reward formula Q A OOP is true there. Furthermore, the regression constraints discussed above hold. For example, let s- k P A Q and es- = s- o (TOP A OOP), and consider the transition to the base state s where s b 1P A Q. It is necessary that there be some labeling of s, es = so+, such that Regr(+, s) is implied by es-. But this is so, because we can take 1c, to be OP A 1OOP. Note that if we had not been able to find such $J, this would mean that es-‘s label did not encode enough history (because we would be unable to determine the correct subsequent label after a move to s). Our algorithm constructs the set of extended phases some- what indirectly, using a two phase approach. Phase I of our algorithm constructs label sets for each state, Is(s), contain- ing PLTL formulas that might be relevant to future reward. The elements of Zs( s) will not necessarily be the labels them- selves, but are the ingredients out of which labels are con- structed. In a certain sense (to be discussed in Section 4.3) it does not matter if we add “too many” (or too strong) formulas to Is(s), so there are in fact several distinct implementations of Phase I. But as we have just seen, regression should be used to impose constraints on label sets. If ~,75 E Is(s), so that C$ might be (part of) the label of a extension es, then Regr(4, s) must be implied by the annotation of any state es- from which es is reachable. Given that Phase I is correct (i.e., it finds all formulas that might be relevant), we can restrict attention to extended states whose labels are combinations of the formulas in Is(s), asserting that some are true and others false. Formally: Definition 4.5 If Y is a set of PLTL formulas, the atoms of Y,denoted ATOMS(Y), is the set of all conjunctions that can be formed from the members of Y and their negations. E.g., if’u = {q A Op,p}, then ATOMS(Y) = {(q A Op) Ap, ‘(q A @P> A P, (q A @P> A ‘P, l(a A @P) A 1~). Thus, the labels extending s will belong to ATOMS(ZS(S)). In general, however, many of these atoms will be inconsistent, or simply not reachable given the set of feasible trajectories in the original NMRDP. Rather than performing theorem proving to check consistency, we will generate the extended states we require in a constructive fashion, by explicitly con- sidering which states are reachable from a start state. This is Phase II of our algorithm. To illustrate, suppose that we have determined Zs(st ) = {Q A OOP, T A OP, I A OP, P}, and that sr b 1Q A P. There is only one atom over Zs( st ) that can be true at s1 in the (length 1) trajectory (sl), namely: f = l(Q A @Of’) A ‘(T A @P) A ‘(1 A OP) A P. We thus include st of in ES. (Note that st of can be logi- cally simplified, to s1olOP.) >From this extended state we consider, for each successor state s2 of sr, which atom over ~2’s label set is true in the trajectory (st , ~2). Again, this will be unique: for instance, if s1 can succeed itself, we obtain a new extended state sr of’ where f’ = l(Q A OOP) A (T A OP) A +I A OP) A P (This also can be simplified, in this case to s2oOP.) For any action a such that Pr(sr , a, ~2) = p > 0, we assert that Pr(st of, a, s20f’) = p. By adding extended states to ES in this way, we will only add extended states that are reachable and whose history is meaningful. For instance, we see that while C$ = Q A OOP is in Zs(st), no label that makes 4 true at st will be reachable (recall st makes Q false). This effectively eliminates 4 from consideration at st . The algorithm is described in Figure 1. We defer a dis- cussion of Phases I and III until Section 4.3. Note, however, that an easy implementation of Phase I is to set Is(s), for all s, equal to lJdiEQ Subformulas where @ is the set of reward formulas. All the results in this section apply to any suitable choice of Is(.), including this one. The MDP GES generated by the algorithm is an expansion of Gs. To show this, it is useful to define a more general concept of which GES is an instance: Definition 4.6 GES = (ES, A, Z&s) is a sound annotation of Gs = (S, A, R) if each state es E ES is of the form so f for s E S and some PLTL formula f, and: 1. Fixing r(so f) = s, there exists o : S I--+ ES such that clauses [I] and [2] of De$nition 4. I hold. 2. Let (so0 fo,sI of,, . . . , s,o fn), n 1 0, be such that a(so) = soofo. Then: (so, s1, - * - , %> b fn This definition is similar to our definition of expansion (Defn. 4. l), except that we give the extended states a partic- ular form: annotations using PLTL formulas. Furthermore, instead of requiring that annotations summarize enough his- tory for the purposes of determining rewards, we no longer care why GES has the annotations it does; we only insist that whatever history is recorded in these annotations be accurate. Because of its generality, the notion of sound annotation may have other applications. However, for our purposes we must make one more assumption: that GES’S labels are informative enough to determine rewards. efinition 4.7 GES determines rewards over a set of reward formulas @ ifi for all es = sof E ESand all & E a,, f determines q+. Handling Uncertaintv 1165 Phase I Find label sets: Choose any Is : 5’ I+ subsets ofPLTL, such that: For all s E S, all f E ATOMS ( ZS( s)), and all formulas 4: If C# E ip, the set of reward formulas for Gs, then: f determines 4 If 4 E Zs(s’), where Pr(s, a, s’) > 0 for some a E A, then: f determines Regr( 4, s’) Note. See text for more discussion of this phase. However, k”b(S> = U4i Ea Subformulas is always suitable. Phase II Generate GES: 1. Foralls E Sdo: (a) Find f E ATOMS(~~(S)) such that (s) k f. Note. Such an atom exists and is unique. (b) Add sof to ES. Note. This will be the start state corresponding to s. (c) Mark sof unvisited. 2. While there exists an unvisited state es E ES, es = sof do: (a) For all s’ such that Pr(s, a, s’) > 0 for some a do: i. Find f’ E ATOMS(~S(S’)) such that f + Regr( f’, s’). ii. If s’o f’ $! ES then add s’o f’ to ES and mark it unvisited. iii. Set Pr(s0 f, a, s’o f’) equal to Pr(s, a, s’), for all a. 3. Fores= sof in ES, set &,s(es) to &,,@{r; : f + &}. 4. Set Pr(s0 f, a, s’o f’) = 0 for all transition probabilities not previously assigned. Phase III Minimization: See Section 4.3 for discussion. Note. This phase is not always necessary. Figure 1: Algorithm to find Annotated Expansion of Gs Proposition 4.8 If GEs = (ES, A, REP) is sound annotation of Gs that determines rewards over a, and REs(so f) = C+ieQ (ri : f k +i>, then GEM is an expansion of Gs. The key to understanding our algorithm is realizing that it is designed to generate a MDP that satisfies the conditions of this proposition. Thus, by the results of Section 4.1, we have succeeded in our goal of finding an equivalent MDP GES for any NMRDP Gs whose rewards are given using a TERF. In particular, we have the following key result: Theorem 4.9 Let Gs be an NMRDP whose rewardfunction is given by a TERI;: over a set offormulas 0. The Expansion Algorithm of Figure I constructs an MDP GES that is an expansion of Gs. Once this expansion GES is constructed, an optimal policy for the MDP GES can be computed using standard techniques. The correspondence presented in Section 4.1 shows that an agent executing this policy will behave optimally with re- spect to the original NMRDP. We note that the labels in GES determine the history that must be kept track of during policy execution. In particular, suppose we are given a policy 7r’ defined on the extended space to apply to the NMRDP, and the process starts in state SO. We take the extended state to be SO’S unique start state es0 and perform r’(eso) = a. An ob- servation of the resulting state SI is made. The dynamics of the extended MDP ensure that there is a unique es1 extending s1 that is reachable from es0 under action a. Thus, we next execute action n’(esl), and proceed as before. Note that we can keep track of the extended state that we are currently in even though we only get to directly observe base states. 4.3 Other Properties of the Algorithm In this section, we very briefly discuss some of the other interesting issues raised by the expansion algorithm. We begin by examining Phase I. As already noted, one possible implementation is Is(s) = Zssi,,b(s); i.e., the label sets consisting of all subformulas of a,. An advantage of this choice is that Phase I becomes trivial, with complexity 0 (L) , where L = c4iEQ length(&) is a bound on the number of subformulas we generate. Furthermore, we can bound the size of GES. Since there are at most zL atoms over Is(s), each base state can receive at most this number of distinct labels. Thus GES can be at most this factor larger than Gs (although Phase II does not usually generate all conceivable labels.) The exponential here may seem discouraging, but there are simple, natural examples in which this number of historical distinctions is required for implementing an optimal policy. For instance, for the reward formula O”P, we need to keep track of when P was true among the previous n steps, leading to 2” distinct annotations. Nevertheless, the main disadvantage of Is,,,,(.) is that it can lead to unnecessarily fine distinctions among histories, so that GES as produced by Phase II is not guaranteed to be minimal (in the sense of having as few states as possible among valid expansions of Gs). If minimality is important, a separate step after Phase II is required. Fortunately, min- imizing GES can be performed using a variant of standard algorithms for minimizing finite state automata [HU79]. We defer discussion to the full paper, but note that the complexity of doing this is only polynomial in the size of GEs. Thus, so long as the intermediate GES produced by Phase II is of manageable size, minimization is fairly straightforward.6 A second implementation of Phase I constructs label sets Is, with “weaker” formulas, subject to the stated require- ments. More precisely, we initially set Is,(s) = a,, for all s. Then, so long as we can find s, s’, such that s’ is reachable from s and { Regr(4, s’) : C$ E Is, (s’)} g Is,(s), we add (Regr(4,s’) : q5 E Zs,(s’)) to Is,(s). We iterate un- til this terminates -which it will, so long as we are careful not to add different (but logically equivalent) formulas twice to Is,(s). This procedure ensures the necessary properties of Is(.). For many natural examples of reward formula, this process terminates quickly, generating small label sets. The major reason for considering Is,(.) is that GES, as constructed subsequently by Phase II, is then guaranteed to have minimal size. But Is, (a) has a serious drawback as well: Phase I can potentially become very complex. The number of iterations until termination can be exponential (in the size of the reward formulas) and the size of the label sets can grow double-exponentially. Perhaps the optimal strategy, then, is to begin to implement Phase I using Is,(.), but if any reward 61f GES is much larger than necessary, Phase II’s complexity could cause difficulties. 1166 Planning formula proves troublesome, to then revert to the subformula technique at that point. We conclude by noting that Phase II is, in comparison, unproblematic. Since each extended state is visited exactly once the complexity is linear in the size of the final answer (i.e., the size of GES.) Furthermore, none of the operations performed in Phase II are difficult. Steps 1 .a and 2.a.i appear to involve theorem-proving, but this is misleading. Step 1.a is actually just model checking (over what is, furthermore, a very short trajectory) and in this particular case can be done in time proportional to ‘&ls(s) length(+). Step 2.a.i can also be performed quickly; the details depend on exactly how Phase I is implemented, but in general (and in particular, for the two proposals discussed above) enough book-keeping information can be recorded during Phase I so that 2.a.i can be performed in time proportional to IZs( s) I. Again, space limitations prevent us from providing the details. In conclusion, the annotation algorithm appears to be quite practical. The potential exists for exponential work (relative to the size of Gs) but this is generally the case exactly when we really do need to store a lot of history (i.e., when GES is necessarily large). 5 Concluding Remarks While MDPs provide a useful framework for DTP, some of the necessary assumptions can be quite restrictive (at the very least, requiring that some planning problems be encoded in an unnatural way). We have presented a technique that weakens the impact of one of these assumptions, namely, the requirement of Markovian (or state-based) reward. The main contributions of this work are a methodology for the natural specification of temporally extended rewards, and an algorithm that automatically constructs an equivalent MDP, allowing standard MDP solution techniques to be used to construct optimal policies. There are a number of interesting directions in which this work can be extended. First, similar techniques can be used to cope with non-Markovian dynamics, and can also be used with partially-observable processes. In addition, other tem- poral logics (such as more standard forward-looking logics) and process logics can potentially be used in a similar fashion to specify different classes of behaviors. Another interesting idea is to use compact representations of MDPs to obviate the need for computation involving in- dividual states. For instance, Bayes net representations have been used to specify actions for MDPs in [BDG95], and can be exploited in policy construction. Given an NMRDP specified in this way, we could produce new Bayes net action descriptions involving an expanded set of variables (or propo- sitions) that render the underlying reward process Markovian, rather than expanding states explicitly. Finally, our technique does not work well if the expanded MDP is large, which may be the case if a lot of history is necessary (note that this is inherent in formulating such a problem as an MDP, whether automatically constructed or not). The complexity of policy construction is typically dom- inated by the size of the state space. An important direction for future work is to combine policy construction with state space expansion. The hope is that one can avoid generating many expanded states using dominance arguments particular to the reward structure of the given NMRDP. Acknowledgements The work of Fahiem Bacchus and Craig Boutilier was sup- ported by the Canadian government through their NSERC and IRIS programs. We thank the anonymous referees for their thoughtful re- views. It is unfortunate that space limitations prevented us from responding to several of their valuable suggestions. References [AH901 R. Alur and T. Henzinger. Real-time logics: complexity and expressiveness. LICS-90, Philadelphia, 1990. [BD94] C. Boutilier and R. Dearden. Using abstractions for decision-theoretic planning with time constraints. AAAZ-94, pp.10161022, Seattle, 1994. [BDG95] C. Boutilier, R. Dearden, and M. Goldszmidt. Exploit- ing structure in policy construction. IJCAZ-95, pp. 1104-l 111, Montreal, 1995. [Be1571 R. E. Bellman. Dynamic Programming. Princeton Univer- sity Press, Princeton, 1957. [BP951 C. Boutilier and M. L. Puterman. Process-oriented plan- ning and average-reward optimality. IJCAI-95, pp. 1096-l 103, Montreal, 1995. [DK89] T. Dean and K. Kanazawa. A model for reasoning about persistence and causation. Camp. Znrel., 5: 142-150, 1989. [DKKN93] T. Dean, L. P. Kaelbling, J. Kirman, and A. Nicholson. Planning with deadlines in stochastic domains. AAAZ-93, pp.574- 579, Washington, D.C., 1993. [Drug91 M. Drummond. Situated control rules. KR-89, pp.103- 113, Toronto, 1989. [EmegO] E. A. Emerson. Temporal and modal logic. In J. van Leeuwen, ed., Handbook Theor: Comp. Sci., Vol.B, pp.997-1072, 1990. [GK91] P Godefroid and F. Kabanza. An efficient reactive planner for synthesizing reactive plans. AAAZ-91, pp.640-645, 1991. [HH92] P. Haddawy and S. Hanks. Representations for decision- theoretic planning: Utility functions for deadline goals. KR-92, pp.71-82, Cambridge, 1992. [How601 R. A. Howard. Dynamic Programming and Markov Pro- cesses. MIT Press, Cambridge, 1960. [HU79] J. E. Hopcroft and J. D. Ullman. Introduction to Automata Theory, Languages and Computation. Addison-Wesley, 1979. [Kab90] F. Kabanza. Synthesis of reactive plans for multi-path environments. AAAZ-90, pp. 164-l 69, 1990. [KHW94] N. Kushmerick, S. Hanks and D. Weld. An algorithm for probabilistic least-commitment planning. AAAZ-94, pp. 1073- 1078, Seattle, 1994. [LDK95] M. L&man, T. L. Dean and L. P. Kaelbling. On the com- plexity of solving Markov decision problems. UAI-95, pp.394- 402, Montreal, 1995. [Put941 M. L. Puterman. Markov Decision Processes: Discrete Stochastic Dynamic Programming. Wiley, New York, 1994. [Sch87] M. J. Schoppers. Universal plans for reactive robots in unpredictable environments. IJCAZ-87,1039-1046, Milan, 1987. [TR94] J. Tash and S. Russell. Control strategies for a stochastic planner. AAAZ-94, 1079-1085, Seattle, 1994. Handling Uncertainty 1167 | 1996 | 172 |
1,813 | uting Optimal Policies for Compact Representations Craig Boutilier and David Poole Department of Computer Science University of British Columbia Vancouver, BC V6T 124, CANADA cebly@cs.ubc.ca, pooleQcs.ubc.ca ecision Abstract Partially-observable Markov decision processes provide a gen- eral model for decision theoretic planning problems, allowing trade-offs between various courses of actions to be determined under conditions of uncertainty, and incorporating partial ob- servations ma& by an agent. Dynamic programming algo- rithms based on the belief state of an agent can be used to construct optimal policies without explicit consideration of past history, but at high computational cost. In this paper, we discuss how structured representations of system dynamics can be incorporated in classic POMDP solution algorithms. We use Bayesian networks with structured conditional prob- ability matrices to represent POMDPs, and use this model to structure the belief space for POMDP algorithms, allowing irrelevant distinctions to be ignored. Apart from speeding up optimal policy construction, we suggest that such representa- tions can be exploited in the development of useful approxi- mation methods. 1 Introduction Recent interest in decision-theoreticplanning (DTP) has been spurred by the need to extend planning algorithms to deal with quantified uncertainty regarding an agent’s knowledge of the world and action effects, as well as competing objectives [9,7,4, 161 (see [2] for a brief survey). A useful underlying semantic model for such DTP problems is that of partially observable Markm decision processes (POMDPs) [ 61. This model, used in operations research [ 17, 121 and stochastic control, accounts for the tradeoffs between competing objec- tives, action costs, uncertainty of action effects and observa- tions that provide incomplete information about the world. However, while the model is very general, these problems are typically specified in terms of state transitions and obser- vations associated with individual states--even specifying a problem in these terms is problematic given that the state space grows exponentially with the number of variables used to describe the problem. Influence diagrams (IDS) and Bayesian networks (BNs) [ 10, 141 provide a much more natural way of specifying the dynamics of a system, including the effects of actions and ob- servation probabilities, by exploiting problem structure and independencies among random variables. As such, prob- lems can be specified much more compactly and naturally 1168 Planning [8,4,16]. In addition, algorithms for solving IDS can exploit such regularities for computational gain in decision-making. Classic solution methods for POMDPs within the OR com- munity, in contrast, have been developed primarily using explicit state-based representations which adds a sometimes unwanted computational burden. However, unlike ID algo- rithms, for which policies grow exponentially with the time horizon, POMDP algorithms offer concepts (in particular, that of belief state) that sometimes alleviate this difficulty. In this paper we propose a method for optimal policy con- struction, based on standard POMDP algorithms, that ex- ploits BN representations of actions and reward, as well as tree [4] or rule [16] representations within the BN itself. In this way, our technique exploits the advantages of classic POMDP and ID representations and provides leverage for approximation methods. In Section 2, we define POMDPs and associated notions, at the same time showing how structured representations, based on BNs (augmented with tree-structured conditional probability tables), can be used to specify POMDPs. In Sec- tion 3, we describe a particular POMDP algorithm due to Monahan [ 121, based on the work of Sondik [ 171. In Sec- tion 4, we describe how we can incorporate the structure captured by our representations to reduce the effective state space of the Monahan algorithm at any point in its computa- tion. Our algorithm exploits ideas from the SPI algorithm of [4] for fully observable processes. In Section 5 we suggest that our method may enable good approximation schemes for POMDPs. 2 OMDPs and Structured Representations In this section we build upon the classic presentation of POMDPs adopted in much of the OR community. We refer to [ 17, 11, 61 for further details and [ 12, 51 for a survey. We describe the main components of POMDPs and related concepts. However, by assuming that problems are specified in terms or propositional (or other random) variables, we are able to describe how structured representations, in particular, decision trees or if-then rules, can be used to describe these components compactly. We begin with a (running) example. Example Imagine a robot that can check whether a user wants coffee and can get it by going to the shop across the From: AAAI-96 Proceedings. Copyright © 1996, AAAI (www.aaai.org). All rights reserved. Figure 1: Action Networks for (a) GetC and (b) TestC street. The robot is rewarded if the user wants coffee WC and has coffee HC, but is penalized if HC is false when WC is true. The robot will also get wet W if it is raining R when it goes for coffee, unless it has its umbrella U. We can imagine a number of other tasks here as well. Although the robot can check on the weather, grab its umbrella, etc., we focus on two actions: getting coffee GetC and checking whether the user wants coffee by means of a quick inspection TestC. 2.1 System Dynamics We assume a finite set of propositions P that describe all relevant aspects of the system we wish to control. This in- duces a finite state space S = 2” consisting of all possible assignments of truth values to P. There is a finite set of actions J! available to the agent or controller, with each ac- tion causing a state transition. We assume the system can be modeled as a POMDP with a stationary dynamics (i.e., the effects of actions do not depend on the stage of the pro- cess). For simplicity we assume all actions can be taken (or attempted) at all states. While an action takes an agent from one state to another, the effects of actions cannot be pre- dicted with certainty; hence (slightly abusing notation) we write Pr((s2]sr, a) to denote the probability that s2 is reached given that action a is performed in state 81. This formulation assumes the Markov property for the system in question. One can represent the transition probabilities associated with action a explicitly using a ISI x ISI probability ma- trix. However, the fact that ISI increases exponentially with the number of problem characteristics ]P I generally requires more compact representation; thus we represent an action’s effects using a “two-slice” (temporal) Bayes net [8]: we have one set of nodes representing the state prior to the action (one node for each variable P), another set representing the state after the action has been performed, and directed arcs repre- senting causal influences between these sets (see Figure 1). We require that the induced graph be acyclic. For simplicity we assume also that arcs are directed only from pre-action to post-action nodes. ’ See [8,4] for details. The post-action nodes have the usual conditional proba- bility tables (CPTs) describing the probability of their values ‘We often denote post-action variables by P’ instead of P to prevent confusion. Causal influences between post-action variables should be viewed as rami@ufion.r and will complicate our algorithm slightly, but only in minor detail. given the values of their parents, under action a. We assume that these CPTs are represented using a decision tree, as in [4] (or if-then rules as in [IS]). These are essentially com- pact function representations that exploit regularities in the Cl%. We will exploit the compactness and structure of such representations when producing optimal policies. We denote the tree for variable P under action a by Tree(P’la).2 Example Figure 1 (a) illustrates the network for action GetC. The network structure shows, for instance, that the truth of IV’, whether the robot is wet after performing GerC, depends on the values of R, U and W prior to the ac- tion. The matrix for W’ quantifies this dependence; and Tree(W’]GetC) illustrates the more compact representa- tion (the leaf nodes indicate the probability of W’ after GetC given the conditions labeling its branch: left arcs denote true and right arcs false). We elaborate on the Qbs variable below. 2.2 Observations Since the system is partially observable, the planning agent may not be able to observe its exact state, introducing another source of uncertainty into action selection. However, we assume a set of possible observations 0 that provide evidence for the true nature of (various aspects of) the state. In general, the observation at any stage will depend stochastically on the state, the action performed and its outcome. We assume a family of distributions over observa- tions, For each si, sj, uk such that I+(+ Isi, uk) > 0, let Pr( q Isi, ah, sj) denote the probability of observing 06 when action ok is executed at state si and results in state sj. (As a special case, a fully observable system can be modeled by assuming 0 = s and Pr(o&?i,ok, sj) = 1 iff 01 = si.) We assume for simplicity that the observation probability de- pends only on the action and starting state, not the resulting state; that is, Pr(od]si, ok, sh) = Pr(o~lsj, ak, sh) for each Si, Sj. 3 To represent observation probabilities compactly, we add a distinguished variable Ohs to each action network that repre- sents the observations possible after performing that action. We use Ohs(a) to denote the set of possible observations given a.4 The variables that influence the observation are in- dicated by directed arcs, and this effect is described, as above, using a decision tree. We note that complex observations may also be factored into distinct observation variables (e.g., should the agent get information pertaining to propositions P and & by performing one action, two distinct variables Qbq and Qbs2 might be used); we ignore this possibility here. 2The network structure is not strictly necessary: the parent of a post-action node can be determined from its CPT or decision tree (see, e.g., Poole’s [ 151 rule-based representation of Bayes nets). 3This is a natural assumption for information-gathering actions, but others am possible; e.g., Sondik’s [17] original presentation of POMDPs assumes the observation depends only on the resulting state. This assumption makes our algorithm somewhat simpler to describe; but it can generalized (see Section 4). 4These are similar to observation variables in influence diagrams [lo]; however, there am no emanating information arcs. Handling Uncertainty 1169 Figure 2: Reward Function Network Example The variable Ohs in Figure l(a) takes on a sin- gle value (Null), obtained with certainty when GetC is executed (i.e., the action provides n.o feedback). More interesting is the action Z&C shown in Figure l(b). Al- though it has no effect on the state variables (we assume persistence), it is useful as an information gathering ac- tion: the value of the variable Obs (Yes or No) is strongly dependent on whether the user wants coffee. Should the value Yes be observed, our robot may be quite confident the user does, in fact, want coffee (see below). 2.3 Rewards The final component needed to describe a PGMDP is a real- valued rewardfunction R that associates rewards or penalties with various states: R(s) denotes the relative goodness of being in state 5. We also assume a cost function C(a, s) denoting the cost of taking action a in state s. The reward (cost) function can be represented in a structured fashion using a value node and decision tree describing the influence of various combinations of variables on rewards (as with tree- structured CPTs). Leaves of the tree represent the reward associated with the states consistent with the labeling of the corresponding branch. Example Figure 2 shows the reward function for our prob- lem, indicating that the reward for a particular state is influenced only by the truth of the propositions IV, WC and HC. A similar representation for action cost can be used. In this example action costs are constant: a cost of 1.0 for GetC and 0.5 for TestC is assumed. The sets of actions, states and observations, the associated transition and observation probabilities, and the reward and cost functions, make up a POMDI? We now turn our attention to the various concepts used in decision-making. 2.4 Policies We focus onJinite-horizonproblems here: given a horizon of size n an agent executes n actions at stages 0 through n - 1 of the process, ending up in a terminal state at stage n. The agent receives reward R(s) for each state s passed through at stages 0 through n (its trajectory). A plan or policy is a function that determines the choice of action at any stage of the system’s evolution. The value of a policy is the expected sum of rewards accumulated (incorporating both action costs and state rewards and penalties). A policy is optimal if no other policy has larger value. In choosing the action to perform at stage k of the process, the agent can rely only on its knowledge of the initial state SO (whether it knows the state exactly, or had an initial distribu- tion over states), and the history of actions it performed and observations it received prior to stage Ic. Different action- observation histories can lead an agent to choose different actions. Thus, a policy can be represented as a mapping from any initial state estimate, and &stage history, to the ac- tion for stage Ic + 1. This is roughly the approach adopted by solution techniques for IDS [lo]. However, an elegant way to treat this problem is to maintain a current belief state, and treat policies as mapping over from belief states to actions. 2.5 Belief States A belief state 7r E A(S) is a probability distribution over states. The probability 7ri assigned to state si by T is the degree of belief that the true (current) state of the system is Si. Given some state of belief zk estimating the system state at stage k of the decision process, we can update our belief state based on the action ak taken and observation ok made at stage k to form a new belief state ?ykfl characterizing the state of the system at stage k+ 1. Once we have ?yle+t in hand, the fact that a”, ok: and ?yk gave rise to it can be forgotten. We use T(n, a, o) to denote the transfomation of the belief state T given that action a is performed and observation o is made: it is defined as T(n, a, O)i = c sj(zS P++j, a, si)Pr(silsj, a>lri c Bj ,#1r ES J+(OlSj 9 a, eJwQ ISj, a>q T(n, a, o)i denotes the probability that the system is in state i once a, o are made, given prior belief state T. The new belief state T(n, a, 0) summarizes all informa- tion necessary for subsequent decisions, accounting for all past observations, actions and their influence on the agent’s estimate of the system state. This is the essential assump- tion behind classical POMDP techniques: at any stage of the decision process, assuming 7rk accurately summarizes past actions and observations, the optimal decision can be based solely on 7rk - history (now summarized) can be ignored [ 17 3. Intuitively, we can think of this as converting a par- tially observable MDP over the original state space S into a fully observable MDP over the belief space t3 (the set of belief states 7r). A belief state may be represented using a vector of ISI probabilities; but structured representations are possible. We do not pursue these here, since most POMDP solution algo- rithms do not use a belief state to construct a policy. 2.6 Value Functions State Value Functions: A state valuefunction VS : S + R associates a value W(s) with each state s. This reflects the expected sum of future rewards the agent will receive, assuming some fixed policy or sequence of actions in the future. In addition, a state Q-function & : S x A + Iw denotes the value Q(s, a) of performing an action a in state s, assuming future value is dictated by a fixed course of action [18]. In particular, let VS’ and Q” be the k-stage-to- go value and Q-functions. If the function VS”-’ is known, then Bellman’s [l] optimality equation ensures that Qk(Si, a)=C(a, S~)+~(si)+~,j~~~(sjlsi,a)vs’E-‘(sj) (1) vh) = n-=ce~{@(Si, a)) (2) 1170 Planning 4 al #q 0 IO *2 jt 10 0 a e 3 4 Figure 4: Structured Domination Testing Figure 3: Piecewise Linear, Convex Value Function Intuitively, once the agent has determined a course of action for the last k - 1 stages of the process (giving rise to vS”-‘), Equation 1 determines the value of executing action a at any state. In the case of fully observable MDPs, this forms the basis of a dynamic programming algorithm that can be used to optimize the choice of action according to Equation 2. We can represent value and Q-functions using decision trees in precisely the same manner as reward functions (e.g., Figure 2). Figure 5 illustrates just such value and Q-trees. In fact, as we will see below, we can apply these equations directly to such structured representations. Belief State Value Functions: Unfortunately, in the case of POMDPs, determining the best action for individual states is not often helpful, for the agent will typically not know the exact state. However, the assignment of value to states via value and Q-functions can also be viewed as an assignment of value to belief states. In particular, any state value function VS induces a value function over belief states: Si ES Following Monahan [ 121 we call these cu-functions. The value of a belief state is the weighted sum of the individual state values; thus, such a-functions our linear functions over belief states. Q-functions can be applied similarly to belief states. Finally, we note that a value tree or Q-tree can be used to represent a linear value function over belief states; when interpreted this way, we call these a-trees. In the sequel, we assume that a-functions are represented by a-trees. In determining optimal policies for POMDPs, we need to represent the optimal (k-stage-to-go) value functions V : A(s) + R for belief states. Clearly, a-functions, being linear, are quite restrictive in expressiveness. However, a key observation of Sondik [17] is that optimal value functions are piecewise linear and convex (p.1.c.~ over the belief space. In other words, we can represent the optimal (k-stage-to-go) value function for any POMDP as a set N of a-functions, with V(X) = max{cu(n) : a E N) (We will see exactly why this is so in the next section.) As a graphical illustration of this p.1.c. representation, consider Figure 3. Assume a single proposition Q (two states q=W and the three &unctions cyt , a2, a3, all represented as trees. Each a-tree determines a linear value function for any belief state (e.g., cri takes its highest value at belief state n(q) = 0;7r@) = 1). The set ((~1, ~22, 03) corresponds to the p.1.c. value function indicated by the thick line. Dominated a-functions: Finally, we note that certain el- ements of a set N of cu-functions may contribute nothing to the induced p.1.c. value function, namely, those elements that are stochastically dominated. For instance, 03 in Figure 3 is dominated by one of cyt or cy2 at all points in the belief space. Monahan [ 121 suggests that such dominated elements be detected by means of a simple linear program and elimi- nated from N (see also [5]). Once again, the use of a-trees can in many cases considerably reduce the size of these LIPS, which normally involve variables for each state. For exam- ple, to consider whether the tree cy4 dominates cyg, as shown in Figure 4, the required LP need only have variables corre- -- sponding to the propositions AB, AB, AC and AC, rather than ISI variables. 3 Computation of Optimal Policies We now describe how to use the ideas above to to determine optimal policies for POMDPs. We begin by presenting the intuitions underlying Monahan’s [ 121 variant of Sondik’s [ 17] algorithm, and how the p.1.c. nature of value functions is exploited. We describe how our compact tree representations can be exploited in the next section. Given a POMDP, we want to determine a policy that se- lects, for any belief state ?T, and k > 0 within the problem horizon, the optimal action to be performed. Intuitively, Pol(?r, k) E d is the best action available to the agent as- suming its state of belief is r and there are k stages of the process remaining. Unfortunately, representing such a func- tion can be problematic, since the set of belief states B is a ISI-dimensional continuous space. However, Sondik’s key observation that k-stage-to-go value functions are p.l.c., and thus finitely representable, also provides a means to finitely represent policies (albeit indirectly). Intuitively, the determi- nation of the “pieces” of the the k-stage-to-go value function will attach actions to each of these pieces. To determine the best action to be performed for a given belief state ?r, the action associated with the “maximal piece” of the value func- tion for n will be the best action. Thus, actions are associated with various regions of the belief space, regions determined by the value function itself. To see this, we first note that with zero stages-to-go the agent has no action choice to make, and the expected value of being in any belief state is given by the a-function determined by immediate reward R; that is, V’(T) = T . R. Thus, V” is a linear function of K. We call this single a-function o”. The computation of V’ depends only on V” and illustrates why the value functions remain p.1.c. The value of perform- ing any action a in a given state s is given by &(a, s) , as defined in Equation 1, using R (or V”) as the terminal value. Handling Uncertainty 1171 Figure 5: Generating Explanation of Future Value then those states have identical expected future value and need not be distinguished in the function QQ. We construct the tree ok so that only these relevant distinctions are made. Construction of &a proceeds abductively: given the tree a, we want to generate the conditions that, prior to the per- formance of action a, could cause the outcome probabilities (with respect to the partitions induced by cw) to vary. We proceed in a stepwise fashion, “explaining” each of the in- terior nodes of a! in turn, beginning with the root node and proceeding recursively with its children. It is important to remember that all of the propositions in cy refer to the state at stage k, and that each of the propositions in Q. refer to stage k + 1. These propositions are related to each other via the state-transition trees for action a. Space precludes a full exposition of the method-we refer to [4] for details of this method (applied to fully observable MDPs)-so we present a simple example. Example To illustrate this process, consider the following example, illustrated in Figure 5. We take the immediate reward function (see Figure 2) to be a tree a0 (the initial value tree), and we wish to generate the expected future value tree for stage 1 assuming action GetC is taken and that cy” determines value at stage 0. We begin by explaining the conditions that influence the probability of WC’ under GetC (Step 1 of Figure 5). This causes Tree( WC’IGetC) to be inserted into the tree cy: as indicated by Figure 1, WC’ is not affected by the action GetC, and thus remains true or false with certainty. The leaves of this partial tree denote the probability of WC’ being true after the action given its value (WC) before the action. We then explain HC’ (Step 2). Since the initial value tree asserts that HC is only relevant when WCis true, the new subtree Tree(HC’IGetC) is added only to the left branch of the existing tree, since WC’ has probability zero on the right. Again, the probabilities labeling the leaves describe the probability of the variable in question afrer the action, while the labels on interior nodes of the branches re- late the conditions before the action under which these probabilities are valid. This becomes clear in Step 3, where we consider the conditions (prior to GetC) that af- fect the occurrence of W’ (wet) after GetG: the relation (Tree(W’IGetC)) is complex, depending on whether the robot had an umbrella and whether it was raining. This final tree has all the information needed to compute ex- pectedfuture value at each leaf-the probabilities at each leaf uniquely determine the probability of landing in any I 3.0 I -1.4 A A 3.0 2.0 -1.4 -2.4 c$: lheforck4c c&yTreeforTatC Figure 6: C.Ptrees with 1 Stage-to-go partition of initial value tree under GetC. Finally, we note that to get the true expected value (not just future value), we must add to each of these trees both the current state value R(s) and the action cost C(a, s). This will generally require the simple addition of cost/reward to the values labeling the leaves of the current tree, though occasionally a small number of additional distinctions may be required. Figure 6 shows the expected (total) value tree for GetC obtained by adding R(s) and C(a, s) to the future value tree of Figure 5. Figure 6 also shows the tree for TestC. 4.2 Incorporating Observations To account for observations, every element of N” must corre- spond to a given action choice a and an observation strategy that assigns a vector in N”-’ to each o E Ohs(a). We now consider the problem of generating the actual o-tree corre- sponding to action a and the strategy assigning cuj E Nk-’ to the observation oj. Since the conditions that influence the probability of a given observation affect expected future value (since they affect the subsequent choice of a-vector with k - I stages- to-go), the new tree cy must contain these distinctions. Thus cy is partially specified by Tree(Obslu), the observation tree corresponding to action a. Recall that the branches of this tree correspond to the conditions relevant to observation probabil- ity, and the leaves are labeled with the probability of making any observation oj. To the leaves of Tree(Obsla) we add the weighted sum of the explanation trees (see also [ 161). More specifically, at each leaf of Tree(Obsla) we have’ a set of possible (nonzero probability) observations; for exposition, assume for some leaf these are oi and oj. IJnder the condi- tions corresponding to that leaf, we expect to observe oi and oj with the given probabilities Pr(oi) and Pr(oj), respec- tively. We thus expect to receive the value associated with the explanation tree for Cyi with probability Pr(oi) 9 and that for oj with probability Pr(oj). We thus take the weighted sum of these trees and add the resulting merged tree to the appropriate leaf node in Tree(Obsla).5 ‘Computing the weighted sum of these trees is relatively straight- forward. We first multiply the value of each leaf node in a given tree by its corresponding probability. To add these weighted trees together involves constructing a smallest single tree that forms a partition of the state space that subsumes each of the explanation trees. This can be implemented using a simple tree merging opera- Handling Uncertainty 1173 C A Ya OS Ya 0.1 NoO.2 NoO.9 stcpl Figure 7: New a-tree for Stage n - 2 Example Consider the following example illustrated in Fig- ure 7. We assume that trees ill and ~2, the trees for GetC and TestC in Figure 6, are elements of N’. We consider generating the new tree (Y to be placed in N2 that corre- sponds to the action TestC and invokes the strategy that associates cyt with the observation Yes and CQ with the ob- servation No. We begin by using the observation tree for TestC: the observation probability depends only on WC (see Step 1 of Figure 7). We then consider the weighted combination of the trees ~1 and 02 at each leaf: to the leaf WC we add the tree 0.8~~ + 0.2~~2 and to m we add 0. lcrt + 0.9~~2. This gives the “redundant” tree in the middle of Figure 7. We can prune away the inconsistent branches and collapse the redundant nodes to obtain the final tree a, shown to the right. We note that this simple combination of trees is due in part to the dependence of observations on only the pre-action state (as is the “separation” in Equation 3). This allows the direct use of Tree(Obsla) in assessing the influence of observations on the values of pre-action states. However, should observa- tions depend instead on the post-action state as is usual in the POMDP literature [ 17,6], our algorithm is complicated only in slight detail. In this case, Tree(Obsla) refers to variables in the state following the action, (recall we are interested in the values of states prior to the action). Generating the prob- ability of the observations based on pre-action variables is, however, a simple matter: we simply generate an explanation for the observation in a manner similar to that described in Section 4.1 (though, in fact, much less complicated). The standard explanation trees are then merged together within this slightly more complicated tree instead of Tree(Obs]a). 4.3 Generation of Nk and Pruning The algorithm for construction of the structured value func- tion proceeds exactly as Monahan’s algorithm in the previous section. The substantial difference is that we start with a tree- structured initial reward function as the sole a-tree at stage 0, and generate collections Nk of a-trees rather than sim- ple (e.g., vector-represented) a-functions. ‘Ihe process de- scribed above involves some overhead in the construction of explanation trees and piecing them together with observation probabilities. We note, however, that we need not generate the trees for &,t o for each observation strategy individu- ally. This tree depends only on a and c&, not on o. Thus, tion (for example, see [4] where similar tree merging is used for a different purpose). In terms of rules [16], this effect is obtained by explaining the conjunction of the roots of the trees. we need only construct ldllN"l such trees; the IdllOllN"I different trees in N”+’ are simply different weighted com- binations of these (corresponding to different observational strategies). Further savings are possible in piecing together certain strategies (e.g., if OS, associates the same vector with each observation, the explanation tree for a can be used directly). One can prune away dominated a-trees from Nk, as sug- gested by Monahan. As described in Section 2.6, this too exploits the structured nature of the a-trees. Finally we note that most POMDP algorithms are more clever about generating the set of possible o-vectors. For example, Sondilc’s algorithm does not enumerate all possible combinations of observational strategies and then eliminate useless vectors. We focus here on Monahan’s approach be- cause it is conceptually simple and allows us to illustrate the exact nature of structured vector representations and how they can be exploited computationally. We are currently investi- gating how algorithms that use more direct vector generation can be adapted to our representation. The Witness algorithm [6] appears to be a promising candidate in this respect, for the LPs used to generate “promising” o-vectors are amenable to the representations described here. 4.4 Executing Policies Given Nk and a belief state ?r, we can determine the optimal action with k stages-to-go by choosing an o E N” such that T l LY is maximal, and carrying out the action associated with cy. We can then make our observations, and use Hayes rule to update our belief state. We are then ready to repeat and choose an action for the next stage. The structured representation of value functions, which will generally be compact, can aid policy execution as well. This will be especially true if the belief state is itself rep- resented in a structured way. The expected value of belief state 7r is the sum of the values at the leaves of the o-tree multiplied by the probabilities of the leaves. The probability at each leaf is the probability of the conjunction of propo- sitions that lead to it (which can be derived from the belief state). Moreover, this also specifies which probabilities are required as part of the belief state (and which may be ig- nored). For instance, if it is discovered in the generation of the value function that certain variables are never relevant to value, these distinctions need not be made in the belief state of the agent. 5 Approximation Methods While the standard vector representation of o-functions re- quires vectors of exponential size (in the number of proposi- tions), computing with decision trees allows one to keep the size of the representation relatively small (with potentially exponential reduction in representation size). However, our example illustrates the natural tendency for these trees to become more “finegrained” as the horizon increases. De- pending on the problem, the number of leaves can approach (or reach) the size of the state space. In such cases, the overhead involved in constructing trees may outweigh the marginal decrease in effective state-space size. 1174 Planning However, an additional advantage of tree (or related) rep- resentations is the ease with which one can impose approx- imation schemes. If the a-tree makes certain distinctions of marginal value, the tree can be pruned by deleting inte- rior nodes corresponding to these distinctions. Replacing the subtrees rooted at U in tree ai of Figure 6 by a midpoint value introduces a (maximum) error of 0.5 in the resulting approximate value function. This may be acceptable given the shrinkage in the representation size it provides. This contraction has the effect of reducing the size of new trees generated for subsequent stages, as well. In addition, the er- ror introduced can be tightly controlled, bounded and traded against computation time. 6 In this sense, tree-based repre- sentations provide an attractive basis for approximation in large discrete problems. A major difficulty with Monahan’s algorithm is the fact that the number of (unpruned) cu-functions ~0~s exponentially with the horizon: Nk contains (IdllOl) pieces. Of course, pruning dominated a-functions can help, but does not reduce worst-case complexity. The methods above address the size of a-trees, but not (apart from pruning) their number. A second advantage of the tree-based representation, and approximation schemes based upon it, is the possibility of greatly reducing the number of a-trees needed at each stage. By blurring or ignoring certain distinctions, the number of dominated vectors (hence the amount of pruning) may be increased. In addition, “approximate domination” testing can be aided: for example, if one tree has strictly worse values than another except for slightly better values in one small region of the state space, it could be pruned away. Again, the compactness of the a-trees can be exploited in such tests, 1 as in Section 2.6. Indeed, this complements certain work that reduces the number of &unctions, such as [ 131 .7 These suggestions are, admittedly, not developed completely at this point. However, a firm grasp of optimal decision making with structured representations provides a sound basis for further investigation of structured approximation methods. 6 Concluding Remarks We have sketched an algorithm for constructing optimal poli- cies for POMDPs that exploits problem structure (as exhib- ited by rules or decision trees) to reduce the effective state space at various points in computation. The crucial aspect of this approach is the ability to construct the conditions relevant at a certain stage of the process given the relevant distinctions at the following stage. This merging of planning and opti- mization techniques (and related approaches) should provide significant improvements in policy construction algorithms. Of great interest are extensions of this work to algorithms that enumerate “vectors” (in our case, trees) in a more direct fashion (rather than by exhaustive enumeration and elimina- tion), as well as empirical evaluation of the overhead of tree %ee [3] on this type of pruning. ‘In [ 131, a continuous approximation of the value function is adjusted via gradient descent on the Bellman error; but there is one adjustable parameter per state. A (dynamic) tree-based representa- tion of the value function may be exploited here. construction. In addition, the development of approximation methods such as those alluded to above is an important step. Acknowledgements: Thanks to Tony Cassandra, Leslie Kaelbling, Michael I&man and Ron Parr for their help- ful discussion and comments. This research supported by NSERC Grants 0GP0121843 andOGP0044121. References VI PI [31 [41 PI WI r71 PI PI [101 r111 WI 1131 1141 WI W53 I171 WI R. E. Bellman. Dynamic Programming. Princeton, 1957. C. Boutilier, T. Dean, and S. Hanks. Planning under uncer- tainty: Structural assumptions and computational leverage. 3rd Eur. Workshop on Pluming, Assisi, 1995. C. Boutilier and R. Dearden. Approximating value trees in structured dynamic programming. ML-%, to appear, 1996. C. Boutilier, R. Dearden, and M. Goldszmidt. Exploiting structure in policy construction. IJCAI-95, pp.1 104-1111, Montreal, 1995. A. R. Cassandra. Optimal policies for partially observable Markov decision processes. TR CS-94- 14, Brown Univ., Prov- idence, 1994. A. R. Cassandra, L. l? Kaelbling, and M. L. Littman. Acting optimally in partially observable stochastic domains. AAAI- 94, pp. 1023-1028, Seattle, 1994. T. Dean, L. l? Kaelbling, J. Kirman, and A. Nicholson. Plan- ning with deadlines in stochastic domains. AAAJ-93, pp.574- 579, Washington, D.C., 1993. T. Dean and K. Kanazawa. A model for reasoning about persistence and causation. Camp. Intel., 5(3): 142-150, 1989. T. Dean and M. Wellman. Pkmning and Control. Morgan Kaufmann, 1991. R. A. Howard and J. E. Matheson. Influence diagrams. R. A. Howard and J. Matheson, eds., The Principles and Applica- tions of Decision Analysis, pp.720-762, 1981. W. S. Lovejoy. Computationally feasible bounds for partially observed Markov processes. Op. Res., 39: 162- 175, 199 1. G. E. Monahan. A survey of partially observable Markov decision processes: Theory, models and algorithms. Mgmt. Sci., 28: 1-16, 1982. R. Parr and S. Russell. Approximating optimal policies for partially observable stochastic domains. IJCAI-95, pp. 1088- 1094, Montreal, 1995. J. Pearl. Probabilistic Reasoning in Intelligent Systems: Net- works of Pbusible Inference. Morgan Kaufmann, 1988. D. Poole. Probabilistic Horn abduction and Bayesian net- works. Art. Intel., 64(1):81-129, 1993. D. Poole. Exploiting the rule structure for decision making within the independent choice logic. UAI-9.5, pp.454-463, Montreal, 1995. R. D. Smallwood and E. J. Sondii. The optimal control of partially observable Markov processes over a finite horizon. Op. Res., 21:1071-1088, 1973. C. J. C. H. Watkins and l? Dayan. Q-Learning. Mach. Learn- ing, 8:279-292, 1992. Handling Uncertainty 1175 | 1996 | 173 |
1,814 | A Qualitative Model for Temporal Reasoning with Incomplete Information Hector Geffner* Departamento de Computaci6n [Jniversidad Simbn Bolivar Aptdo 89000, Caracas 1080-A, Venezuela Abstract We clevelop a qualitative framework for temporal reasoning with incomplete information that features a modeling language based on rules and a seman- tics based on infinitesimal probabilities. The frame- work relates logical and probabilistical models, and accommodates in a natural way features that, have been found problematic in other models like non- determinism, action qualifications, parallel actions, and abduction to actions and fluents. Introduction Logic, probabilities and dynamic systems are standard frameworks for reasoning with time but are not always good modeling languages. This has led in recent years to the development8 of alternative languages, more suit- able for modeling, that can be thought, of as one of t,wo types. Translation. languages, on the one hand, aim to provide ways for specifying logical, probabilistic and determinist,ic dynamic systems by means of shorter and more intuitive descriptions (e.g., (Pednault 1989; Dean & Kanazawa 1989; Gelfond & Lifschitz 1993)). Default languages, on the other, aim to extend classi- ca.1 logic with the ability to express the expected effects of actions and the ecpecfed evolution of fluents (e.g. (McCarthy 1986)). In this paper we develop a model for temporal rea- soning that is hybrid in the sense that it features a language based on defaults and a semantics based on ‘approximate’ Markov Processes. More precisely, the user describes the dynamics of the domain of in- terest in terms of default, rules, and the defaults get ma,pped into a Markov Process with probabilities re- placed by order-of-magnitude a.pproximations. This results in a framework that, relates logical and proba- bilistic models and accommodates in a natural way fea- tures that have been found problematic in other mod- els like non-determinism, action qualifications, parallel actions, and abduction to both fluents and actions. *Mailing address from TJS and Europe: Hector Geffner, Bamco CC3 144-00, P.O.BOX 02-5322, Miami Florida 33102-5.322, USA. E-mail: hector@usb.ve. 1176 Planning Dynamic Systems Dynamic systems can often modeled by means of a transition function f that maps states si and inputs ua into unique successor states si+.l = f( si , zli) (Padulo Rr Arbib 1974; Dean & Wellman 1991).l The language for actions developed by Gelfond and Lifschitz (1993) is a language for specifying systems of this sort by means of rules of the form : A causes B if Cy (1) Rules such as these are understood as constraints over the function f that must map states sa where B holds into states s;+l where C holds when the input u,i is A. Under the assumption that, B and c are conjunc- tions of literals, and that all atoms (except actions) persist by default, these rules determine the function f completely. The semantics of Gelfond’s and Lifschitz’s language is given in terms of such funct#ions. Roughly, a literal I,i follows from a sequence of actions ~10, Q.I, . . . and observations 01,02, . . . when Li is true in all the state- space trajectories so, sl, . . . , that are compatible with the rules (i.e., si+l = f( si, ai)) and the observations (si satisfies oi ) . Gelfond and Lifschitz model is not affected by the difficulties reported by Hanks and McDermott (1986) because the transition function f provides the right se- mantic structure for interpreting defaults in this con- text. Persistence defaults - which are present in the model even if they are not encoded explicitly by means of rules - are regarded constraints on the possible transitions from one state to the immediate successor states, indepen*dent of both future and past, and th,e actual observations. Other models based on a similar idea are Baker’s (1991) and Sandewall’s ( 199 1). Markov recesses The model above assumes that, the dynamics of the system is deterministic in the sense that knowledge of the state and the inputs is sufficient to predict the ‘This is for systems that are discrete-time, time- invariant and deterministic; see (Padulo & Arbib 1974). From: AAAI-96 Proceedings. Copyright © 1996, AAAI (www.aaai.org). All rights reserved. future with complete certainty. For the cases where this assumption is not good a different class of models has been developed in which the state and the inputs predict the future behavior of the system with some known probability (Howard 1971). Formally, a state si and an input, u; determine now a set of possible successor states .si+l with probabilities P(.si+l Isi, 14,:). Then, under the assumptions that fu- ture inputs do not affect past, states (the causality prin- ciple) and that the future is independent, of the past given the present (Markovian assumption), the prob- ability of each state trajectory so, ~1, . . . , SN given a sequence of inputs UO, ul, . . . , UN-1 is given by the equation: P( SO) . . . , l sNI?dO, . . . , UN-l) = P(+o)-~~ P(&+l Isi, t&) i= 0 (2) When a set of observations 0 is obtained, this prob- ability is multiplied by a normalizing constant if the trajectory satisfies the observations, and by zero oth- erwise. The probability of a proposition is simply the sum of the probabilities of the trajectories that, make the proposition true. Models of this type, known as Markov Processes, are significantly more expressive than deterministic mod- els in which transition probabilities can only be zero or one. This generality comes at the price of specifying and rom.puting with such models. For this reason, at- tempts to use probabilistic dynamic models in AI have focused on the development of languages that trade off some of the expressive power of probabilistic models for the benefit of simple rule-based specifications (Dean Pr Kanazawa 1989; Hanks & McDermott 1994). In this paper we develop a different type of prob- abilistic temporal model based on rules that, may be adequate when exact probabilities are not needed and the distinction between likely and unlike consequences suffices. The key concepts are two: an abstraction of Markov Processes in which probabilities are replaced by their order-of-magnitude approximations and a way to specify systems of that sort by means of pnrtial and iwompletc sets of rules. We consider each idea in turn. Qualitative Markov Processes The order-of-magnitude of a probability tneasure p rel- ative to a small parameter c can be defined as the smallest hteger K(P) such that p < en(p). For exam- ple, if 6 = 0.2, the order-of-magnitude of pl = 0.5 and p2 = 0.01 are I = 0 and ~c(pz) = 2 respectively. In- terestingly, as shown by Spohn (19$8), as the param- eter c is made smaller and smaller, in the limit, the order-of-magnitude measures K obey a calculus given by the axioms:2 (3) which is structurally similar to the calculus of proba- bilities, with products replaced by sums, sums by min- imizations, etc. Spohn refers to the K measures as degrees of surprise or disbelief as lower K measures stand for higher proba- bilities and higher K measures stand for lower probabil- ities. In particular, a. proposition p is deemed plausible if K(P) = 0 and implausible or disbelieved if KC(P) > 0. Since the axioms rule out two complementary propo- sitions from being disbelieved at the same time, p is accepted or believed when its negation is disbelieved, i.e., if I > 0. The appeal of Spohn’s K-calculus is that, it, combines the basic intuitions underlying probability theory (con- text dependence, conditionalization, etc.) with the no- tion of plain belief. The beliefs sanctioned by K func- tions are plain and revzsable in the sense that, p can be believed given q, and yet lp can believed given q and something else. Indeed, the funct,ion K expresses a preference relations on worlds in which a world 20 is preferred to w’ if K.(U) < K( ZU’). Frotn this point of view, the criterion K(lplq) > 0 for accepting p given q is nothing else but, an abbreviation of the standard condition in non-monotonic logics that require p to be true in all maximally preferred worlds that satisfy q (e.g., (Lehmann k Magidor 1988)). Goldszmidt and Pearl (1992) were the first to ex- ploit the dual connection of the K-calculus to probabil- ities and non-monotonic reasoning, showing how some problems in causal default reasoning could be solved by using K functions that, satisfy a stratification condi- tion analogous to the condition that, defines Probabilis- tic Bayesian Networks (Pearl 1988). They refer to K measures as qualitative probabilities, and to stratified K functions as Qualitative Bayesian Networks (Gold- szmidt & Darwiche 1994) In the same perspective, we consider in this paper Qualitative Markov Processes, defined as the K. func- tions for which the plausibility of trajectories so, . . . , sN given the inputs ug, . . . , UN-1 is given by the qual- itative version of (2): N-l K(s(), . . .,sNl?&j,. . ., UN-l) = +o)+ x ‘+i+l 1% W) i=o (4) The model for temporal reasoning below is a language for specifying systems of this sort by tneans of partial and incomplete sets of rules. ‘We also assume unsatisfiable. h:(p) = 00 and ~(qlp) = 0 when p is Handling Uncertainty 1177 The Proposed Model Language We deal with temporal models or theories specified by means of three type of constructs: temporal rules, ob- servations, and what we call completion functions. The first two are familiar and their precise syntax will be given below. The third one is less familiar and will be introduced in the next section. Intuitively, rules will be restrictions on the possible state transitions, observations will be restrictions on the actual trajec- tories, and completion functions will be functions that fill out missing information to determine the plausibil- ity of state transitions uniquely. The syntax of temporal rules and observations pre- sumes a finite set P of time-dependent primitive propo- sitions (atoms) p, q, r, . . . , and a time set T given by the non-negative integers 0, 1, . . . . We will refer to the language comprised of the propositions in P closed under the standard propositional connectives as ,C, and use the symbols A, B, etc. to denote arbitrary formu- las in L. We will also use the symbol L to refer to literals in & (atoms or their negations) and -L to refer to their complements. The temporal rules are default rules of the form A - L saying that if A is true at time i, then by default L will be true at time 2’+ 1 for any i E T. Each rule has a priority which is represented by a non-negative integer: the higher the number, the higher the priority. The idea is that when two rules say different things about the same literal, the higher priority rule prevails. Non-deterministic rules (Sandewal 1991) are accom- modated by expressions of the form A --+ p(~p and are understood as a sh,orthand for the pair of rules A - p and A - lp. Non-deterministic rules express that A sometimes makes p true and sometimes makes p false (e.g., dropped-cup - breaksllbreaks). [Jnless other- wise specified, the priority of rules is assumed to be zero (lowest priority). Finally, the observations are formulas that have been observed to be true at some specific time points. We use the notation p(i) to express that0 the primitive proposition p is true at time i and call such expres- sions temporal propositions. We refer to the language that results from closing such temporal propositions under the standard propositional connectives as LT. & will thus be the language of the observations and the conclusions that we may want to draw from them. We will call the formulas in CT the temporal formSu- Zas. Thus, if CL, b and c are primitive propositions u(2) V a(3) > c(4) V lb( 1) will be a valid temporal for- mula and hence a possible conclusion or observation. Semantics A set of temporal rules augmented with a comple- tion function will determine one specific Qualitative Markov Process represented by a particular K func- tion. A conclusion C will then follow from a set of 1178 Planning observations 0 if K(+IO) > 0. We make this precise by defining the states, trajectories, and the form of the K function. The states are truth assignments to the primitive propositions that determine the truth value of all for- mulas in Z,. The notation si , sj , etc. will be used to denote states at times i, j, etc., while state-space tru- jectories will be denoted as so, $1, . . . , si, . . . . We will say that, a trajectory satisfies a temporal propo- sition p(i) if the state si in the trajectory satisfies the primitive proposition p. Following the standard inter- pretation of the propositional connectives, trajectories represent logical interpretations over the temporal lan- guage CT assigning a truth-value to all temporal for- mulas, and hence, to all observations. For example, a trajectory SO, ~1, . . . with a state .$I that satisfies p and q, and a state s2 that satisfies only p, will satisfy the tem.poral formulas p( 1) 3 ~(2) and lq(2), but, not p(1) 2 q(2) or lq( 1) V lp( 1). Our dynamic systems will have 710 inputs. Actions, which in other frameworks are represented as inputs, will be represented here in the state. Thus to express that a switch was toggled at time 1: = 5, we will simply say that toggZe(5) was observed. With no inputs, the transition plausibilities that characterize our Markov Processes will simplify from K(si+l Isi, ui) to K( si+l Isi). Later on we will show that by representing inputs as observations we are not giving up the ‘causality prop- erty’ by which actions should not affect past states (Padulo Rt Arbib 1974). Instead we will gain the abil- ity to deal naturally with parallel actions, action qual- ifications and abduction to actions. Transition Plausihilities. We are left to deter- mine the prior and transition plausibilities ~(sl:) and ~(si+l Isi) from the information provided by the user. Let us say that a state s makes a rule A - L active in the theory when s satisfies A but s does not satisfy B for any conflicting rule B --L with higher priority. Let us also use the notation Li to stand for temporal literals like p(i) or -p(i) and say that Li+l is supported by a sta,te si when a rule A - L is active in sd. Then the proposed mapping from rules to transition plau- sibilities ~(si+l Isi) can be understood in terms of the following assumptions: Literals Li+l and L:+, that are logically independent are conditionally independent given the past, ,s;.~ L i+l is not disbelieved when si supports L;+l L i+l is completely disbelieved when si supports 4+1 but not Li+l The plausibilities of two complementary literals Li+l and -Li+l that, are not supported by si are in.de- pendent of si “Two literals are logically independent if the truth of one does not constraint the truth of the other. Assumption 1 excludes the possibility Gons and translates into the identity: of rmijicn- (5) where L ranges over the literals that, are true in si+l. Assumptions 2 and 3 are consequences of the default) reading of the rules. 4 The last assumption is the most, important and follows from assuming that0 the past in- fluenwes th.e future only through the active rules; i.e., same active rules mean same transition plausibilities. These assumptions impose restrictions on the type of Qualitative Markov Processes that can be expressed, yet with the exception of Assumption 1, we have found them reasonable in domains where predictions can be explained qualitatively in terms of rules and prior judgements. Later on we will show that received mod- els for temporal reasoning that do not deal with rami- fications embed these and other assumptions. The assumptions determine the following mapping from rules to plausibilities: K( Li+1 ISi, = 1 0 when si supports Li CC? when .si supports -Li but not Li n(L) when .si supports neither (6) We call this model of interaction the Noisy-Rule model in analogy to the Noisy-OR model used in Bayesian Networks (Pearl 1988) In this model, the parameters n(L) and 7r(- L) determine the plausibilities of the literals L and -L in. the absen.cc of reas0n.s to belicae in. either one of them (see (Geffner 1996) for a different application of this model). The function 7r is what, we call the completion. function and must be such that for each literal L, n(L) must be a non-negative integer and either 7r( L) or X(W L) must be zero (i.e., K is a plausibility function over L and N-L). For literals Li+l and -Li+l that are not supported by cony state si, e.g., the literals Lo referring to the initial state, r( Li+l ) and ~(wLi+l) encode the prior plausibilities ~(Li+l ) and ~(wLi+l) respectively.” Two completion functions that we will find useful are the grounded and uniform functions. The first makes 7r( lp) = 0 and r(p) = 1 for each primitive proposition p, expressing that in the absence of rea- sons for or against, p, -p is assumed more plausible (like in the Closed World Assumption). The second makes a(p) = n( -p) = 0, expressing that in the ab- sence of reasons for or against p, the literals p and -p are assumed equally plausible. Later on we will show that, some familiar systems embed assumptions that fit naturally with these functions. 4 We could make Assumption 3 less extreme by replacing ‘complete disbelief’ (infinite K) by ‘partial disbelief’ (non- zero 6). Yet this weaker condition would make the specifi- cation of the resulting models more complex. 5This is because in that case &(L,+l) = min,, K(L,+I Is*) and ~(Li+l Is,) = n(L) . As an illustration, given the rules q - p and r ---+ p. the grounded completion function produces a Noisy- OR type of model in which p is certain given q or r and -p is more likely than p when q and r are false (below s+ and ST stand for the states that make q V r true or false respectively). K(p;+Js+) = 0 K( -pi+1 I.$) = 00 K(Pi+l(si) = 1 /c(-p,+&y = 0 Summary The proposed model works as follows. The user pro- vides the rules and the completion function and from (6) we get the plausibilities K( Lo) and K( Li+l Is;) for all literals and states. These plausibilities are combined by means of (5) to yield the prior and transition plau- sibilities K(s~) and ~(si+l Isi), which plugged into (4) give us the plausibility of any trajectory, and hence, of any formula (3). To determine whether a temporal for- mula Cy follows from the observations 0 we check then whether ~($10) > 0, where K(%‘IO) is the differ- ence between the plausibilities of the most plausibility trajectories that satisfy XAO and 0 respectively (3). Example. Consider the expressions ‘if a block is pushed it moves’, ‘if a block is pushed and is held, it may not move’, and ‘if a very heavy block is pushed, it does not move’, represented by the rules: a-p; aAb - pi-p ; a A q - -p in increasing order of priority. We also consider a rule q - q capturing the persistence of the property ‘heavy’, and a grounded completion function 7r where for every positive literal q, 7r(-q) = 0 and n(q) = 1 (i.e., atoms are assumed false by default). We want to determine whether p( 1) follows from a(0); i.e., whether a block will move if pushed. We will use the fact that for the completion function above ~(si+l Isi), when finite, is equal to the number of atoms x true in s;+l that are not supported by si. This also applies to the initial states so where no atom is sup- ported and hence where K(SO) is equal to the number of atoms true in so. Consider now the trajectory t = SO, ~1, . . . that only makes two atoms true: a(O) and p( 1). We want to show that t is the single most plausible trajec- tory compatible with a(O). From the considerations above, rc(so) = 1, and since so supports p( 1) but no other atom, ~(~11~0) = 0. This means that K(t) = 1, as all states ~1, ~2, . . . support no atoms and hence K(Si+l ISi) = 0 for all i > 1. We need to show that for any other trajectory t’ = sl,, s’1, . . . satisfying a(O), K;(t’) > 1. This is actually straightforward as any state sb compatible with a(O) but different than SO will have a plausibility K(s~) > 1, and similarly, any state s:+~ different than si+l will have a plausibility K(S! z+llsi) > 0. Thus, t is the single most, plausible trajectory compatible with a.(O), and hence, a( 0) implies p( 1). Handling Uncertainty 1179 This scenario provides an example of a projection. Examples of parallel actions, non-determinism, action qualifications and abduction can all be obtained by using similar arguments in slightly different, settings. For instance, if both a(O) and b(0) are observed (i.e., the block is pushed while held), neither p( 1) nor -p( 1) will be predicted (as both literals are supported by the sta.tes so that, make cr.(O) A b(0) true and q(0) false). Similarly, if the observation q(5) is added (the block is very heavy), -p( 1) will be predicted. Finally, if the rule lc1 - lp is added, a(O) will follow from p( 1) (‘the blocked moved, therefore it was pushed’). Action Theories In this section we will specialize the framework laid out above by introducing some common assumptions about, actions and fluents t,hat facilitate the specifica- tion and processing of temporal theories. These as- sumptions are: 1) every primitive proposition repre- sents either an action or a fluent, 2) flue&s persist by default, 3) actions are exogenous, 4) actions occur with low probability, and 5) changes occur only in the presence of actions (that are not necessarily known). Formally, this means that 1) every rule will be a persistence rule or an action rule, 2) persistence rules will be of the form p - p and -p - -11 (actually we assume one such pair of rules for every fluent p), 3) ac- tion symbols cc do not occur in the head of the rules, 4) actions are unlikely, i.e., n(u) > 0, and 5) action rules have priority over persistence rules, and actions symbols are not negated in the body of such rules. We will also assume that all observations are liter&. Theories of this type are similar to some of the the- ories considered in the literature (e.g., (Mfond’s and Lifschitz‘s) yet they allow for non-determinism, arbi- trary plausibility function over fluents, abduction to actions and fluents, action qualifications, and parallel actions. We will call such theories, action theories. We mention briefly three main properties of action theories. First, in spite of representing actions as part of the state, actions remain independent of past states in compliance with the so-called causality principle (Padulo & Arbib 1974)):6 Proposition 1 In action. Uleories, aciions urc in- dependent of past states, i.e., if ai denotes lihe occurrence of a numOber of uctions at time i, ~(~S~~.Si--l,u~-~,u~,u~+l,. . .) = K(S&s&l,U&l). Second, all uncertainty in action theories is summa- rized by the prior plausibility of actions and the prior plausibility of fluents: Proposition 2 In action theories, th.e plansibiliZy measure of a trujectory t = SO, .$I, . . . , when, finide, “Actually, since actions are part of the state, actions affect the present state yet they do not affect the present st,ate of jkerats. is given by the sum of Zhe prior plausibiliiies of the uc- tions that occur in t and the prior plausibilities of the literal fEuen,Ss that occw in, ~0:~ 4) = c w + c c ‘44 LESO 2 nEst The last property we mention is that only the rel- ative prior plausibilities of actions and Auents matter when the theories are predictive (i.e., when there are no surprises). For such theories, any two completion functions 7r and 7r’ that order complementary literals in the same way, i.e., 7r(L) < 7$-L) iff n’(L) < 7r’(wL), will yield the same behavior. Definition 1 An action theory is predictive given a set of obserzraliow 0 un.d a completion fun*ction. T. if K(OIOA, 00) = 0, where OA E 0 refer-s to the observed actions an,d 00 C_ 0 refers 2h*e observed fIuent.s al liimc i = 0. Proposition 3 The conclusions suncfioned by a pre- dictive action theory are not uflected by changes in th.e completion fun,ction th,at preserve the plausibility or- derin$g of complementury laterals. This means that in these theories the exact value of n(L) is irrelevant as long as n(L) > 0, because in such case 7r(m L) < n(L). For this reason, in such cases it is sufficient to determine whether each positive literal is true by default, false by default, or undecided. The grounded and uniform completion functions, for exam- ple, make the second and third choices respectively. Related Models The semun.tics of the model draws from approaches in the literature that exploit the double connection of Spohn’s K functions to non-monotonic reasoning and probabilities (Goldszmidt, & Pearl 1992; Goldszmidt Rr Darwiche 1994). The latter work in particular deals with temporal reasoning and is based on structures similar to Bayesian Networks (Pearl 1988) in which conditional probabilities P( -1.) are replaced by condi- tional plausibilities ti(.I.) provided by the user. The lan.guuge of this model, on the other hand, draws from temporal logics like Gelfond’s and Lifs- chitz’s (1993) that do not handle uncertainty. We want to show in this section that the proposed model pro- vides a natural generalization of such logics by repre- senting uncertainty explicitly in the form of completion functions. We focus here on Gelfond’s and Lifschitz’s logic only; equivalent formalizations are discussed in (Kartha 1993). For simplicity, and without any loss of generality, we consider domain descriptions with actions rules ‘A causes B if C’ and initial conditions ‘initially L’ ‘This is beta use fluent literals in SO and actions any- where are not supported and hence for them K+(Z) = x(z), while for all other fluent literals ICI(L+IIS~) = 0 or q-L+1 1%) = 00. 1180 Planning only. Given a domain description D. we define TD as the action theory with rules A A B - c and observa- tions Lo (all action rules have the same priority, and persistence rules for fluents are implicit as in any ac- tion theory). The relation between D and TD is then as follows: Proposition 4 Let D be a con.sistenl dom.nin descrip- ti~n..~ Then (1 value proposition ‘L after Ao, . . . A, ’ is entailed by D according to Gelfond and Lifschitz i-8 th.e literal Ln,+l follows from TD an.d the actions Ai( i=O,..., n, under th.e uniform completion function. In other words, Gelfond’s and Lifschitz’s model can be understood in this framework in terms of two as- sumptions: that, complementary fluents are equally plausible, and all actions are implausible a priori. The advantage of the model proposed is that, these assump- tions are explicit, and can be modified by a change in the completion function 7r (e.g., a grounded completion functions for fluents, for example, leads to the behavior characteristic of negation as failure). This actually ex- plains why we can accommodate action qualifications and represent, actions in the state. If we only had the uniform completion function we would get a behavior monoton.ic in the set, of actions, very much like Gel- fond’s and Lifschitz’s model yields a monotonic behav- ior in the observa.tions. Summary We have developed a qualitative model for tempo- ra.1 reasoning that relates logical and probabilistic ap- proaches, and handles non-determinism, actions qual- ifications, parallel actions and abduction in a natural wa,y. The model is limited in other ways such as in its inability to deal with ramifications. We hope to ad- dress this limitation in the future by making the map- ping from rules to transition plausibilities sensitive to the domain constraints. We have also developed in- ference procedures that we plan to include in the full version of this paper. Acknowledgments. Many of these ideas originated in conversations with Yoav Shoham. I also want to thank Blai Bonet, Nir Friedman, Joe Halpern for useful discussions, and the anonymous AAAI reviewers for useful comments. References Baker, A. 1991. Nonmonotonic reasoning in the framework of the situation calculus. AIJ 49:5-23. Dean, T., and Kanazawa, K. 1989. A model for rea- soning about persistence and causation. Computa- tion,al Intelligence 5: 142-150. Dean, T., and Wellman, M. 1991. Planning and Con- trol. Morgan Kaufmann. Geffner, H. 1996. A formal framework for causal mod- eling and argumentation. In Proceedings FA PR ‘96. Bonn. Springer Verlag. Gelfond, M., and Lifschitz, V. 1993. Representing action and change by logic programs. J. of Logic Pro- gramming 17:301-322. Goldszmidt, M., and Darwiche, A. 1994. Action net- works. In Proceedin*gs Spring AAAI Sym.posium. on Decision- Theoretic Planning. Goldszmidt, M., and Pearl, J. 1992. Rank-based sys- tems. In Proceedings KR-92, 661-672. Morgan Kauf- mann. Hanks, S., and McDermott, D. 1986. Default reason- ing, non-monotonic logics, and the frame problem. In Proceedin.gs A AA I-86, 328-333. Hanks, S., and McDermott, D. 1994. Modeling a dynamic and uncertain world I: symbolic and proba- bilistic reasoning about change. AIJ 66( 1): l-55. Howard, R. 1971. Dynamic Probabilistic System.s- Volume I: Markov Models. New York: Wiley. Kartha, G. 1993. Soundness and completeness re- sults for three forma.lizations of action. In Proceedings IJCAI- 93, 724-729. Lehmann, D., and Magidor, M. 1988. Rational logics and their models. Dept. of Computer Science, Hebrew University, Israel. McCarthy, J. 1986. Applications of circumscription to formalizing commonsense knowledge. Artificial In- telligence 28:89-116. Padulo, L., and Arbib, M. 1974. System Theory. Hemisphere Publishing Co. Pearl, J. 1988. Probabilistic Reasoning in Intelligent Systems. Los Altos, CA.: Morgan Kaufmann. Pednault, E. 1989. ADL: Exploring the middle ground between Strips and the situation calculus. In Proceedings IiR-$9, 324-332. Sandewal, E. 1991. Features and fluents. Technical R,eport, R-91-29, CS Department, Linkoping 1Jniver- sity, Linkoping, Sweden. Spohn, W. 1988. A general non-probabilistic theory of inductive reasoning. In Proceedings 4th Workshop on Uncertainty, 315-322. ‘D is consistent if no pair of rules associated with the same action have antecedents that are jointly satisfiable and consequents that are jointly unsatisfiable. Handling Uncertainty 1181 | 1996 | 174 |
1,815 | On the Size of Reactive Plans Peter Jonsson and Christer BkkstrSm Department of Computer and Information Science Linkiiping University, S-581 83 Linkoping, Sweden {petej,cba}@ida.liu.se Abstract One of the most widespread approaches to reactive planning is Schoppers’ universal plans. We propose a stricter definition of universal plans which guaran- tees a weak notion of soundness not present in the original definition. Furthermore, we isolate three dif- ferent types of completeness which capture different behaviours exhibited by universal plans. We show that universal plans which run in polynomial time and are of polynomial size cannot satisfy even the weakest type of completeness unless the polynomial hierarchy collapses. However, by relaxing either the polynomial time or the polynomial space requirement, the con- struction of universal plans satisfying the strongest type of completeness becomes trivial. Introduction In recent years reactive planning has been proposed as an alternative to classical planning, especially in rapidly changing, dynamic domains. Although this term has been used for a number of more or less related approaches, these have one thing in common: There is usually very little or no planning ahead. Rather the idea is centered around the stimulus-response principle-prompt reaction to the input. One of the most well-known methods for reactive planning is the universal plans by Schoppers (1987). A universal plan is a function from the set of states into the set of op- erators. Hence, a universal plan does not generate a sequence of operators leading from the current state to the goal state as a classical planner; it decides after each step what to do next based on the current state. Universal plans have been much discussed in the literature. In a famous debate (Ginsberg 198913; Schoppers 1989; Ginsberg 1989a; Schoppers 1994), Ginsberg criticised the approach while Schoppers de- fended it l. Based on a counting argument, Gins- berg claims that almost all (interesting) universal plans takes an infeasibly large amount of space. Schopper’s ‘This list is not exhaustive. Other authors, such as Chapman (1989), h ave joined the discussion. However, it seems that the main combatants have been Schoppers and Ginsberg. defence has, to a large extent, built on the observa- tion that planning problems are structured. Accord- ing to Schoppers, this structure can be exploited in order to create small, effective universal plans. We refrain from going into the details of this debate and merely note that both authors have shown great in- genuity in their argumentation. However, from the standpoint of formal rigour, these papers do not settle the question. One of the few papers that treats uni- versal plans from a formal, complexity-theoretic point of view is the paper by Selman (1994). He shows that the existence of small (polynomially-sized) universal plans with the ability to generate minimal plans im- plies a collapse of the polynomial hierarchy. Since a collapse of the polynomial hierarchy is widely con- jectured to be false in the literature (Johnson 1990; Papadimitriou 1994), the existence of such universal plans seems highly unlikely. It should be noted that this result holds even for severely restricted problems such as the blocks-world. In our opinion, one of the problems with universal plans is the over-generality of the definition. This gen- erality makes formal analysis hard or even impossible. Therefore, we begin this paper by giving a stricter def- inition of universal plans, a definition that embodies the notion of soundness. In addition, we supply three different types of completeness. These notions of com- pleteness capture different desirable properties of uni- versal plans. For example, A-completeness states that if the problem has a solution, then the universal plan will find a solution in a finite number of steps. The main result of this paper is that universal plans which run in polynomial time and are of polynomial size can- not satisfy even this weakest type of completeness2. However, by relaxing either the polynomial time re- quirement or the polynomial space requirement, it be- comes trivial to construct universal plans that satisfy the strongest type of completeness. Also in this case, the result holds for severely restricted problems. The organisation of the paper is as follows: We begin by defining the basic STRIPS formalism and formally 2Under the assump tion that the polynomial hierarchy does not collapse. 1182 Planning From: AAAI-96 Proceedings. Copyright © 1996, AAAI (www.aaai.org). All rights reserved. define universal plans and various restrictions on them. We continue by showing that small, fast universal plans cannot be complete even in a very weak sense. The paper is concluded with a brief discussion of the results. Basic Formalism We base our work in this paper on the propositional STRIPS formalism with negative goals (Bylander 1994), which is equivalent to most other variants of proposi- tional STRIPS (Backstrom 1995). Definition 1 An instance of the PSN planning prob- lem is a quadruple II = (P, 0, Z, G) where e P is a finite set of atoms; e 0 is a finite set of operators where o E 0 has the form Pre + Post where Pre is a satisfiable conjunction of positive and negative atoms in P, respectively called the posi- tive preconditions (pre+(o)) and the negative pre- conditions (pre-(0)); Post is a satisfiable conjunction of positive and negative atoms in P, respectively called the posi- tive postconditions (add(o)) and the negative post- conditions (deZ(o)); E P denotes the initial state; e and G = (C?,G-) denote the positive and negative goal, respectively, satisfying G+, 6’ E P and g+ n 6- = 0. A PSN structure is a tuple @ = (P, 0) where P is a set of atoms and 0 is a set of operators over P. We denote the negation of an atom by overlining it. As an example, the operator o defined as F =+ q,?; satisfies pre+(o) = 0, pre-(o) = {p}, add(o) = {q} and deZ(o) = {r}. Definition 2 Given a set of operators 0, we define the set of all operator sequences over 0 as Seqs(U) = (0) U {(o);wlo E 0 and w E Seqs(U)}, where ; is the sequence concatenation operator. A sequence (01,. . . , on> E Seqs(U) of operators is called a PSN plan (or simply plan) over II. We can now define when a plan solves a planning instance. Definition 3 The ternary relation Valid s Seqs(U) x 2p x (2’ x 2’) is defined s.t. for arbitrary to1 , . . . ,on) E Seqs(0) and S,T+,T’ E P, VaZid((ol,. . . , on), S, (T+ , T’)) iff either 1. n=O,T+&SandT’nS=0or 2. n > 0, pre+(ol) C S, pre-(01) n S = 0 and VaZid((o2,..., on), (S-dez(ol))Uadd(ol), (T+, T-)). A plan (01,. . . , on> E Seqs(0) is a solution to II iff VaZid((ol,. . . 9 on)J, (G+, 47)). We define the planning problems that we will consider as follows. - - Definition 4 Let II = (P, U,Z, (@, B-)) be a given PSN instance. The plan generation problem (PG) is to find some o E Seqs(U) s.t. w is a solution to II or answer that no such w exists. The bounded plan generation problem (BPG) takes an integer K 2 0 as additional parameter and the object is to find some o E Seqs(U) s.t. w is a solution to II of length 5 K or answer that no such w exists. Universal Plans Universal plans are defined as follows in the literature (Ginsberg 198913). A universal plan is an arbitrary function from the set of possible situations S into the set of primitive actions A. Using the terminology we have adopted in this paper results in the following equivalent definition. Definition 5 Given a PSN structure @ = (P, U), a universal plan is a function from the set of states 2p into the set of operators 0. This very general notion of universal plans is difficult to use as a basis for formal analyses. We would like, for example, to discuss the issuses of correctness and resource consumption. In the sequel, we will try to classify universal plans in greater detail. For a given PSN structure @ = (P, 0) let S = 2’, SL = 2p U {I} and U+ = UU(ol, or}. Here I is a new state denoting undefinedness and 01, or are two “special” operators. These operators are not to be considered as operators in the sense of Definition 1 but rather as two com- pletely new symbols without internal structure. The special operators will be used by the universal plans for “communication with the environment”. The fol- lowing definition is needed for defining soundness of universal plans. Definition 6 Let QPI = (P,U) be a PSN structure. The update operator $ : Sl x U+ + Sl is defined as follows: I $0 = I for all 0 E U+ . Let S E S. If 0 is a standard operator then S@ o = (S - deZ( 0)) U add(o) iff pre+(o) C SApre’(o)nS = 0. Otherwise, S@o = 1. If o is not a standard operator then S $01 = I and s@oT = S. An operator o E U+ is admissible in a state S E SI iff S $0 # 1. We can now refine our notion of universal plans. Definition 7 Let ip = (P, 0) be a PSN structure and let 5; be a goal over P. A sound universal plan UQ for the goal 6 is a function that maps SA to U+ such that 1. for every S E SL , if UG (S) = o E 0 then o is admis- sible in S; 2. for every S E so, UG(S) = or iff S Satisfies 6; The first point in the definition says that if the univer- sal plan generates an operator, then this operator is executable in the current state. This restriction seems to have been tacitly assumed in the literature. The Handling Uncertainty 1183 second point tells us that the special operator OT is generated if and only if the universal plan is applied to a state satisfying the goal state. Thus, OT is used by Ug to report success. The reason for introducing the operator 0~ is to avoid the generation of new op- erators when the current state satisfies the goal state. The special operator OL, on the other hand, indicates that the universal plan cannot handle the current state. This can, for instance, be due to the fact that the goal state is not reachable from the current state. Observe that no operator is admissible in i so UQ must gener- ate 01 whenever applied to 1. Henceforth, we will use the term universal plan as an abbreviation for sound universal plan. We continue by defining four properties of universal plans, For a universal plan U,J we use the notation U,f(S) to denote the operator UQ(SK) where Sl = S and &+I = SK $ U@K)- Definition 8 A universal plan UG for a PSN structure @=(P,U)is PT poly-time iff UG can be implemented as a deter- ministic algorithm that runs in polynomial time in the size of a,; Ps poly-space iff Us; can be implemented ministic algorithm A satisfying asa deter- 1. the size of A is polynomially bounded by the size & and 2. the size of the space used by A is polynomially bounded by the size of @; A acceptance-complete iff for every S E S such that (P, 0, S g) is solvable there exists an integer K such that U$(S) = 0~; R rejection-complete iff for every S E S such that (P, 0, S, G) is not solvable there exists an integer K such that Ut (S) = 01. Universal plans satisfying some subset of the restric- tions PT, Ps, A and R are named by combining the corresponding letters. For example, a PTAR universal plan is poly-time, acceptance-complete and rejection- complete. The definition of poly-time should be quite clear while the definition of poly-space may need fur- ther explanation, The first part of the definition en- sures that UG can be stored in a polynomially-bounded memory. The second part guarantees that any compu- tation will use only a polynomially-bounded amount of auxiliary memory. Hence, we can both store and run the algorithm in a memory whose size is bounded by a polynomial in the size of @. This restriction excludes algorithms using extremely large fixed data structures as well as algorithms building such structures during run-time. For the sake of brevity, we use the terms A- and R-completeness for acceptance- and rejection- completeness, respectively. A minimal requirement on universal plans is that they are A-complete so we are guaranteed to find a solution within a finite number of steps if there is one. Observe that if an A-complete universal plan is not R-complete then v,“(S) can dif- fer from 01 for all K if 6 is not reachable from S. R-completeness is, thus, desirable but not always nec- essary. In domains such as the blocks-world, where we know that a solution exists in advance, R-completeness is of minor interest. To have R-completeness without A-completeness is useless since we can trivially con- struct universal plans satisfying PT,sR for all prob- lems. Simply let UG (S) = 01 for all S E Sl. This R- complete universal plan can trivially be implemented as a poly-time and poly-space deterministic algorithm. In certain applications, we need a stronger form of R-completeness. Definition 9 A universal plan UQ for a PSN structure (P, 0) is strongly rejection-complete (R+) iff for every S E S such that (P, 0, S, G) is not solvable, UQ(S) = O.L* The motivation for introducing strong R-completeness is simple. If the universal plan outputs operators, we cannot know whether they will lead to a solution or not. Executing such operators is not advisable, since we may wish to try planning for some alternative goal if there is no solution for the first one. However, ex- ecuting the “invalid” operators may prevent us from reaching the alternative goal. From a complexity-theoretic point of view, it can be argued that universal plans have to be both poly-time and poly-space to be feasible in practice. This is a hard restriction since by dropping any of the polynomial- ity requirements, constructing universal plans become easy. Theorem IQ For every PSN structure Q = (P, 8) and goal state 6 over P there exist universal plans UQ and Ub satisfying PTAR+ and PsAR+, respectively. Proof: Construction of UG: We define a function f : SL 4 u+ as follows. For each I< 2 1 and S E S such that (P,U,S,G) h as a shortest solution of length K, choose an o E 0 such that (P, 0, S $ o, (?) has a shortest solution of length K - 1. Denote this operator OS and let 1 01 if (P, 0, S, G) is not solvable f(S) = oT s Satisfies 5! OS otherwise Clearly, for every S E S there exists an integer K such that if (P, 0, S, 6) is solvable then UC(S) = 0~. Otherwise, Ug(S) = 01. Consequently, f is both A- complete and strongly R-complete. The proposed con- struction of the function f is obviously of exponen- tial size. However, it can be arranged as a balanced decision tree of depth IPJ and, hence, be accessed in polynomial time. Consequently, we have constructed UC Construction of U’ : it Consider a forward-chaining PSN planning algorit m P that is sound, complete and generates shortest plans. We modify the algorithm to 1184 Planning output only the first operator of the plan that leads from S to g. Since a plan might be of exponential size this cannot necessarily be implemented in polynomial space. However, we can guess the plan one operator at a time and compute the resulting state after each action, using only polynomial space. Hence, this modi- fied planner can be represented by a non-deterministic algorithm using polynomial space. Thus, by Savitch’s theorem (Savitch 1970), it can also be represented by a deterministic algorithm that uses polynomial space. This modified planner can be the same for all prob- lems simply by giving the PSN structure <p and the goal state G as additional inputs. Hence, it is of con- stant size, i.e its size does not depend on the size of the given PSN structure. Consequently, we can disregard the size of the planner and we have constructed a poly- space universal plan. (Observe that the soundness of P implies soundness of UG if we modify Ub to generate oT whenever the current state satisfies the goal state.) The planner P is complete and generates minimal plans. Hence, if the shortest plan from the current state S to the goal state 6 is of length L, the length of the shortest plan from S @ Ui (S) to 6 is L - 1. By this observation and the fact that P is complete, A-completeness of Ui follows. Finally, if there is no plan from the current state to the goal state, the planner will fail to generate even the first operator. In this case we simply output 01 and strong R-completeness follows. 0 It is crucial that the planner used in the previous the- orem generates shortest plans. Otherwise, we cannot guarantee A-completeness. We illustrate this with a small, contrived example. Example 11 Consider the following PSN structure <f, = (P, 0) = ({p,q}, {p+, q+,q-)) where the oper- ators are defined as follows: p+ = (F j p), q- = (q j q) and Q+ = @ * q). algorithm A that generates the plan wi = (q-, p+) for II1 and w2 = (q+ , p+) for II2. A universal plan UG based on A would then satisf;y vQ(Z1) = q+ and UG(Z2) = !I-. Consequently, UG (Xl) = q+ for odd K and Ug(Zl> = q- for even K. In other words, the universal plan will toggle q forever. Hence, UG is not A-complete. For planning problems such that BPG3 can be solved in polynomial-time, we can construct universal plans satisfying PT,~AR+ by Theorem 10. For planning problems such that PG is polynomial but BPG is not, the theorem does not apply. This method for constructing universal plans is pointed out by Sel- man (1994) but he does not explicitly state that gen- 3Recall that BPG and PG denote the bounded and unbounded plan generation problem respectively. erating the shortest plan is necessary. The question whether we can construct PT,~AR+ universal plans for problems where PG is polynomial but BPG is not re- mains open. Non-Existence of T,SA universal In order to show that PT,~A universal plans do not ex- ist for all PSN planning ‘problems, we will use advice- taking Turing machines (Johnson 1990). Advice- taking TMs are an alternative way of describing non- uniform circuits, which is the approach adopted by Sel- man (1994). Definition 12 An advice-taking Turing machine is a TM 2” that has associated with it a special “advice or- acle” A, which is a (not necessarily computable) func- tion. Let z be an arbitrary input string and let 1x1 denote the size of x. When T is applied to x, a spe- cial “advice tape” is automatically loaded with A( lx]) and from then on the computation proceeds as normal, based on the two inputs, x and A(IxI). An advice- taking Turing machine uses polynomial advice iff its advice oracle satisfies IA(n)1 5 p(n) for some fixed polynomial p and all nonnegative integers n. The class P/poly is the set of languages defined by polynomial- time advice-taking TMs with polynomial advice. Advice-taking TMs are very powerful. They can, for instance, compute certain undecidable functions. De- spite their apparent power, it is highly unlikely that all problems in NP can be solved by P/poly TMs. Theorem 13 (Karp & Lipton 1982) If NP c P/poly then the polynomial hierarchy collapses into E;. E; is a complexity class in the second level of the poly- nomial hierarchy (Johnson 1990). Collapse of the poly- nomial hierarchy is widely conjectured to be false in the literature (Johnson 1990; Papadimitriou 1994). Our proofs rely-on the following construction. Lemma 14 Let Y=n be the set of all 3SAT (Garey & Johnson 1979) instances with n variables. For every n, there is a PSN structure 0, = (P, 8) and a goal state gn such that for every F E Fn, there exists an ZF with the following property: HF = (P, 8, ZF, 6,) is a planning instance which is solvable iff F is satisfiable. Furthermore, any solution to HF must have a length less than or equal to 8n3 + 2n. Proof: Let U = (~1,. .., un} be the set of vari- ables used by the formulae in Fn. Observe that there can only be (2n)3 different clauses in any formula in Fn. Let C = {Cl,. . . , Csra3} be an enumeration of the possible clauses over the variable set U. Let P = {T(i),F(i),C(j)ll < i 5 n, 1 5 j 5 8n3}. The atoms will have the following meanings: T(i) is true iff the variable ui is true, F(i) is true iffthe variable ui is false and C(j) is true iff the clause Cj is satisfied. For each variable ui, two operators are needed: @ T(i),F(i) * T(i), Handling Uncertainty 1185 . T(i), F(i) + F(i). That is, T(i) can be made true iff F(i) is false and vice versa. In this fashion, only one of T(i) and F(i) can be true. For each case where a clause C(j) E C contains a variable ui, the first operator below is needed: for a negated variable lui, the second operator is needed: * T(i), Co * C(j), l F(i), c(j) a C(j). We specify the goal such that & = (@, Q;) = (Wl,... , Cs,,s), 0). Let F E F. We want to construct an initial state ZF such that II = (p, 0, ZF, &) is solv- able iff F is satisfiable. Let TF = { C( j)lC( j) 4 F}. Clearly, every C(j) can be made true iff a satisfying as- signment for F can be found. Finally, it is easy to see that any solution to HF must be of length 5 8n3 + 2n since we have exactly 8n3 + 2n atoms and each atom can be made true at most once. Cl Lemma 15 If, for every integer n 1 1, there exists a polynomial advice function that allows us to solve HF for all F E F’ in polynomial time, then the polynomial hierarchy collapses into X;. Proof: Suppose HF is solvable iff F has a satisfying truth assignment, then NP E P/poly so, by Theorem 13, the polynomial hierarchy collapses into X;. 0 We can now prove our main theorem. Theorem 16 If there exists a universal plan Ug, sat- isfying PT,sA for O,, n 2 1, then the polynomial hi- erarchy collapses into Et. Proof: Assume UQ, to be a PT,sA universal plan for 0,. Consider the algorithm A in Figure 1. Ug, is sound so it must generate an operator that is admissi- ble in the given state or generate one of the special op- erators 01 , 0~. Hence, by Lemma 14, the repeat loop can iterate at most 8n3+2n times before o equals either 01 or 0~. We have assumed that Ug, is a polynomial- time algorithm so algorithm A runs in polynomial time. We show that algorithm A accepts iff F has a satisfy- ing truth assignment. The if-part is trivial by noting that if F has a satisfying truth assignment then the algorithm accepts by A-completeness. For the only-if part, assume that the algorithm accepts. Then UG,, has returned the operator OT when applied to some state S. By Definition 7, UG, (S) = or iff S satisfies Gn. Consequently, F is satisfiable by Lemma 14. Hence, the algorithm accepts iff F is satisfiable and rejects iff F is not satisfiable. Furthermore, UQ,, is a polynomial advice function since we have restricted UQ, to be of polynomial size and the theorem follows by Lemma 15. The generality of this theorem has to be emphasize& Recall that an advice is an arbitrary function from the size of the input. This function does not even have to be computable. Hence, there does not exist any mech- anism whatsoever that is of polynomial size and can be accessed in polynomial time with the ability to solve Algorithm A. Input: A 3SAT formula F with n variables. s + ZF repeat 0 c hL(S) S-S@0 until 0 E {OI,OT} if 0 = OT then accept else reject Figure 1: The algorithm used in the proof of Theorem 16. problems like those exhibited in the previous theorem. Methods that have been proposed to reduce the size of universal plans, such as the variables introduced by Schoppers (1994)) cannot change this fact. Moreover, observe that Theorem 16 applies even to a class of severely restricted PSN structures. The restric- tions are, among others, that all delete-lists are empty and each operator has at most two preconditions. Since the delete-lists are empty, this restricted class is in NP (Bylander 1994). C onsequently, it is a class with considerably less expressive power than the general PSN planning problem which is PSPACE-complete (un- der the plausible assumption that NP#PsPAcE). Yet, PT,sA universal plans do not exist for this class of planning problems. Note that this is not caused by the existence of exponentially-size minimal plans since all minimal plans in this class are polynomially bounded. Finally, we would like to compare Theorem 16 with a negative result by Selman (1994). Theorem 17 Unless NPEP/poly, there exists a blocks-world planning goal for which there is no PT,sA universal plan for generating the minimal sequence of operators leading to the goal. It is important to note the difference between this theo- rem and Theorem 16. Where Selman shows that PT,sA universal plans cannot generate minimal plans under certain conditions, we show that there are cases when they cannot generate any plans at all. Discussion The results in this paper should not be interpreted too negatively. What they tell us is that naive approaches to universal planning will not work. In particular, we cannot hope for efficient universal plans solving arbi- trary planning problems. However, we question only the efficiency of universal plans. We do not claim uni- versal plans to be inferior to classical planners in all aspects. It is, for instance, highly probable that uni- versal planning can offer great advantages over classical planning in rapidly changing, dynamic domains. Thus, one of the challenges for the future is to characterize which planning problems can be efficiently solved by universal plans. We have seen that if a problem can be solved optimally in polynomial time, then there is 1186 Planning an efficient universal plan solving it. Almost certainly, there are other interesting classes of planning problems that can be solved by small, fast universal plans. Another question to be answered in the future is how to make universal planning more powerful. Sev- eral approaches are conceivable. One would be to give universal plans access to random sources-thus making universal planning probabilistic. Recent research has shown that probabilistic algorithms can be surprisingly efficient for certain types of problems. To mention one example, the probabilistic GSAT algorithm (Selman, Levesque, & Mitchell 1992) for satisfiability testing of propositional formulae has shown good performance in empirical studies. Another extension would be to allow universal plans to have an internal state; that is, the output of the universal plan is not only dependent on the current state, but also on previous states. Univer- sal plans with internal states have been studied briefly by Selman (1994). Th e results are unfortunately not encouraging. Universal planning should also be compared with incremental planning (Ambros-Ingerson & Steel 1988; Jonsson & Backstrom 1995). The idea behind incre- mental planning is to have a planner that can output valid prefixes of the final plan before it has finished planning. It has been argued that this method could considerably bring down the time lost in planning, es- pecially in dynamic domains, where replanning has to occur frequently. This motivation is almost exactly the same as the motivation for introducing universal plans (or reactive planning in general). Here we have a spec- trum of different approaches to planning ranging from classical planning which first computes the complete plan and then executes it, via incremental planning, where chunks of the plan are generated and executed in an interleaved fashion, to universal planning, where just one operator at a time is generated and immedi- ately executed. Conclusions We have proposed a stricter definition of universal plans which guarantees a weak notion of soundness not present in the original definition. In addition, we have identified three different types of completeness which capture different behaviours exhibited by uni- versal plans. A-completeness guarantees that if there exists a plan from the current state to the goal state, then the universal plan will find a solution in a finite number of steps. R-completeness is the converse of A- completeness, i.e. if there does not exist a plan from the current state to the goal state, then the univer- sal plan will report this after a finite number of ap- plications. R+-completeness is a stronger version of R-completeness, stating that if there does not exist a plan from the current state to the goal state, then the universal plan will report this after one application. We show that universal plans which run in polynomial time and are of polynomial size cannot be A-complete unless the polynomial hierarchy collapses. However, by dropping either the polynomial time or the poly- nomial space requirement, the construction of A- and R+-complete universal plans becomes trivial. References Ambros-Ingerson, J. A., and Steel, S. 1988. Integrat- ing planning, execution and monitoring. In Proc. 7th (US) Nat ‘1 Conf. on Artif. Intell. (AAAI-88), 83-88. BSickstrijm, C. 1995. Expressive equivalence of plan- ning formalisms. Artif. Intell. 76( l-2):17-34. Bylander, T. 1994. The computational complexity of propositional STRIPS planning. Artif. Intell. 69:165- 204. Chapman, D. 1989. Penguins can make cake. AI Mug. 45-50. Garey, M., and Johnson, D. 1979. Computers and Intractability: A Guide to the Theory of NP- Completeness. New York: Freeman. Ginsberg, M. L. 1989a. Ginsberg replies to Chapman and Schoppers. AI Mag. 61-62. Ginsberg, M. L. 1989b. Universal planning: An (al- most) universally bad idea. AI Mug. 40-44. Johnson, D. S. 1990. A catalog of complexity classes. In van Leeuwen, J., ed., Handbook of Theoretical Computer Science: Algorithms and Complexity, vol- ume A. Amsterdam: Elsevier. chapter 2, 67-161. Jonsson, P., and Backstrom, C. 1995. Incremental planning. In Ghallab, M., and Milani, A., eds., New Trends in AI Planning: Proc. 3rd Eur. WS. Planning (EWSP’95). Assisi, Italy: 10s Press. Karp, R. M., and Lipton, R. 1982. Turing machines that take advice. Enseign. Math 28:191-209. Papadimitriou, C. H. 1994. Computational Complex- ity. Reading, MA: Addison Wesley. Savitch, W. J. 1970. Relationships between nondeter- ministic and deterministic tape complexities. Journal of Computer and System Sciences 4(2):177-192. Schoppers, M. J. 1987. Universal plans for reactive robots in unpredictable environments. In Proc. 10th Int ‘1 Joint Conf. on Artif. Intell. (IJCAI-87), 1039- 1046. Schoppers, M. J. 1989. In defense of reaction plans as caches. AI Mug. 51-62. Schoppers, M. 1994. Estimating reaction plan size. In Proc. 12th (US) Nat’1 Conf. on Artif. Intell. (AAAI- 94), 1238-1244. Selman, B.; Levesque, H.; and Mitchell, D. 1992. A new method for solving hard satisfiability prob- lems. In Proc. 10th (US) Nat’1 Conf. on Artif Intell. (AAAI-921, 440-446. Selman, B. 1994. Near-optimal plans, tractability, and reactivity. In Proc. 4th Int ‘1 Conf. on Prin- ciples of Knowledge Repr. and Reasoning (KR-94), 521-529. Handliig Uncertainty 1187 | 1996 | 175 |
1,816 | Is “early commitment” in plan generation ever a good idea? David Joslin Computational Intelligence Research Laboratory 1269 University of Oregon Eugene, OR 97403 joslin@cirl.uoregon.edu Abstract Partial-Order Causal Link planners typically take a “least-commitment” approach to some deci- sions (notably, step ordering), postponing those decisions until constraints force them to be made. However, these planners rely to some degree on early commitments in making other types of decisions, including threat resolution and oper- ator choice. We show why existing planners cannot support full least-commitment decision- making, and present an alternative approach that can. The approach has been implemented in the Descartes system, which we describe. We also provide experimental results that demonstrate that a least-commitment approach to planning can be profitably extended beyond what is done in POCL and similar planners, but that taking a least-commitment approach to every planning decision can be inefficient: early commitment in plan generation is sometimes a good idea. Introduction The “least-commitment” approach to plan generation has, by and large, been successful where it has been tried. Partial-Order Causal Link (POCL) planners, for example, typically take a least-commitment approach to decisions about step ordering, postponing those de- cisions until constraints force them to be made. How- ever, these planners rely to some degree on early com- mitments for other decisions, including threat resolu- tion and choice of an operator to satisfy open con- ditions. An obvious question is whether the least- commitment approach should be applied to every plan- ning decision; in other words, is early commitment ever a good idea? An obstacle to addressing this question experimen- tally arises from the way in which POCL (and similar) planners manage decision-making. They take what we call a passive postponement approach, choosing one de- cision at a time to focus on, and keeping all the other, postponed decisions on an “agenda.” Items on the agenda play no role in planning until they are selected for consideration, despite the fact that they may im- pose constraints on the planning process. Martha E. Pollack Department of Computer Science and Intelligent Systems Program University of Pittsburgh Pittsburgh, PA 15260 pollack@cs.pitt.edu In this paper, we present experimental evidence of the efficiency penalty that can be incurred with pas- sive postponement. We also present an alternative ap- proach, active postponement, which has been imple- mented in the Descartes system. In Descartes, plan- ning problems are transformed into Constraint Satis- faction Problems (CSPs), and then solved by applying both planning and CSP techniques. We present ex- perimental results indicating that a least-commitment approach to planning can be profitably extended be- yond what is done in most planners. We also demon- strate that taking a least-commitment approach to ev- ery planning decision can be inefficient: early commit- ment in plan generation is sometimes a good idea. Passive postponement POCL algorithms use refinement search (Kambham- pati, Knoblock, & Yang 1995). A node N (a partial plan) is selected for refinement, and a flaw F (a threat or open condition) from N is selected for repair. Suc- cessor nodes are generated for each of the possible re- pairs of F. All other (unselected) flaws from the parent node are inherited by these successor nodes. Each flaw represents a decision to be made about how to achieve a goal or resolve a threat; thus, each unselected flaw rep- resents a postponed decision. We term this approach to postponing decisions passive postponement. Deci- sions that are postponed in this manner play no role in planning until they are actually selected. Passive postponement of planning decisions can in- cur severe performance penalties. It is easiest to see this in the case of a node that has an unrepairable flaw. Such a node is a dead end, but the node may not be recognized as a dead end if some other, repairable flaw is selected instead: one or more successor nodes will be generated, each inheriting the unrepairable flaw, and each, therefore, also a dead end. In this manner, a sin- gle node with a fatal flaw may generate a large number of successor nodes, all dead ends. The propagation of dead-end nodes is an instance of a more general problem. Similar penalties are paid when a flaw that can be repaired in only one way- a “forced” repair-is delayed; in that case, the forced 1188 Planning From: AAAI-96 Proceedings. Copyright © 1996, AAAI (www.aaai.org). All rights reserved. repair may have to be repeated in multiple successor nodes. Passive postponement also means that inter- actions among the constraints imposed by postponed decisions are not recognized until all the relevant deci- sions have been selected for repair. The propagation of dead-end nodes is not just a theoretical problem; it can be shown experimentally to cause serious efficiency problems. We ran UCPOP (Penberthy & Weld 1992) on the same test set of 49 problems from 15 domains previously used in (Joslin & Pollack 1994)) with the default search heuristics pro- vided by UCPOP. As in the earlier experiments, 32 of these problems were solved by UCPOP, and 17 were not, within a search limit of 8000 nodes generated. We counted the number of nodes examined, and the num- ber of those nodes that were immediate successors of dead-end nodes, i.e., nodes that would never have been generated if fatal flaws were always selected immedi- ately. For some problems, as many as 98% of the nodes examined were successors of dead-end nodes. The av- erage for successful problems was 24%) and for unsuc- cessful problems, 48%. See (Joslin 1996) for details. One response to this problem is to continue passive postponement, but to be smarter about which deci- sions are postponed. This is one way to think about the Least-Cost Flaw Repair (LCFR) flaw selection strat- egy we presented in (Jo&n & Pollack 1994). Indeed, LCFR is a step in the direction of least-commitment planning, because it prefers decisions that are forced to ones that are not. However, even with an LCFR-style strategy, postponed decisions do not play a role in rea- soning about the plan until they are selected. Because of this, LCFR can only recognize forced decisions (in- cluding dead-ends) that involve just a single flaw on the agenda, considered in isolation. When a decision is forced as a result of flaw interactions, LCFR will not recognize it as forced, and thus may be unable to take a least-commitment approach. Overview of active postponement The active postponement approach to planning recog- nizes that flaws represent decisions that will eventu- ally have to be made, and that these postponed de- cisions impose constraints on the plan being devel- oped. It represents these decisions with constrained variables whose domains represent the available op- tions, and posts constraints that represent correctness criteria on those still-to-be-made decisions. A general- purpose constraint engine can then be used to enforce the constraints throughout the planning process. To illustrate active postponement .for threat resolu- tion, consider a situation in which there is a causal link from some step A to another step B. Assume a third step C, just added to the plan, threatens the causal link from A to B. A constrained variable, Dt, is in- troduced, representing a decision between promotion and demotion, i.e., Dt E {p, d}, and the following con- straints will be posted: (Dt = p) --f @er(C, B) (Dt = d) --+ before(C, A) The decision about how to resolve the threat can then be postponed, and made at any time by binding Dt. Suppose that at some later point it becomes impos- sible to satisfy after(C) B). The constraint engine can deduce that Dt # p, and thus Dt = d, and thus C must occur before A. Similarly, if the threat becomes unresolvable, the constraint engine can deduce that Dt E {p, d} cannot be satisfied, and thus that the node is a dead end. This reasoning occurs automatically, by constraint propagation, without the need to “select” the flaw that Dt represents. When a threat arises, all of the options for resolv- ing that threat are known; other changes to the plan may eliminate options, but cannot introduce new op- tions. Active postponement for goal achievement is more complex, because the repair options may not all be known at the time the goal arises: steps added later to the plan may introduce new options for achieving a goal. To handle this, we allow the decision variables for goal achievement (termed cuusuI variables) to have dy- namic domains to which new values can be added. We notate dynamic domains using ellipses, e.g., D, E {. . .) represents the decision about how to achieve some goal g, for which there are not yet any candidate establish- ers. An empty dynamic domain does not indicate a dead end, because new options may be added later; it does, however, mean that the problem cannot be solved unless some new option is introduced. Suppose that at a later stage in planning, some new step F is intro- duced, with g as an effect. An active postponement planner will expand the domain so that D, E {F, . . .}. Constraints will be posted to represent the conditions, such as parameter bindings, that must be satisfied for F to achieve g. Least-commitment planning The active-postponement approach just sketched has been implemented in the Descartes planning system. Descartes provides a framework within which a wide variety of planning strategies can be realized (Joslin 1996). In this section, we provide the details of the algorithm used by Descartes to perform fully “least- commitment” planning: LC-Descartes. LC-Descartes (Figure 1) can be viewed as perform- ing plan generation within a single node that contains both a partial plan II and a corresponding CSP, C. II contains a set of steps, some of which have been only tentatively introduced. C contains a set of vari- ables and constraints on those variables. Variables in C represent planning decisions in II. For example, each precondition p of a step in II will correspond to a causal variable in C representing the decision of how to achieve p. Parameters of steps in II will correspond to variables in C representing the binding options. Search 1189 LC-Descartes(N = (II,C), A) 1. If some variable v E C has an empty static do- main, then fail. 2. Else, if some variable v E C has an empty dy- namic domain, then (a) Let p be the precondition corresponding to v. (b) Restricted expansion. For each operator o E A with an effect e that can achieve p, call Expund(N,o) to tentatively add o to II, and the corresponding constraints and variables to C. (c) Convert v to have a static domain (i.e., commit to using one of the new steps to achieve p.) (d) Recursion: Return L C-Descurtes(N, A) 3. Else, call Solve(C); if it finds a solution to the CSP, then return that solution. 4. Else, (a) Unrestricted expansion. For each o E A, call Expund(N,o) to tentatively add a new step. (b) Recursion: Return L C-Descurtes(N, A). Figure 1: LC-Descartes algorithm Expand(N = (II,C), a) 1. For each threat that arises between the new step, cr, and steps already in the plan, II, expand the CSP, C, to represent the possible ways of resolv- ing the threat. 2. For each precondition, p, of step CT, add a new causal variable to C whose dynamic domain in- cludes all the existing steps in II that might achieve p. Add constraints for the required bind- ings and temporal ordering. 3. For each precondition, p, in II that might be achieved by 0, if the causal variable for p has a dynamic domain, add cr to that domain, and add constraints for the required bindings and tempo- ral ordering. 4. For any step ~11 in II, add the constraint that if CT and cu are actually in the plan, they occur at different times. 5. Add step CT to II. 6. Apply constraint reduction techniques (Tsang 1993) to prune impossible values from domains. Figure 2: Expand algorithm Initially, LC-Descartes is called with a node to which only pseudo-steps for the initial and goal state, Si and S,, have been added. Following the standard planning technique, Si’s effects are the initial conditions, and Ss’s preconditions are the goal conditions. The third argument to LC-Descartes is the operator library, A. LC-Descartes first checks whether any variable has an empty static domain, indicating failure of the plan- ning process. If failure has not occurred, it checks whether any causal variable o has an empty dynamic domain, indicating a forced decision to add some step to achieve the goal corresponding to v. In this case, it invokes the Expand function (see Figure 2), to add a tentative step for each possible way of achieving the goal, postponing the decision about which might be used in the final plan. As Expand adds each new, ten- tative step to the plan, it also expands the CSP (1) to resolve threats involving the new step, (2) to al- low any step already in the plan to achieve precon- ditions of the new step, (3) to allow the new step to achieve preconditions of steps already in the plan, and (4) to prevent the new step from occurring at the same time as any step already in the plan. This approach to goal achievement generalizes multi-contributor causal structures (Kambhampati 1994). If no variable has an empty domain, LC-Descartes invokes Solve, which applies standard CSP search tech- niques to try to solve the CSP. At any point, the CSP has a solution if and only the current set of plan steps can be used to solve the planning problem. A solution of the CSP must, of course, bind all of the variables, satisfying all of the constraints. Binding all of the variables means that all planning decisions have been made-the ordering of plan steps, parameter bindings, etc. Satisfying all of the constraints means that these decisions have been made correctly. In Solve, dynamic domains may be treated as static since what we want to know is whether the current set of plan steps are sufficient to solve the planning prob- lem; for this reason, Solve uses standard CSP solution techniques. We will assume that Solve performs an exhaustive search, but this is not required in practice. If Solve is unsuccessful, then LC-Descartes performs an unrestricted expansion, described in the example below. The algorithm continues in this fashion until failure is detected or a solution is found. A short example. Consider the Sussman Anomaly Blocks World problem, in which the goal is to achieve on(A,B), designated gl, and on(B,C), designated g2, from an initial state in which on(C,A), on(A,l’ubZe), and on(B, Table). The initial CSP is formed by calling Expand to add the pseudo-actions for the initial and goal states. At this point, the only dynamic variables will be the causal variables for gl and 92; call these DL?l and Dgz. Since neither goal can be satisfied by any step currently in the plan (i.e., they aren’t satisfied in the initial state), both of these variables have empty domains. There are no empty static domains, but both D,J and D,z have empty dynamic domains. LC-Descartes selects one of them; assume it selects D,j. It per- forms a restricted expansion to add one instance of each action type that might achieve this goal. For this example, suppose we have only one action type, pu- ton(?x, ?y, 2~)) that takes block ?z off of ?z and puts ?a: on ?y. The new puton step will be restricted so that it 1190 Planning must achieve on(A,B), which here means that the new step is puton(A,B, Zz1); call this step S1. Adding this step means that D,j = S1. (The domain of Dgl is no longer dynamic because a commitment is made to use the new step; this also causes S1 to be actually in the plan, not just tentatively.) New variables and con- straints will also be added to the CSP for any threats that are introduced by this new step. The resulting CSP again has a variable with an empty dynamic domain, D,z, corresponding to the goal on(B,C). A restricted expansion adds the step pu- ton(B, C, ?z2), and adds this step (S2) to the domain of D 92. Threat resolution constraints and variables are again added. Some of the preconditions of S1 are po- tentially satisfiable by S2, and vice versa, and the cor- responding variables will have their dynamic domains expanded appropriately. At this point, the CSP turns out to have no variables with empty domains. This indicates that us fur us can be determined by CSP reduction alone, it is possible that the problem can be solved with just the current set of plan steps. Solve is called, but fails to find find a solution; this indicates that, in fact, the current plan steps are not sufficient. The least-commitment approach demands that we do only what we are forced to do. The failure to find a solution told us that we are forced to expand the plan, adding at least one new step. Unlike the restricted expansions, however, we have no information about which precondition(s) cannot be satisfied, and there- fore, do not know which action(s) might be required. That is, there are no empty dynamic domains, so ev- ery condition has at least one potential establisher, but conflicts among the constraints prevent establishers be- ing selected for all of them. Under these conditions, the least-commitment ap- proach requires that we add new steps that can achieve any precondition currently in the plan. In LC-Descartes, this is an unrestricted expansion, and is accomplished by adding one step of each type in the operator library. In the current example, this is sim- plified by the fact that there is only one action type. The new step, S3, will be puton(?x3, ?y3, ?23), and as before, variables and constraint will be added to allow S3 to achieve any goal it is capable of achieving, and to resolve any threats that were introduced by the addition of S3. The CSP still has no variables with empty domains, so SoZve is again called. This time the standard solution to the Sussman anomaly is found. As promised by its name, LC-Descartes will only commit to decisions that are forced by constraints. However, this least-commitment behavior requires un- restricted expansions, which are are potentially very in- efficient. They introduce steps that can be used for any goal; note that unlike the first two steps, which were constrained to achieve specific goals, step S3 above has no bound parameters. Even worse, a new step of each action type must be introduced. In addition, detecting 0 100 L 90 g 80 2 70 4 60 50 & 40 E 30 ; 20 g 10 0 1 2 3 4 5 6 7 8 9 10111213141516 Number of blocks Figure 3: Blocks World, random problems the need for an unrestricted expansion is much more laborious than checking for variables with empty do- mains, since it requires that the exhaustive search for a solution. Solve function Experimental results We begin with a special type of domain, one that has only a single action type. Although unlikely to be of fail an practical interest, such domains allow us to test the limits of LC-Descartes. If unrestricted expansions are a problem even in single-action domains, we know that they will be prohibitive in more realistic domains. Figure 3 shows experimental results on randomly generated Blocks World problems, using the single- action encoding given in the previous section. Prob- lems were generated using the problem generator pro- vided in the UCPOP 2.0 distribution. The number of blocks was varied from 3 to 16, with 10 problems at each level, for a total of 140 problems. UCPOP, LCFR and LC-Descartes were run on the test problems, with a search limit of 300 CPU seconds on a SPARCsta- tion 20. We report the percentage of problems solved within the CPU limit, and not the number of nodes generated, because LC-Descartes does all its planning within a single node. UCPOP solved all of the 4-block problems, but its performance fell off rapidly after that point. It dropped below 50 percent at six blocks, and solved no problems of more than eight blocks. LCFR’s performance fol- lowed a similar pattern. LC-Descartes started to fail on some problems at seven blocks, and dropped be- low 50 percent at ten blocks. It solves problems of up to fifteen blocks, and fails to solve any problems larger than that. Roughly speaking, on a given set of problems LC-Descartes performed about as well as UCPOP or LCFR performed on problems with half as many blocks. (Doubling the CPU limit did not change the results appreciably.) In interpreting this result, note that the difficulty of blocks world problems in- creases exponentially with the number of blocks. Also note that because it takes a fully least-commitment approach, in a single-action domain LC-Descartes will always generate minimal-length plans, something not guaranteed by either UCPOP or LCFR. Search 1191 We investigated what LC-Descartes is doing on the larger problems (see (Joslin 1996) for details), and saw that virtually all of its work is in the form of restricted expansions. Of the 58 successful plans for five-block problems and larger, the average number of steps in each plan was 7.3; of these, only an average of 1.1 steps were added via unrestricted expansions. 31% of these 58 problems were solved with only restricted expansions, another 39% were solved with only one unrestricted expansion. In other words, where LC- Descartes was successful, it was because active post- ponement allowed it to exploit the structure of the problem enough to either reach a solution directly, or at least get very close to a solution. Success depended on avoiding unrestricted expansions. EC-Descartes. These results led us to conjecture that the least-commitment approach should be taken at all points except those at which LC-Descartes per- forms unrestricted expansions. We explored this idea by modifying Descartes to make some early commit- ments: EC-Descartes. EC-Descartes still differs sig- nificantly from other planning algorithms, which make early commitments at many points. In POCL plan- ners, for example, threat resolution generates sepa- rate successor nodes for promotion and demotion; each node represents a commitment to one step ordering. If both promotion and demotion are viable options, how- ever, then a commitment to either is an early commit- ment. EC-Descartes avoids this kind of early commit- ment by posting a disjunctive constraint representing all of the possible options, and postponing the decision about which will be used to resolve the threat. EC-Descartes and LC-Descartes behave identically except at the point at which LC-Descartes would per- form an unrestricted expansion. There, EC-Descartes instead generates more than one successor node. Its objective is to exchange one node that has become under-constrained, and so difficult to solve, for a larger number of nodes that all have at least one variable with an empty domain, static or dynamic. In each of these branches, EC-Descartes then returns to the least- commitment approach, until a solution is found or the problem again becomes under-constrained. We implemented two versions of EC-Descartes that achieve this objective. The first, EC(l), adopts the simple strategy of selecting the dynamic domain vari- able with the smallest domain; because only causal variables have dynamic domains, these will be early commitments about action selection. EC( 1) generates two successor nodes. In one, all values are removed from the selected variable’s domain, forcing a restricted expansion in that node. In the other successor node, the domain is made static. These early commitments are complete; the latter commits to using some step currently in the node, and the former commits to using some step that will be added later. (If EC(l) fails to find a variable with a dynamic domain, it uses EC(2)‘s CPU t EC(I) 7.2 --q-q- 6.7 Problem LC Problem LC EC( 1) EC(2) 1 1 * * 6 7.2 6.7 2 2 31.5 31.5 142.4 3 3 * (2) * (2) 94.7 94.7 4 4 8.7 (1) 8.7 (1) 4.8 4.8 5T1 5 5 6 6 “%’ 3”;;$) 5.2 5.2 155.2 155.2 9;8 7 7 9.0 (1) 9.0 (1) 7.4 7.4 6.2 8 8 4.0 (0) 4.0 (0) 3.6 3.6 9 9 5.8 (1) 5.8 (1) 5.7 5.7 3;6 imes are in imes are in seconds; * = exceeded time 142.4 * 5.1 9.8 * 6.2 3.6 * seconds; * = exceeded time imit Figure 4: Early- and least-commitment strategy instead.) The second version of EC-Descartes performs early commitments in a manner analogous to the “divide and conquer” technique sometimes used with CSPs. EC(2) performs early commitment at the same time as EC(l), but it selects a static variable with minimal domain size, and then generates two successor nodes, each in- heriting half of the domain of the selected variable. In that both EC(l) and EC(2) select decisions with min- imum domain size (within their respective classes of variables), both bear some resemblance to LCFR. Figure 4 shows CPU times for LC-Descartes and both versions of EC-Descartes on a set of nine prob- lems from the DIPART transportation planning do- main (Pollack et al. 1994); it also shows in parentheses the number of unrestricted expansions performed by LC-Descartes. EC(l) solved all of the problems, while EC(2) failed on three problems, and LC-Descartes failed to solve four, hitting a search limit of 600 CPU seconds. On all but the “easiest” problem (# 8)) LC- Descartes needs to resort to at least one unrestricted expansion, and it failed on all the problems on which it performed a second unrestricted expansion. Al- though this domain only has three action types, the added overhead of carrying unneeded (tentative) steps, and all of the associated constraints, is considerable. Not surprisingly (at least in retrospect) the fully least- commitment approach loses its effectiveness rapidly af- ter the transition to unrestricted expansions occurs. The relative advantage of EC(l) over EC(2) suggests that early commitments on action selection are partic- ularly effective. Related work Work related to LCFR includes DUnf and DMin (Peot & Smith 1993), b ranch-l/branch-n (Currie & Tate 1991), and ZLIFO (Schubert & Gerevini 1995). DMin, which enforces ordering consistency for post- poned threats, could be viewed as a “weakly active” approach. To a lesser extent, even LCFR and ZLIFO could be thought of as using “weakly active” postpone- ment, since enough reasoning is done about postponed decisions to detect flaws that become dead ends. Virtually all modern planners do some of their work by posting constraints, including codesignation con- 1192 Planning straints on possible bindings, and causal links and tem- poral constraints on step ordering. Allen and Koomen (Allen & Koomen 1990) and Kambhampati (Kamb- hampati 1994) generalize the notions of temporal and causal constraints, respectively. Planners that make more extensive use of constraints include Zeno (Penberthy & Weld 1994) and O-Plan (Tate, Drabble, & Dalton 1994). Zeno uses constraints and temporal intervals to reason about goals with deadlines and continuous change. O-Plan makes it pos- sible for a number of specialized “constraint managers” to work on a plan, all sharing a constraint representa- tion that allows them to interact. Both Zeno and O- Plan maintain an agenda; Descartes differs from them in its use of active postponement. Previous work that has used constraints in a more active sense during plan generation includes (Stefik 1981; Kautz & Selman 1992; Yang 1992). MOLGEN (Stefik 1981) posts constraints on variables that rep- resent certain kinds of goal interactions in a partial plan. These constraints then guide the planning pro- cess, ruling out choices that would conflict with the constraint. Descartes can be seen as taking a similar constraint-posting approach, but extending it to apply to all decisions, not just variable binding, and placing it within a more uniform framework. Kautz and Selman have shown how to represent a planning problem as a CSP, given an initial user-selected set of plan steps; if a solution cannot be found using some or all of those steps, an expansion would be required, much like an unrestricted expansion in LC-Descartes. WATPLAN (Yang 1992) uses a CSP mechanism to resolve conflict among possible variable bindings or step orderings; its input is a possibly incorrect plan, which it transforms to a correct one if possible. WATPLAN will not extend the CSP if the input plan is incomplete. Conclusions The Descartes algorithm transforms planning problems into dynamic CSPs, and makes it possible to take a fully least-commitment approach to plan generation. This research shows that the least-commitment ap- proach can be profitably extended much further than is currently done in POCL (and similar) planners. There are, however, some fundamental limits to the effectiveness of the least-commitment approach; early commitments are sometimes necessary. In particular, one can recognize that constraints have ceased to be effective in guiding the search for a plan, and at that point shift to making early commitments. These early commitments can be viewed as trading one node whose refinement has become difficult for some larger number of nodes in which constraints force restricted expan- sions to occur, i.e., trading one “hard” node for several “easy” nodes. One direction for future research will be to look for more effective techniques for making this kind of early commitment. Acknowledgements. This research has been sup- ported by the Air Force Office of Scientific Re- search (F49620-91-C-0005)) Rome Labs (RL)/ARPA (F30602-93-C-0038 and F30602-95-l-0023)) an NSF Young Investigator’s Award (IRI-9258392)) an NSF CISE Postdoctoral Research award (CDA-9625755) and a Mellon pre-doctoral fellowship. References Allen, J., and Koomen, J. 1990. Planning using a tem- poral world model. In Readings in Planning. Morgan Kaufmann Publishers. 559-565. Currie, K., and Tate, A. 1991. O-plan: The open planning architecture. Art. Int. 52:49-86. Joslin, D., and Pollack, M. E. 1994. Least-cost flaw repair: A plan refinement strategy for partial-order planning. In Proc. AAAI-94, 1004-1009. Joslin, D. 1996. Passive and active decision postpone- ment in plan generation. Ph.D. dissertation, Intelli- gent Systems Program, University of Pittsburgh. Kambhampati, S.; Knoblock, C. A.; and Yang, Q. 1995. Planning as refinement search: A unified frame- work for evaluating design tradeoffs in partial-order planning. Art. Int. 76( l-2):167-238. Kambhampati, S. 1994. Multi-contributor causal structures for planning: a formalization and evalu- ation. Art. Int. 69( l-2):235-278. Kautz, H. and Selman, B. 1992. Planning as Satisfia- bility. Proc. ECAI-92 Vienna, Austria, 1992, 359-363. Penberthy, J. S., and Weld, D. 1992. UCPOP: A sound, complete, partial order planner for ADL. In Proc. 3rd Int. Conf. on KR and Reasoning, 103-114. Penberthy, J. S., and Weld, D. 1994. Temporal planning with continuous change. In Proc. AAAI-94, 1010-1015. Peot, M., and Smith, D. E. 1993. Threat-removal strategies for partial-order planning. In Proc. AAAI- 93, 492-499. Pollack, M. E.; Znati, T.; Ephrati, E.; Joslin, D.; Lauzac, S.; Nunes, A.; Onder, N.; Ronen, Y.; and Ur, S. 1994. The DIPART project: A status report. In Proceedings of the Annual ARPI Meeting. Schubert, L., and Gerevini, A. 1995. Accelerating partial order planners by improving plan and goal choices. Tech. Rpt. 96-607, Univ. of Rochester Dept. of Computer Science. Stefik, M. 1981. Planning with constraints. Art. Int. 16:111-140. Tate, A.; Drabble, B.; and Dalton, J. 1994. Reasoning with constraints within 0-Plan2. Tech. Rpt. ARPA- RL/O-Plan2/TP/6 V. 1, AIAI, Edinburgh. Tsang, E. 1993. Foundations of Constraint Satisfac- tion. Academic Press. Yang, Q. 1992. A theory of conflict resolution in planning. Art. Int. 58( l-3):361-392. Search 1193 | 1996 | 176 |
1,817 | ming, Propositional Logic, and Stochastic Search Henry Kautz and Bart Selman AT&T Laboratories 600 Mountain Avenue Murray Hill, NJ 07974 {kautz, selman)@research.att.com http://www.research.att.com/“{kautz, selman} Abstract Planning is a notoriously hard combinatorial search problem. In many interesting domains, current plan- ning algorithms fail to scale up gracefully. By combin- ing a general, stochastic search algorithm and appro- priate problem encodings based on propositional logic, we are able to solve hard planning problems many times faster than the best current planning systems. Although stochastic methods have been shown to be very effective on a wide range of scheduling problems, this is the first demonstration of its power on truly challenging classical planning instances. This work also provides a new perspective on representational issues in planning. Introduction There is a widespread belief in the AI community that planning is not amenable to general theorem-proving techniques. The origin of this belief can be traced to the early 1970’s, when work on plan generation usin first-order, resolution theorem-proving (Green 1969 7 failed to scale up to realistically-sized problems. The relative success of the STRIPS system (Fikes and Nils- son 1971) established the basic paradigm for practi- cally all subsequent work in planning. Planning is viewed as a systematic search through either a state- space or through a space of partial plans. Different representations are used for actions and for states or fluents. Control strategies are not discussed in terms of general rules of inference, but rather in terms of rules for establishing and protecting goals, detecting conflicts between actions, and so forth. The results described in this paper challenge this be- lief. We have applied general reasoning systems to the task of plan synthesis, and obtained results that are competitive with, and in many cases superior to, the best specialized planning systems. Why was this pos- sible? We believe that the lesson of the 1970’s should not have been that planning required specialized algo- rithms, but simply that first-order deductive theorem- proving does not scale well. By contrast, the past few years have seen dramatic progress in the size of prob- lems that can be handled by propositional satisfiabil- ity testing programs (Trick and Johnson 1993, Selman 1995). In particular, new algorithms based on random- ized local search (Selman et al. 1992) can solve certain classes of hard problems that are an order of magni- tude larger than those that can be solved by older ap- 1194 Planning proaches. Therefore, our formalization of planning is based on propositional satisfiability, rather than first- order refutation. We ran experiments with both one of the best sys- tematic satisfiability algorithms (“tableau”, by Craw- ford and Auton (1993)) and one of the best stochas- tic algorithms (“Walksat”, by Selman et al. (1994; 1996)). All task-specific information was given a uni- form clausal representation: the inference engines had no explicit indication as to what stood for a goal or what stood for an operator. This meant that the solvers were not constrained to perform a strict back- ward or forward chaining search, as would be done by most planning systems. Far from being a disadvantage, this greatly adds to the power of approach, by allow- ing constraints to propagate more freely and thus more quickly reduce the search space. (The idea of viewing planning as general constraint satisfaction rather than directional search has also been explored by other au- thors; see, for example, Joslin and Pollack (1995).) The notion of formalizing planning as propositional reasoning immediately raises certain questions. Plan- ning is a notoriously hard problem. In fact, the general plan-existence problem for STRIPS-style operators is PSPACE-complete (Bylander 1991, Erol et al. 1992, Backstrom 1992). How, then, is it possible to formulate planning as only an NP-complete problem? This dif- ficulty disappears when we realize that the PSPACE hardness result only holds when the potential solutions can be of exponential length. If we are only interested in polynomial-length plans, then planning is indeed NP-complete. Many other planning systems can be viewed as spe- cialized propositional reasoning engines. A surpris- ingly efficient recent planning system is Graphplan, developed by Blum and Furst (1995). Graphplan works in two phases: in the first, a problem stated using STRIPS notation is converted to a data struc- ture called a “planning graph”. In the second, the graph is systematically searched for a solution. The planning graph is in fact a propositional representa- tion. In some of our experiments, we directly converted planning graphs into sets of clauses, and then applied Walksat or tableau. For other experiments we devel- oped by hand even more compact and efficient clausal encodings of the problems. As we will see, it was often the case that Walksat dramatically outperformed both the general and specialized systematic search engines. From: AAAI-96 Proceedings. Copyright © 1996, AAAI (www.aaai.org). All rights reserved. The success of stochastic local search for planning may come as a surprise. Although local search has been successfully applied to scheduling problems (Adorf and Johnston 1990, Minton et al. 1990, 1992)) it has seen little use for planning. Some authors (Kautz and Sel- man (1992), Crawford and Baker (1994)) have sug- gested that planning (finding a partially-ordered set of operators that achieve a goal) and scheduling (as- signing times and resources to a given, fixed set of operators) require different control mechanisms, and that planning is inherently a systematic process Our present success can be mainly attributed to two factors: first, the greater speed and power of Walksat over ear- lier local search satisfiability algorithms (e.g., GSAT (Selman et al. 1992)); and second, our use of better problem encodings - including “compiling away” plan operators, and extending a technique from Blum and Furst for encoding partially-ordered plans with parallel actions. Our results appear to be the first convincing evidence that stochastic local search is indeed a pow- erful technique for planning. We will discuss techniques for encoding planning problems as propositional SAT in some detail below. Our experience has been that the search for domain axiomatizations with better computational properties has led us to valuable insights at the representational level. For example, we will describe one encoding we used that “compiles away” any explicit propositions that stand for actions, leaving only fluents. While this encoding was initially motivated by a concern for re- ducing the number of different propositions in the fi- nal formula, it turned out to also enable a particularly simple and elegant solution to the frame problem with parallel actions. It is important to note that we are emphatically not suggesting that control knowledge should be “mixed-in” with declarative information, as occurs in logic programming. Instead, we are suggest- ing that it can be advantageous to try to optimize the gross statistical properties of an axiomatization when developing or choosing between declarative represen- tations. This paper is organized as follows After a short pre- view of the results, we discuss general approaches to planning as satisfiability, and particular encoding tech- niques. We then present experimental results drawn from several domains, including logistics problems, the “rocket” domain, and the blocks world. We compare the performance of both systematic and stochastic al- gorithms on different kinds of SAT encodings to the performance of Graphplan, and cite comparisons of Graphplan with the well-known Prodigy (Carbonell et al. 1992, Stone et al. 1994) and UCPOP (Penberthy and Weld 1992) systems. Preview of Results Before we describe our approach in detail, we will first highlight some of our main experimental results. In or- der to evaluate our method, we considered planning do- mains that lead to serious computational difficulties in traditional planners. Barrett and Weld (1994) discuss various characteristics of such domains. In general, the hardest planning domains contain intricate inter- actions between planning operators, and various types of goal and subgoal interactions. These interactions complicate the order in which the goals and subgoals should be established, and make it difficult to select the right operator for establishing a goal. Real-world domains often contain both sources of computational difficulties. In our experiments, we focussed on two natural do- mains: the “rocket” domain (Blum and Furst 1995) and the “logistics” domain (Veloso 1992). Blum and Furst showed that Graphplan outperforms Prodigy and UCPOP on the rocket problems. We extended this problem somewhat to make it more challenging for Graphplan. The logistics domain can be viewed as yet a further extension of the rocket domain, making it even harder. We also considered several relatively large blocks world problems, because even small blocks world instances are often already surprisingly hard for traditional planners. Table 1 gives the results on some of the hardest in- stances we c0nsidered.i From the last column, it is clear that using a stochastic method (Walksat) and a direct SAT encoding, we can solve these instances two or more orders of magnitude faster than Graphplan. Walksat actually found the optimal (i.e., shortest pos- sible plans) for these problems. Thus, for example, it found an optimal 36 step plan to the blocks world prob- lem “bw1arge.d”. This instance contains 19 blocks and has multiple stacks in both the initial and goal state. We not aware of any other planning algorithm that can solve instances this size without incorporating domain- specific search control knowledge. The table also contains our results of running on the SAT encodings derived from Graphplan’s planning graphs. The first two instances are again solved sig- nificantly faster than by using Graphplan itself. The SAT encodings for the last two instances became too large for our SAT procedures. Our SAT encoding is more compact than the original Graphplan represen- tation, but Graphplan can handle larger internal data structures than can our SAT procedures. The planning graph for “bw1arge.b” contains 18,069 nodes and has over one million exclusion relations. This is just within Graphplan’s reach, taking over 7 hours; on our state- base encoding, Walksat takes only 22 seconds. Walk- sat9s solution was proved optimal using the systematic algorithm tableau. Interestingly, although tableau is able to show that there is no shorter solution, it can- not actually find the solution itself! This show how stochastic and the systematic methods can comple- ment one another. Planning as Satisfiability While planning has traditionally been formulated as deduction in first-order logic (Green 1969, McCarthy and Hayes 1969, Pednault 1988, Allen 1991), Kautz Search 1195 61 pro em time rap pan actions T encoding systematic stochastic stochastic rocket-ext.a 7134 520 4.4 4.7 0.1 1ogistics.c 13165 - 23,040 240 1.9 bwlarge. b 9118 27,115 - - 22 bw1arge.d 18/36 - - - 937 Table 1: Preview of experimental results. Times in seconds. A long dash (-) indicates that the experiment was terminated after 10 hrs with no solution found. and Selman (1992 propositional satis B formalized planning in terms of ability. In this framework, a plan corresponds to any model (i.e., truth-assignment) that satisfies a set of logical constraints that represent the initial state, the goal state, and domain axioms. Time consists of a fixed, discrete number of instances. A proposition corresponds either to a time-varying con- dition (a fluent) holding at a particular instant (e.g., on(A ,B, 3)), or to an action that begins to occur at the specified instance and ends at the following instance (e.g., pickup(A,3)). G eneral constraints over facts and actions are written as axiom schemas, which are then instantiated for the objects and number of time instances used by a particular problem. The maximal length of a plan is thus fixed at instantiation time; if this quantity is not known in advance, it is straightfor- ward to perform a binary search on instantiations of various sizes, to find the smallest for which a solution is found. (For example, if the optimal plan length is 7, the search would proceed through plans of length 2, 4, 8 (plan found) 9 6 (no plan found) 9 and finally 7.) The satisfiability approach can be directly imple- mentcd using SAT algorithms, that in general have much better scaling properties than deductive FOL theorem provers. Another advantage is its expressive power. It is easy to represent arbitrary constraints over intermediate states (not just the initial and goal states), and over the structure of the plan itself. For example, to assert that every pickup is immediately followed by a stack, one could write a schema like pickup(x,i) > !ly.stack(x,y,i+l). It is quite hard to represent these kinds of constraints in STRIPS. Finally, because it a “real logic” as op- posed to STRIPS the relationships between predicates can be stated explicitly, and it is unnecessary to distin- guish “primitive” from “derived” predicates. For ex- ample, most STRIPS-style operators for planning han- dle the predicates clear and on separately, whereas in our framework one can simply assert clear(x,i) E lEly.on(y,x,i) The domain axioms for the satisfiability approach are in general stronger than those used by the deduc- tive framework, because it is necessary to rule out all “unintended9’ models. We will describe several ways this can be done: (i) encodings derived from the plan- ning graphs of Graphplan (Blum and Furst 1995); (ii) the linear encodings of Kautz and Selman (1992); and (iii) general state-based encodings, which incorporate the best features of the previous two. We refer to the both the linear and state-based encodings as “direct” encodings. 1196 Planning Graphplan-based Encodings As mentioned above, the Graphplan system (Blum and F’urst 1995) works by converting a STRIPS-style spec- ification into a planning graph. This is an ordered graph, where alternating layers of nodes correspond to grounds facts (indexed by the time step for that layer) and fully-instantiated operators (again indexed by the time step). Arcs lead from each fact to the operators that contain it as a precondition in the next layer, and similarly from each operator to its effects in the next layer. For every operator layer and every fact there is also a no-op “maintain” operator that simply has that fact as both a precondition and “add” effect. A solution is a subgraph of the planning graph that contains all the facts in the initial and goal layers, and contains no two operators in the same layer that con- flict (i.e., one operator deletes a precondition or an effect of the other). Thus, a solution corresponds to a partially-ordered plan, which may contain several op- erators occuring at the same time step, with the se- mantics that those operators may occur in any order (or even in parallel). For planning problems that can take advantage of this kind of parallelism, the planning graph can have many fewer layers than the number of steps in a linear solution - and therefore be much smaller. A planning graph is quite similar to a propositional formula, and in fact, we were able to automatically con- vert planning graphs into CNF notation. The transla- tion begins at goal-layer of the graph, and works back- ward. Using the “rocket” problem in Blum and Furst (1995, Fig. 2) as an example (where “load(A,R,L,i)” means “load A into R at location L at time i”, and “move (R, L , P , i.)” means “move R from L to P at time ;“), the translation is: the initial state holds at layer 1, and the goals hold at the highest layer; each fact at level i implies the disjunction of all the operators at level i - 1 that have it as an add-effect; e-g., in(A,R,3) > (load(A,R,L,2) V load(A,R,P,2)V maintain(in(A,R) ,2)) operators imply their preconditions, e.g., load(A,R,L,2) > (at(A,L,l) A at(R,L,l)) conflicting actions are mutually exclusive; e.g., lload(A,R,L,2) V lmove(R,L,P,2) Graphplan uses a set of rules to propagate the effects of mutually exclusive actions, leading to additional exclu- siveness constraints. In our logical formulation these additional constraints are logically implied by the orig- inal formulation. Linear Encodings Kautz and Selman (1992) described a set of sufficient conditions for ensuring that all models of the domain axioms, initial, and goal states correspond to valid plans. These were: e an action implies both its preconditions and effects; o exactly one action occurs at each time instant; e the initial state is completely specified; o classical frame conditions for all actions (i.e., if an action does not change the truth condition of fact, then the fact remains true or remains false when the action occurs). Intuitively, the first condition makes sure that actions only occur when their preconditions hold, and the “sin- gle action” and frame axioms force any state that fol- lows a legal state to also be a legal state. Models un- der this encoding correspond to linear plans; as the number of operators in a plan increases, these encod- ings become very large. Kautz and Selman observed that the number of propositional variables can be sig- nificantly reduced by replacin t certain predicates that take two or more arguments plus a time-index argu- ment) with ones that take a single argument (plus a time-index). For example, instead of the predicate move (x , y , z , i) (meaning “move block x from y to z at time i”), and dest they used three predicates, object, ination, with the correspondence source, move(x,y,z,i) - (object(x,i> A source(y,i)A destination(z,i)) When instantiated, this yields 0(3n2) propositions rather than O(n*) propositions. This technique can also be viewed as a kind of “lifting.” The blocks world problems described below use these kind of linear en- codings State-Based Encodings The ability to express partially-ordered plans by a sin- gle model gives Graphplan a powerful performance ad- vantage. On the other hand, we have seen that the STRIPS-style input notation has many expressive lim- itations. We have developed a methodology that we call “general state-based encodings”, which enjoys the advantages of the two previous approaches, as well as incorporating further representational refinements. We use the term “state-based” because it emphasizes the use of axioms that assert what it means for each individual state to be valid, and gives a secondary role to the axioms describing operators. For example, in the blocks world, the state axioms assert that only one block can be on another, every block is on something, a block cannot both be clear and have something on it, etc. In the logistics domain, state axioms include assertions that each transportable object can only be in a single truck, and that a truck is only at a single location. Given that the state axioms force each state to be internally consistent, it turns out that only a rela- tively small number of axioms are needed to describe state transitions, where each transition can be the result of the application of any number of mutually non-conflicting actions. These axioms describe what it means for a fact to change its truth value between states. One way to do this is to write axioms about the possible actions that could account for each change. For example, in the logistics domain, if an instance of in goes from false to true, then the object must have been loaded: (+n(x,y,i) A in(x,y,i+l)) > 3z.load(x,y,z,i) This style of axiom can be seen as an instance of the “domain specific” frame axioms described by Haas (1987) and Schubert (1989). Note that classical frame axioms of the type used above for linear encodings are not included - in fact, they are inconsistent with par- allel actions. These axioms are also similar to the “backward-chaining” axioms used in the Graphplan encodings above. The Graphplan example axiom can be rewritten as (lmaintain(in(A,R) ,2) A in(A,R,3)) > (load(A,R,L,2)Vload(A,R,P,2)). This formula can be identified as an instance of the general schema, once the dummy maintain proposi- tion is replaced by its precondition, in(A,R,2). Fi- nally, axioms are added that assert that actions entail both their preconditions and effects, and that conflict- ing actions are mutually exclusive. As described thus far, this approach has greater ex- pressive power than the Graphplan encodings, but is no more compact. However, the number of proposi- tional variables in this form of encoding can be signifi- cantly decreased by using the trick of reducing the ar- ity of predicates, as described in the previous section. Furthermore, many of the axioms relating actions to their preconditions and effects can be safely eliminated, because the strong state consistency axioms propa- gate the consequences of the remaining assertions. This process of eliminating propositions and simpli- fying axioms can be carried to the extreme of com- pletely eliminating propositions that refer to actions! Only fluents are used, and the axioms directly relate fluents betweens adjacent state. We have done this for the logistics domain, a relatively complex domain that involves moving packages between various loca- tions using trucks and airplanes. The STRIPS-style formalization requires operators such as load-truck, unload-truck, drive-truck, load-airplane, etc. On the other hand, instead of using explicit load ax- ioms, we use a single schema that relates the predicates at and in: at(obj ,loc,i) > at(obj ,loc,i+l)V 3x E truck U airplane. in(obj ,x,i+l)A at(x,loc,i)A at(x,loc,i+l) In English, this simply asserts that if an object is at a location, it either remains at that location or goes into some truck or plane that is parked at that location. Another schema accounts for the state-transitions as- sociated with unloading, by asserting that an object in a vehicle either stays in the vehicle, or becomes Search 1197 at the location where the vehicle is parked. Interest- ingly, no additional transition axioms at all are needed for the vehicle movement operators, drive-truck and fly-airplane, in this domain. The state validity ax- ioms alone ensure that each vehicle is always at a single location. A solution to a state-based encoding of a planning problem yields a sequence of states. The “missing” actions are easily derived from this sequence, because each pair of adjacent states corresponds to the (easy) problem of finding a unordered plan of length l0 (In the most general case, even finding unordered plans of length 1 is NP-hard; however, in domains we have ex- amined so far, including the logistics and blocks world domains, there is a linear-time algorithm for finding such plans.) The initial motivation for developing this purely state-based representation was pragmatic: we wished to find very compact logical encodings, of a size that could be handled by our SAT algorithms. We achieved this goal: for example, we can use our stochas- tic algorithm to solve state-based encodings of logistic problems that cannot be solved by any other domain- independent planner of which we are aware. (For an example of high-performance planning using domain- dependent control heuristics for the blocks world, see Bacchus and Kabanza (1995).) Beyond these compu- tational concerns, the encodings are interesting from a purely representational standpoint. There are no ex- plicit frame axioms, or axioms about preconditions and effects, or axioms about conflicts between actions; ev- erything is subsumed by simple, uniform relationships between fluents. These axiomatizations appear at least as “natural” as situation-calculus or STRIPS formal- izations, and avoid many of the traditional problems those approaches encounter. The experiments reported in this paper do not in- volve an automatic way of deriving state-based encod- ings from a STRIPS-style problem specification. The encodings we used in our experiments were created by hand, based on our understandin of the semantics of the various benchmark domains 8 which were, indeed, described by STRIPS operators). A separate paper (Kautz et al. 1996) describes our initial results on au- tomating the process of compiling away the operators for a given domain. However, one could equally well take a state-based description of a domain as primary, and then add actions to the axioms through meaning postulates. Experiments: Systematic versus Stochastic Search In this section, we will discuss our experimental results. We first compare the various encoding schemes with respect to the cost of finding a plan. We then show that the solutions we obtained are optimal, by showing that no shorter plans exist. To solve our SAT encodings, we consider both a systematic and a stochastic method. Tableau, the systematic procedure, is based on the Davis-Putnam procedure, and was developed by Crawford and Au- ton (1993). It’s one of the fastest current complete SAT procedures (Trick and Johnson 1993; Dubois et al. 1996). Walksat, the stochastic procedure, is a de- scendant of GSAT, a randomized greedy local search method for satisfiability testing (Selman et al. 1992, Selman et al. 1994, 1996). Such stochastic local search methods have been shown to outperform the more tra- ditional systematic methods on various classes of hard Boolean satisfiability problems. Note, however, that these procedures are inherently incomplete: they can- not prove that a formula is unsatisfiable. Walksat operates as follows. It first picks a ran- dom truth assignment, and randomly selects one of the clauses in the SAT instance that is not satisfied by the assignment. It then flips the truth assignment of one of the variables in that clause, thereby satisfying the clause. However, in the process, one or more other clauses may become unsatisfied. Therefore, in deciding which variable to flip from the clause, Walksat uses a greedy bias that tends to increase the total number of satisfied clauses. Specifically, the bias picks the vari- able that minimizes the number of clauses that are sat- isfied by the current assignment, but which would be- come unsatisfied if the variable were flipped. Because the bias can lead the algorithm into local minima, per- formance is enhanced if the bias is not always applied. The best rule appears to be to always apply the bias if there is a choice that would make no other clauses become unsatisfied; otherwise, randomly apply it half the time. The procedure keeps flipping truth values until a satisfying assignment is found or until some predefined maximum number of flips is reached. In Sel- man et al. (1994, 1996), it was shown that this method significantly outperforms basic GSAT, and other local search methods such as such as simulated annealing (Kirkpatrick et al. 1983). Finding Plans Table 2 gives the computational cost of solving several hard planning problems. We consider two SAT encod- ings for each instance, one Graphplan-based and the other direct (linear or state-based). For our SAT en- codings, we give both the timings for the systematic tableau method and for the stochastic Walksat proce- dure. We compare our results to those of the Graph- plan system. As mentioned in the preview of results, we con- sidered hard instances from the rocket and the logis- tics domains (Blum and F’urst 1995, Veloso 1992), as well as the blocks world. We noted that Graphplan has been shown to outperform Prodigy and UCPOP on the rocket problems. The logistics domain is a strictly richer environment than the rocket domain.2 In the column marked with “time/actions”, we give the length of the plan found in terms of the number of time steps. Since we allow for parallel (indepen- dent) actions, we also give the total number of actions that will lead us from the initial state to the goal. We created a state-based encoding for rocket and logistics problems, and for the blocks world used the original 2Preliminary data indicate that Graphplan, and thus our algorithms, out P erform UCPOP on the logistics do- main, as expected Friedman 1996). However, it is im- portant to note that UCPOP is a regression planner, and certain state-based notions are inaccessible or obscure to it. UCPOP may well prove superior on other domains, in which reasoning is more causal, and less related to topo- logical notions. 1198 Planning problem rocket-ext.a rocket-ext.b 1ogistics.a logistics. b 1ogistics.c bw1arge.a bwlarge. b bw1arge.c bw1arne.d time I actions ---Tpi- 7/30 11154 13147 13165 6112 9118 14128 18/36 Graphplan +yiz-#% 1,701 2,337 2,891 6,743 3,382 2,893 4,326 - 5,779 11.5 18,069 27,115 - - Gr vars 1,103 1,179 1,782 2,069 2,809 5,772 phplan-Based syst. stoch. 4.4 4.7 2.8 21 6.9 6.4 Yi3 23,061 262 - - - - - - - - vars 331 351 828 843 1,141 459 1,087 3,016 6.764 Direct Table 2: The computational cost of finding plans for several hard planning problems. For each instance, the optimal (minimal length) plan was found. Times in seconds. A long dash (-) indicates that the experiment was terminated after 10 hrs with no solution found, or, when we do not give the number of variables or the number of nodes, it means that the problem instance was too large to fit into main memory. The rocket and logistic direct encodings are state-based, and the bw (blocks world) direct encodings are linear. linear encodings from Kautz and Selman (1992). Be- fore applying the solvers, all of the SAT instances were first simplified by a linear-time algorithm for unit prop- agation, subsumption, and deletion of unit clauses. Ta- ble 2 gives the number of variables in each instance afrer simplification. The results for “rocket-ext.a” show the general trend. The direct encodings are the most compact, and can be solved many times faster than the Graphplan- based SAT encodings, which is in turn are more effi- cient than extracting the plans directly from the plan- ning graphs, using the Graphplan system.3 We also see that stochastic search (Walksat; see col- umn marked ‘%toch.“) often outperforms systematic search order o I tableau; see column marked “syst.“) by an magnitude. Especially striking is the perfor- mance of Walksat on the state-based encodings (last column). These results strongly suggest that stochastic methods combined with efficient encoding techniques are a promising method for solving challenging classi- cal planning problems. As the instances become harder, the difference in performance between Walksat on the direct encodings and the other approaches becomes more dramatic. For example, see “1ogistic.c” and “bw-1arge.d.” As we will discuss in the next section, all problems were solved to optimality. Thus, for the blocks world instance, bw-large.d, we found the minimal length plan of 36 operations (pickup/putdown/stack/unstack) from the initial state to the goal state. Only Walksat on the linear encoding could synthesize this plan. The prob- lem involves 19 blocks, with 4 stacks in the initial and 3 stacks in the goal state. Note that we did not en- code any special search control knowledge (such as, “move a block directly to a goal position, if possible”). To get a better feel for the computational difficulty of this problem, let us briefly consider some of the formal computational properties of the blocks world domain. Optimal blocks world planning was shown to be NP- complete in 1991, but a plan within a factor two of optimal can be obtained in polynomial time (Gupta 30ur timings do not include the time needed for gen- erating the planning graph or for constructing the SAT encodings. On the harder instances, those times are just a fraction of what it takes to solve the planning graph or the SAT problems. and Nau 1991, 1992, Chenoweth 1991). Gupta and Nau (1992) give an algorithm for finding such approx- imate solutions. The basic idea is to first move blocks to the table and then build up the goal stacks. Gupta and Nau’s approximation algorithm would generate a plan with 58 operations for our instance, requiring 12 via-the-table moves. (Some blocks don’t have to be moved to the table. Note that a via-the-table move generally involves an unstack, a putdown a pickup, and a stack operation.) Selman (1994) shows that it’s unlikely that we can find a better polytime approx- imation algorithm: All the difficulty lies in deciding how one can avoid via-the-table moves by making di- rect stack-to-stack moves. To do so, one has to deter- mine which stack-to-stack moves to make and in what order. Walksat manages to eliminate 11 of the 12 via- the-table moves - leaving an optimal plan of 36 steps with only a single, unavoidable via-the-table move! We do not know of any other planning system that can op- timally solve unrestricted blocks world problems of this size without using any kind of domain-specific control knowledge. Despite the fact that the blocks world do- main is somewhat artificial, we are encouraged by our results because we believe that the rich interactions be- tween operator and sub-goal sequencing, which makes the domain relatively hard, is also quite likely to be found in more practical domains, such as, for example, the softbot planning domain (Etzioni and Weld 1994). (Indeed, in the next phase of this project, we hope to apply our methods to the softbot domain.) Finally, from the columns that give the number of nodes and number of variables, we see that direct en- codings (and in particular, the state-based encodings) result in a significant reduction of the number of vari- ables in the problem instances. Our Graphplan-based SAT encodings also have fewer variables than the num- ber of nodes in the corresponding planning graph, be- cause of the unit-propagation simplification described above. Although our results are quite promising for stochas- tic methods, we do not mean to suggest that these methods will always outperform systematic ones. In fact, we have done some preliminary experiments on one of the artificial domains (D’S’) studied in Barrett and Weld (1994 and in Blum and Furst (1994). We considered the b raphplan-based encoding, and found Search 1199 that the Graphplan system itself scales better than our SAT approach using either Walksat or tableau. The special structure of the domain, which is specif- ically designed to check the sequencing of operators, appears to steer Walksat repeatedly in the wrong di- rection. Tableau performs poorly because it performs a depth-first search, and the domain appears to require a breadth-first approach. * State-based encodings may again give better results on this domain. We also ob- tained some promising results on these instances using SAT encodings based on McAllester and Rosenblitt’s (1991) “causal” planning formulation. In general, we expect that systematic and stochastic methods will complement each other - each having different rel- ative strengths depending on the domain. In the next section we’ll discuss another way in which these meth- ods complement each other. Proving Optimality To show that the plans we found in our previous exper- iments are optimal, we now show that no shorter plan exists. Table 3 gives our results. This time we can only use methods that systematically explore the space of all possible plans up to a certain size, because we have to demonstrate that shorter plans do not exist. From the table, we see that in this case using tableau on the Graphplan-based SAT encoding is not very ef- fective (except for “1ogistics.a”). Neither the Graph- plan system nor tableau with direct SAT encodings strictly dominate one another; the former is superior on the logistics problems, and the latter on the blocks world problems. None of the methods could show the inconsistency of “1ogistics.c” when using at most 12 time steps. There- fore, to show the optimality of a 13 step solution to “1ogistics.c” 9 we constructed “1ogistics.b” as a strictly smaller subproblem. Graphplan was able to show that this problem does not have 12 step solution. It follows that “1ogistics.c” does have a 12 step solution either. In general, our results suggests that it’s harder to show the non-existence of a plan up to a certain length than it is to find such a plan if it exists. This kind of asymmetry has also been observed in several other problem domains (Selman 1995). The issue is closely related to the practical difference between solving NP and co-NP complete problems. Tableau can show the infeasibility of a 17 time slot (34 stack/unstack) solution for “bw1arge.d” 9 while Walksat can find a 18 time slot (36 stack/unstack) plan (Table 2). No systematic approach could find the feasible solution. This demonstrates how stochastic and systematic methods are complementary: one can be used for plan synthesis and the other to determine lower-bounds on the plan length. Conclusions We have shown that for solving hard lems from several challenging domains, planning prob- our approach of 4For a discussion of issue first search in variations of see Dechter and Rish (1994) of depth-first version breadth- the Davis-Putnam procedure, using linear or state-based axiomatizations and a gen- eral, stochastic satisfiability algorithm (Walksat) out- performs some of the best specialized planning algo- rithms by orders of magnitude. Furthermore, Walk- sat is often superior to good general (tableau) and specialized (Graphplan) systematic search engines on SAT encodings derived from STRIPS-style operators. These results challenge the common assumptions in AI that planning requires specialized search techniques and that planning is an inherently systematic process. Of course, we are not ruling out the possibility that in other domains some of the specialized planning sys- tems could prove superior. This is an important issue for further research. We have also shown that systematic and local search algorithms complement each other well in the planning as satisfiability framework. Systematic algorithms can be used to provide a lower-bound on the length of solu- tion plans, and then stochastic algorithms can be used to find the actual solutions. It is interesting to ob- serve that in certain cases systematic algorithms are better at proving infeasibility than at finding solutions to problems instances of comparable size. Finally, our experiments with different SAT en- codings of planning problems indicates that much progress can be made by considering novel kinds of axiomatizations. In particular, our experience sug- gests that axiomatizations that concentrate on states and fluents can be more compact and easier to solve than approaches that directly encode STRIPS-style state-changing operators. Furthermore, these state- based encodings are interesting from a representational standpoint, and appear to provide clean and elegant ways to handle parallel actions and frame conditions. References Adorf, H.M., Johnston M.D. (1990). A discrete stochas- tic neural network algorithm for constraint satisfaction problems. Proc. of the Int. Joint Conf. on Neural Net- works, San Diego, CA, 1990. Allen, J. (1991). Planning as temporal reasoning. Proc. K&89, Cambridge, MA, 1991. Bacchus, F. and Kabanza, F. (1995). Using temporal logic to control search in a forward chaining planner. Proc. E WSP-95, 157-169. Backstrom, C. (1992). Computational complexity of rea- soning about plans, Ph.D. thesis, Linkoping University, Linkoping, Sweden. Barrett, A. and Weld, D. (1994). Partial-order planning: evaluating possible efficiency gains. Artificial Intelli- gence, 67:71-112, 1994. Blum, A. and Furst, M.L. (1995). Fast planning through planning graph analysis. Proc. IJCAI-95, Montreal, - - Canada. Bylander, T. (1991). Complexity results for planning. - Proc. IJCAj-91, Sidney, Australia, 274-279. - - Carbonell, J. 9 Blythe J., Etzioni, O., Gil, Y., Joseph, R., Kahn, D., Knoblock, C., Minton, S., Perez, A., Reilly, S., Veloso, M., Wang, X (1992). Prodigy 4.0: the manual and tutorial. CMU, CS Tech. Report CMU-CS-92-150. Chenoweth, S.V. (1991). On the NP-hardness of the blocks world. Proc. AAAI-91, Anaheim, CA, 623-628. Crawford, J.M. and Auton, L.D. (1993) Experimental Re- sults on the Cross-Over Point in Satisfiability Problems. Proc. AAAI-93, Washington, DC, 21-27. Crawford, J. and Baker, A.B. (1994). Experimental re- sults on the application of satisfiability algorithms to 1200 Planning scheduling problen 1s. Proc. AAAI-94, Seattle, WA. problem rocket ,ext .a rocket -ext. b 1ogistics.a logistics. b 1ogistics.c bw1arge.a bwlarge. b bw1arge.c bw1arge.d time 6 6 :; 5,:: 8116 13126 17134 G raphplan Graphplan-Based St&e-Based I SAT Encoding I Table 3: Showing the infeasibility of shorter plans. Times in seconds. Dechter, R. and Rish, I. (1994). Directional resolution: the Davis-Putnam procedure, revisited. Proc. KR-94, Bonn, Germany. Dubois, 0. , Andre, P., Boufkhad, Y., and Carlier, J. (1996). A-SAT and C-SAT. Dimacs Series in Discrete Mathematics and Theoretical Computer Science. (to ap- pear) Erol, K., Nau, D.S., and Subrahmanian, V.S. (1992). On the complexity of domain-independent planning. Proc. AAAI-92, 381-386. Etzioni, 0. and Weld, D. S. (1994). A softbot-based inter- face to the internet. Comm. ACM, July 1994. Fikes, R.E. and Nilsson, N.J. (1971). STRIPS: A new ap- proach to the application of theorem proving to problem solving. Artificial Intelligence, 2(3/4), 189-208. Friedman, M. (1996). Personal communication. Green, C. (1969). Application of Theorem Proving to Problem’Solving. In*&oc. IJCAI-69, Washington, D.C., 1969. 219-239. Gupta and Nau (1991). Complexity results for blocks- world planning. Proc. AAAI-91, Anaheim, CA, 629- 633. Gupta and Nau (1992). On the complexity of blocks-world planning. Artificial Intelligence, 56, 139-403. Haas, A. (1987). The case for domain-specific frame ax- ioms. In The Frame Problem in Artificial Intelligence, Proceedings of the 1987 Workshop, F.M. Brown, ed., Lawrence, KS, 1987. Morgan Kaufmann Publishers, Los Altos, CA. Joslin, D. and Pollack, M. (1995). Passive and Active De- cision Postponement in Plan Generation. In the Euro- pean Workshop on Planning (EWSP), Assisi, Italy, Sept. 1995. Kautz, H. and Selman, B. (1992) Planning as Satisfiability. Proc. ECAI-92, Vienna, Austria, 1992, 359-363. Kautz, H., McAllester, D., and Selman, B. (1996). Encod- ing Plans in Propositional Logic. In preparation. Kirkpatrick, S., Gelatt, C.D., and Vecchi, M.P. (1983). Optimization by simulated annealing. Science, 220 (1983) 671-680. McCarthy, J. and Hayes, P. (1969). Some philosophical problems from the standpoint of artificial intelligence. In Machine Intelligence 4, D. Michie, ed., Ellis Horwood, Chichester, England, 1969, page 463ff. McAllester, D. and Rosenblitt, D. (1991). Systematic non- linear planning. Proc. AAAI-91, Anaheim, CA. Minton, S., Johnston, M.D., Philips, A.B., and Laird, P. (1990) Solving large-scale constraint satisfaction an scheduling problems using a heuristic repair method. Proc. AAAI-90, 1990, 17-24. Minton, S., Johnston, M.D., Philips, A.B., and Laird, P. (1992) Minimizing conflicts: a heuristic repair method for constraint satisfaction and scheduling problems. Ar- vars syst. I 4.5 849 3.5 1,415 80.1 1,729 - 2,353 - 4,939 - - I - 51886 1 25,289 tijicial Intelligence, (58)1-3, 1992, 161-205. Pednault, E. (1988). Synthesizing plans that contain ac- tions with context-dependent effects. Computational In- telligence, 4(4):356-372, 1988. Penberthy, J. and Weld, D. (1992). UCPOP: A sound, complete, partial order planner for ADL. In the Proc. KR-92, Boston, MA, 103-114. Davis, M., Logemann, G., and Loveland, D. (1962). A machine program for theorem proving. Comm. ACM, 5, 1962, 394-397. Schubert, L. (1989). Monotonic Solution of the Frame Problem in .the Situation Calculus: an Efficient Method for Worlds with Fully Specified Actions. In Knowledge Representation and ljefiasible Reasoning, H. Kyburg, R. Loui, and G. Carlson, eds. Selman, B. (1994). Near-Optimal Plans, Tractability, and Reactivity. Proc. KR-94, Bonn, Germany, 1994, 521- 529. Selman, B. (1995). Stochastic Search and Phase Transi- tions: AI Meets Physics. Proc. IJCAI-95, Montreal, Canada, 1995. Selman, B. , Kautz, H., and Cohen, B. (1994). Noise Strategies for Local Search. Proc. AAAI-94, Seattle, WA, 1994, 337-343. Selman, B., Kautz, H., and Cohen, B. (1996) Local Search Strategies for Satisfiability Testing. Dimacs Series in Discrete Mathematics and Theoretical Computer Sci- ence. (to appear) Selman, B., Levesque, H., and Mitchell, D. (1992). A New Method For Solving Hard Satisfiability Problems. Proc. AAAI-92, San Jose, CA, 1992, 440-446. Stone, P., Veloso, V., and Blythe, J. (1994). The need for different domain-independent heuristics. In AIPS94, pages 164-169, Chicago, 1994. Trick, M. and Johnson, D. (Eds.) (1993) Proc. DIMACS Challenge on Satisfiability Testing. Piscataway, NJ, 1993. (DIMACS Series on Discr. Math.) Veloso, M. (1992). Learning by analogical reasoning in general problem solving. Ph.D. Thesis, CMU, CS Techn. Report CMU-CS-92-174. Search 1201 | 1996 | 177 |
1,818 | Finding 0 al Solutions to the Twenty-Four Puzzle Richard E. Korf and Larry A. Taylor Computer Science Department University of California, Los Angeles Los Angeles, Ca. 90024 korf@cs.ucla.edu, ltaylor@cs.ucla.edu Abstract We have found the first optimal solutions to random instances of the Twenty-Four Puzzle, the 5 x 5 ver- sion of the well-known sliding-tile puzzles. Our new contribution to this problem is a more powerful admis- sible heuristic function. We present a general theory for the automatic discovery of such heuristics, which is based on considering multiple subgoals simultane- ously. In addition, we apply a technique for pruning duplicate nodes in depth-first search using a finite- state machine. Finally, we observe that as heuristic search problems are scaled up, more powerful heuris- tic functions become both necessary and cost-effective. I I I I I IO 11 12 13 14 Introduction 20 21 22 23 24 The sliding-tile puzzles, such as the Eight and Fifteen Puzzle, have long served as testbeds for heuristic search in AI. A square frame is filled with numbered tiles, leaving one position empty, called the blank. Any tile that is horizontally or vertically adjacent to the blank can be slid into the blank position. The task is to rearrange the tiles from some random initial configu- ration into a particular goal configuration, ideally or optimally in a minimum number of moves. The state space for the Eight Puzzle contains over lo5 nodes, the Fifteen Puzzle space contains about 1013 nodes, and the Twenty-Four Puzzle contains almost 1O25 nodes. Figure 1: The Twenty-Four Puzzle in its goal state threshold for each succeeding iteration is the minimum total cost, f(n) = g(n)+h(n), of all nodes on the fron- tier of the previous iteration. The algorithm continues until a goal node is chosen for expansion. Due to its small search space, optimal solutions to the Eight Puzzle can be found with breadth-first search. We first found optimal solutions to the Fif- teen Puzzle using Iterative-Deepening-A* (IDA*) and the Manhattan distance heuristic function (Korf 1985). IDA* is a variant of the well-known A* algorithm (Hart, Nilsson, and Rafael 1968), which runs in space that is linear in the maximum search depth, rather than exponential. IDA* proceeds in a series of depth- first search iterations, starting from the initial state. Each path is explored until a node n is reached where the number of moves from the initial state, g(n), plus the heuristic estimate of the number of moves neces- sary to reach the goal state, h(n), exceeds a threshold for that iteration. The threshold for the first iteration is the heuristic estimate for the initial state, and the The Manhattan distance heuristic is computed by taking each tile, counting the number of grid units to its goal location, and then summing these values for all tiles. Since only one tile can move at a time, Manhat- tan distance never overestimates the number of moves needed to solve a given problem. Given such an admis- sible heuristic function, IDA* is guaranteed to return an optimal solution, if one exists. IDA* with the Manhattan distance heuristic can solve random instances of the Fifteen Puzzle (Korf 1985). In spite of considerable work on this problem in the last decade, however, nobody has solved a signifi- cantly larger version of the puzzle. Note that the state space of the Twenty-Four Puzzle is almost a trillion times larger than that of the Fifteen Puzzle. We present the first random Twenty-Four Puzzle in- stances for which optimal solutions have been found. Ten random solvable instances were generated, and so far we have found optimal solutions to all but one. 1202 Planning From: AAAI-96 Proceedings. Copyright © 1996, AAAI (www.aaai.org). All rights reserved. Three factors have contributed to this limited success. The first is simply faster computers. The Sun Ultra Spare workstation that these experiments were run on is about 70 times faster than the DEC 2060 that the Fifteen Puzzle was originally solved on. The second is a technique we developed for pruning duplicate nodes in depth-first search (Taylor and Korf 1993). Finally, we have discovered more powerful heuristic functions for this problem. The most important contribution of this paper, however, is a new theory that allows these heuristics to be automatically learned and applied. All examples in this paper refer to the Twenty-Four Puz- zle, where positions are labelled by the tiles that oc- cupy them in the goal state shown in Figure 1. Improved Admissible Heuristics Linear Conflict Heuristic The first significant improvement to Manhattan dis- tance was the linear-conflict heuristic (Hansson, Mayer, and Yung 1992). It applies when two tiles are in their goal row or column, but are reversed relative to their goal positions. For example, if the top row of the puzzle contains the tiles (2 1) in that order, to reverse them, one of the tiles must move down out of the top row, to allow the other to pass by, and then back up. Since these two moves are not counted in the Manhat- tan distance of either tile, two moves can be added to Manhattan distance without violating admissibility. As another example, if the top row contains the tiles (3 2 1) in that order, four more moves can be added to the Manhattan distance, since every pair of tiles is reversed, and two tiles must move out of the row temporarily. Furthermore, a tile in its goal position may be in both a row and a column conflict. Since the extra moves required to resolve a row conflict are vertical moves, and those required by a column conflict are horizontal, both sets of moves can be added to the Manhattan distance, and still preserve admissibility. This addition to the Manhattan distance heuristic reduces the number of nodes generated by IDA* on the Fifteen Puzzle by roughly an order of magnitude. The additional complexity of computing the linear conflicts slows down node generation by about a factor of two, however, for a net improvement of a factor of five. Effi- ciently computing this heuristic involves precomputing and storing all possible permutations of tiles in a row or column, and incrementally computing the heuristic value of a child from that of its parent. Last Moves Heuristic The next enhancement to the heuristic is based on the last moves of a solution, which must return the blank to its goal position, the upper-left corner in this case. Thus, the last move must either move the 1 tile right, or the 5 tile down. Therefore, immediately before the last move, either the 1 or 5 tile must be in the upper- left corner. Since the Manhattan distance of these tiles is computed to their goal positions, unless the 1 tile is in the left-most column, its Manhattan distance will not accommodate a path through the upper-left corner. Similarly, unless the 5 tile is in the top row, its Man- hattan distance will not accommodate a path through the upper-left corner. Thus, if the 1 tile is not in the left-most column, and the 5 tile is not in the top row, we can add two moves to the Manhattan distance, and still preserve admissibility. While two moves may seem like a small improve- ment, it can be added to about 64% of random Twenty- Four Puzzle states. The effect of two additional moves is to save an entire iteration of IDA*. Since each it- eration of IDA* on the Twenty-Four Puzzle can gen- erate up to ten times as many nodes as the previous iteration, saving an iteration can result in an order of magnitude savings in nodes generated. We can extend the same idea to the last two moves. If the last move is made by the 1 tile, the next-to-last move must either move the 2 tile right, or the 6 tile down. Similarly, if the last move is made by the 5 tile, the next-to-last move must either move the 6 tile right, or the 10 tile down. Considering the last two moves can add up to four moves to the Manhattan distance. Extending this idea to the last three moves was not cost effective on the Twenty-Four Puzzle. To benefit from both the linear conflict and last moves enhancements, and maintain admissibility, we must consider their interactions. For example, assume that the 1 tile is not in the left-most column, and the 5 tile is not in the top row. If the 1 tile is in its goal col- umn, and in a column conflict with another tile, then the two additional moves added by the linear conflict could be used to move the 1 tile left, allowing it to pass through the upper-left corner. Similarly, if the 5 tile is in its goal row, and in a row conflict, the two addi- tional linear conflict moves could be used to move it up and hence through the upper-left corner. Thus, if ei- ther of these conditions occur, we can’t add two more moves for the last move, since that may count twice moves already added by the linear conflict heuristic. Similarly, any additional moves added for the last two moves must also be checked against linear conflicts in- volving the 2, 6, and 10 tiles. In general, whenever more than one heuristic is being used, we must com- pute their interactions to maintain admissibility. Relation to Bidirectional Search The reader may notice that a heuristic based on the last moves in the solution is related to bi-directional search. The most effective form of bidirectional heuristic search is called perimeter search (Dillenburg and Nelson 1994) (Manzini 1995). A limited breadth-first search back- ward from the goal state is performed, and the nodes on the perimeter of this search are stored. IDA* is then run from the initial state, with heuristic calculations made to determine the minimum distance to any state on the perimeter. This heuristic value is then added to the distance from the initial state to the given node, Search 1203 plus the distance from the perimeter to the goal state, for a more accurate admissible heuristic. In a unidirectional search, the heuristic function is always computed to a single goal state. As a result, the heuristic calculation can be optimized to take ad- vantage of this. With any form of bidirectional search, however, the heuristic must be calculated between ar- bitrary pairs of states, reducing the opportunities for optimization. While (Manzini 1995) reports speedups of up to a factor of eight on the Fifteen Puzzle us- ing his improved perimeter search, he uses only the Manhattan distance heuristic function. It’s not clear if similar results could be achieved with a more complex heuristic such as linear conflict. Corner-Tiles Heuristic The next enhancement to our heuristic focuses on the corners of the puzzle. For example, if the 3 tile is in its goal position, but some tile other than the 4 is in the 4 position, the 3 tile will have to move temporarily to correctly position the 4 tile. This requires two moves of the 3 tile, one to move it out of position, and another to move it back. If the 3 tile is involved in a row conflict, then two moves will already be counted for it, and no more can be added. It can’t be involved in a column conflict if it’s in its goal position. The same rule applies to the 9 tile, unless the 9 is involved in a column conflict. In fact, if both the 3 and 9 tiles are correctly positioned, and the 4 tile is not, then four moves can be added, since both the 3 and 9 tiles will have to move to correctly position the 4. This rule also applies to the 15, 19, 21, and 23 tiles. It applies to the 1 and 5 tiles as well, but the interac- tion of this heuristic with the last moves heuristic is so complex that to avoid the overhead of this calculation, we exclude the 1 and 5 tiles from the corner heuristic. The corner-tile heuristic can potentially add up to twelve additional moves to the Manhattan distance, two for each of the six tiles adjacent to three of the corners. These extra moves require that at least one of these six tiles be in its goal position, a situation that only occurs in about 22% of random states. A search for the goal, however, does not generate a random sam- ple of states, but is biased toward states that are close to the goal, or at least appear to the heuristic to be close. In other words, the search is trying to correctly position the tiles, and hence this heuristic adds extra moves much more often than would be expected from a random sample of states. In summary, we have considered three enhancements to the Manhattan distance heuristic, based on linear conflicts, the last moves, and the corner tiles. The last two are introduced here for the first time. A New Theory of Admissible While these enhancements result in a much more powerful heuristic, they appear to be a collection of domain-specific hacks. Furthermore, integrating the enhancements together into an admissible heuristic seems to require even more domain-specific reasoning. However, all these heuristics can be derived from a gen- eral theory that is largely domain-independent, and the heuristics can be automatically learned and applied. While we would like to be able to claim that these heuristics were discovered from the general theory, in reality the theory was discovered after the fact. The classic theory of admissible heuristic functions is that they are the costs of optimal solutions to sim- plified problems, derived by removing constraints from the original problem (Pearl 1984). For example, if we remove the condition that a tile can only be moved into the blank position, the resulting problem allows any tile to move to any adjacent position at any time, and allows multiple tiles to occupy the same position. The number of moves to optimally solve this simpli- fied problem is the Manhattan distance from the initial state to the goal state. While this theory accounts for many heuristics for many problems, it doesn’t explain any of the above enhancements to Manhattan distance. Automatically Learning the An alternative derivation of Manhattan distance is based on the original problem, but focuses on only one tile at a time. For each possible location of each indi- vidual tile, we perform a search to correctly position that tile, ignoring all other tiles, and only counting moves of the tile in question. In this search, a state is uniquely determined by the position of the tile of interest and the position of the blank, since all other tiles are equivalent. Since the operators of the sliding- tile puzzle are invertible, we can perform a single search for each tile, starting from its goal position, and record how many moves of the tile are required to move it to every other position. Doing this for all tiles results in a table which gives, for each possible position of each tile, its Manhattan distance from its goal position. Then, noticing that each move only moves one tile, for a given state we add up the Manhattan distances of each tile to get an admissible heuristic for the state. Of course, we don’t really need to do the search in this case, since we can easily determine the values from the problem, but we presented it in this way to eliminate as much domain-specific reasoning as possible, and replace it with domain-independent search. The value of this reconstruction of Manhattan dis- tance is that it suggests a further generalization. The above formulation considers each tile in isolation, and the inaccuracy of the resulting heuristic stems from ig- noring the interactions between the tiles. The obvious next step is to repeat the above process on all possible pairs of tiles. In other words, for each pair of tiles, and each combination of positions they could occupy, per- form a search to their goal positions, and count only moves of the two tiles of interest. We call this value the pairwise distance of the two tiles from their goal locations. A state of this search consists of the posi- 1204 Planning tions of the two tiles and the position of the blank, since all other tiles are equivalent. Again for efficiency, for each pair of tiles we can perform a single search starting from their goal positions, with the blank also in its goal position, and store the pairwise distances to all other positions. The goal of this search is to find the shortest path from the goal state to all possible positions of the two tiles, where only moves of the two tiles of interest are counted. We can do this with a best-first search, counting only these moves. Since states of these searches are only distinguish- able by the positions of the two tiles and the blank, the size of these search spaces is O(n3), where n is the number of tiles. There are O(n2) such searches to per- form, one for each pair of tiles, for a time complexity of O(n5). The size of the resulting table is 0(n4), for each pair of tiles in each combination of positions. For almost all pairs of tiles and positions, their pair- wise distances equal the sum of their Manhattan dis- tances from their goal positions. However, there are three types of cases where the pairwise distance ex- ceeds the combined Manhattan distance. The first is when the two tiles are in a linear conflict. The second is when the two tiles are 1) a tile in its goal position adjacent to a corner, and 2) the tile that either belongs in, or that happens to be in, the corresponding corner. The third case is tiles 1 and 5, which are adjacent to the blank position in the goal state. The reason their pair- wise distance may exceed their combined Manhattan distances is that the backwards pairwise search starts from the goal state, and hence the first move is to move the 1 or the 5 tile into the corner. Thus, computing all the pairwise distances by a simple search “discov- ers” Manhattan distance along with all three of the heuristic enhancements described above, with very lit- tle domain-specific reasoning. No other enhancements are discovered by the pairwise searches. Applying the Heuristics The next question is how to automatically handle the interactions between these heuristics to compute an ad- missible heuristic estimate for a particular state. As- sume that we have precomputed all the pairwise tile distances and stored them in a table. Given a particu- lar state, we look up all the pairwise distances for the current positions of the tiles. To compute the over- all heuristic, we then partition the tiles into groups of two, and sum the corresponding pairwise distances, in a way that maximizes the resulting heuristic value. To see this problem more clearly, represent a state as a graph with a node for each tile, and an edge between each pair of tiles, labelled with their pairwise distance. We need to select a set of edges from this graph, so that no two edges are connected to a common node, and the sum of the labels of the selected edges is maximized. This problem is called the maximum weighted match- ing problem, and can be solved in O(n3) time, where n is the number of nodes (Papadimitriou and Steiglitz 1982). Thus, this approach to heuristic generation can be automated, and runs in polynomial time. Higher-Order Heuristics Unfortunately, the pairwise distances do not account for the full power of the heuristic enhancements de- scribed above. For example, consider the linear con- flicts represented by the tiles (3 2 l), in that order in the top row. The linear conflict heuristic would add four moves to the Manhattan distance of these tiles, since all pairs are reversed, and two of the tiles must move out of the row. The pairwise distance of each pair of these tiles is two moves plus their Manhattan dis- tances. The graph representation of this situation is a triangle of tiles, with each edge of the triangle having weight two, ignoring the Manhattan distances. The maximum matching on this graph only contains one edge, with a total weight of two, since any two edges have a node in common. Thus, the pairwise distances capture only part of the linear conflict heuristic. As another example, consider the corner-tile heuris- tic, and a state in which the 3 and 9 tiles are correctly positioned, but the 4 tile is not. The corner heuristic would add four moves to the Manhattan distance of the 4 tile, since both the 3 and 9 tiles must move to correctly place the 4 tile. The graphical representation of this situation consists of an edge between the 3 and 4 tiles, and an edge between the 9 and 4 tiles, each with a label of two, if we ignore the Manhattan distance. Since both these edges include the 4 tile, we can only select one of them, for an addition of only two moves. Finally, while the pairwise distances capture the en- hancement due to the last move of the solution, it doesn’t capture the last two moves, since these involve the 2, 6, and 10 tiles, in addition to the 1 and 5 tiles. In order to capture the full power of these heuris- tics, we extend the idea of pairwise distances to include triples of tiles, quadruples, etc. The linear conflict ex- ample of (3 2 1) requires us to consider all three tiles together to get four additional moves. If we consider each corner tile together with both adjacent tiles, we get the full power of the corner-tile heuristic. Finally, the last-two-moves enhancement requires considering all five tiles that may be involved. The correspond- ing matching problem is hypergraph matching, where a single edge “connects” three or more nodes, and un- fortunately is NP-Complete. Thus, we may have to rely on a greedy approach to the higher-dimensional matching problem, and a lower heuristic value. As we consider higher-order heuristics, the complexity of the learning search, the size of the lookup table, and the complexity of the matching all increase, in return for more accurate heuristic values. We believe this is a general theory for the discov- ery and implementation of admissible heuristic func- tions. All combinatorial problems involve solving mul- tiple subgoals. Many admissible heuristics are con- structed by considering the solution to each individual Search 1205 subproblem in isolation, and ignoring the interactions with other subproblems. We are proposing heuristics based on the simultaneous consideration of-two, three, or more subgoals. As another example, consider a job- shop scheduling problem. There are a set of jobs to be performed, and a collection of machines with which to accomplish them. Each machine can only process a single job at a time. One way to derive a lower bound on the optimal solution is to consider the resources required by each job individually, and sum this over all jobs, ignoring resource conflicts between the jobs. Following our approach, one would consider all pairs of jobs, compute the resources required for each pair, and then compute the total resources by summing these values for a pairwise partition of the jobs. last move was Up, however, the only allowable move is another Up move. Similarly, if the last move was Down, the only allowable move is another Down move. This finite-state machine can only generate a single path to each point of the grid, and hence a depth-first search controlled by this machine runs in time O(d2), which is the same as a breadth-first search. These finite-state machines can be automatically learned, from a small breadth-first search to discover duplicate operator strings. In this case, a breadth-first search to depth two is sufficient to learn all the dupli- cate strings to construct the above machine. Once the machine is constructed, there is almost no overhead to using it to control the depth-first search. This technique can be applied to other problems, such as the sliding tile-puzzles. After rejecting inverse operators, the next shortest cycle in the sliding-tile puzzles is twelve moves long, corresponding to rotating the tiles in a 2 x 2 square. Using a breadth-first search, a finite-state machine for the Twenty-Four Puzzle was constructed with over 619,000 states. This machine is then used to control a depth-first search, rejecting op- erators that lead to duplicate nodes. The effect of this duplicate pruning is to reduce the asymptotic complex- ity of a depth-first search from 0(2.368d) to 0(2.235d). While this may seem like a small improvement, in the two easiest problems reported below, duplicate prun- ing decreased the running time of IDA* by factors of 2.4 and 3.6, with the larger improvement coming on the harder problem. Pruning Duplicate Nodes While the main concern of this paper is the heuristic functions, we also used another orthogonal technique to significantly speed up the experiments. Any depth-first search, such as IDA*, will generate the same node multiple times on a graph with cycles. For example, consider a square grid problem space, with the moves Up, Down, Left, and Right, each mov- ing one unit in the indicated direction. Since there are four moves from every state, the asymptotic complex- ity of a depth-first search to depth d is O(4d). How- ever, there are only O(d2) distinct states at depth d in a grid, and a breadth-first search, which stores all nodes generated and checks for duplicates, will run in O(d2) time. The difference in complexity between the breadth-first and depth-first search in this example il- lustrates the magnitude of this problem. In the grid example, the operator pairs Left-Right and Up-Down are inverses of each other. Any good depth-first search implementation will remember the last operator applied, and never immediately apply its inverse. This can be done by encoding the last opera- tor applied as the state of a finite-state machine. The machine has five states, an initial transient state and four recurrent states, one for each last move. Each arc of the machine represents an operator, except that the inverse of the last move is excluded. This reduces the complexity of the depth-first search from O(4d) to O(3d), a significant reduction, but still far from the O(d2) complexity of the breadth-first search. This idea can be carried further, and is described in detail in (Taylor and Korf 1993). Ideally, we would like to allow only one path to each node in the grid. This can be done by first making all Left or Right moves, if any, followed by a single turn, and then all Up moves or all Down moves, if any. These rules can also be enforced by a five-state finite-state machine. The initial state allows all four operators, and each resulting state encodes the last move applied. If the last move was to the Right, all moves are allowed except a move to the Left. Similarly, if the last move was to the Left, all moves are allowed except a move to the Right. If the Experimental Results We implemented IDA *, taking full advantage of the Manhattan distance, linear conflict, last-two-moves, and corner-tile heuristics, as well as the finite-state machine pruning. Since we were concerned with ef- ficiency, our implementation was specialized to these heuristics and their interactions, rather than using a general table lookup and matching algorithm. As a first test of our program, we ran it on 100 randomly generated solvable instances of the Nineteen Puzzle. The Nineteen Puzzle is the 4 x 5 sliding-tile puzzle, and its state space contains about 101’ states. All the puzzle instances were solved optimally, and the average solution length was 71.5 moves, as compared to an average solution length of 52.6 moves for the Fifteen Puzzle. The average number of node generations per problem instance was almost a billion, which is compa- rable to those generated by IDA* on the Fifteen Puzzle using just the Manhattan distance heuristic. To our knowledge, these are the first random Nineteen Puzzle problem instances to be solved optimally. We then turned our attention to the Twenty-Four Puzzle. Ten random solvable instances were generated. Since there is enormous variation in the time to solve these problems, different iterations of IDA* were inter- leaved on different problem instances, in order find and solve the easier ones first. To date, nine of these prob- 1206 Planning No. Initial State Nodes Generated Optimal Sol. 1 17 1 20 9 16 2 22 19 14 5 15 21 0 3 24 23 18 13 12 7 10 8 6 4 11 8,110,532,608 100 2 14 5 9 2 18 8 23 19 12 17 15 0 10 20 4 6 11 21 1 7 24 3 16 22 13 18,771,430,922 95 3 7 13 11 22 12 20 1 18 21 5 0 8 14 24 19 9 4 17 16 10 23 15 3 2 6 82,203,971,683 108 4 18 14 0 9 8 3 7 19 2 15 5 12 1 13 24 23 4 21 10 20 16 22 11 6 17 83,573,198,724 98 5 2 0 10 19 1 4 16 3 15 20 22 9 6 18 5 13 12 21 8 17 23 11 24 7 14 221,769,436,018 101 6 16 5 1 12 6 24 17 9 2 22 4 10 13 18 19 20 0 23 7 21 15 11 8 3 14 523,772,060,498 96 7 21 22 15 9 24 12 16 23 2 8 5 18 17 7 10 14 13 4 0 6 20 11 3 1 19 792,795,062,385 104 8 6 0 24 14 8 5 21 19 9 17 16 20 10 13 2 15 11 22 1 3 7 23 4 18 12 1,415,436,865,760 97 9 3 2 17 0 14 18 22 19 15 20 9 7 10 21 16 6 24 23 8 5 14 11 12 13 3,033,449,077,924 113 10 2314024179202121810132213114166571281519 >3,000,000,000,000 2 112 Table 1: Twenty-four puzzle problem instances, nodes generated, and optimal solution lengths lems have been solved optimally, with a lower bound established for the remaining one. Table 1 shows all ten problem instances, sorted by difficulty. For the solved problems, we give the number of nodes generated and the optimal solution length, and for the unsolved one we give lower bounds on these values. The tiles are listed from left to right and top to bottom, with 0 rep- resenting the blank. In this notation, the tiles of the goal state in Figure 1 would be listed in numerical or- der. The average optimal solution length for these ten problems is at least 102.4 moves. The code was writ- ten in C, runs on a Sun Ultra Spare workstation, and generates about a million nodes per second. The eas- iest problem took about two hours and 15 minutes to solve, and the most difficult solved problem took over a month. To date, the remaining unsolved problem has run for over a month. These are the first random Twenty-Four Puzzle instances to be solved optimally. Conclusions We have found the first optimal solutions to random instances of the Twenty-Four Puzzle, a problem with almost 1O25 states. The branching factor is 2.368, and the optimal solutions average over 100 moves long. We implemented IDA* on a state-of-the-art workstation, with a more powerful admissible heuristic function, and a method for pruning duplicate nodes in depth- first search. The most important contribution of this paper is a new general theory for the automatic dis- covery and application of admissible heuristics. In- stead of considering individual subgoals in isolation, our approach considers two or more subgoals simulta- neously. This theory allows one to automatically dis- cover Manhattan distance, along with the linear con- flict, last moves, and corner-tile enhancements to it, with nothing more than small searches of the problem space. By considering three or more subgoals at a time, even more powerful heuristics can be derived. A more powerful heuristic function increases the time per node generation by a polynomial amount. On the other hand, it generally decreases the effec- tive branching factor by a small amount, yielding an asymptotic improvement. For small problems, more powerful heuristics may not be cost effective, since one doesn’t search deep enough to overcome the poly- nomial overhead. As machines get faster and larger problems are addressed, however, seeming small im- provements in a heuristic function eventually become cost effective. Thus, as problem size increases, it be- comes both necessary and cost-effective to encode more knowledge of the problem in the form of improved heuristics. We have developed an approach to doing this automatically. Acknowledgements This work was supported by NSF Grant IRI-9119825, and a grant from Rockwell International. References Dillenburg, J.F., and P.C. Nelson, Perimeter search, Artificial Intelligence, Vol. 65, No. 1, Jan. 1994, pp. 165-178. Hansson, O., A. Mayer, and M. Yung, Criticizing so- lutions to relaxed models yields powerful admissible heuristics, Information Sciences, Vol. 63, No. 3, 1992, pp. 207-227. Hart, P.E., N.J. Nilsson, and B. Raphael, A formal basis for the heuristic determination of minimum cost paths, IEEE Transactions on Systems Science and Cybernetics, Vol. 4, No. 2, 1968, pp. 100-107. Korf, R.E., Depth-first iterative-deepening: An opti- mal admissible tree search, ArtificiaZ Intelligence, Vol. 27, No. 1, 1985, pp. 97-109. Manzini, G., BIDA*: An improved perimeter search algorithm, Artificial Intelligence, Vol. 75, No. 2, June 1995, pp. 347-360. Papadimitriou, C.H., and K. Steiglitz, Combinatorial Optimization: Algorithms and Complexity, Prentice- Hall, Englewood Cliffs, N.J., 1982. Pearl, J. Heuristics, Addison-Wesley, Reading, MA, 1984. Taylor, L., and R.E. Korf, Pruning duplicate nodes in depth-first search, Proceedings of the National Con- ference on Artificial Intelligence (AAAI-93), Wash- ington D.C., July 1993, pp. 756-761. Search 1207 | 1996 | 178 |
1,819 | Linear Time Near-Optimal Planning in the Blocks World John Slaney Automated Reasoning Project Australian National University Canberra, ACT 0200, Australia John.Slaney@anu.edu.au Abstract This paper reports an analysis of near-optimal Blocks World planning. Various methods are clarified, and their time complexity is shown to be linear in the num- ber of blocks, which improves their known complexity bounds. The speed of the implemented programs (ten thousand blocks are handled in a second) enables us to make empirical observations on large problems. These suggest that the above methods have very close aver- age performance ratios, and yield a rough upper bound on those ratios well below the worst case of 2. F’ur- ther, they lead to the conjecture that in the limit the simplest linear time algorithm could be just as good on average as the optimal one. Motivation The Blocks World (BW) is an artificial planning do- main, of little practical interest. Nonetheless, we see at least two reasons for examining it in more detail. In the first place, for good or ill, BW is by far the most extensively used example in the planning liter- ature. It often serves for demonstrating the merit of domain-independent techniques, paradigms and plan- ners. See (Bacchus & Kabanza 1995; Kautz & Selman 1992; 1996; Schoppers 1994) for recent examples. In or- der to assess the benefits of these approaches and the significance of the claims formulated in the literature, it is therefore necessary to know certain basic facts about BW, such as what makes optimal’ BW planning hard (Gupta & Nau 1992)) how it may best be approx- imated and what BW-specific information our systems must be able to represent and use in order to cope with it. In the cited papers, for example, Bacchus and Ka- banza show how specific methods for near-optimal BW planning can be encoded in a general system, while Kautz and Selman exhibit domain-independent tech- niques that dramatically improve performance for BW. These facts should not be misinterpreted as showing that such systems are really effective for problems like BW unless they match the best domain-specific ones, ‘In the following, optimal planning denotes the prob- lem of finding a plan of minimal length, and near-optimal planning the problem of finding a plan of length at most k times the minimal, for some constant factor k. 1208 Planning Sylvie ThiGbaux IRISA Campus de Beaulieu 35042 Rennes Cedex, Prance Sylvie.Thiebaux@irisa.fr both in time complexity and in solution quality. We do not suppose that the cited authors are themselves confused on this point, but as long as little is known about the behavior of BW-specific methods, such mis- interpretation of their claims is dangerously easy. The second motivation for studying BW arises from research on identifying tractable classes of planning problems. We note that within some restricted classes of domain-independent formalisms, such as SAS+-US or the restriction of STRIPS to ground literals and op- erators with positive preconditions and one postcon- dition, planning is tractable while optimal planning and even near-optimal planning are not (Backstrom & Nebel 1995; Bylander 1994; Selman 1994).2 However, near-optimal planning is tractable for certain domains that are too sophisticated to be encoded within such classes. BW is one such domain. This suggests that there is more to learn by focusing first on tractable near-optimal planning in the domain-dependent set- ting, where the specific features responsible for in- tractability are more easily identified and coped with. Indeed, the identification of tractable subclasses of SAS+ originated from the careful examination of a sim- ple problem in sequential control. BW appears then as a good candidate for identifying in a similar way a class of planning problems for which near-optimal planning is tractable. Again, this requires that we first acquire detailed knowledge of near-optimal BW planning, keep- ing in mind that it has many properties that are not necessarily shared by other applications. Although we hope that our investigations will help towards this second goal, our direct concern in this paper is with the first one, i.e. improving the current knowledge of BW to be used for assessment purposes. We shall focus on the performance in time complexity and average solution quality of polynomial time near- optimal algorithms for BW planning. Various methods for near-optimal BW planning within a factor of 2 exist, for which we shall take (Gupta & Nau 1991; 1992) as sources. However, we find that these methods are 2The intractability of near-optimal planning for SAS+-US follows directly from the corresponding intractability result for the mentioned subclass of STRIPS (Selman 1994) and from the inclusion of this latter subclass in SAS+-US. From: AAAI-96 Proceedings. Copyright © 1996, AAAI (www.aaai.org). All rights reserved. nowhere clearly formulated and that little is known about their performance. The paper makes the following contributions. The first part formulates those methods and shows that they can all be implemented to run in time linear in the number of blocks. This improves the cubic up- per bound given in (Gupta & Nau 1992). The speed of the implemented programs (10000 blocks in under a second) also makes it possible to look at the plans produced by the algorithms on large problems. The second part then, is devoted to experimental results. We first introduce a technique for producing truly random BW problems, which is a nontrivial task. Experiments on these random problems give us a rough upper bound of around 1.2 on the average performance ratios of the near-optimal algorithms, and suggest that when the number of blocks is large, it makes little dif- ference which of these algorithms (more sophisticated or trivial) is used, because all produce plans of length close to twice the number of blocks on average. Fur- ther, though optimal BW planning is NP-equivalent (Gupta & Nau 1992) and though there is a hard lower bound on absolute performance ratios tractably achiev- able (Selman 1994), the experiments lead to the con- jecture that on average and in the limit, linear time algorithms could be just as good as the optimal one. Definitions Before presenting the algorithms, we shall enter some definitions. We assume a finite set x3 of blocks, with TABLE as a special member of D which is not on any- thing. Noting that the relation ON is really a function, we write it as a unary S (for ‘support’), where S(X) picks out, for block 2, the block which z is on. Thus S is a partial function from ~\{TABLE} to B, injective except possibly at TABLE and such that its transitive closure is irreflexive. We refer to the pair (a, S) as a part-state, and identify a state of BW with such a part-state in which S is a total function. For a part-state 0 = (a, S) and for any a and b in a, we define: ON,(U, b) iff S(a) = b, CLEAR,(U) iff either a = TABLE or 13b (oN,(~, a)), ABOVE~ as the transi- tive closure of ON,,, and POSITION,(U) as the sequence (a :: POSITION,(S(U))) if S(u) exists and (a) other- wise. That is, the position of a block is the sequence of blocks at or below it. We refer to the position of a clear block as a tower. A tower is grounded iff it ends with the table. Note that in a state (as opposed to a mere part-state) every tower is grounded. A BW planning problem over B is a pair of states ((a, Si), (a, Sz)). In problem (I, G), I is the initial state and G is the goal state. Here we consider only problems with completely specified goal states. A move in state o = (a,S) is a pair m = (a, b) with a E B\{TABLE} and b E B, such that CLEAR,(U), CLEAR,(~) and -ON, (a, b). The result of m in o is the state RES(m,a) = (B, S’) where S’(u) = b and S’(z) = S(z) for z E X~\{TABLE, u}. initial state goal state Figure 1: BW Planning Problem A plan for BW problem (I, G) is a finite sequence (ml,... , mP) of moves such that either I = G and p=Oorelsemi isamoveinIand (m2,...,mP) isa plan for ( RES(mi, I), G) in virtue of this definition. We say that a block whose position in I is differ- ent from its position in G is misplaced in (I, G), and that one which is not misplaced is in position. Next, we say that a move (u,b) is constructive in (1,G) iff a is in position in (RES((U, b),l),G). That is, iff a is put into position by being moved to b. Once a block has been moved constructively it need never be moved again in the course of solving the problem. If no constructive move is possible in a given problem, we say that the problem is deudZocked.3 In that case, for any misplaced block bi there is some block b2 which must be moved before a constructive move with bi is possible. Since the number of blocks is finite the se- quence (bi , b2 . . .) must eventually loop. The concept of a deadlock, adapted from that given in (Gupta & Nau 1992), makes this idea precise. A deadlock for BW problem (1, G) over t3 is a nonempty subset of B that can be ordered (dl, . . . , dk) in such a way that: where B(I,G) h @ E POSITIONS # POSITIONG(U) A POSITION1 (b) # POSITIONG (b) A 3a: # TABLE (ABOVE&Z) A ABOVEG(U,Z)) E.g., the problem in Figure 1 is deadlocked, the dead- locks being {a} and {a, d}.4 It is easy to see that if B(I,G)(u, b) then in any plan for (I, G), the first time b is moved must precede the last time a is moved. A deadlock being a loop of the B(I,G) relation, at least one block in each deadlock must be moved twice. The first move of this block may as well always be to the ta- ble, so as to break deadlocks without introducing new ones. What makes optimal BW planning hard is to choose those deadlock-breaking moves so that it pays in the long term (Gupta & Nau 1992). 3By slight abuse of notation, we allow ourselves to speak of moves as constructive in a state, rather than in a prob- lem, of a state rather than a problem as deadlocked and so forth, leaving mention of the goal to be understood. *To see this, note that B~I,G) (a, d) taking the third block in the definition to be x = e, that B~I,G) (d, a) taking x = c, and that B(I,G) (a, a) taking x = b. Search 1209 Near-Optimal BW Planning There is a nondeterministic algorithm which solves BW problems optimally in polynomial time (Gupta & Nau 1991; 1992). It consists basically of a loop, executed until broken by entering case 1: 1. If all blocks are in position, stop. 2. Else if a constructive move (a, b) exists, move a to b. 3. Else nondeterministically choose a misplaced clear block not yet on the table and move it to the table. In the course of their discussion, Gupta and Nau also note three deterministic polynomial time algorithms which approximate optimality within a factor of 2. US The first and simplest is one we have dubbed US (Unstack-Stack). It amounts to putting all mis- placed blocks on the table (the ‘unstack’ phase) and then building the goal state by constructive moves (the ‘stack’ phase). No block is moved by US unless it is misplaced, and no block is moved more than twice. Every misplaced block must be moved at least once even in an optimal plan. Hence the total number of moves in a US plan is at worst twice the optimal. GNl Another algorithm which is usually better in terms of plan length than US (and never worse) is the simple deterministic version of the above optimal one given on pages 229-230 of (Gupta & Nau 1992). It differs from the nondeterministic version just in choosing urbitruriZy some move of a misplaced clear block to the table whenever no constructive move is available. We call it GNI for Gupta and Nau. GN2 Yet another algorithm is suggested (Gupta & Nau 1991) though with no details of how it may be achieved. This one uses the concept of a deadlock. We call it GN2 and it is the same as GNl except that the misplaced clear block to be moved to the table is chosen not completely arbitrarily but in such a way as to break at least one deadlock. That is: 3. Else arbitrarily choose a clear block which is in a deadlock and move it to the table. Gupta and Nau (1992, p. 229, step 7 of their algorithm) say that in a deadlocked state every misplaced clear block is in at least one deadlock. If this were true, GNI and GN2 would be identical. It is false, however, as may be seen from the example in Figure 1. Block f is not in any deadlock, and so can be chosen for a move to the table by GNl but not by GN2. It is possible for GNl to produce a shorter plan than GN2 (indeed to produce an optimal plan) in any given case, though on average GN2 performs better because it never completely wastes a non-constructive move by failing to break a deadlock. In order for GN2 to be complete, in every deadlocked state there must be at least one clear block which is in a deadlock. In fact, we can prove the stronger result that in every deadlocked state there exists a deadlock consisting entirely of clear blocks. We now sketch the proof, since it makes use of the notion of a A sequence which will be needed in implementing GN2. Let 0 = (B,S) b e a state that occurs during the attempt to solve a problem with goal state G = (x3, SC) and suppose the problem 7r = (a, G) is deadlocked. Let b be misplaced and clear in 0. Consider POSITIONG (b), the sequence of blocks which in G will lead from b down to the table. Let c be the first (highest) block in this sequence which is already in position in CT (c may be the table, or SC(~), or somewhere in between). In the goal state, either b or some block below b will be on c. Let us call this block d. What we need to do, in order to advance towards a constructive move with b, is to put d on c. This is not immediately possible because there is no constructive move in 0, so either c is not clear or else d is not clear. If c is not clear, we must move the blocks above it, starting with the one at the top of the tower which contains c in 0. If c is clear, we should next move the clear block above d in 0. This is how the function 6, is defined: S,(b) = z such that CLEAR, (~)and ABOVE&, d) if CLEAR,(C) ABOVE,(x,c) if ICLEAR, If b is in position or not Clear or if SG (b) is in position and clear, let 6,(b) be undefined. Now let A,(b) be the sequence of blocks obtained from b by chasing the function 6, : A&r(b))) if S,(b) exists otherwise The point of the construction is that if S,(b) exists, then CLEAR&&(~)) and &(b,&(b)). To see why the latter holds, note that either c or d is below &(b) in LT and below b in the goal. Now for any misplaced clear b, if A,(b) is finite then there is a constructive move in 0 using the last block in A,(b), while if it is infinite then it loops and the loop is a deadlock con- sisting of clear blocks. This loop need not contain b of course: again Figure 1 shows an example, where A, (d) is (d,a,a,. . .). Cur suggestion for a way of implementing GN2, therefore, is to replace the original clause 3 with: 3. Else arbitrarily choose a misplaced clear block not on the table; compute its A sequence until this loops; detect the loop when for some x in A,, &(x) occurs earlier in A,; move x to the table. Linear time algorithms We now show how to implement all of us, GNI and GN2 to run in time linear in the number n of blocks. This improves on the known complexity of these al- gorithms. The original (Gupta & Nau 1992) did not mention any bound better than O(n3) for near opti- mal BW planning, though O(n2) implementations have been described by other authors (Chenoweth 1991; Bacchus & Kabanza 1995). 1210 Planning function INPOS (b : block) : boolean if b = TABLE then return true if not Examinedb then Examinedb t true if Sib # Sgb then lnpositionb + false else lnpositionb t lNPOS(Sib) return 1 n Positionb procedure US () INIT() for each b E ~\{TABLE) do if Clearb then UNSTACK(b) for each b E Z~\{TABLE} do STACK(b) procedure INIT () Plan t ( ) for each b E Z~\{TABLE} do clearb + true Examinedb t false for each b E D\{TABLE} do INPOS(b) if sib # TABLE then Clearsib t false procedure UNSTACK (b : block) if (not InPositionb) and (Sib # TABLE) then (local) c + sib MOVE( (b, TABLE)) UNSTACK(c) procedure MOVE ((a, b) : move) Plan t ((a, b) :: Plan) if Si, # TABLE then ClearSi t true if b # TABLE then clearb + false InPosition, + (Sg, = b) and lnpositionb else InPosition, t (sg, = TABLE) Si, t b procedure STACK (b : block) if not Inpositionb then STACK(Sgb) MoVE((b, %b)) Figure 2: The US Algorithm How to make US linear The key to making US a linear time algorithm is to find a way to compute which blocks are in position in O(n), and to execute this computation only once in the course of the problem solution. We do this by means of a combination of recursion and iteration, as shown in Figure 2. The algorithm makes use of several variables associated with each block b: Clearb True iff b is clear in the current state. InPositionb True iff b is already in position. Examinedb True iff InPositionb has been determined. sib Block currently below b. %b Block below b in the goal. During initialization, INPOS can be called at most twice with any particular block b as parameter: once as part of the iteration through the blocks and at most once recursively from a call with the block above b. Hence the number of calls to INPOS is bounded above by 2n. Similar considerations apply to the recursive STACK and UNSTACK procedures. The stored infor- mation is updated in constant time by MOVE. How to make GNl linear For GNU, we add organizational structure to the prob- lem representation. At any given time, each block has a status. It may be: 1. ready to move constructively. That is, it is misplaced but clear and its target is in position and clear. 2. stuck on a tower. That is, it is misplaced, clear and not on the table, but cannot move constructively because its target is either misplaced or not clear. 3. neither of the above. More variables are now associated with each block. One records the status of this block, while others de- note the blocks (if any) which are on this one currently and in the goal. Initialising and updating these does not upset the O(n) running time. To make it possible to select moves in constant time, the blocks of status 1 and 2 are organized into doubly linked lists, one such list containing the blocks of each status. Inserting a block in a list and deleting it from a list are constant time operations as is familiar. The next block to move is that at the head of the ‘ready’ list unless that list is empty, in which case it is the block at the head of the ‘stuck’ list. If both lists are empty, the goal is reached. When a block a moves, it changes its status as well as its position. At most three other blocks may also change status as a result of the move: the block cur- rently below a (if any), the block that will be on a in the goal (if any), and the block which in the goal will be on the block currently below a (if any). Hence the number of delete-insert list operations is at worst lin- ear in the number of moves in the plan, and so in the number of blocks. Nothing else stands in the way of a linear time implementation of GNl. How to make GN2 linear GN2, however, is a different matter. To implement GN2 via A sequences, it is necessary to compute S,(b) for various blocks b, and to achieve linear time there must be both a way to do this in constant time and a way to limit the number of 6 calculations to a constant number per block. On the face of it, neither of these is easy. To find 6, (b) it is necessary to know which is the highest block in position in the goal tower of 13 and to know which is the clear block above a given one. These items of information change as moves are made, and each time such an item changes for one block it changes for all the O(n) blocks in a tower, so how can those changes be propagated in constant time? Moreover, when a deadlock is to be broken, a new A sequence has to be computed, as many blocks may have moved since the last one was computed, thus changing 6. Computing a A sequence appears to be irreducibly an O(n) problem, and since O(n) such sequences may be needed, this appears to require GN2 to be of O(n2) even if S,(b) can somehow be found in constant time. Search 1211 cl a GY&l’ I I-I ‘Who is on top?’ ‘It is a’ Figure 3: ‘Who lives at the top of the tower?’ The first trick that begins to address these difficul- ties is to note that whatever changes in a tower of blocks, one thing does not change: the block on the table at the bottom of the tower. We call the bottom block in a tower the concierge for that tower. Now if we want to know who lives at the top of the tower, we can ask the concierge. When a block comes or goes at the top of the tower, only the concierge need be informed (in constant time) and then since every block knows which is its concierge, there is a constant time route from any given block to the information as to which block above it is clear (see Figure 3). Not only the towers in the initial (or current) state have concierges, but so do the towers in the goal state. These keep track of which block in their tower is the highest al- ready in position. Additional variables associated with each block b denote its initial and goal concierges. In case b is a concierge, there are more variables denoting the clear block above it, and the highest block already in position in its goal tower. Through the concierges, there is a a constant time route from b to the c and d required to define S,(b). The procedure for initialising the additional variables is closely analogous to that for determining which blocks are in position and can be executed in linear time for the same reason. Updating them when a move is made takes constant time. Next, the key to managing the A sequences is that although 6 may change the B, relation is indestruc- tible except by moving the blocks involved. That is, if B(,,G)(x, y) then that relationship persists in the sense that in all future states 6, B(~,G)(x, y) unless x or y has moved in getting from 0 to 8. Moreover, if B, (x, y) then x cannot move constructively until y has moved at least once. Now, let p = (bl , . . . , bk) be a non-looping sequence of clear blocks which are stuck on towers, each except the last linked to its succes- sor by the relation B,. At some point, bk may cease to be stuck and become ready to move, but no other block in ,O can change its status until bk actually moves. Thus the ,O sequence may dwindle, and even become null, as moves are made, but it always remains a single sequence-it never falls into two pieces as would hap- pen if a block from the middle of it changed status- and because B, is indestructible ,O remains linked by 1212 Planning p sequence loops Remove f: good cl a Remove c: bad Figure 4: How [not] to break a deadlock B,. For the algorithm, then, we maintain such a se- quence p, constructed from parts of A sequences as follows. Initially, p is null. If the problem becomes deadlocked, first if /? is null then it is set to consist just of the block at the head of the ‘stuck’ list, and then it is extended by adding S,(bk) to the end of it. This is done repeatedly until the sequence threatens to loop because 6, of the last block b, is already in ,f3. At that point b, is chosen to break the deadlock. It is important not to choose S,(b,) for this purpose, since that could result in breaking ,O into two pieces (see Fig- ure 4). Each addition to /3 takes only constant time, and any given block can be added to the sequence at most once. Therefore maintenance of the p sequence requires only linear time. Pseudo-code for our implementations of both GNU and GN2 is given in the technical report (Slaney & Thikbaux 1995). Generating Random I3 W-Problems In designing the experiments to be presented in the next section, we needed a supply of uniformly dis- tributed random BW problems, (pairs of BW states). Every state of the chosen size must have the same prob- ability of being generated, otherwise the sample will be skewed and experiments on the average case may be biased. Note that, contrary to what might have been expected, generating such random BW states is not en- tirely trivial. Nai’vely incorporating a random number generator into an algorithm for producing the states does not work: typically, it makes some states more probable than others by a factor exponential in the number of blocks. The unconvinced reader is invited to try by hand the case n=2 with a view to generating the three possible states each with a probability of l/3 (most methods give probabilities l/2, l/4 and l/4). To build a BW state of size n, we start with the part-state containing n ungrounded towers each con- sisting of a single block, and extend this part-state progressively by selecting at each step an arbitrary un- grounded tower and placing it on top of another tower (already grounded or not) or putting it on the table. The difficulty is that the probabilities of these place- ments are not all the same. 0.8 0.6 0.4 0.2 Figure 5: Average CPU time (in seconds) 4+ the The solution is to count the states that extend a 1 ungrounded ones. Let this be g( 4 + part-state in which there are r grounded towers and next step in the construction process, 1, r).5 there At will be 4 ungrounded towers, since one ungrounded tower will have been placed on something. The probability that the selected tower will be on the table in an ar- bitrary reachable state is g(4, r + l)/g(+ + 1,~) while each of the other possible placements has probability 9k4 al(4 + 1’4. The recursive definition of g is quite simple. First g(0,~) = 1 for all r, since if 4 = 0 then every tower is already grounded. Now consider g(4 + 1,~). In any reachable state, the first ungrounded tower is either on the table or on one of the b-+ r other towers. If it goes on the table, that gives a p&t-state with 4 ungrounlded towers and with r + 1 grounded ones. If it goes on another tower, that leaves 4 ungrounded towers and r grounded ones. In sum: SIC4 + 1,7) = 9(4,7)(4 + 7) + $?(A 7- + 1) An equivalent iterative definition is: In fact, for present purposes it is hard to work with g directly, as the numbers rapidly become too large. For example, g(lOO,O) N 2.4 10164. It is better to work with the ratio R(+, 7) = g(4, r + 1) / g(+, 7) since this is always a fairly small real number lying roughly in the range 1.. . &.” El ementary calculation shows that R(O,7) = 1 for all r and that R((j + 1, r) = w?4 m + 7- + 1 + WA 7 + 1)) 4+7-+wo-) 5As a special case, note that g(n,O) is the number of BW states of size n. 61n the limit as 4 becomes much larger than r it con- verges to a. dn the other hand, where r is much larger than 4 the limiting value is 1 (Slaney & Thihbaux 1995). A corollary of these results is that the average number of towers in a state of n blocks converges to 6. 2 U.5 GNl 1.9 GN2 1.6 l.sl-...*. .'*""'. 2000 4000 6000 8000 10000 Figure 6: Average plan length: number of moves Experimental number of blocks With a random BW states generator using R to calcu- late the relevant probabilities,7 we generated between 1000 and 10000 random BW problems for each of the 160 sizes we considered, up to n = 10000 blocks. With this test set of some 735000 problems, we observed the average performance of US, GNU and GN2. The graphs in this section are continuous lines passing throigk all data points. Average Run Times Figure 5 shows the average runtimes of the near- optimal algorithms as a function of n. As will be clear from the next experiment, there is no significant differ- ence betweeen the average and worst cases runtimes. Times were obtained on a Sun 670 MP under Solaris 2.3. Naturally, since nobody really wants to convert one BW state into another, the program speeds are not important in themselves. The point of this experiment was just to confirm empirically the theoretical claims that linear execution time may be attained. Average Plan Length The length of the plan produced by all near-optimal al- gorithms, as well as the optimal plan length, is 2n - 2 in the worst case. Figure 6 shows that the average case approaches this worst case quite closely. As ex- pected, US gives the longest plans on average and GN2 the shortest, but for large numbers of blocks it makes little difference which algorithm is used. In particu- lar, it is evident from the graphs that the algorithms will give very close (and lengths in the limit. maybe identical) average plan The immediate questions raised by the present re- sults are whether all of the algorithms converge to the same average plan length and if so whether this lim- iting figure is 2n. For what our opinion is worth conjecture positive answers to both questions. , we 7The generator is made available by the first author at http://arp.auu.edu.au/arp/jks/bwstates.html. Search 1213 l:~~~---z;~ 20 40 60 SO 100 120 140 Figure 7: Average perf. ratio: plan length optimal plan length Average Performance Ratio The absolute performance ratio of the near-optimal al- gorithms is 2 in the limit (for a description of the worst- case instances, see (Slaney & Thiebaux 1995)). Figure 7 shows the averugk performance ratios. The optimal plan lengths were determined by running an optimal BW planner (Slaney & Thiebaux 1995). This graph contains a real surprise: the average per- formance of US does not degrade monotonically but turns around at about n = 50 and begins to improve thereafter. The explanation seems to be that the plan lengths for US quickly approach the ceiling of 2n, at which point the optimal plan lengths are increasing more quickly than the US ones because the latter are nearly as bad as they can get. We expect that the other near-optimal algorithms would it were possible to observe the for high enough values of n. exhibit similar curves if length of optimal plans One result readily available from the graphs is an upper bound around 1.23 for average performance ra- tios. However, Figures 6 and 7 together suggest that the limiting value will be well below this rough upper bound. Our open question following this experiment is whether the optimal plans tend to a length of 2n in the limit. If they do, then not only the near-optimal algo- rithms, but even the ‘baby’ algorithm of unstacking all blocks to the table before building the goal, are opti- mal on average in the limit. A positive answer would be implied if the number of singleton deadlocks tended to n, but on investigating this figure experimentally we found that it appears to be only around 0.4n. Conclusion and Future Work By presenting linear time algorithms for near-optimal BW planning within a factor of 2, this paper has closed the question of its time complexity. We hope that both this result and the algorithms we have presented will contribute to clarification and thus to the understand- ing of BW planning needed for assessment purposes, and that the present paper will not be seen a report on how to stack blocks fast. merely as While the time complexity questions have been closed, a number of open questions remain concerning the average solution quality of these algorithms. We do not believe that further experimentation alone will an- swer those questions, because this would require gener- ating optimal BW plans for very large problems, which is impossible on complexity grounds. On the other hand, the theoretical investigation does not appear easy either. Indeed, even to prove (Slaney & Thiebaux 1995) that the average length of the plans produced by the ‘baby’ algorithm converges to 2(n - fi) we needed nontrivial mathematics involving complex analysis and the theory of Laguerre polynomials. Therefore, we consider that we have stopped at a convenient point. It appears likely that extensions of the present investigations will belong less directly to planning and increasingly to pure number theory. The most important future direction is to address the sec- ond goal mentioned in the introduction of this paper: to exploit our investigations in identifying a class of problems for which near-optimal planning is tractable, and to generalize them to related phenomena in plan- ning domains of greater practical interest. References Bacchus, F., and Kabanza, F. 1995. Using temporal logic to control search in a forward chaining planner. In Proc. EWSP-95, 157-169. Backstrom, C., and Nebel, B. 1995. Complexity re- sults for SAS+ planning. Computational Intelligence ll(4). Bylander, T. 1994. The computational complexity of propositional STRIPS planning. Artificial htelhgence 69: 165-204. Chenoweth, S. 1991. On the NP-hardness of blocks world. In Proc. AAAI-91, 623-628. Gupta, N., and Nau, D. 1991. Complexity results for blocks world planning. In Proc. AAAI-91, 629-633. Gupta, N., and Nau, D. 1992. On the complexity of blocks world planning. Artificial Intelligence 56:223- 254. Kautz, H., and Selman, B. 1992. Planning as satisfi- ability. In Proc. ECAI-92, 359-363. Kautz, H., and Selman, B. 1996. Pushing the en- velope: Planning, propositional logic, and stochastic search. In Proc. AAAI-96. In this issue. Schoppers, M. 1994. Estimating reaction plan size. In Proc. AAAI-94, 1238-1244. Selman, B. 1994. Near-optimal plans, tractability, and reactivity. In Proc. K&94, 521-529. Slaney, J., and Thiebaux, S. 1995. Blocks world tamed: Ten thousand blocks in under a second. Tech- nical Report TR- ARP- 17-95, Automated Reasoning Project, Australian National University. ftp://arp.anu.edu.au/pub/techreports. 1214 Planning | 1996 | 179 |
1,820 | Incorporating Opponent Models into Adversary Search David Carmel and Shaul Markovitch Computer Science Department Technion, Haifa 32000, Israel carmel@cs.technion.ac.il shaulm@cs.technion.ac.il Abstract This work presents a generalized theoretical frame- work that allows incorporation of opponent models into adversary search. We present the M* algorithm, a generalization of minimax that uses an arbitrary op- ponent model to simulate the opponent’s search. The opponent model is a recursive structure consisting of the opponent’s evaluation function and its model of the player. We demonstrate experimentally the po- tential benefit of using an opponent model. Pruning in M* is impossible in the general case. We prove a sufficient condition for pruning and present the cup* algorithm which returns the M* value of a tree while searching only necessary branches. Introduction The minimax algorithm (Shannon 1950) has served as the basic decision procedure for zero-sum games since the early days of computer science. The basic assump- tion behind minimax is that the player has no knowl- edge about the opponent’s decision procedure. In the absence of such knowledge, minimax assumes that the opponent selects an alternative which is the worst from the player’s point of view. However, it is quite possible that the player does have some knowledge about its opponent. Such knowl- edge may be a result of accumulated experience in playing against the opponent, or may be supplied by some external source. How can such knowledge be ex- ploited? In game theory, the question is known as the best response problem - looking for an optimal response to a given opponent model (Gilboa 1988). However, there is almost no theoretical work on using opponent models in adversary search. Korf (1989) outlined a method of utilizing multiple-level models of evaluation functions. Carmel and Markovitch (1993) developed an algorithm for utilizing an opponent model defined by an evaluation function and a depth of search. Jansen (1990) describes two situations where it is important to consider the opponent’s strategy. One is a swindle position, where the player has reason to believe that the opponent will underestimate a good move, and will therefore play a poorer move instead. Another 120 Agents situation is a trap position, where the player expects the opponent to overestimate and therefore play a bad move. Another situation, where an opponent model can be beneficial, is a losing position (Berliner 1977). When all possible moves lead to a loss, an opponent model can be used to select a swindle move. The goal of this research is to develop a theoreti- cal framework that allows exploitation of an opponent model. We start by defining an opponent model to be a recursive structure consisting of a utility func- tion of the opponent and the player model (held by the opponent). We then describe two algorithms, M* and M&xzsJ~ g eneralizations of minimax that can use an opponent model. Pruning in M* is not possible in the general case. However, we have developed the a@+ algorithm that allows pruning given certain con- straints over the relationship between the player’s and the model’s strategies. The Af* algorithm Let S be the set of possible game states. Let ; :s4 2’ be the successor function. Let cp : - S be an opponent model that specifies the opponent’s selected move for each state. The A4 algorithm takes a position s E S, a depth limit d, a static evaluation function f : S - %, an arbitrary opponent model cp, and returns a value. i fG9 dg M(s, 4 f 9 4 = s,y$,(f (4) d=l max (M(v(s’), d-2, f, v)) d > 1 s’ea( s) The algorithm selects a move in the following way. It generates the successors of the current position. It then applies cp on each of the successors to obtain the opponent’s response and evaluates each of the resulting states by applying the algorithm recursively (with re- duced depth limit). It then selects the successor with the highest value. The algorithm returns the static evaluation of the input position when the depth limit is zero. Note that A4 returns a value. M can also be defined to return the state with the maximal value instead of From: AAAI-96 Proceedings. Copyright © 1996, AAAI (www.aaai.org). All rights reserved. the value itself. To simplify the discussion, from now on we will assume that M returns either the state or the value according to the context. Let M&J, Id) be the regular minimax algorithm that searches to depth d using an evaluation function fs. Minimax uses itself with d - 1 and -fs as an opponent model, therefore it can be defined as a special case of M: M&o) 4-f) (4 = WY dYfO7 M&fo),d-1)) Assume that the player uses an evaluation function fi. A natural candidate to serve as the model ‘p in the M algorithm is M* with another evaluation function fe. We call this special case of M the M1 algorithm: MiPl ,fo),d) (4 = ws’dYflYM&J),d-l))- The M1 algorithm works by simulating the oppo- nent’s minimax search (to depth d - 1) to find its se- lected move and evaluates the selected move by calling itself recursively (to depth d - 2). 9 (a) (b) Figure 1: The search trees spanned by calling Ml. Part (a) shows the two calls to minimax for determining the opponent’s choice. Part (b) shows the recursive calls to M1 for evaluating the nodes selected by the opponent. Figure 1 shows an example of a search to depth 3 performed by the M1 algorithm. Part (a) shows the two calls to minimax to simulate the opponent’s search. Note that the opponent is a maximizer. Part (b) of the figure shows the two recursive calls to Ml applied to the boards selected by the opponent. Note that while the opponent’s minimax procedure believes that in node e the player selects the move leading to node Ic, the player actually prefers node j. We can define the M” algorithm, for any n, to be the M algorithm with cp = Mn-l. Mn can be formally defined as follows: Mia(&z,...,fo),d)(s~ = M(s’ fn’ 4 Mig+,fo),61)) Thus, a player using the M1 algorithm assumes that its opponent uses MO (minimax), a player using the M2 algorithm assumes that its opponent uses Ml, etc. We will define the M* algorithm that includes every Mn algorithm as a special case. M* receives as a param- eter a player which includes information about both the player’s evaluation function and its model of the opponent. Definition 1 A player is a pair defined us follows: 1. Given an evaluation function f, P = (f, NIL) is a player (with a modeling-level 0). 2. Given an evaluation function f and a player 0 (with modeling level n - l), P = (f, 0) is a player (with a modeling level n). The first element of a player is culled the player’s strategy and the second element is culled the opponent model. Thus, a zero-level modeling player, (fo , NIL), is one that does not model its opponent. A one-level model- ing player, (fly (fo, NIL)), is one that has a model of its opponent, but assumes that its opponent is a zero- level modeling player. A two-level modeling player, (f27 (fl, (fo, NW), is one that uses a strategy f2, and has a model of its opponent, (f 1, (fo, NIL)). The op- ponent’s model uses a strategy j-1 and has a model, (fo, NIL), of the player. The recursive definition of a player is in the spirit of the Recursive Modeling Method by Gmytrasiewicz, Durfee and Wehe (1991). M* receives a position, a depth limit, and a player, and outputs a move selected by the player and its value. The algorithm generates the successor boards and sim- ulates the opponent’s search from each of them in order to anticipate its choice. This simulation is achieved by applying the algorithm recursively with the opponent model as the player. The player then evaluates each of its optional moves by evaluating the outcome of its opponent’s reaction by applying the algorithm recur- sively using its own strategy. f2=8 I2=4 f2=4 f-2=7 f2=-6 fk 1 c?=lO f2=2 fl=-6 fl=6 fl=-8 fl z-7 fl=7 fl=-2 fl=-4 fl=O Kk4 RI=-8 NklO ffk 3 N)=-4 iI)= 4 II,= 4 In= 6 Figure 2: The set of recursive cJls generated by calling M*(a, 3, (fz(fl, fo))). Each call is written next to the node it is called from. The dashed lines show which move is selected by each call. Figure 2 shows an example of a search tree spanned by M*(a, 3, fs(fl, fe)). The numbers at the bottom are the static values of the leaves. The recursive calls applied to each node are listed next to the node. The dashed lines indicate which move is selected by each recursive call. The player simulates its opponent’s search from nodes b and c. The opponent simulates the player by using its own model of the player from nodes d and e. At node d the model of the player used by the oppo- nent (fo) selects node h. The opponent then applies its Negotiation & Coalition 121 fi function to node h and concludes that node h, and therefore node d, are worth -6. The opponent then applies the player’s model (fs ) to node e , concludes that the player select node j, applies its own function (fi ) to node j, and decides that node j, and therefore node e, are worth -8. Therefore, the opponent model, when applied to node b, selects the move that leads to node d. The player then evaluates node d using its own criterion (fz). It applies M* to node d and concludes that node d, and therefore node b, are worth 8. Simu- lation of the opponent from node c yields the selection of node g. The player then evaluates g according to its own strategy and finds that it is worth 10 (the value of node n). Note that when the opponent simulates the player from node g, it wrongly assumes that the player selects node o. Therefore, the player selects the move that leads to c with a value of 10. Note that using a regular minimax search with f2 would have resulted in selecting the move that leads to node b with a value of 7. The formal listing of the M* algorithm is shown in figure 3. Procedure M’ (pas, d, (fpt, 0)) z,",= 0 then return (NIL,f,l(pos)) max:-value + -co s + o(pos) for each s E S ifd = 1 then pl-v + fpl(s) else (op-b, op-v) + M’ (s, d - 1,O) (pl-b,pl-v) + M’ (OP-b, d - 2, (fpz, if pl-v > max-value max-value +- pl-v max-board - s return (max-board, max-value) Figure 3: The M* algorithm The Mn algorithm calls M” when n reaches 0. How- ever, note that M* does not contain such a call. This will work correctly when the modeling level is larger than the search depth. If the modeling level of the original player n is smaller than the depth of the search tree d, we replace the O-level modeling player fo by d-n ifo, (-fo, (fo, . . a>‘. A one-pass version of M* It is obvious that the M* algorithm performs multiple expansions of parts of the search tree. We have de- veloped another version of the M* algorithm, called M* l-pass) that expands the tree one time only, just as minimax does. The algorithm expands the search tree in the same manner as minimax. However, node values are propagated differently. Whereas minimax propagates only one value, M* propagates n + 1 val- ues, (V,, . . . , VO). The value V;: represents the merit of the current node according to the i-level model, fa. M* I-pass passes values associated with the player and values associated with the opponent in a different man- ner. In a player’s node (a node where it is the player’s turn to play) ‘, for values associated with the player (KY K-2,. * -)Y V;: receives the maximal Vi value among its children. For values associated with the opponent (K-1, G-3, * * .> , K receives the Vi value of the child that gave the maximal value to K-1 . For example, the opponent believes (according to the model) that the player evaluates nodes by I/n-z. At a player’s node, the opponent assumes that the player will select the child with maximal Vn-2 value. Therefore, the value of the current node for the opponent is the V, - 1 value of the selected child with the maximal Vn-2 value. At an opponent’s node, we do the same but the roles of the opponent and the player are switched. V[2]= 8 V[l]=-6 d V[O]= 4 x f2=8 k-4 fl=-6 fl= 6 m=4 f&-8 f2=4 f2= I fl=-8 fl=-7 ffkI0 fo=3 Figure 4: The value vectors propagated by M;--pass. This is the same tree as the one shown in Figure 2. Figure 4 shows an example for a tree spanned by M*- . This is the same tree as the one shown in Fi&Fgsi. Let us look at node e to understand the way in which the algorithm works. Three players evaluate node e: the player, the opponent model, and the op- ponent’s model of the player. The opponent’s model of the player evaluates all the successors using evalu- ation function f0 and selects board j with a value of 10. The opponent model knows that it has no effect on the decision taken at node e, since it is the player’s turn to move. It therefore assigns node e the value of j, selected by the player model, using its own evalu- ation function f 1, (f l(j) = -8). The player actually prefers node k with the higher f2 value (f2( k) = 7). Thus, the vector propagated from node e is (7, -8,10). Note that the values in the vectors correspond to the results of the recursive calls in figure 2. Figure 5 lists the MT-pass algorithm. Properties of n/r* The following theorem shows that M* and MTepass return the same value. Theorem 1 Assume that P is a n-level modeling player. Let (v, b) = M*(pos, d, P), and let (V[n]) = MTepass (pos, d, P). Then v = V[n]. ‘Traditionally such a node is called a MAX node. How- ever, we assume that both players are maximizers. 122 Agents Procedure MTGPass (~08, d, Un, (fn-I, (. . , fo) . .>>I if d = 0 then return (fn(pos), . . . , fo(pos)) else S +- o(pos) v + (- cKl,...,---00) for each s E S WCC-V + M1+--pass(s,d- l,(fn,(fn-l,...,fo)...)) for each i associated with current player if succ-V[i] > V[i] then V[i] + succ-V[i] if i < n then V[i + l] + succ-V[i + l] return (V[n], . , V[d]) Figure 5: MT--pass: A version of the M” algorithm that performs only one pass over the search tree The proof for all the theorems in this paper can be found in (Carmel & Markovitch 1996). It seems as though MTspass is always prefered over M* because it expands less nodes. MTspass expands each node in the tree only once, while M* re-expands many nodes. However, while MTspass performs less expansions than M*, it may perform more evaluations. An upper limit on the number of node expansions and evaluations in M* and MTBpass is given in the following theorem. Theorem 2 Assume that M* and Mrepass search a tree with a uniform branching factor b and depth d. 1. The number of calls to the expansion function by M* is bounded by (b + l)d-l. The number of calls by h/i* l-pass is s. 2. The number of calls to evaluation functions by M” is bounded by (b + l)d. the number of calls by MTepass is dbd. The theorem implies that M* and MTWpass each has an advantage. If it is more important to reduce the number of node expansions than the number of evalu- ation function calls then one should use MT- P ass. Oth- erwise, M* should be used. For example, or d = 10 and b = 30 and a one-level modeling player, M* ex- pands 1.3 times more nodes than Mrspass but M;-pass performs 1.5 times more calls to evaluation functions. Note that when the set of evaluation functions consist of the same features (perhaps with different weights), the overhead of multiple evaluation is reduced signifi- cantly. We have already shown that M* is a generalization of minimax. An interesting property of M* is that it al- ways selects a move with a value greater or equal to the value of the move selected by minimax that searches the same tree with the same strategy. Theorem 3 Assume that M* and Minimax use the same evaduation function f. Then Minimaz(pos, d, f) 5 iW*(pos, d, (f, 0)) for any op- ponent model 0. The theorem states that if you have a good reason to believe that your opponent’s model is different than yours, you could only benefit by using M* instead of minimax. The reader should note that the above theo- rem does not mean that M* selects a move that is bet- ter according to some objective criterion, but rather a subjectively better move (from the player’s point of view, according to its strategy). If the player does not have a reliable model of its opponent, then playing minimax is a good cautious strategy. Adding pruning to One of the most significant extensions of the minimax algorithm is the o/3 pruning technique. Is it possible to add such an extension to M* as well? Unfortu- nately, if we assume a total independence between fi and fo, it is easy to show that such a procedure cannot exist. Knowing a lower bound for the opponent’s eval- uation of a node does not have any implications on the value of the node for the player. A similar situation arises in MAXN, a multi-player game tree search algo- rithm. Luckhardt and Irani (1986) conclude that prun- ing is impossible in MAXN without further restrictions about the players’ evaluation functions. Korf (1991) showed that a shallow pruning for MAXN is possible if we assume an upper bound on the sum of the players’ functions, and a lower bound on every player’s func- tion. The basic assumption used for the original CYP al- gorithm is that fi + fo = 0 (the zero-sum assump- tion). This assumption is used to infer a bound on a value of a node for a player based directly on the oppo- nent’s value. A natural relaxation to this assumption is Ifi + fol 5 23. This assumption means that while fi and -fo may evaluate a board differently, this differ- ence is bounded. For example, the player may prefer a rook over a knight while the opponent prefers the opposite. In such a case, although the player’s value is not a direct opposite of the opponent’s value, we can infer a bound on the player’s value based on the opponent’s value and B. The above assumption can be used in the context of the M?-pass algorithm to determine a bound on Vi + V;:- 1 at the leaves level. But in order to be able to prune using this assumption, we first need to determine how these bounds are propagated up the search tree. Lemma 1 Assume that A is a node in the tree spanned by MTopass. Assume that ,!?I,. . . , Sk are its successors. If there exist non-negative bounds Bo, . , . , B,, such that for each successor Sj, and for each model i, II+, [i] + Vs, [i - l] 1 2 Bi . Then, for each model 1 5 i 5 n, ]V~[i]+v~[i- 111 5 Bi +2. B&l. Based on lemma 1, we have developed an algorithm, a/?* 1 that can perform a shallow and deep pruning assuming bounds on the absolute sum of functions of the player and its opponent model. crp* takes as in- put a position, a depth limit, and for each model i, a strategy fi, an upper bound Ba on lfi + fi-1 I, and a Negotiation & Coalition 123 cutoff value oi . It returns the M* value of the root by searching only those nodes that might affect this value. The algorithm works similarly to the original cyp al- gorithm, but is much more restricted as to which sub- trees can be pruned. The ap* algorithm only prunes branches that all models agree to prune. In regular cup, the player can use the opponent’s value of a node to determine whether it has a chance to attain a value that is better than the current cutoff value. This is based on the opponent’s value being exactly the same as the player’s value (except for the sign). In CY~* , the player’s function and the opponent’s function are not identical, but their difference is bounded. The bound on Vi + Vi-1 depends on the distance from the leaves level. At the leaves level, it can be directly computed using the input Bi . At distance d, the bound can be computed from the bounds for level d - 1 as stated by lemma 12. A cutoff value CQ for a node v is the highest current value of all the ancestors of 2, from the point of view of player i. ok is modified at nodes where it is player i’s turn to play, and is used for pruning where it is player i - l’s turn to play. At each node, for each i associated with the player whose turn it is to play, o; is maximized any time Vi is modified. For each i such that i - 1 is associated with the current player, the algorithm checks whether the i player wants its model (the i - 1 player) to continue its search. fl=8 fl= 9 fl=4 m=-6 m=-9 fo=-5 Ifl+mi52 Figure 6: An example of pruning performed by cup*. Figure 6 shows a search tree where every leaf I sat- isfies the bound constraint Ifi + f*(a)1 5 2. This bound allows the player to perform a cutoff of branch g, knowing that the value of node c for the opponent will be at least -5. Therefore, its value will be at most 7 for the player. The ap* algorithm is listed in figure 7. The following theorem proves that ap* always returns the M* value of the root. Theorem 4 Let V = cup* (pos, d, P, (-00, . . . , -00)). Let V’ = ll!f* l-pasJ(p~~, d, P’) where P’ is P without the bounds. Assume that for any leaf d of the game tree 2~j3* computes the bound B for each node. However, a table of the B values can be computed once at the begin- ning of the search, since they depend only on the B, and the depth. Procedure a/3*(pos, d, ((fn, bn)(. . . , (fo, bo)) . . .>>(@n, ” , ao)) iJ,“,= 0 then return (fn(po8), , fo(pos)) B + ComputeBounds(d, (bn, . . , bo)) v + (-00,. . . , -00) s + a(pos) for each s E S succ-V + @*(s,d - 1, ((fn, bn)(. . , (fo, a,>>. .>)(a,, . . , (Yo)) loop for each i associated with current player if succ-V[i] > V[i] then V[i] + succ-V[i] if i < n then V[i + l] + succ-V[i + l] 0, + max(Qr, V[i]) if for every i not associated with current player [a, 2 B[i] - V[i - l]] then return (V[n], , V[dl) return (V[n], . , . , V[d]) Procedure ComputeBounds( 4 (bn, . . ifd= 0 then return (bn, . . , bo) ,bd) else succ_B + ComputeBounds( d - 1, (b, , . . , bo) ) loop for each i associated with current player B[i] t succ-B[i] + 2 succ-B[i - l] if i < n then B[i + l] + succ-B[i + l] return B Figure 7: The cup* algorithm spanned from position pos to depth d, 1 fi (a)+ fiel (/)I 5 bi. Then, V = V’. As the player function becomes more similar to its opponent model (but with an opposite sign), the amount of pruning increases up to the point where they use the same function where cup* prunes as much as cup. Experimental study: The potential benefit of M* We have performed a set of experiments to test the potential merit of the M* algorithm. The experiments involve three players: MSTAR, MM and OP. MSTAR is a one-level modeling player. MM and 0P are regular minimax players (using a/? pruning). The experiments were conducted using the domain of checkers. Each tournament consisted of 800 games. A limit of 100 moves per game was set during the tournament. In the first experiment all players searched to depth 4. MM and M* used one function while 0P used an- other with equivalent power. MSTAR knows its op- ponent’s function while MM implicitly assumes that its opponent uses the opposite of its own function. Both MSTAR and MM, limited by their own search depth, wrongly assume that OP searches to depth 3. The following table shows the results obtained. Wins Draws Losses Points MM vs. OP 94 616 90 804 MSTAR vs. OP 126 618 56 870 The first row of the table shows that indeed the two evaluation functions are of equivalent power. The sec- 124 Agents ond row shows that MSTAR indeed benefited using the extra knowledge about the opponent. In the second experiment all players searched to depth 4. The three players used the same evalua- tion function f(b,pl) = Mat(b,pd) - & . Tot(b) where Mat(b, pa) returns the material advantage of player pl and Tot(b) is the total number of pieces. However, while M* knows its opponent’s function, MM implic- itly assumes that its opponent, o, uses the function f(b, 0) = -f(b,pZ) = -Mut(b,pl) + & * Tot(b) = Mat (b, o) + & . Tot(b) . Therefore, while OP prefers ex- change positions, MM assumes that it prefers to avoid them. The following table shows the results obtained. Wins Draws Losses Points MM vs. OP 171 461 168 803 MSTAR vs. OP 290 351 159 931 This result is rather surprising. Despite the fact that all players used the same evaluation function, and searched to the same depth, the modeling program achieved a significantly higher score. We repeated the last experiment replacing the AI* rwpass algorithm with a@*and measured the amount of pruning. The bound used for pruning is jfr + fa 1 = pczt(b,pZ) - & * Tot(b) + Aht(b,o) - & * Tot(b)1 5 2 . Tot(b). While the average number of leaves-per- gave for a search to depth 4 by MT- ass was 723, ap* managed to achieve an average of lgf leaves-per-move. (w/? achieved an average of 66 leaves-per-move. The last results raise an interesting question. As- sume that we allocate a modeling player and an un- modeling opponent the same search resources. Is the benefit achieved by modeling enough to over- come the extra depth that the non-modeling player can search due to the better pruning? We have tested the question in the context of the above ex- periment. We wrote an iterative deepening versions of both a@ and ap* and let the two play against each other with the same limit on the number of leaves. Wins Draws Losses Points al3* vs. OP 226 328 246 780 This result is rather disappointing. The benefit of modeling was outweighed by the cost of the reduced pruning. In general this is an example of the known tradeoff between knowledge and search. The question of when is it worthwhile to use the modeling approach remains open and depends mostly on the particular nature of the search tree and evaluation functions. Conclusions This paper describes a generalized version of the mini- max algorithm that can utilize different opponent mod- els. The iU* algorithm simulates the opponent’s search to determine its expected decision for the next move, and evaluates the resulted state by searching its asso- ciated subtree using its own strategy. Experiments performed in the domain of checkers demonstrated the advantage of A&* over minimax. Can the benefit of modeling outweigh the cost of reduced pruning? We don’t have a conclusive answer for that question. We expect the answer to be dependent on the particular domain and particular evaluation functions used. There are still many remaining open questions. How does an error in the model effect performance? How can a player use an uncertain model? How can a player acquire a model of its opponent? We tackled these questions but could not include the results here due to lack of space (a full version of the paper that in- cludes these parts is available as (Carmel & Markovitch 1996) .) In the introduction we raised the question of utilizing knowledge about the opponent’s decision procedure in adversary search. We believe that this work presents a significant progress in our understanding of opponent modeling, and can serve as a basis for further theoret- ical and experimental work in this area. eferences Berliner, H. 1977. Search and knowledge. In Proceed- ing of the International Joint Conference on Artificial Intelligence (IJCAI 77), 975-979. Carmel, D., and Markovitch, S. 1993. Learning models of opponent’s strategies in game playing. In Proceedings of the AAAI Fald Symposium on Games: Planning and Learning, 140-147. Carmel, D., and Markovitch, S. 1996. Learning and using opponent models in adversary search. Technical Report CIS report 9609, Technion. Gilboa, I. 1988. The complexity of computing best response Automata in repeated games. Journal of economic theory 45:342 -352. Gmytrasiewicz, P. J.; Durfee, E. H.; and Wehe, D. K. 1991. A decision theoretic approach to coordinating multiagent interactions. In Proceedings of the Twelfth International Joint Conference on Artificial Intelli- gence (IJCAI 91), 62 - 68. Jansen, P. 1990. Problematic positions and specu- lative play. In Marsland, T., and Schaeffer, J., eds., Computers, Chess and Cognition. Springer New York. 169-182. Korf, R. E. 1989. Generalized game trees. In Proceed- ing of the International Joint Conference on Artificial Intelligence (IJCAI 89), 328-333. Korf, R. E. 1991. Multi-player alpha-beta pruning. Artificial Intelligence 48, 99-111. Luckhardt, C. A., and Irani, K. B. 1986. An algorith- mic solution of n-person games. In Proceeding of the Ninth National Conference on Artificial Intelligence (AAAI-86), 158-162. Shannon, C. E. 1950. Programming a computer for playing chess. Philosophical Magazine, 41, 256-275. Negotiation & Coalition 125 | 1996 | 18 |
1,821 | Planning for Temporally Extended Fahiem Bacchus Dept. of Computer Science University of Waterloo Waterloo, Ontario Canada, N2L 3Gl Introduction One of the features that distinguishes intelligent agents is their flexibility: generally they have the ability to accomplish a task in a variety of ways. Such flexibility is necessary if the agent is to be able to accomplish a variety of tasks under a range of conditions. Yet this flexibility also poses a problem: how do we communicate to such an agent the task we want accomplished in a sufficiently precise manner so that it does what we really want. In the areaof planning, methods and algorithms are studied by which, given information about the current situation, an intelligent agent can compose its primitive abilities so as to accomplish a desired task or goal. The afore mentioned problem then becomes the problem of designing sufficiently expressive and precise ways of specifying goals. Much of the work in planning has dealt with goals specified as conditions on a final state. For example, we might specify * This work was supported by the by the Canadian government through their NSERC and IRIS programs. Fahiem Bacchus also wishes to thank the University of Toronto for hosting his sabbatical leave during which much of this work was accomplished. Froduald Kabanza Dept. de Math et Informatique Universite de S herbrooke S herbrooke, Quebec Canada, JlK 2Rl a goal as a list of literals. The intent of such goals is that the agent should find a plan that will transform the current situation to a configuration that satisfies all of the literals in the goal. Any plan that achieves such a satisfying final state is deemed to be correct. However, there are many important constraints we might wish to place on the agent’s behavior that simply cannot be expressed using these semantics for goals. The importance of specifying such constraints on the agent’s plans has been recognized. For example, Weld and Etzioni [WE941 present strong arguments for looking beyond the simple achievement of a final state, and suggest two additional constraints on plans, a notion of don’t-disturb and restore. In this paper we present a richer formalism for specify- ing goals that borrows from work in verification [MP92], and develop a planning algorithm for generating plans to achieve such goals. Our formalism suggests a different way of viewing goals in planning. Instead of viewing goals as characterizing some set of acceptable final states and a plan as being correct if it achieves one of these states, we will view a goal as specifying a set of acceptable sequences of states and a plan as being correct if its execution results in one of these sequences. As we will show our formalism for goals subsumes the suggestions of Weld and Etzioni, except that instead of viewing don’t-disturb and restore as constraints on plans, we view them as simply being additional goals. Our formalism allows us to specify a wide range of tem- porally extended goals. This range includes classical goals of achieving some final state: goals with temporal deadlines; safety and maintenance goals like those discussed by Weld and Etzioni and others [HH93]; and quantified goals (both universally and existentially quantified). Furthermore, our formalism is a logical language that carries with it a precise, and quite intuitive, semantics. This latter is important, as without a precise semantics for our goals we will not be able to analyze and verify exactly what it is our agents will be accomplishing. Temporally extended goals have previously been exam- ined in the literature. Haddawy and Flanks [HH93] have provided utility models for some types of temporally ex- tended goals. Kabanza et al. [Kab90, GK9 1, BKSD95] have developed methods for generating reactive plans that achieve temporally extended goals, as has Drummond [Dru89]. Plan- Temporal Reasoning 1215 From: AAAI-96 Proceedings. Copyright © 1996, AAAI (www.aaai.org). All rights reserved. ning systems and theories specifically designed to deal with temporal constraints (and sometimes other metric resources) have also been developed [Ver83, Wl188, AKRT91, CT91, Lan93, PW94]. The main difference between these previous works and what we present here, lies in our use of a temporal logic that supports a unique approach to computing plans, an approach based on formula progression. The method of formula pro- gression lends itself naturally to the specification and uti- lization of domain dependent search control knowledge. As shown in our previous work [BK95], the approach of do- main dependent search control offers considerable promise, and has motivated our approach to dealing with temporally extended goals. The other works that have constructed tem- poral planners have utilized complex constraint management techniques to deal with temporal information. In [Kab90, GK9 l,lBKSD95] similar temporal logics and similar notions of formula progression have been utilized. In this case the main difference is that here we address classical plans, i.e., finite sequences of actions, while these works have concentrated on generating reactive plans, i.e., mappings from states to actions (sometimes called universal plans). Reactive plans have to specify an on-going interac- tion between an agent and its environment, and thus pose a quite distinct set of problems. To generate plans that achieve the goals expressed in our formalism we present a planning algorithm that uses the logical mechanism of formula progression. This notion was previously utilized in our TLPLAN system [BK95]. In fact we have implemented the planning algorithm by extending the TLPLAN system. TLPLAN is planning system whose key feature is that it is able to utilize domain dependent search control information.. This control is expressed in a temporal logic that is a limited form of the logic presented here, and it is utilized by the planner via the mechanism of formula progression. The planning algorithm we develop is sound and complete and as we will demonstrate it is able to generate a range of interesting plans. Further work is required, however, to evaluate the planner’s performance on realistic planning problems. In the rest of the paper we will first provide the details of the logic we propose for expressing goals. This logic is a tempo& logic that is based on previous work by Alur et al. [AFH91]. We then present our approach to planning, provide examples to demonstrate the range of goals that our system can cope with, and discuss the heuristic adequacy of our approach to planning. Finally, we close with some conclusions and discussion of future work. Expressing goals in MlTL We use a logical language for expressing goals. The logic is based on Metric Interval Temporal Logic developed by Alur et al. [AFH91], but we have extended it to allow first-order quantification. Syntax We start with a collection of n-w predicate (including equal- ity and the predicate constants TRUE and FALSE) function and constant symbols, variables, and the connectives 1 (not) and A (and). We add the quantifiers Y and 3 and the modal operators 0 (next) and U (until). From this collection of symbols we generate MITL, the language we use to express goals. MlTL is defined by the traditional rules for gener- ating terms, atomic formulas, and Boolean combinations, taken from ordinary first-order logic. In addition to those formula formation rules we add: (1) if 4 is a formula then so is 04; (2) if & and & are formulas and I is an interval then so is & Ur & (the syntax of intervals is defined below); and (3) if a(z) is an atomic formula in which the variable d: is free, and 4 is a formula then so are ~[z:Q(z)] 4, and 3[2:c+)] q5* Notice that in our language we use bounded quantification. The atomic formula ar is used to specify the range over which the quantified variable ranges. The precise semantics are given below. The syntax of intervals is as one would expect. The al- lowed intervals are all intervals over the non-negative real line, and we specify an interval by giving its two endpoints, both of which are required to be non-negative numbers. To allow for unbounded intervals we allow the right endpoint to be 00. For example, [0, 00) specifies the interval of num- bers a: such that 0 5 2, (5.1,6. l] specifies the interval 5.1. < x < 6.1, and [5,5] specifies the interval 5 < x 5 5 (i.e., the point x = 5). Although non-negative interva.ls are the only ones allowed in the formulas of MITL, in the semantics and algorithms we will need to utilize shifted intervals and to test for negative intervals. For any interval I, let I + T be the set of numbers x such that x - T E I, I - T be the set of numbers x such that x + T E I, and I < 0 be true iff all numbers in I are less than 0. For example, (5,001 + 2.5 is the new interval (7.5, oo), (0,2) - 2.5 is the new interval (-2.5, -0.5), and (-2.5, -0.5) < 0 is true. Finally, we introduce 3 (implication), and v (disjunction) as standard abbreviations. We aIso introduce the temporal modalities eventually 0 and always 0 as abbreviations with W f TRUE eS, 4, and 014 q TO&J. We will also abbreviaeeintervalsoftheform (T, 00) and [0, T),e.g., O(+,,) will be written as O,,. and • L~,~I as 00. Finally, we will often omit writing the interval [0, oo],k.g., we will write 41qo,fxl]42~ 44 62.l Semantics We intend that goals be expressed as sentences of the lan- guage MITL. As hinted in the introduction such formulas are intended to specify sets of sequences of states. Hence, it should not be surprising that the underlying semantics we as- sign to the formulas of MITL be in terms of state sequences. ‘The temporal modalities with the interval [0, CQ] correspond precisely to the traditional untimed modalities of Linear Temporal Logic [EmegO]. 1216 Planning A model for MITL is a timed sequence of states, M = bo 7”‘7 sn , . . .). In particular, a model is an infinite se- quence of states, and each state is a first-order model over a fixed domain D. That is, each state si assigns a denotation for each predicate and function symbol over the domain 6). Furthermore, there is a timing function 7 that maps each state sa in M to a point on the non-negative real line such that for all i, T(si) 5 T( si+l) and for all real numbers r there exists an i such that T(si) > r. This means that time is only required to be non-decreasing, not strictly increasing. Time can stall at a single point for any finite number of states. Eventually, however, time must increase without bound. Let V be a variable assignment, i.e., a mapping from the variables to elements of D; 4, &, and 42 be formulas of MITL; and M be an MITL model. The semantics of MlTL are then defined by the following clauses. e (M, si, V) j= 4, when 4 is atemporal (i.e., contains no temporal modalities) and quantifier free, iff (s; , V) b 4.2 * (4 si, V) I= 04 iff(W si+l, V) I= 4. e (M, si, V) + 41 UI ~$2 iff there exists sj with T(s~) E I + T( si) such that (M, sj , V) k #Q and for all Sk with i<k<jwehave(M,sk,V)+&. e (M, si, V) k V[x:a( x)] 4 iff for all d E D such that (si, V(x/d)) I= a(x) we have (4 si, V(x/d)) l= 4 e (M, si 9 V) j= 3[x:a(x)] 4 iff there exists d E D such that (Si, V(x/d)) I= a(X) ad (M, si 3 V(x/d)) I= d- It is not difficult to show that any formula of MITL that has no free variables, called a sentence of MITL, has a truth value that is independent of the variable assignment V. Given a sentence 4 of MITL we say it is true in a model M, M k 4, iff (W SO) I= 9. Since sentences of MITL are either true or false on any individual timed sequence of states, we can associate with every sentence a set of sequences: those sequences on which it is true. We express goals as sentences of MITL, hence we obtain our desired semantics for goals: a set of acceptable sequences. Discussion Intuitively, the temporal modalities can be explained as fol- lows. The next modality 0 simply specifies that something must be true in the next state. Its semantics do not depend on the time of the states. It is important to realize, however, that what it requires to be true in the next state may itself be a formula containing temporal modalities. MITL gets its expressive power from its ability to nest temporal modalities. The until modality is more subtle. The formula & U15,~l c$~, for example, requires that 42 be true in some state whose time is between 5 and 7 units into the future, and that dl be true in all states until we reach a state where #Jo is true. The eventually modality thus takes on the semantics that 014 requires that 4 be true in some state whose time lies in the 2Note that 8; is a first-order model, so the relationship “(SiyV) k 4" d fi d is e ne according to the standard rules for first- order semantics. interval I, and 014 requires that 4 be true in all states whose time lies in I. Turning to the clauses for the bounded quantifiers we see that the range of the quantifier is being restricted to the set of domain elements that satisfy ar. If Q is true of all domain individuals, then the bounded quantifiers become equiva- lent to ordinary quantification Similarly, we could express bounded quantification with ordinary quantifiers using the syntactic equivalences ‘v’[x:a( x)] 4 q VX.CU( x) 3 4 and 3[xc:a(x)] gt z 3x.a(x) A 4. We have defined MITL to use bounded quantification because we will need to place finite- ness restrictions on quantification when we do planning. lanning Planning Assumptions and Restrictions Now we turn to the problem of generating plans for goals expressed in the language MITL. First we specify the as- sumptions we are making. (1) We have as input a complete description of the initial state. (2) Actions preserve this completeness. That is, if an action is applied to a completely described state, then the resulting state will also be com- pletely described. (3) Actions are deterministic; that is, in any world they must produce a unique successor world. (4) Plans are finite sequences of actions. (5) Only the agent who is executing the plan changes the world. That is, there are no other agents nor any exogenous events. (6) All quantifier bounds, i.e., the atomic formulas a(x) used in the defmi- tion of quantified formulas, range over afinite subset of the domain. These assumptions allow us to focus on a particular exten- sion of planning technology. They are essentially the same assumptions as made in classical planning. For example, the assumption that actions preserve completeness is implied by the standard STRIPS assumption. It is possible to weaken our assumptions of completeness. Incomplete state descriptions will suffice as long as they are complete enough to (1) determine the truth of the precon- ditions of every action and (2) determine the truth of all atemporal subformulas of the goal formula. The price that is paid however is efficiency, instead of a database lookup, theorem proving may be required to determine the truth of these two items. However, more conservative notions of in- completeness like locally closed worlds [EGW94] could be utilized in our framework without imposing a large compu- tational burden. Also, it should be made clear that restricting ourselves to deterministic actions does not mean actions cannot have con- ditional effects. In fact, the planner we implemented handles full ADL conditional actions [Ped89] including actions with disjunctive and existentially quantified preconditions. Han Correctness Given a goal g expressed as a sentence of MlTL we want to develop a method for generating plans that satisfy g* Sen- tences of MITL are satisfied by the timed state sequences described above. Hence, to determine whether or not a plan satisfies g we must provide a semantics for plans in terms of the models of MITL. Temporal Reasoning 1217 IIIPU~S: A state s;, with formula label 4, and a time duration A to the successor state. Output: A new formula 4+ representing the formula label of the successor state. Algorithm Progress(&s;,A) Case 1. C/I contains no temporal modalities: if& j=tj (b+ :=TRuE else 4+:=FALSE 2. q5 = $1 A& 4+ := Progress(& , 3; , A) A Progress( 42 9 si 3 A) 3. c$=T#J,: (b+ := 4Yogress(&, si, A) 4. $=O#,: ++ :=t#q 5. 4 =41 U&2: ifI< ~+:=FuE elseif E I 4+ := Progress(&, si7 A) v (Progress(~l,si,A)~~l UI-A 42) else Progress(&,si,A) ~41 UI-A 42 6. 4 = V[~:~] 41: (5+ := l\~c:sico~c~3 progres$h(z/c), si, A) 7. # = 3[~:~]#1: 4+ := Vj,:,iCatcJj progress(h(~/c), si, A) Table 1: The progression algorithm. Since actions map states to new states, any finite sequence of actions will generate a finite sequence of states: the states that wouldarise as the plan is executed. Furthermore, we will assume that part of an action’s specification is a specification of its duration, which is constrained to be greater than or equal to 0. This means that if we consider so to commence at time 0, then every state that is visited by the plan can be given a time stamp. Hence, a plan gives rise to a finite timed sequence of states-almost a suitable model for MITL. The only difficulty is that models of MITL are infinite sequences. Intuitively, we intend to control the agent for some finite time, up until the time the agent completes the execution of its plan. Since we are assuming that the agent is the only source of change, once it has completed the plan the final state of the plan idles, i.e., it remains unchanged. Formally, we define the MITL model corresponding to a plan as follows: Definition 1 Let plan P be the finite sequence of actions tal , . . . , a,). Let S = (so, . . . , sn) be the sequence of states such that si = ai (si- 1), and so is the initial state. S is the sequence of states visited by the plan. Then the MITL model corresponding to P and SO is defined to be lso 7*--Y %7&t*.- }, i.e., S with the final state s, idled, where 7’(si) = T(si- 1) + duration(ai), 0 < i 5 n, T( SO) = 0, and the time of the copies of sn increases without bound. Therefore, every finite sequence of actions we generate corresponds to a unique model in which the final state is idling. Given a goal expressed as a sentence of MITL we can determine, using the semantics defined above, whether or not the plan satisfies the goal. Definition 2 Let P be a plan, g be a goal expressed as a formula of MITL, SO be the initiaI state, and M be the model corresponding to P and SO. P is a correct pEan for g given soiffM kg. 1218 Planning Generating Plans We will generate plans by adopting the methodology of our previous work [BK95]. In particular, we have constructed a forward-chaining planning engine that generates linear se- quences of actions, and thus linear sequences of states. As these linear sequences of states are generated we incremen- tally check them against the goal. Whenever we can show that achieving the goal is impossible along a particular se- quence we can prune that sequence and all of its possible extensions from the search space. And we can stop when we find a sequence that satisfies the goal. The incremental checking mechanism is accomplished by the logical progres- sion of the goal formula. Formula Progression The technique of formula progres- sion works by labeling the initial state with the sentence representing the goal, call it 9. For each successor of the initial state, generated by forward chaining, a new formula label is generated by progressing the initial state’s label us- ing the algorithm given in Table 1. This new formula is used to label the successor states. This process continues. Every time a state is expanded during planning search each of its successors is given a new label generated by progression. Intuitively a state’s label specifies a condition that we are looking for. That is, we want to find a sequence of states starting from this state that satisfies the label. The purpose of the progression algorithm is to update this label as we extend the state sequence. It takes as input the current state and the duration of the action that yields the successor state. The logical relationship between the input formula and output formula of the algorithm is characterized by the fol- lowing proposition: Proposition 3 Let M = (SO, ~1, . . .) be any MITL model. Then, we have for any formula C$ of MITE, (M, si) b 4 if and only if(M, si+l) k hgress(~, si, 7(si+l) - ‘T(s~)). This proposition can easily be proved by utilizing the def- inition of MITL semantics. Say that we label the start state, so, with the formula 4, and we generate new labels using the progression al- gorithm. Furthermore, say we find a sequence of states, s = (s, 2, s2 , . . .), starting at state s that satisfies S’S label. Then a simple induction using Proposition 3 shows that the sequence leading from SO to s followed by the sequence S, i.e., (so,. . . , S, 2, s2,. . .), satisfies 4. The progression al- gorithm keeps the labels up to date: they specify what we are looking for given that we have arrived where we are. From this insight we can identify two important features of the formula progression mechanism. First, if we find any state whose idling satisfies its label, we have found a correct ph. Proposition 4 Let (SO, ~1, . . . , sn) be a sequence of states generated by forward chainingfrom the initial state so to sn. For each state si let its label be l(si). Let the labels of the states be computed via progression, i.e., for each state si in the sequence e(Si+l) = prOgreSS(t(Si),Si, T(Si+l) - T(Si))* Inputs: A state s, and a formula 4. Output: True if the state sequence (5, s, . . .), where time increases without bound, satisfies C#J. False otherwise. Algorithm Idle(&s) Case 1. 4 contains no temporal modalities: ipe” 4 return TRuE return FALSE 2. 4 = $1 A&: return Idle( 41, s) A Idle( 42, s) 3. c#c+,: return 4dle(& , s) 4. i$=oc$,: return Idle(& , S) 5. (b=(hu142: ifI<O RtUrU FALSE eke if 0 E I return Idle(&, S) else return Idle( &, s) A Idle( 42, s) 6. 4 = V[=a]&: return A~c,,~~,,,Idle(~l(~/c),s) 7. 4 = 3[=+#~: returnV~e:.C~(c,~Idle(~l(elc),s) Table 2: The idling algorithm. ThenM=(so ,,..., sn,sn ,...) /=e(sO)ifS(s,,s,,...) b +?a>. The proof of this proposition follows directly from Propo- sition 3. Since .t(so) is a formula specifying the goal, this propo- sition shows that the plan leading to s, satisfies the goal. Hence, if we have a method for testing for any state s and any formula 4 E MITL whether or not (s, s, a. *> + 4, we have a termination test for the planning algorithm that guarantees soundness of the algorithm. We will describe an appropriate method below. Furthermore, as long as the search procedure used by the algorithm eventually examines all finite sequences of states the planning algorithm will also be complete. The second feature of formula progression is that it allows us to prune the search space without losing completeness. As we compute the progressed label we simplify it by processing all TRUE and FALSE subformulas. For example, if the label 4 A TRUE is generated we simplify this to 4. If any state receives the label FALSE we can prune it from the search space, thus avoiding searching any of its successors. From Proposition 3 we know that this label specifies a requirement on the sequences that start at this state. No sequence can satisfy the requirement FALSE, hence no sequences starting from this state can satisfy the goal and this state and its successors can be safely pruned from the search space. Termination As indicated above, we can detect when a plan satisfies the goal if we can detect when an idling state satisfies its label. This computation is accomplished by the algorithm given in Table 2. Proposition 5 Ide( 4,s) returns TRUE if and only if ( s, s, o . .) b qk That is, Idle detects ifan idling state satisfies a formula. The Planning Algorithm Given the pieces developed in the previous sections we specify the planning algorithm pre- sented in Table 3. The algorithm labels the initial state with the goal and searches among the space of state-formula pairs. We test for termination by running the Me algorithm on the Inputs: An initial state so, and a sentence g E MITL specifying the goal. Returns: A plan P consisting of finite sequence of actions. Algorithm Pla.n(g,s) 1. (&en +- ((9, so)). 2. While Open is not empty. 2.1 (4, s) t Remove an element of Open. 2.2 if Idle(4, s) Return ((4,s)). 2.3 Successors t Expand(s). 2.4 For all (s+ ?a) E Successors 2.4.1 c$+ t Progress(#+ s,duration(a)). 2.4.2 if $+ # FALSE 2.4.2.1 Parent((4+, s+)) t(4, s). 2.4.2.2 Open tOpen U {(s+,q3+)}. Table 3: The planning algorithm. state’s formula. To expand a state-formula pair we apply all applicable actions to its state component, returning all pairs containing a successor state and the action that produced that state (this is accomplished by Expand(s)). We then compute the new labels for those successor states using the Progress algorithm. It should be noted that we cannot treat action sequences that visit the same state as being cyclic. If we are only looking for a path to a final state, as in classical planning, we could eliminate such cycles. Goals in MITL, however, can easily require visiting the same state many times. Nevertheless, we can view visiting the same state-formula pair as a cycle, and optimize those-cycles using the standard techniques.3 Intuitively, when we visit the same state-formula node we have arrived at a point in the search were we are searching for the same set of extensions to the same state. Proposition 6 The planning algorithm is sound and com- plete. That is, it produces a plan that is correctfor g given SO (Definition 2), and so long as nodes are selected from Open in such a manner that every node is eventually selected, it willfind a correct plan if one exists. This proposition follows from the tion test (Proposition 4). soundness of our termina- We have implemented the planning algorithm as an ex- tension of the TLPLAN system [Bac95]. This allowed us to utilize many of the features already built into the TLPLAN system, including full support of the ADL formalism [Ped89] for specifying actions. Example and Empirical Results Types of Goals The domain we used is a variant of the classical STRIPS robot rooms domain [FN71]. The configuration of the rooms is illustrated in Figure 1. In this domain there are objects and a robot, which can be located at any of the 2 locations in the corridor, Cl or C4, or any of the 4 rooms Rl, . . . , R4. The robot can move between connected locations, it can 3For example, we can eliminate that node or search from it again if the new path we have found to it is better than the old path. These considerations will determine how we decide to set Parent((d+, a+)) in step 2.4.2.1 Temporal Reasoning 1219 T Precondition 1 Adds I Deletes * open(?d) close(?d) grasp( ?o) release( ?o move(?x, ?y) at robot, ?a: connects(?d, ?xG, ?y) closed(?d) door at(robot, ?x) connects(?d, ?x, ?y) opened door at(robot, ?x) at(?o, ?x) handempty object(?o) holding (?o) at(robot, ?x) connects(?d, ?x, ?y) opened opened closed ?d cZosed(?d) opened holding(?o) handempty handempty hoZdzng(?o) at(robot, ?y) at(robot, ?x) holding(?o) holding(?o) =9 at(?o, ?y) * at(?o, ?x) Table 4: Operators for Robot Room domain. open and close doors (indicated as gaps in the walls), and it can grasp and carry one object at a time. The operators that characterize its capabilities are shown in Table 4. In this table variables are preceded by a question mark “?“. Also, the move operator is an ADL operator with conditional effects. For all objects that the robot is holding it updates their position. This is indicated in Table 4 by the notation fi + e in the add and delete columns: the literal e is added or deleted if fr holds. The duration of most of the actions is set to 1. Our implementation allows us to set the duration of an action to be dependent on the instantiation of its parameters. In particular, we set the duration of move (2, y) to be 1, except for move(C1, C4) which has duration 3. Any initial state for this domain must specify the location of the robot and the existence and location of any objects in the domain. It must also specify whether each door is opened’ or closed. The doors connect the rooms to each other and to the corridor locations, and a set of connects relations must be specified, e.g., connects(D1, Cf, Rl). Door DP connects the corridor location Cl and Rl, door 04 connects C4 and R4, and the doors D;j connect rooms Ri and Rj (6 j E -L&3)). Finally, the two corridor locations are connected by a “corridor” which is always “open”. So literals of the form connects(corridor, CI, C4), and opened(corridor), must also be present in the initial state description. I I I 1 I Rl I R2 I R3 I R4 I Cl c4 Figure 1: Robot Room domain Classical Goals: Classical goals can easily be encoded as untimed eventualities that hold forever. For example, the classical goal {at(robot, Cl), at(obj1, R4)) expressed as a set of literals, can be encoded as the MITL formula 00 (at(robot, Cl) A at(obj1, R4)). Any classical goal can II be encoded in this manner. Given the semantics of plans as idling their final state, this formula will be satisfied by a plan only if the final state satisfies the goal. More generally we can specify a classical “achieve a fi- nal state” goal by enclosing any atemporal formula of our language in an eventuality. We can specify disjunctive goals, negated conditions, quantified goals, etc. The formula O(El[z:object(z)] at(z, R4) V at(robot, R4)), for example, specifies the goal state where some object or the robot is in room R4. Safety and Maintenance Goals: In lJVE94] Weld and Et- zioni discuss the need for safety conditions in plans. Such conditions have also been studied in the verification literature [IvlP921. MITL can express a wide range of such conditions. Maintenance goals (e.g., [KEI93# which involve keeping some condition intact, are very similar. Weld and Etzioni propose two specific constructions, don’t-disturb and restore, as a start towards the general goal of expressing safety conditions. Both of these constructions are easily encoded as goals in MITL. Don’t-disturb specifies a condition #(z). A plan is defined to satisfy a don’t-disturb condition if dluring its execution no instantiation of d(z) changes truth value. Such conditions are easily specified by conjoining the formula VZ.~(Z) + 04(x) to the original goal4 For example, the god OO(at(robot, Cl) A at(obj1, R4)) A V[z:opened( z)] Oopened(z), can only be satisfied by a plan that does not disturb any open doors. Restore also specifies a condition 4(z). A plan satisfies a restore condition if it tidies up after it has finished. That is, at the end of its plan it must append a new plan to restore the truth of all instantiations of 4 (2) that held in the initial state. We can specify restore goals in MITL by conjoining the formula VZ.~(Z) =P 004(z), which specifies that the final state of the plan must satisfy all instantiations of 4 that held 4We must app ro p riately rewrite VX.C#I(X) in terms of bounded quantification. Also it is not difficult to see that multiple variables in C#J can be handled by additional quantifiers. Similar remarks hold for encoding restore. 1220 Planning in the initial state.5 Notice that the semantic distinction between restore and don’t-disturb goals is made clear by our formalism. Restore goals use 00 while don’t-disturb goals use 0. That is, restore goals allow the violation of 4 during the plan, as long as these conditions are eventually restored in the final state. Both of these conditions are limited special cases. MITL can express much more than this. For example, say that we want to constrain the robot to close doors that it opens. We cannot place a don’t-disturb condition closed(z), as this would prohibit the robot from moving into rooms where the doors are closed. If we specify this as a restore condition, the robot might leave a door opened for a very long time until it has finished the rest of its plan. In MITL, however we can use the formula 0 (V[z, y, z:connects(z, 2, y)] (1) at(robof, 2) A closed(z) A Oopen(z) j OOat(robot, y) A OOOclosed(z)) This formula specifies that if the robot opens a closed door (closed(x) A O(open(z))), then it must go through the door (OOat(robot, 3)) and then it must close the door (OOOcZosed(z)). Hence, the robot is forced to be tidy with respect to doors: it only opens doors for the purpose of mov- ing through them, and it closes the doors it opens behind it. Timing Deadlines: MlTL is also capable of expressing goals with timing conditions. For example 0 > 1od requires the condition 4 be achieved within ten time units. Empirical Results We have tested different goals from each of the cate- gories mentioned above. Most of the plans were generated from the initial state in which at(objl, Rl), at(obj2, R2), at(robot, Cl), handempty, object(objl), object(obj2), and all of the doors are opened. G1: From this initial state we set the goal to be OO(at(robot, Cl) A at(objl, R2)). This corre- sponds to the classical goal (at(robot, Cl), at(objl, R2)). The planner generates the plan: move(C1, Rl), gmsp(objP), move(R1, R2), reIease(objl), move(R2, Rl), move (RP, Cl). It took the planner 22 sec., expanding 636 worlds to find this ~lan.~ G2: From the same initial state we set the goal to be 00(3[z:object(z)] at(z, R3) A handempty). Now theplan- ner generates the plan: move(Cf, Rl), move(R1, R2), grasp(O2), move(R2, R3), rejease(02). In this case it has generated a plan for a quantified goal. This plan takes the planner 3 sec., expanding 126 worlds to find the plan. 5 When we add this formula as a conjunct to the original goal we force the planner to find a plan that satisfies the restore. If we want to give restore conditions lower priority, as discussed in m94], we could resort to the techniques of replanning suggested there. ‘Timings are taken on a SPARC station 20, and a breadth first strategy was used so as to find the shortest plans. G3: Now we change the initial state so all of the doors are closed. We set the goal to be Oa(at(robot, Cl) A at(objl, R2)) conjoined with Formula 1. This is simply a classical goal with an additional constraint on the robot to ensure it closes doors behind it. For this goal the planner generates the plan open(Dl), move(C1, Rl), close(Dl), grasp(Ol), open(D12), move(R1, R2), close(D12), reIease(Ol), open(D12), move(R2, Rl), close(D12), open(Dl), move(R1, Cl), close(D1). This plan took the planner 77 sec., expanding 1571 worlds, to find. 64: We reset the initial state to one where all of the doors are open and set the goal to be a>aoat(objl 9 R4) A q >,at(obj2, R3) A V[z:opened(z)] Oop&ed(z). This is a gc%l with a tight deadline. The robot must move directly to 622 and move obj2 to R3. If it stops to grasp objP along the way it will fail to get obj2 into R3 on time. Also we conjoin a subgoal of not closing any open doors. As we will discuss below this safety constraint acts as a form of search con- trol, it stops the planner pursing useless (for this goal) close actions. The planner generates the plan: move(C1, Rl), move(R1, R2), grasp(O2), move(R2, R3), release(O2), move(R3, R2), move(R2, Rl), grasp(Ol), move(R1, R2), move (R2, R3), move (R3, R4). This plan took the planner 8 sec., expanding 284 worlds, to find. G5: If we change the time deadlines in the previ- ous goal and set the goal it to be O>gat(objl, R4) A q >,,at(obj2, R3) A V[z:opened(z)] oop&zed(z) Theplan- ne? generates the plan: move(C1, Rl), gmsp(Ol), move(R1, R2), move(R2, R3),move(R3, R4), release(Ol), move(R4, R3), move(R3, R2), grasp(O2), move( R2, R3). It took the planner 120 sec. to find this plan, expanding 1907 worlds on the way. Search Control Although our planner can generate an interesting range of plans, by itself it is not efficient enough for practical prob- lems. For example, when it is only given the goal of achiev- ing some final state, it has to resort to blind search to find a plan. Similarly, it has no special mechanisms for planning for quantified goals, it simply searches until it finds a state satisfying the goal. Safety goals offer better performance, as such goals prune the search space of sequences that falsify them. This is why we included safety conditions on open doors in the fourth and fifth tests above: they allow the plan- ner to find a plan faster. Again for goals with complex timing constraints, the planner does not utilize any special temporal reasoning. The major advantage of our approach lies in the ability of the planner to utilize domain dependent search control information. Such information can be expressed as formulas of MITL and conjoined with the goal. We have explored this approach to search control in [BK95] where we demonstrate that is often possible to construct polynomial time planners using quite simple search control knowledge. We know of no other approach to increasing the efficiency of planners Temporal Reasoning 1221 that has been able to produce polynomial time behavior in these domains. As a simple illustration of the power of this using search control consider the following trivial search control formula: 0 ( V[z:at(robot, cc)] l(Olat(robot, cc) A OOat(robot, cc)) A V[z:object(z)] l(-holding(z) A OhoZding(x) A OOlhoZding(z))) If we conjoin this formula with any other goal, the planner will prune sequences in which (1) the robot grasps an object and then immediately releases it, and (2) the robot moves away from a location and then immediately moves back. For this domain these sequences serve no purpose even in plans where the robot must visit the same state more than oncee7 Conjoining this formula with the example goals given above we obtain the following speedups. 0 Example 1 Time 1 World 1 New-Time 1 New-Worlds 0 The columns give the planning time and the number of worlds expanded, before and after we add the search control formula. Note in particular, the speedups obtained on the harder prob- lems. Furthermore, it should be noted that this is only the simplest and most obvious of control formulas for this do- main. References [AFH91] Rajeev Alur, Tomas Feder, and Thomas Henzinger. The benefits of relaxing punctuality. In Tenth Annual ACM Symposium on Principles of Distributed Comput- ing (PODC I991), pages 139-152,199l. [AKRT91] J. Allen, H. Kautz, Pelavin R., and J. Tenenberg. Rea- soning about Plans. Morgan Kaufmann, San Mateo, CA, 1991. [Bac95] Fahiem Bacchus. Tlplan (version 2.0) user’s manual. Available via the URL ftp://logos.uwaterloo.ca:/pub/bacchus/tlplan- manual.ps.Z.1995. [BK95] Fahiem Bacchus and Froduald Kabanza. Using tem- poral logic to control search in a forward chaining planner. l[n Proceedings of the 3rd European Work- shop on Planning, 1995. Available via the URL ftp://logos.uwaterloo.ca:/pub/tlplan/tlplan.ps.Z. [BKSD95] M. Barbeau, F. Kabanza, and R. St-Denis. Synthesizing plant controllers using real-time goals. h Proc. Thir- teenth International Joint Conference on Artificial In- telligence (IJCAI ‘95), pages 79 l-798,1995. [CT911 K. Currie and A. Tate. O-plan: the open planning architecture. Artificial Intelligence, 52:49-86,199 1. ‘In general, in or der to achieve some timed goals we may need to allow the robot to wait. But, in that case it is more effective to introduce a specific wait action and still outlaw pointless cycles. 1222 Planning D-u891 [EGW94] [EmegO] rm711 [GK9f] [HH93] [ Kab90] [Lan93] [RIP921 [Ped89] [PW94] [Sch87] [Ver83 J [WE941 WilSS] M. Drummond. Situated control rules. In Proc. First International Conference on Principles of Knowledge Representation and Reasoning (KR ‘89), pages 103- 113. Morgan Kaufmann, 1989. 0. Etzioni, K. Golden, and D. Weld. Tractable closed world reasoning with updates. In Principles of Knowl- edge Representation and Reasoning: Proc. Forth Inter- national Conference (KR ‘94), pages 178-189,1994. E. A. Emerson. Temporal and modal logic. In J. van Leeuwen, editor, Handbook of Theoretical Computer Science, Volume B, chapter 16, pages 997-1072. MIT, 1990. Richard Fikes and Nils Nilsson. Strips: A new ap- proach to the application of theorem proving to prob- lem solving. Artijcial Intelligence, 2: 189-208,197 1~ P. Godefroid and E Kabanza. An efficient reactive plan- ner for synthesizing reactive plans. %n Proc. National Conferenceon Artijkial Intelligence (AAAI ‘9P), pages 64%645,199l. P. Haddawy and S. Hanks. Utility models for goal- directed decision-theoretic planners. Technical Report 93-06-04, University of Washington, 1993. Technical Report. F. Kabanza. Synthesis of reactive plans for multi-path environments. In Proc. National Conference on Artifi- cial Intelligence (AAAI ‘90), pages 164-l 69,199O. A. Lansky. Localized planning with diversified plan construction methods. Technical Report T.R. FIA-93- 17, NASA Ames Research Center, 1993. Technical Report. Zohar Manna and Amir Pnueli. The temporal logic of reactive and concurrent systems: Specication. Springer-Verlag, New York, 1992. E. Pednault. ADL: Exploring the middle ground be- tween STRIPS and the situation calculus. In Proc. First International Conference on Principles of Knowledge Representation and Reasoning (KR ‘89), pages 324- 332,1989. J. Scott Penberthy and Daniel Weld. Temporal planning with continuous change. In Proc. National Conference onArtijkialIntelligence(AAAI’94), pages lOlO-1015. Morgan Kaufmann, 1994. M. J. Schoppers. Universal plans for reactive robots in unpredictable environments. In Proc. Tenth Interna- tional Joint Conference on Artificial Intelligence (IJ- CAI ‘87), pages 1039-1046,1987. S. Vere. Planning in tune: Windows and durations for activities and goals. IEEE Trans. on Pattern Analysis and Machine Intelligence, 5,f983. Daniel Weld and Oren Etzioni. The 6rst law of robotics (a call to arms). In Proc. National Conference on Arti- ficial Intelligence (AAAI ‘94), pages 1042-1047,1994. D. Wilkins. Practical Planning. Morgan Kaufmann, San Mateo, CA, 1988. | 1996 | 180 |
1,822 | A Cost-Directed Planner: Preliminary Report Eithan Ephrati Martha E. Pollack Marina Milsht ein AgentSoft Ltd. and Computer Science Department Computer Science Department Department of Mathematics and Intelligent Systems Program University of Pittsburgh and Computer Science University of Pittsburgh Pittsburgh, PA 15260, USA Bar Ilan University Pittsburgh, PA 15260, USA marisha@cs.pitt.edu Ramat Gan 52900, ISRAEL pollack@cs.pitt.edu tantushQsnnlight.cs.biu.ac.il Abstract We present a cost-directed heuristic planning al- gorithm, which uses an A* strategy for node se- lection. The heuristic evaluation function is com- puted by a deep lookahead that calculates the cost of complete plans for a set of pre-defined top-level subgoals, under the (generally false) as- sumption that they do not interact. This ap- proach leads to finding low-cost plans, and in many circumstances it also leads to a significant decrease in total planning time. This is due in part to the fact that generating plans for subgoals individually is often much less costly than gen- erating a complete plan taking interactions into account, and in part to the fact that the heuristic can effectively focus the search. We provide both analytic and experimental results. Introduction Most of the work on search control for planning has been based on the assumption that all plans for a given goal are equal, and so has focused on improving plan- ning efficiency. Of course, as has been recognized in the literature on decision-theoretic planning (Williamson & Hanks 1994; Haddawy & Suwandi 1994), the solu- tions to a given planning problem are not necessarily equal: some plans have lower execution cost, some are more likely to succeed, and so on. In this paper, we present a cost-directed heuristic planner, which is capable of finding low-cost plans in domains in which actions have different costs associ- ated with them. Our algorithm performs partial-order causal-link (POCL) planning, using an A* strategy. The heuristic evaluation function is computed by a deep lookahead that calculates the cost of complete plans for a set of pre-defined top-level subgoals, under the (generally false) assumption that those subgoals do not interact. The essential idea is to treat a set of top- level subgoals as if they were independent, in the sense of (Korf 1987). For each of these subgoals, we indepen- dently find a subplan that is consistent with the current global plan, i.e., the partial plan in the current node. The sum of the costs of these subplans is then used as the heuristic component, h’, of the A* algorithm. The overall estimate f’ (= g + h’, where g is the cost of the actions in the global partial plan) is used to determine which node to work on next. This contrasts with most POCL planners, which perform node selection using a shallow heuristic, typically a function of the number of steps and flaws in each node. Our approach leads to finding low-cost plans, and in many circumstances it also leads to a significant de- crease in total planning time. This is due in part to the fact that generating plans for subgoals individu- ally is often much less costly than generating a com- plete plan, taking interactions into account(Korf 1987; Yang, Nau, & Hendler 1992). Moreover, while focus- ing on lower-cost plans, the heuristic function effec- tively prunes the search space. The use of the deep evaluation in node selection can outweigh the marginal additional complexity. The Algorithm We model plans and actions similarly to other POCL systems, except that we assume that each action has an associated cost. We also assume that the cost of a plan is equal to the sum of the costs of its consituent actions. At the beginning of the planning process, the global goal is partitioned into n exhaustive and mu- tually disjoint subgoal sets, which should be roughly equivalent in complexity. For simplicity in this paper we assume that each subgoal set is a singleton, and just speak about top-level subgoals, sgi, for 1 5 i 5 n; we call the set of all these top-level subgoals SG . At a high level, our algorithm is simply the following: Until a solution has been found do: 1. Select ’ a node p representing a partial global plan with minimal f’ value. 2. Select a flaw in p to refine. For each successor node pi generated: - Set the actual cost function g(pi) to be the actual cost of p plus the cost of any new action added ‘The choose operator denotes non-deterministic choice, and hence, a possible backtrack point; the select operator denotes a heuristic choice at a point at which back-tracking is not needed. Temporal Reasoning 1223 From: AAAI-96 Proceedings. Copyright © 1996, AAAI (www.aaai.org). All rights reserved. in the refinement. (If the refinement is a threat resolution, then g(pi) = g(p)). - Independently generate a complete plan to achieve each of the original subgoals sgi. Each such subplan must be consistent with the par- tial global plan that pi represents, i.e., it must be possible to incorporate it into pi without vi- olating any ordering or binding constraints. Set h’(pi) to the sum of the costs of the complete subplans generated. A formal definition of the algorithm is given in Fig- ure 1. It relies on the following definitions, which are similar to those of other POCL algorithms, except for the inclusion of cost information for actions, and g and h’ values, and subgoal partitions, for plans: Definition 1 (Operator Schemata) An operator schema is a tupbe 4 T, V, P, E, B, c t where T is an action type, V is the list of free variables, P is the set of preconditions for the action, E is the set of ef- fects for the action, B is the set of binding constraints on the variables in V, and c is the cost.2 A copy of some action a with fresh variables will be denoted by Fresh(a). Given some action instance, s, we wibb re- fer to its components by T(s), V(s), P(s), E(s), B(s), and c(s) . Definition 2 (Plan) A plan is a tupbe + S, C, O,l3, &I, SG,g, h’ +, where S is a set of the plan steps, C is a set of causal links on S, 0 is a set of ordering constraints on S, I3 is a set of bindings of variables in S, 4 is the set of open conditions, T is the set of unresolved threats, SG is the partition of A induced by the initial partition of top-bevel subgoals, g is the accu- mulated cost of the plan so far, and h’ is the heuristic estimate of the remaining cost. -l The initial input to the planner is: (4 so, %03,@, {so < s,), 0, G, 0, SG, 0,O ~3, where SO is the dummy initial step, s, is the dummy final step, G is the initial set of goals, and SG is a partition of G. The algorithm also accesses an operator library A. As described above, the algorithm iteratively refines nodes in the plan space until a complete plan has been generated. Its main difference from other POCL plan- ners is its computation of heuristic estimates of nodes. (See the boxed parts of the algorithm.) The algorithm maintains a queue of partial plans, sorted by f’; on each iteration, it selects a node with minimal f’. Dur- ing refinement of that node, both the g and h’ comp- nents must be updated. Updating g is straightforward: whenever a new step is added to the plan, its cost is added to the existing g value (Step 2(a)ii). Updating h’ occurs in Step 4, in which subplans are generated for each of the top-level subgoals. 2To maintain an evaluation function that tends to un- derestimate, the cost of a step with uninstantiated vari- ables is heuristically taken to be equivalent to the cost of its minimal-cost grounded instance. CostDirectedPlanSearch (Q) While Q # 0 Select the first plan, p=+S,L,0,23,A,I,SG,g,h’%,in Q 1. [Termination] If A = 7 = 0 return p. 2. [Generate Successors] Let 0’ = I’ = b’ = s’ = 0, and Select either: (a) An open condition (d,Sd) (in A of p), and Choose an establisher: i. [An Existing Global Step] s’ E S and b’, s.t. e E E(8’), (e GsUb’ d), and ( 3’ < Sd) iS consistent with 0, ii. ;i N ew step1 9’ = Fresh(a) such that a E A and b’, s-t. e E E(a), (e EBub’ d). -1 Let 0’ = (3’ < L?d), I’ = ((S’, d, ad)}. (b) An unresolved threat (st, se, d, Sd) E 7 of p, and Choose either: i. [Demotion] If (st < se) is consistent with 0, let 0’ = ((at < se)}, or ii. [PrOmOtiOd If (st > sd) is consistent with 0, let 0’ = ((St > Sd)}. If there is no possible establisher or no way to resolve a threat, then fail. 3. [Update Plan1 Let S’ = S U s’, L’ = L U I’, 0’ = 6 u 0’, 8’ = 23 U b’, A’ = (d \ d) U P(s’). Update 7 to include new threats. 4. [Update Heuristic Value] h’ = c 89* ESG SubPlan( 4 S’, L’, O’, B’, 0, sgi , 0 +). i 5. [Update Queue] Merge the plan 4 S’, L’, O’, B’,d’, SG, g, h’ F back into Q sorted by its g + h’ value. Figure 1: The Search for a Global Plan There are two alternatives for subplanning. In this paper, we assume that subplanning is done by a fairly standard POCL algorithm, performing best first search. The subplanning process is invoked from the main program (in Step 4) with its steps, links, and ordering and binding constraints initialized to their equivalents in the global plan. The set of open condi- tions is initialized to sgi, which consists of the open conditions associated with the ith original subgoal: any conditions from the original partition that remain open, plus any open conditions along paths to mem- bers of sgi.3 Essentially, when subplanning begins, it is as if it were already in the midst of planning, and had found the global partial plan; it then forms a plan for the set of open conditions associated with a top- level subgoal. As a result, the plan for the subgoal will be consistent with global plan. If a consistent subplan cannot be found, then the global plan is due to fail Subplanning can thus detect dead-end paths. 3The main algorithm tags each establisher chosen in step 2a with the top-level goal(s) that it supports. We _ _ __ _ --- have omitted this detail from the Figure 1 to help improve readibility. 1224 Planning The subplanning process keeps track of the actual cost of the complete subplan found, and returns that value to the main algorithm. An alternative method for subplanning would be to recursively call the global planning algorithm in Fig- ure 1. This would be likely to further reduce the amount of time spent on each node, because it would amount to assuming independence not only among top- level goals, but also among their subgoals, and their subgoals’ subgoals, and so on. On the other hand, it would lead to a less accurate heuristic estimate, and thus might reduce the amount of pruning. We are conducting further experiments, not reported in this paper, to analyze this trade-off. Complexity We next analyze the complexity of the cost-directed planning algorithm. Let b be the maximum branching factor (number of possible refinements for each node), and let d be the depth of search (number of refine- ments performed to generate a complete plan). Then, as is well known, the worst-case complexity of plan- ning search is O(@). D uring each iteration of the cost- directed algorithm, the most promising partial plan is refined, and, for each possible refinement, a complete subplan for each of the n elements of the original sub- q~=r goal set (SG) is generated. Let bi and di denote the - breadth and depth of subplanning for the subgoal sg,, and let & = maxi bi, and let (ii = maxi di (1 < i < n). Then the complexity of each subplanning step in the al- gorithm (Step 4) is O(n x (ii)‘;). So in the worst case, if there is no pruning, the overall complexity of the search is O(bd x n x (ii)“). Because & 5 b, and & 5 d, the absolute worst case complexity is O(n x b2d). However, for many planning domains, & and t& are likely to be smaller than b and d. As noted by Korf (Korf 1987, p.68), and by Yang et ab. (Yang, Nau, & Hendler 1992), planning separately for n subgoals will tend to reduce both b and d by a factor of n, i.e., bi x i and di x $. Note that reduction in search depth is due to the fact that, if there were no interac- tions among the subgoals, an overall plan would have length equal to the sum of the lengths of the plans for the subgoals. Of course, positive interactions will decrease the length of the overall plan, while negative interactions will increase it. We assume that the ef- fects of negative interactions will be at least as great as the effects of positive interactions, which appears to be the case in many domains. To obtain the maximum benefits of planning for in- dividual subgoals separately, the subgoals must be of roughly equal “complexity” to one another. If virtu- ally all of the work of the top-level planning problem is associated with a single subgoal, then planning sep- arately for that subgoal will be almost as costly as planning for the entire set of subgoals. We therefore also assume planning domains in which it is possible to partition subgoals into sets of equal complexity. For domains for which this does not hold, it may still be possible to use the cost-directed planning algorithm, but it would require invoking it recursively for sub- planning, as described earlier. We next consider the effect of pruning. The heuris- tic function in the A* search reduces the complexity of the search from O(bd) to O(hd) where h is the heuris- tic branching factor (Korf 1987). Thus, for planning problems with the properties mentioned above (bal- anced positive and negative interactions; capable of being paritioned into subgoals with roughly equal com- plexity) the overall complexity of the cost-directed al- gorithm is O(hd x n x (i)t). Cost-directed search for these problems will thus consume less time than a full breadth-first planner as long as the following inequality holds:4 n/(n - 1)logh 5 logb - logn Of course, no POCL planning algorithms actually use breadth-first node selection; this inequality simply pro- vides a baseline for theoretical comparison. Later, we provide experimental comparison of our algorithm with best-first and branch-and-bound control strategies. Branching Factor = 3. nepch = 9 d/n)) - b**d . . . . . .9s Factor (h/b) Figure 2: Typical Search Space Complexity Figure 2 illustrates the inequality, comparing the complexity of cost-directed search with a breadth-first search (the level plane) as a function of the number of subgoals (n) and the pruning effect (h/b). We use a planning problem with a branching factor of 3 and a depth of 9 as an example. As the figure demonstrates, there exists a region in which the cost-directed planner performs better, namely the area in which the values of the curve fall below the level plane. In general, the ef- fectiveness of the cost-directed planner increases with the pruning effect and the number of subgoals being used. Note that this analysis assumes that the pruning fac- tor is independent of the number of subgoal partitions. 4The complete derivation is given in (Ephrati, Pollack, & Miishtein 1996). Temporal Reasoning 1225 In reality, the pruning factor is inversely related to the number of subgoals-the higher the number of sub- goals that are being used, the less accurate the heuris- tic evaluation will tend to become. Thus, for a specific problem, there exists some domain-dependent decom- position into subgoals which will optimize the perfor- mance of cost-directed search. Caching Subplans In the algorithm as so far presented, all subplans are recomputed on each iteration. Often, however, previ- ously generated subplans may remain compatible with the current global partial plan, and it may thus make sense to cache subplans, and to check whether they are still consistent before attempting to regenerate them. An added advantage of caching the subplans is that the top-level planner can then at tempt to re-use subplan steps; this helps maintain the accuracy of the heuristic evaluation function. We therefore modify the original algorithm by at- taching to each global plan a set of subplans (P = -pl,- - .,EJ) instead of just a partition of subgoals. Then, following each refinement of the global plan, the set of subplans is checked for consistency (with the newly refined global plan) and only the subplans that are incompatible are regenerated. Although caching subplans may significantly reduce the number of subplans generated, it can also increase space complexity; details of this trade-off are given in (Ephrati, Pollack, & Milshtein 1996). In all the exper- iments reported below, subplans were cached. Experimental Results The complexity analysis above demonstrates the po- tential advantages of our approach. To verify the ad- vantages in practice, we conducted a series of experi- ments comparing the cost-directed search with caching against the standard UCPOP algorithm with the built- in default search control, and againt UCPOP with a control mechanism that finds the minimal cost plan using branch-and-bound over action costs (henceforth called B&B). Effects of Subgoal Independence In the first experiment, our aim was to study the per- formance of the algorithms in domains in which the top-level subgoals had different levels of dependence. We therefore constructed three synthetic operator sets in the style of (Barrett & Weld 1994): l Independent Subgoals: Barrett and Weld’s BjD”Sn (p. 86). sn means that it takes n steps to achieve each top-level subgoal. Do means that the delete set of each operators is empty, so all the top-level subgoals are independent. @j means that there are j different operators to achieve each precondition. 1226 Planning Heavily Dependent Subgoals: Barrett and Weld’s 6j D”S” * (p. 93). As above, there are j different op- erators to achieve each precondition, and each top- level subgoal requires an n-step plan. Dm* refers to a pattern of interaction among the operators. There is a single operator that must end up in the middle of the plan, because it deletes all the preconditions of the operators in stages 1 through k for some k, as well as all the effects of the operators in stages k + 1 through n. In addition, for each stage, the opera- tors for the ith subgoal delete all the preconditions of the same-stage operators for subgoals j < i. Con- sequently, there is only a single valid linearization of the overall plan; the dependency among top-level subgoals is heavy. Moderately Dependent Subplans: A variant of Bar- rett and Weld’s 6jDmSn (p. 91). Here again there are j different operators to achieve each precondi- tion, and each top-level subgoal requires an n-step plan. In addition, the union of delete lists of the k + 1st stage operators for each subgoal sgi delete abb of the preconditions for the kth stage operators for all other sgj, j # i. Because these preconditions are partitioned among the alternative kth stage op- erators, there are multiple valid linearizations of the overall plan. Although the top-level subgoals inter- act, the dependency isn’t as tight, given the multiple possible solutions. One thing to note is that the individual subplanning tasks in our experiments are relatively easy: the in- teractions occur among the subplans for diRerent sub- goals. Of course, this means that planning for UCPOP and B&B is comparatively easy as well. Further exper- iments are being conducted using operator sets with interactions within each subplan. For the first experiment, we used problems with 4 top-level subgoals, and built the operator sets so that each top-level subgoal required a plan of length 3 (i.e., we used the S3 version of each of the operator sets). The actions were randomly assigned costs between 1 and 10. Finally, we varied the number of operators achieving each goal (the j in 6j) to achieve actual branching factors, as measured by UCPOP, of about 3 for the case of independent subgoals, about 2 for medium dependent subgoals, and about 1.5 for heavily dependent subgoals. The results are shown in Table 1, which lists o the number of nodes generated and the number of nodes visited (these numbers are separated by an arrow in the tables); * the CPU time spent in planning.5 5For certain problems that required a great deal of mem- ory, the garbage collection process caused Lisp to crash, re- porting an internal error. Schubert and Gerevini (Schubert & Gerevini 1995) encountered the same problem in their UCPOP-based experiments, with problems that required a a the cost of the plans generated (sum of action costs). As can be seen in the table, COST performed best in all cases, not only examining significantly fewer nodes, but taking an order of magnitude less time to find a solution than B&B. Not surprisingly, COST’s advan- tage increases with the degree of independence among the subgoals: the greater the degree of independence among the subplans, the more accurate is COST’s heuristic function, and thus the more effective its prun- ing. However, even in the case of heavy dependence among top-level subgoals, COST performs quite well. same costs, then no pruning will result from a strat- egy based on cost comparison, and the space taken by caching subplans will be enormous. Degree Planner Nodes Time cost UCPOP failure 28980.45 - Low B&B failure 26165.37 - COST 70 + 26 210.44 2167 UCPOP failure 36519.96 - Medium B&B failure 26952.57 - COST 134 + 44 407.84 54 UCPOP failure 14986.58 - High B&B failure 27304.23 - COST failure * Dependency Planner Nodes Time 1 Cost UCPOP failure 39927.89 - Independent B&B 35806 + 12900 2183.21 37 COST 60 + 22 126.13 37 UCPOP failure 40348.54 - Medium B&B 214 failures 9185.691 - 14827 -+ 5198 (succ. only) COST 309 4 146 613.54 37 UCPOP failure * - Heavy B&B 214 failures 12727.83 - 42978 -+ 17567 (sncc. only) COST 780 + 444 3834.517 37 Table 1: Varying dependency (4 medium uniform sub- goals, 3 Steps each, b M 3 for independent, b x 2 for medium, b x 1.5 for heavy) Effects of Uniformity The next thing we varied was the uniformity of action cost: see Tables 2 and 3. For this experiment, we used the independent subgoal operator set described above, with 3 steps to achieve each subgoal, and an actual branching factor of approximately 3. The experiment in Table 2 involved 6 top-level subgoals, while the ex- periment in Table 3 involved 4. In both experiments, we varied the distribution of action costs: in the highly uniform environment, all actions were assigned a cost of 1; in the medium uniform environment, they were randomly assigned a cost between 1 and 10; in the the low uniform environments, they were randomly as- signed a cost between 1 and 100. Both UCPOP and B&B failed to find a solution for problems with 6, and even with 4 independent subgoals within the 150,000 nodes-generated limit. In highly uniform domains, i.e., those in which alI actions had the same cost, our cost-directed algorithm fared no better: it also failed, although by exceeding memory limits. This is not surprising: if all actions have the lot of memory. We do not report results for these problems, but instead put an asterisk (*) in the time column. Also, in some cases, B&B succeeded- on some operator sets, and failed on others. For those cases, we report the average time taken on all runs (which will be an underestimate, as the failed run were terminated after 150,000 generated nodes). We report the number of nodes only for the suc- cessful cases. Finally, we do not report the costs found in these cases, because the high-cost plans are typically the cases that fail. In no case did B&B find a lower-cost plan than COST. Table 2: Varying Uniformity (6 independent subgoals, 3 Steps each, b x 3) Degree Planner Nodes Time cost UCPOP failure 14617.54 - Low B&B 65000 + 23158 12708.66 1761 COST 50 + 18 98.22 1761 UCPOP failure 39927.89 - Medium B&B 35806 + 12900 2183.21 37 COST 60 + 22 126.13 37 UCPOP failure 14528.38 - High B&B failure 26988.26 - COST failure * Table 3: Varying Uniformity (4 independent subgoals, 3 Steps each, b x 3) However, when the environments become less uni- form, we see the payoff in the cost-directed approach. For low uniform environments and a planning problem with 6 subgoals (Table 2), the cost-directed planner finds plans by generating less than 70 nodes, taking about 3.5 minutes-while UCPOP and B&B failed, af- ter generating 150,000 nodes, and taking over 7 hours. Even with medium uniformity, cost-directed planning succeeds quickly, while the other two approaches fail. Similar results are observed for the smaller (4 subgoal) problem (Table 3). Note again that these are very easy problems, given the independence of the top-level subgoals. Indeed, the optimal strategy would have been to plan completely separately for each subgoal, and then simply concate- nate the resulting plans. However, one may not know, in general, whether a set of subgoals is independent, prior to performing planning. The cost-directed search performs very well, while maintaining a general, POCL strategy that is applicable to interacting, as well as in- dependent, goals. Other Factors Effecting Planning Several other key factors that are known to effect the efficiency of planning are the average branching factor, the length of the plan, and the number of top-level subgoals. We have conducted, and are continuing to conduct, experiments that vary each of these factors; so far, our results demonstrate that in a wide range of environments, the COST algorithm performs very well, finding low-cost plans in less time than either UCPOP or B&B (Ephrati, Pollack, & Milshtein 1996). Temporal Reasoning 1227 Related Research Prior work on plan merging (Fousler, Li, & Yang 1992; Yang, Nau, & Hendler 1992) has studied the problem of forming plans for subgoals independently and then merging them back together. Although similarly mo- tivated by the speed-up one gets from attending to subgoals individually, our work differs in performing complete planning using a POCL-style algorithm, as opposed to using separate plan merging procedures. The idea of using information about plan quality during plan generation dates back to (Feldman & Sproull 1977); more recent work on this topic has in- volved the introduction of decision-theoretic notions (Haddawy & Suwandi 1994). Perez studied the prob- lem of enabling a system to learn ways to improve the quality of the plans it generates (Perez 1995). Particu- larly relevant is Williamson’s work on the PYRRHUS system (Williamson & Hanks 1994), which uses plan quality information to find an optimal plan. It per- forms standard POCL planning, but does not termi- nate with the first complete plan. Instead, it computes the plans’s utility, prunes from the seach space any par- tial plans that are guaranteed to have lower utility, and then resumes execution. The process terminates when no partial plans remain, at which point PYRRHUS is guaranteed to have found an optimal plan. What is in- teresting about PYRRHUS from the perspective of the current paper is that, although one might expect that PYRRHUS would take significantly longer to find an optimal plan than to find an arbitrary plan, in fact, in many circumstances it does not. The information pro- vided by the utility model results in enough pruning of the search space to outweigh the additional costs of seeking an optimal solution. This result, although ob- tained in a different framework from our own, bears a strong similarity to our main conclusion, which is that the pruning that results from attending to plan quality can outweigh the cost of computing plan quality. Conclusions We have presented an efficient cost-directed planning algorithm. The key idea underlying it is to replace a shallow, syntactic heuristic for node selection in POCL planning with an approximate, full-depth lookahead that computes complete subplans for a set of top-level subgoals, under the assumption that these subgoals are independent. Our analytical and experimental results demonstrate that in addition to finding low-cost plans, the algorithm can significantly improve planning time in many circumstances. The performance of the algo- rithm is dependent upon the possibility of decompos- ing the global goal into subgoal sets that are relatively (though not necessarily completely) independent, and that are roughly equivalent in complexity to one an- other. The experiments we presented in this paper support the main hypothesis, but are by no means complete. We are continuing our experimentation, in particular, studying domains that involve a greater degree of in- teraction within each subplan. In addition, we are investigating the trade-off between using a complete POCL planner for subplanning, as in the experiments reported here, and recursively calling the main algo- rithm for subplanning. Finally, we are developing tech- niques to make the cost-directed search admissible. Acknowledgments This work has been supported by the Rome Laboratory of the Air Force Material Command and the Advanced Research Projects Agency (Contract F30602-93-C-0038), by the Office of Naval Research (Con- tract N00014-95-l-1161) and by an NSF Young Investiga- tor’s Award (IRS9258392). References Barrett, A., and Weld, D. 1994. Partial-order plan- ning: Evaluating possible efficiency gains. Artificial Intelligence 67( 1):71-112. Ephrati, E.; Pollack, M. E.; and Milshtein, M. 1996. A cost-directed planner. Technical Report, Dept. of Computer Science, Univ. of Pittsburgh, in prepara- tion. Feldman, J. A., and Sproull, R. F. 1977. Decision the- ory and artificial intelligence II: The hungry monkey. Cognitive Science 1:158-192. Fousler, D.; Li, M.; and Yang, Q. 1992. Theory and algorithms for plan merging. Arti$ciab Intelligence 57:143-181. Haddawy, P., and Suwandi, M. 1994. Decision- theoretic refinement planning using inheritance ab- straction. In Proceedings of the Second International Conference on AI Planning Systems, 266-271. Korf, R. E. 1987. Planning as search: A quantitative approach. Artificial Intelligence 33:65-88. Perez, M. A. 1995. Improving search control knoweldge to improve plan quality. Technical Re- port CMU-CS-95-175, Dept. of Computer Science, Carnegie Mellon University. Ph.D. Dissertation. Schubert, L., and Gerevini, A. 1996. Accelerating partial order planners by improving plan and goal choices. Technical Report 96-607, Univ. of Rochester Dept. of Computer Science. Williamson, M., and Hanks, S. 1994. Optimal plan- ning with a goal-directed utility model. In Proceedings of the Second International Conference on Artificial Intelligence Planning Systems, 176-181. Yang, Q.; Nau, D. S.; and Hendler, J. 1992. Merg- ing separately generated plans with restricted inter- actions. Computational Intelligence 8(2):648-676. 1228 Planning | 1996 | 181 |
1,823 | Monitoring the Progress of Anyti Eric A. Hansen and Shlomo Zilberstein Computer Science Department University of Massachusetts Amherst, MA 0 1003 {hansen,shlomo}@cs.umass.edu Abstract Anytime algorithms offer a tradeoff between solution qual- ity and computation time that has proved useful in applying artificial intelligence techniques to time-critical problems. To exploit this tradeoff, a system must be able to determine the best time to stop deliberation and act on the currently available solution. When the rate of improvement of so- lution quality is uncertain, monitoring the progress of the algorithm can improve the utility of the system. This paper introduces a technique for run-time monitoring of anytime algorithms that is sensitive to the variance of the algorithm’s performance, the time-dependent utility of a solution, the ability of the run-time monitor to estimate the quality of the currently available solution, and the cost of monitoring. The paper examines the conditions under which the technique is optimal and demonstrates its applicability. Introduction Anytime algorithms are being used increasingly for time- critical problem-solving in domains such as planning and scheduling (Boddy & Dean 1994; Zilberstein 1996), belief network evaluation (Horvitz, Suermondt, & Cooper 1989; Wellman & Liu 1994), database query processing (Shekhar & Dutta 1989; Smith & Liu 1989), and others. The defining property of an anytime algorithm is that it can be stopped at any time to provide a solution, such that the quality of the solution increases with computation time. This property allows a tradeoff between computation time and solution quality, making it possible to compute approximate solu- tions to complex problems under time constraints. It also introduces a problem of meta-level control: making an opti- mal time/quality tradeoff requires determining how long to run the algorithm, and when to stop and act on the currently available solution. Meta-level control of an anytime algorithm can be ap- proached in two different ways. One approach is to al- locate the algorithm’s running time before it starts and to let the algorithm run for the predetermined length of time no matter what (Boddy & Dean 1994). If there is little or no uncertainty about the rate of improvement of solution quality, or about how the urgency for a solution might change after the start of the algorithm, then this ap- proach can determine an optimal stopping time. Very often, however, there is uncertainty about one or both. For AI problem-solving in particular, variance in solution quality is common (Paul et al. 1991). Because the best stop- ping time will vary with fluctuations in the algorithm’s per- formance, a second approach to meta-level control is to monitor the progress of the algorithm and to determine at run-time when to stop deliberation and act on the currently available solution (Breese & Horvitz 1991; Horvitz 1990; Zilberstein & Russell 1995). Monitoring the progress of anytime problem-solving in- volves assessing the quality of the currently available so- lution, making revised predictions of the likelihood of fur- ther improvement, and engaging in metareasoning about whether to continue deliberation. Previous schemes for run-time monitoring of anytime algorithms have assumed continuous monitoring, but the computational overhead this incurs can take resources away from problem-solving itself. This paper introduces a framework in which the run-time overhead for monitoring can be included in the problem of optimizing the stopping time of anytime problem-solving. It describes a framework for determining not only when to stop an anytime algorithm, but at what intervals to monitor its progress and re-assess whether to continue or stop. This framework makes it possible to answer such questions as: How much variance in the performance of an anytime algorithm justifies adopting a run-time monitoring strat- egy rather than determining a fixed running time ahead of time? How should the variance of an algorithm’s performance affect the frequency of monitoring? Is it better to monitor periodically or to monitor more fre- quently toward the algorithm’s expected stopping time? For a large class of problems, the rate of improve- ment of solution quality is the only source of uncer- tainty about how long to continue deliberation. Ex- amples include optimizing a database query (Shekhar & Dutta 1989), reformulating a belief net before solving it (Breese & Horvitz 1991), and planning the next move in Temporal Reasoning 1229 From: AAAI-96 Proceedings. Copyright © 1996, AAAI (www.aaai.org). All rights reserved. a chess game (Russell & Wefald 1991). For other prob- lems, the utility of a solution may also depend on the state of a dynamic environment that can change unpre- dictably after the start of the algorithm. Examples include real-time planning and diagnosis (Boddy & Dean 1994; Horvitz 1990). For such problems, meta-level control can be further improved by monitoring the state of the environ- ment as well as the progress of problem-solving. We focus in this paper on uncertainty about improvement in solution quality. However, the framework can be extended in a rea- sonably straightforward way to deal with uncertainty about the state of a dynamic environment. We begin by describing a framework for constructing an optimal policy for monitoring the progress of any- time problem-solving, assuming the quality of the currently available solution can be measured accurately at run-time. Because this assumption is often unrealistic, we then de- scribe how to modify this framework when a run-time mon- itor can only estimate solution quality. A simple example is described to illustrate these results. The paper concludes with a brief discussion of the significance of this work and possible extensions. Formal Framework Meta-level control of an anytime algorithm - deciding how long to run the algorithm and when to stop and act on the currently available solution - requires a model of how the quality of a solution produced by the algorithm increases with computation time, as well as a model of the time- dependent utility of a solution. The first model is given by a per$ormance pro$le of the algorithm. A conventional performance profile predicts solution quality as a function of the algorithm’s overall running time. This is suitable for making a one-time decision about how long to run an algorithm, before the algorithm starts. To take advantage of information gathered by monitoring its progress, however, a more informative performance profile is needed that makes it possible to predict solution quality as a function of both time allocation and the quality of the currently available solution. Definition 1 A dynamic performance profile of an anytime algorithm, Pr(qj jqi, At), denotes the probability of getting a solution of quality qj by resuming the algorithm for time interval At when the currently available solution has quality qi. We call this conditional probability distribution a dynamic performance profile to distinguish it from a performance profile that predicts solution quality as a function of running time only. The conditional probabilities are determined by statistical analysis of the behavior of the algorithm. For sim- plicity, we rely on discrete probability distributions. Time is discretized into a finite number of time steps, to . . . t,,, where to represents the starting time of the algorithm and t, its maximum running time. Similarly, solution quality is discretized into a finite number of levels, qo . . . qnr, where qo is the lowest quality level and qm is the highest quality level. Let qstart denote the starting state of the algorithm before any result has been computed. By discretizing time and quality, the dynamic performance profile can be stored as a three-dimensional table; the degree of discretization controls a tradeoff between the precision of the performance profile and the size of the table needed to store it. A dynamic performance profile can also be represented by a compact parameterized function. Meta-level control requires a model of the time- dependent utility of a solution as well as a performance profile. We assume that this information is provided to the monitor in the form of a time-dependent utility function. Definition 2 A time-dependent utility function, U (qi , tk), represents the utility of a solution of quality qi at time tk. In this paper, we assume that utility is a function of time and not of an external state of the environment. This assumption makes it possible to set to one side the problem of model- ing uncertainty about the environment in order to focus on the specific problem of uncertainty about improvement in solution quality. Finally, we assume that monitoring the quality of the cur- rently available solution and deciding whether to continue or stop incurs a cost, C. Because it may not be cost-effective to monitor problem-solving continuously, an optimal policy must specify when to monitor as well as when to stop and act on the currently available solution. For each time step tk and quality level q;, the following two decisions must be specified: 1. how much additional time to run the algorithm; and 2. whether to monitor at the end of this time allocation and re-assess whether to continue, or to stop without moni- toring. Definition 3 A monitoring policy, n(qi, tk), is a mapping frmn time Step tk and quality level qi into a monitoring decision (At, m) such that At represents the additional amount of time to allocate to the anytime algorithm, and m is a binary variable that represents whether to monitor at the end of this time allocation or to stop without monitoring. An initial decision, 7r(qdtatt, to), specifies how much time to allocate to the algorithm before monitoring for the first time or else stopping without monitoring. Note that the variable At makes it possible to control the time interval between one monitoring action and the next; its value can range from 0 to t, - t;, where t, is the maximum running time of the algorithm and t; is how long it has already run. The binary variable m makes it possible to allocate time to the algorithm without necessarily monitoring at the end of the time interval; its value is either stop or monitor. 1230 Planning An optimal monitoring policy is a monitoring policy that maximizes the expected utility of an anytime algorithm. Given this formalization, it is possible to use dynamic programming to compute a combined policy for monitoring and stopping. Dynamic programming is often used to solve optimal stopping problems; the novel aspect of this solution is that dynamic programming is also used to determine when to monitor. A monitoring policy is found by optimizing the following value function: CJ’ Pf’(qjIQir At)U(qj, tk+At) qqi, tk) = max if m = stop, Atom C-j Pr(qj)qi, At)v(qj,tk+At)-C if m = monitor Theorem 1 A monitoring policy that maximizes the above valuefunction is optimal when quality improvement satisfies the Markov property. This is an immediate outcome of the application of dy- namic programming under the Markov assumption (Bert- sekas 1987). The assumption requires that the probability distribution of future quality depends only on the current “state” of the anytime algorithm, which is taken to be the quality of the currently available solution. The validity of this assumption depends on both the algorithm and how solution quality is defined, and so must be evaluated on a case-by-case basis. But we believe it is at least a useful approximation in many cases. Uncertain measurement of quality We have described a framework for computing a policy for monitoring an anytime algorithm, given a cost for moni- toring. Besides the assumption that quality improvement satisfies the Markov property, the optimality of-the policy depends on the assumption that the quality of the currently available solution can be measured accurately by a run-time monitor. How reasonable is this second assumption likely to be in practice? We suggest that for certain types of problems, calculating the precise quality of a solution at run-time is not fess< ble. One class of problems for which anytime algorithms are widely used are optimization problems in which a so- lution is iteratively improved over time by minimizing or maximizing the value of an objective function. For such problems, the quality of an approximate solution is usually measured by how close the approximation comes to an op- timal solution. For cost-minimization problems, this can be defined as Co&( Approximate Solution) Cost(Optima1 Solution) The lower this approximation ratio, the higher the quality of the solution, and when it is equal to one the solution is optimal. The problem with using this measure of solution qual- ity for run-time monitoring is that it requires knowing the optimal solution at run-time. This is no obstacle to using it to construct a performance profile for an anytime algo- rithm, because the performance profile can be constructed off-line and the quality of approximate solutions measured in terms of the quality of the eventual optimal solution. But a run-time monitor needs to make a decision based on the approximate solution currently available, without knowing what the optimal solution will eventually be. As a result, it cannot know with certainty the actual quality of the ap- proximate solution. In some cases, it will be possible to bound the degree of approximation, but a run-time monitor can only estimate where the optimal solution falls within this bound. A similar observation can be made about other classes of problems besides optimization problems. For problems that involve estimating a point value, the difference be- tween the estimated point value and the true point value can’t be known until the algorithm has converged to an exact value (Horvitz, Suermondt, & Cooper 1989). For anytime problem-solvers that rely on abstraction to create approximate solutions, solution quality may be difficult to assess for other reasons. For example, it may be difficult for a run-time monitor to predict the extent of planning needed to fill in the details of an abstract plan (Zilberstein 1996). We conclude that for many problems, the best a run-time monitor can do is estimate the quality of an anytime solution with some degree of probability. Monitoring based on estimated quality When the quality of approximate solutions cannot be accu- rately measured at run-time, the success of run-time moni- toring requires solving two new problems. First, some reli- able method must be found for estimating solution quality at run-time. It is impossible to specify a universal method for this - how solution quality is estimated will vary from algorithm to algorithm. We sketch a general approach and, in the section that follows, describe an illustrative example. The second problem is that a monitoring policy must be conditioned on the estimate of solution quality rather than solution quality itself. To solve these problems, we condition a run-time esti- mate of solution quality on some feature fF of the currently available solution that is correlated with solution quality. When a feature is imperfectly correlated with solution qual- ity, we have also found it useful to condition the estimate on the running time of the algorithm, tk. Conditioning an estimate of solution quality on the algorithm’s running time as well as some feature observed by a run-time monitorpro- vides an important guarantee; it ensures that the estimate will be at least as good as if it were based on running time alone. As a general notation, let Pr(q; If?, tk) denote the prob- Temporal Reasoning 1231 ability that the currently available solution has quality qi when the run-time monitor observes feature f’ after running time tk. In addition, let Pr(f, lqi, tk) denote the probability that the run-time monitor will observe feature f? if the cur- rently available solution has quality Qi after running time tk. Again, these probabilities can be determined from statisti- cal analysis of the behavior of the algorithm. These “partial observability” functions, together with the dynamic perfor- mance profile defined earlier, can be used to calculate the following probabilities for use in predicting the improve- ment of solution quality after additional time allocation At when the quality of the currently available solution can only be estimated. These probabilities can also be determined directly from statistical analysis of the behavior of the algorithm, iithout the intermediite calculations. In either case, these prob- abilities make it possible to find a monitoring poliiy by optimizing the following value function using dynamic pro- gramming. r Cj pf’($‘I.fr, tkc At)u(qj, tk+At) V(&., tk) = max if m = StoPI *t,m I ca Pr(faIfr,tk, At)v(fd, tk+&)-c L ifm = monitor The resulting policy may not be optimal in the sense that it may not take advantage of all possible run-time ev- idence about solution quality, for example, the trajectory of observed improvement. Finding an optimal policy may require formalizing the problem as a partially observable Markov decision process and using computationally inten- sive algorithms developed for such problems (Cassandra, Littman, & Kaelbling 1994). The approach we have de- scribed is simple and efficient, however, and it provides an important guarantee: it only recommends monitoring if it results in a higher expected value than allocating a fixed running time without monitoring. This makes it possible to distinguish cases in which monitoring is cost-effective from cases in which it is not. Whether monitoring is cost- effective will depend on the variance of the performance profile, the time-dependent utility of the solution, how well the quality of the currently available solution can be esti- mated by the run-time monitor, and the cost of monitoring - all factors weighed in computing the monitoring policy. xample As an example of how this technique can be used to deter- mine a combined policy for monitoring and stopping, we apply it to a tour improvement algorithm for the traveling salesman problem developed by Lin and Kernighan (1973). quality Length(Current tour) Length(Optima1 tour) 1.05 + 1.00 1.10 + 1.05 1.20 + 1.10 1.35 + 1.20 1.50 + 1.35 00 + 1.50 L Table 1: Discretization of solution quality. r feature 6 5 4 3 2 1 0 Length(Current tour) Length&ower bound) 1.3 -+ 1.0 1.4 + 1.3 1.5 + 1.4 1.6 + 1.5 1.7 + 1.6 2.0 -+ 1.7 00 -+ 2.0 Table 2: Discretization of feature correlated with solution quality. This local optimization algorithm begins with an initial tour, then repeatedly tries to improve the tour by swapping ran- dom paths between cities. The example is representative of anytime algorithms that have variance in solution quality as a function of time. We defined solution quality as the approximation ratio of a tour, Length(Current tour) Length(Optima1 tour) and discretized this metric using Table 1. The maximum running time of the algorithm was discretized into twelve time-steps, with one time-step corresponding to approxi- mately 0.005 CPU seconds. A dynamic performance pro- file was compiled by generating and solving a thousand random twelve-city traveling salesman problems. The time- dependent utility of a solution of quality qi at time tk was arbitrarily defined by the function u(qi, tk) = looqi - 2otk. Note that the first term of the utility function can be regarded as the intrinsic value of a solution and the second term as the time cost, as defined by Russell and Wefald (199 1). Without monitoring, the optimal running time of the al- gorithmis eight time-steps, with an expected value of 269.2. Assuming solution quality can be measured accurately by the run-time monitor (an unrealistic assumption in this case) and assuming a monitoring cost of 1, the dynamic program- ming algorithm described earlier computes the monitoring policy shown in Table 3. The number in each cell of Ta- ble 3 represents how much additional time to allocate to the 1232 Planning quality -T--- time-step start 1 2 3 4 5 6 7 8 9 10 11 00000000000 1M 1M 1M 1M 1M 1M 1M 1M 1M 1 0 1M 1M 1M 1M 1M 1M 1M IM 1M 1 0 3M 3M 3M 3M 3M 3M 3M 3M 2 1 0 4M 4M 4M 4M 4M 5 4 3 2 1 0 5M 5M 5M 5M 5M 6 5 4 3 2 1 0 Table 3: Optimal policy based on actual solution quality. time-step feature start 1 2 3 4 5 6 7 8 9 10 11 6 000000000 5 1111000000 4 2M 2M 1M 1M 1M 1M 1 0 0 0 3 4M 3M 2M 1M 1M 1M 1M 1M 1 0 0 2 4M 3M 2M 2M 2M 2M 1M 3 2 1 0 1 4M 3M 3M 3M 3M 3M 2M 2M 1M 1 0 5M 4M 3M 3M 3M 3M 3M Table 4: Policy when solution quality is estimated. algorithm based on the observed quality of the solution and the current time. The letter M next to a number indicates a decision to monitor at the end of this time allocation, and possibly allocate additional running time; if no M is present, the decision is to stop at the end of this time allocation with- out monitoring. The policy has an expected value of 303.3, better than the expected value of allocating a fixed running time despite the added cost of monitoring. Its improved performance is due to the fact that the run-time monitor can stop the algorithm after anywhere from 5 to 11 time steps, depending on how quickly the algorithm finds a good result. (If there is no cost for monitoring, a policy that monitors every time step has an expected value of 309.4.) The policy shown in Table 3 was constructed by assuming the actual quality of an approximate solution could be mea- sured by the run-time monitor, an unrealistic assumption because measuring the quality of the current tour requires knowing the length of an optimal tour. The average length of an optimal tour can provide a very rough estimate of the optimal tour length in a particular case, and this can be used to estimate the quality of the current tour. For a travel- ing salesman problem that satisfies the triangle inequality, however, much better estimates can be made by using one of a number of algorithms for computing a lower bound on the optimal tour length (Reinelt 1994). Computing a lower bound involves solving a relaxation of the problem; it is analogous to an admissable heuristic function in search. For a traveling salesman problem that satisfies the triangle inequality, there exist polynomial-time algorithms that can compute a lower bound that is on average within two or three percent of the optimal tour length. For our test, how- ever, we used Prim’s minimal spanning tree algorithm that very quickly computes a bound that is less tight, but still correlated with the optimal tour length. The feature Length(Current tour) Length(Lower bound) was discretized using Table 2. The cost overhead of moni- toring consists of computing the lower bound at the begin- ning of the algorithm and monitoring the current tour length at intervals thereafter. Table 4 shows the monitoring policy given a monitoring cost of 1, when an estimate of solution quality is conditioned on both this feature and the running time of the algorithm. The expected value of the policy is 282.3, higher than for al- locating a fixed running time without monitoring but lower than if the run-time monitor could determine the actual qual- ity of an approximate solution. As this demonstrates, the less accurately a run-time monitor can measure the quality of an approximate solution, the less valuable it is to monitor. When an estimate of solution quality is based only on this feature, and not also on running time, the expected value of monitoring is 277.0. This is still an improvement over not monitoring, but the performance is not as good as when an estimate is conditioned on running time as well. Because conditioning a dynamic performance profile on running time significantly increases its size, however, this tradeoff may be acceptable in cases when the feature used to estimate quality is very reliable. For all of these results, the improved performance predicted by dynamic programming was confirmed by simulation experiments. For the tour improvement algorithm, variance in solution quality over time is minor and the improved performance with run-time monitoring correspondingly small. We plan Temporal Reasoning 1233 to apply this technique to other problems for which variance in solution quality is larger and the payoff for run-time monitoring promises to be more significant. However, the fact that this technique improves performance even when variance is small, solution quality is difficult to estimate at run-time, and monitoring incurs a cost, supports its validity and potential value. Conclusion The framework developed in this paper extends previ- ous work on meta-level control of anytime algorithms. One contribution is the use of dynamic programming to compute a non-myopic stopping rule. Previous schemes for run-time monitoring have relied on myopic compu- tation of the expected value of continued deliberation to determine a stopping time (Breese & Horvitz 1991; Horvitz 1990), although Horvitz has also recommended various degrees of lookahead search to overcome the limi- tations of a myopic approach. Because dynamic program- ming is particularly well-suited for off-line computation of a stopping rule, it is also an example of what Horvitz calls compilation of metareasoning. Another contribution of this framework is that it makes it possible to find an intermediate strategy between contin- uous monitoring and not monitoring at all. It can recognize whether or not monitoring is cost-effective, and when it is, it can adjust the frequency of monitoring to optimize utility. An interesting property of the monitoring policies found is that they recommend monitoring more frequently near the expected stopping time of an algorithm, an intuitive strategy. Perhaps the most significant aspect of this framework is that it makes it possible to evaluate tradeoffs between various factors that influence the utility of monitoring. For example, the dynamic programming technique is sensitive to both the cost of monitoring and to how well the quality of the currently available solution can be estimated by the run- time monitor. This makes it possible to evaluate a tradeoff between these two factors. Most likely, there will be more than one method for estimating a solution’s quality and the estimate that takes longer to compute will be more accurate. Is the greater accuracy worth the added time cost? The framework developed in this paper can be used to answer this question by computing a monitoring policy for each method and comparing their expected values to select the best one. Support for this work was provided in part by the National Science Foundation under grant IRI-9409827 and in part by Rome Laboratory, USAF, under grant F30602-95- l-00 12. References Bertsekas, D.P. 1987. Dynamic Programming: Deter- ministic and Stochastic Models. Englewood Cliffs, N.J.: Prentice-Hall. Boddy, M., and Dean., T. 1994. Deliberation schedul- ing for problem solving in time-constrained environments. Artijkial Intelligence 67~245-285. Breese, J.S., and Horvitz, E.J. 1991. Ideal reformulation of belief networks. Proceedings of the Sixth Conference on Uncertainty in Artificial Intelligence, 129- 143. Cassandra, A.R.; Littman, M.L.; and Kaelbling, L.P. 1994. Acting optimally in partially observable stochastic do- mains. Proceedings of the Twelth National Conference on Artificial Intelligence, 1023-1028. Horvitz, E.J.; Suermondt, H.J.; and Cooper, G.F. 1989. Bounded conditioning: Flexible inference for decisions under scarce resources. Proceedings of the Fifth Workshop on Uncertainty in Artificial Intelligence. Horvitz, E.J. 1990. Computation and Action under Bounded Resources. PhD Thesis, Stanford University. Lin, S., and Kernighan, B.W. 1973. An effective heuristic algorithm for the Traveling Salesman problem. Operations Research 2 1:498-5 16. Paul, C.J.; Acharya, A.; Black, B.; and Strosnider, J.K. 1991. Reducing problem-solving variance to improve pre- dictability. CACM 34(8):80-93. Reinelt, G. 1994. The Traveling Salesman: Computational Solutions for TSP Applications. Springer-Verlag. Russell, S., and Wefald, E. 1991. Do the Right Thing: Studies in Limited Rationality. The MIT Press. Shekhar, S., and Dutta, S. 1989. Minimizing response times in real time planning and search. Proceedings of the Eleventh IJCAI, 238-242. Smith, K.P., and Liu, J.W.S. 1989. Monotonically im- proving approximate answers to relational algebra queries. COMPSAC-89, Orlando, Florida. Wellman, M.P., and Liu, C.-L. 1994. State-space abstrac- tion for anytime evaluation of probabilistic networks. Pro- ceedings of the Tenth Conference on Uncertainty in Arti- ficial Intelligence, 567-574. Zilberstein, S. 1993. Operational Rationality through Compilation of Anytime Algorithms. Ph.D. dissertation, Computer Science Division, University of California at Berkeley. Zilberstein, S. 1996. Resource-bounded sensing and plan- ning in autonomous systems. To appear in Autonomous Robots. Zilberstein S., and Russell S. 1995. Approximate reason- ing using anytime algorithms. In S. Natarajan (Ed.), Im- precise and Approximate Computation, Kluwer Academic Publishers. 1234 Planning | 1996 | 182 |
1,824 | A Linear-Programming A roach to Tern Peter Jonsson and Christer Btickstriim Department of Computer and Information Science Linktjping University, S-581 83 Linkoping, Sweden {petej,cba}Oida.liu.se Abstract We present a new formalism, Horn Disjunctive Lin- ear Relations (Horn DLRs), for reasoning about tem- poral constraints. We prove that deciding satisfiabil- ity of sets of Horn DLRs is polynomial by exhibiting an algorithm based upon linear programming. Fur- thermore, we prove that most other approaches to tractable temporal constraint reasoning can be en- coded as Horn DLRs, including the ORD-Horn algebra and most methods for purely quantitative reasoning. Introduction Reasoning about temporal constraints is an impor- tant task in many areas of AI and elsewehere, such as planning, natural language processing, diagnosis, time serialization in archeology etc. In most ap- plications, knowledge of temporal constraints is ex- pressed in terms of collections of relations between time intervals or time points. Typical reasoning tasks include determining the satisfiability of such collec- tions and deducing new relations from those that are known. The research has largely concentrated on two kinds of formalisms; systems of inequalities on time points (Dechter, Meiri, & Pearl 1991; Meiri 1991; Koubarakis 1992) to encode quantitative information, and systems of constraints in Allen’s algebra (Allen 1983) to encode qualitative relations between time intervals. Some attempts have been made to inte- grate quantitative and qualitative reasoning into uni- fied frameworks (Kautz & Ladkin 1991; Meiri 1991). Since the satisfiability problem is NP-complete for Allen’s algebra the qualitative and unified approaches have suffered from computational difficulties. In response to the computational hardness of the full Allen algebra, several polynomial subalgebras have been proposed in the literature (van Beek & Cohen 1990; Golumbic & Shamir 1993; Nebel & Biirckert 1995). Some of these algebras have later been ex- tended with mechanisms for handling quantitative in- formation. For example the TIMEGRAPH II system (Gerevini, Schubert, & Schaeffer 1993) extends the pointisabde algebra (van Beek & Cohen 1990) with a limited type of quantitative information. Of special interest is the ORD-Horn algebra (Nebel & Biirckert 1995) which, under certain conditions, is the unique maximal tractable subclass of Allen’s algebra. Hence, it would be especially interesting to extend this algebra with quantitative information since the maximality re- sult would carry over to the new algebra, at least with respect to its qualitative expressiveness. To our knowl- edge, no such attempt have been made. Now, to make the topic of reasoning about temporal constraint more concrete, consider the following fic- tious crime scenario. Professor Jones has been found shot on the beach near her house. Rumours tell that she was almost sure of having a proof that P#NP, but had not yet shown it to any of her colleagues. The graduate student Hill is soon to defend his thesis on his newly invented complexity class, NRQPx(a)*, which would unfortunately be of no value were it to be known for certain that P#NP. Needless to say, Hill is thus one of the prime suspects and inspector Smith is faced with the following facts and observations: Professor Jones died between 6 pm and 11 pm, ac- cording to the post-mortem. Mr Green, who lives close to the beach, is certain that he heard a gunshot sometime in the evening, but certainly after the TV news. The TV news is from 7.30 pm to 8.00 pm. A reliable neighbour of Hill claims Hill arrived at home sometime between 9.15 pm and 9.30 pm. It takes between 10 and 20 mins. to walk or run from the place of the crime to the closest parking lot. It takes between 45 and 60 mins. to drive from this parking lot to Hill’s home. The first thing to do is verifying that these facts and observations are consistent, which is obviously the case here. VVe can also draw some further conclusions, like narrowing the time of death to the interval between 8.00 pm and 11 pm, assuming the gunshot heard by mr Green actually was the killing shot. Now, suppose inspector Smith adds the hypothesis that Hill was at the place of the murder at the time of Temporal Reasoning 1235 From: AAAI-96 Proceedings. Copyright © 1996, AAAI (www.aaai.org). All rights reserved. the gunshot, which is only known to occur somewhere in the interval from 8.00 pm to 11.00 pm. If the set of facts and observations together with this hypothesis becomes inconsistent, then inspector Smith can rule out Hill as the murdererl. This problem can easily be cast in terms of a temporal-constraint-reasoning problem, involving both quantitative and qualitative relations over time points, intervals and durations. Unfortunately, it seems like this simple example cannot be solved by any of the computationally tractable methods reported in the lit- erature. It can, however, be solved in polynomial time by the method proposed in this paper. We introduce Horn Disjunctive Linear Relations (Horn DLRs for short) which is a temporal constraint formalism that allows for polynomial-time satisfiabil- ity checking. Horn DLRs subsumes the ORD-Horn algebra and most of the formalisms for encoding quan- titative information proposed in the literature. The approach is rather different from the commonly used constraint network or graph-theoretic approaches. We base our method upon linear programming which proves to be a convenient tool for managing tempo- ral information. Since most of the low-level handling of time points is thus abstracted away, the resulting algorithm is surprisingly simple. We strongly believe that Horn DLRs are useful in other areas of computer science than temporal reasoning. For instance, the proposal for constraint query languages in deductive databases by Kanellakis et al. (1995) has some resem- blance with Horn DLRs. The paper is structured as follows. We begin by giv- ing basic terminology and definitions used in the rest of the paper together with a brief introduction to com- plexity issues in linear programming. We continue by presenting the polynomial-time algorithm for deciding satisfiability of Horn DLRs. The paper concludes with a short discussion of the results. Disjunctive Linear Relations We begin by defining different types of linear relations. Definition 1 Let X = ($1, . . . , zn} be a set of real- valued variables. Let a, @’ be linear polynomials (i.e. polynomials of degree one) over X. A linear disequa- tion over X is an expression of the form cy # p. A lin- ear equality over X is an expression of the form a = p. A linear relation over X is an expression of the form cllrp where r E {<, 5, =, #, 2, >). A convex linear re- lation over X is an expression of the form cwc/3 where rc E {<,I,=, 2, >). A disequational linear relation over X is an expression of the form CE # /?. A dis- junctive linear relation (DLR) is a disjunction of one or more linear relations. Example 2 A typical DLR over {z~,z=~,Q} is (1.2x1+ x2 5 x3 + 5) V (12x3 # 7.5~2) V (x2 = 5). ‘Unfortunately, it seems like Hill will be in need of ju- ridiciaJ assistance. Throughout this paper, we assume all sets of DLRs to be finite. The definition of satisfiability for DLRs is straightforward. Definition 3 Let X = {xl, . . . , x,} be a set of real- valued variables and let R = {RI, . . . , Rk} be a set of DLRs over X. We say that R is satisfiable iff there exists an assignment of real values to the variables in X that makes at least one member of each Ri , 1 5 i 5 Ic, true. It is important to note that we only consider assign- ments of real values, thus assuming that time is linear, dense and unbounded. (We will see that it is sufficient to consider assignments of rational values further on.) We continue by classifying different types of DLRs. Definition 4 Let y be a DLR. C(y) denotes the con- vex relations in y and NC(r) the disequational rela- tions in y. We say that y is convex iff l~VC(r)l = 0 and that y is disequational iff IC(y)l = 0. If y is convex or disequational we say that y is homogenous and other- wise heterogenous. Furthermore, if IC(y)I 5 1 then y is Horn. We extend these definitions to sets of relations in the obvious way. For example, if r is a set of DLRs and all y E l? are Horn, then I’ is Horn. This classification provides the basis for the forthcom- ing proofs. One detail to note is that if a Horn DLR is convex then it is a unit clause, i.e. a disjunction with only one member. For Horn DLRs, we restrict ourselves only to use 5 and # in the relations. This is no loss of gen- erality since we can express all the other relations in terms of these two. For example, a DLR of the form x < y V D can be replaced by the disjunctions (x 5 y V D, x # y V D}. Observe that the resulting set of disjunctions can contain at most twice as many disjunctions as the original one. Hence, this is a poly- nomial time transformation. (Note, however, that this does not hold for general DLRs.) Definition 5 Let A be a satisfiable set of DLRs and let y be a DLR. We say that y blocks A iff for every d E NC(y), A U (d) is not satisfiable. Observe that if A u {y) is satisfiable and y blocks A then there must exist a relation S E C(y) such that A U {S} is satisfiable. This observation will be of great importance in forthcoming sections. Linear rogramming Our method for deciding satisfiability of Horn DLRs will be based on linear programming techniques and we will provide the basic facts needed in this section. The linear programming problem is defined as follows. Definition 6 Let A be an arbitrary m x n matrix of rationals in finite precision and let x = (zi, . . . , x~) be an n-vector of variables over the real numbers. Then an instance of the linear programming (LP) problem is defined by: {min cTx subject to Ax < b} where b is 1236 Planning an m-vector of rationals and c an n-vector of rationals. The computational problem is as follows: 1. Find an assignment to the variables xi, . . . , xn such that the condition Ax < b holds and cTx is minimial subject to these condityons, or 2. Report that there is no such assignment, or 3. Report that there is no lower bound for cTx under the conditions. Analogously, we can define an LP problem where the objective is to maximize cTx under the condition Ax 5 b. We have the following important theorem. Theorem 7 The linear programming problem is solv- able in polynomial time. Observe that the restriction to finite precision is not a restriction in practice since computers are (almost without exception) using finite precision arithmetics. Several polynomial algorithms have been developed for solving LPs. Well-known examples are the algorithms by Khachiyan (1979) and Karmarkar (1984). Despite their theoretical value, it is not at all clear that they out-perform the simplex algorithm which is exponen- tial in the worst case (Klee & Minty 1972). In fact, re- cent theoretical analyses lend support to its favourable average-case performance. (See for example (Smale 1983).) In the following, we assume all coeffecients to be rationals represented in finite precision. Satisfiability of Horn DLRs In this section we present a polynomial algorithm for deciding satisfiability of Horn DLRs. The algorithm can be found in Figure 1. The problem of decid- ing satisfiability for a set of Horn DLRs is denoted HoRNDLRSAT. We begin by exhibiting a simple method for deciding whether a set of convex linear re- lations augmented with one disequation is satisfiable or not. There may be more efficient methods for checking this than the one we propose. However, throughout this paper we will stress simplicity instead of tuning efficiency. Lemma 8 Let A be an arbitrary m x n matrix, b be an m-vector and x = (xi, . . . , x~) be an n-vector of vari- ables over the real numbers. Let a, ,@ be linear poly- nomial over xl,. . .,xn. Deciding whether the system S={Ax<b,a#p} is satisfiable or not is polynomial. Proof: Let cy’ = o - c and ,6’ = p - d where c and d are the constant terms in ~11 and ,6, respectively. Consider the following instances of LP: LPl= {min cy’ - p’ subject to Ax 5 b} LP2= {max CY’ - /3’ subject to Ax 5 b} If LPl and LP2 have no solutions then S is not sat- isfiable. If both LPl and LP2 yield the same optimal value d - c then S is not satisfiable since every solu- tion to LPl and LP2 forces cy to equal p. Otherwise S is obviously satisfiable. Since we can solve the LP problem in polynomial time by Theorem 7, the lemma follows. 0 Before proceeding, we recapitulate some standard mathematical concepts. Definition 9 Given two points x, y E Rn , a convex combination of them is any point of the form z = Xx + (1 - X)y where 0 5 X 5 1. A set S E R” is convex iff it contains all convex combinations of all pairs of points x,y E s. Definition 10 A hyperplane H in R” is a non-empty set defined as {X E R”lalxl+. . .+anxn = b} for some al ,..., a,,bE R. Definition 11 Let A be an arbitrary m x n matrix and b be an m-vector. The polyhedron defined by A and b is the set {x E R” IAx 5 b}. The connection between polyhedrons and convex sets is expressed in the following well-known fact. Fact 12 Every non-empty polyhedron is convex. Consequently, the convex relations in a set of Horn DLRs defines a convex set in R”. Furthermore, we can identify the disequations with hyperplanes These observations motivate the next lemma. in R”. Lemma 13 Let S g Rn be a convex set and let HI , . . . , Hk s R” be distinct hyperplanes. If S C Uf=, Hi then there exists a j, 1 5 j 5 k such that SCHj. Proof: If it is possible to drop one or more hyper- planes from H and still have a union containing S then do so. Let H’ = {Hi, . . . , HA}, m < k, be the resulting minimal set of hyperplanes. Every Hi E H’ contains some point xi of S not in any other Hi E H’. We want to prove that there If this is not the v is only one hyperplane case, consider the line in H’. segment L adjoining x1 and x2. (The choice of x1 and 52 is not important. Every choice of xi and xj, 1 5 i, j 5 m and i # j, would d o equally well.) By convexity, L 2 S. Each Hi E H’ either contains L or meets it in at most one point. But no Ha/ E H’ can contain L, since then it would contain both x1 and x2. Thus each Hi has at most one point in common with L, and the rest of L would not be a subset of UE1 Hi which contradicts that L E S E ULi Hi. cl We can now tie together the results and end up with a sufficient condition for satisfiability of Horn DLRs. Lemma 14: Let I’ be a set of arbitrary Horn DLRs. Let C C I’ be the set of convex DLRs in l? and let D = IDI,.. . , Dk} s I’ be the set of DLRs that are not convex. Under the condition that C is satisfiable, I’ is satisfiable if Dd does not block C for any 1 5 i 5 k. Proof: Pick one disequation di out of every Di such that {C, da) is satisfiable. This is possible since no Di blocks C. We show that I” = {C, dl, . . . , dk} is Temporal Reasoning 1237 algorithm SAT( I’) A c- U(C(y)ly E I? is convex} if A not satisfiable then reject if 3y E r that blocks A and is disequational then reject if 3y E I? that blocks A and is heterogenous then SAWI‘ - m u C(Y)) accept Figure 1: Algorithm for deciding satisfiability of Horn DLRs. satisfiable and, hence, I’ is satisfiable. Assume that 4 = (aa # &). Define the hyperplanes HI, . . . , Hk such that Hi = (x E R” 1 &i(z) = pi(z)}. Since ev- ery {C, di} is satisfiable, the polyhedron P defined by C (which is non-empty and hence convex by Fact 12) is not a subset of any Hi. Suppose I” is not satis- fiable. Then P - &, Hi = 0 which is equivalent with P E Uf=, Hi. By Lemma 13, there exists a Nj, 1 < j < k such that S C_ Hj. Clearly, this contradicts our iniZia1 assumptions. cl It is important to note that the previous lemma does not give a necessary condition for satisfiability of Horn DLRs. We claim that the algorithm in Figure 1 cor- rectly solves HORNDLRSAT in polynomial time. To show this, we need an auxiliary lemma which is a for- mal version of an observation made in the second sec- tion of this paper. Lemma 15 Let I? be a set of Horn DLRs and let C & I’ be the set of convex DLRs in I’. If there exists a heterogenous DLR y E I’ such that y blocks C, then I’ is satisfiable iff (I’ - (7)) U C(y) is satisfiable. Proof: if: Trivial. only-if: If I’ is satisfiable, then y has to be satisfiable. Since y blocks C, C(y) must be satisfied in any solution 0f r. Cl We can now prove the soundness and completeness of SAT. Lemma 16 Let I’ be a set of Horn DLRs. If SAT(I’) accepts then l? is satisfiable. Proof: Induction over n, the number of heterogenous DLRs in I’. Basis step: If n = 0 and SAT(r) accepts then the formulae in A are satisfiable and there does not exist any y E I’ that blocks A. Consequently, I’ is satisfiable by Lemma 14. Induction hypothesis: Assume the claim holds for n = k, k 2 0. Induction step: I’ contains k + 1 heterogenous DLRs. If SAT accepts in line 5 then (I’ - {y}) u C(y), which contains k heterogenous DLRs, is satisfiable by the in- duction hypothesis. By Lemma 15, this is equivalent with I’ being satisfiable. If SAT accepts in line 6 then 1238 Planning there does not exist any disequational or heterogenous y E r which blocks A. By Lemma 14, this means that I’ is satisfiable. cl Before proving the completeness of SAT we need the following lemma. Lemma 17 Let I’ be a set of Horn DLRs. Let C E I’ be the set of convex DLRs in I’. If there exists a disequational DLR y E I’ that blocks C then I’ is not satisfiable. Proof: In any solution to r, the relations in C U (y} must be satisfied. Since 7 is disequational and blocks C this is not possible and the lemma follows. 0 Lemma 18 Let r be a set of Horn DLRs. If SAT(r) rejects then l? is not satisfiable. Proof: Induction over n, the number of heterogenous DLRs in r. Basis step: If n = 0 then SAT can reject in lines 3 and 4. If SAT rejects in line 3 then, trivially, I? is not satisfiable. If SAT rejects in line 4 then there exists a disequational y E I’ that blocks A. Hence, l? is not satisfiable by Lemma 17. Induction hypothesis: Assume the claim holds for n = k, k > 0. Indu&‘on step: I’ contains k + 1 heterogenous DLRs. If SAT rejects in line 3 then l? is not satisfiable. If SAT rejects in line 4 then l? is not satisfiable by Lemma 17. If SAT rejects in line 5 then (l? - (y}) U C(y), which contains k heterogenous DLRs, is not satisfiable by the induction hypothesis. By Lemma 15, this is equivalent with I? not being satisfiable. 0 Finally, we can show that SAT is a polynomial-time al- gorithm and, thus, show that HORNDLRSAT is poly- nomial. Theorem 19 HORNDLRSAT is polynomial. Proof: By Lemmata 16 and 18, it is sufficient to show that SAT is polynomial. The number of recur- sive calls is bounded by the number of heterogenous DLRs in the given input. By Lemma 8, we can in poly- nomial time decide whether a linear inequality system with one disequation is satisfiable. Since we need only check a polynomial number of such systems in each re- cursion, the theorem follows, cl We conclude this section with a discussion about whether HORNDLRSAT can be effeciently solved on parallel computers. The complexity class NC consists of the problems that can be solved with a polynomial number of processors in polylogarithmic time and it is often argued that NC captures our intuitive notion of problems satisfactorily solved by parallel computers. Recall that the satisfiability problem for propositional Horn clauses (HORNSAT) is P-complete under log- space reductions (Greenlaw, Hoover, & Russo 1993). Clearly, it is trivial to reduce HORNSAT to HoRNDLR- SAT in log-space. Since HORNDLRSAT is polynomial, it follows that it is P-complete as well. This implies that HORNDLRSAT is not in NC and, hence, there does not exist parallel algorithms for HORNDLRSAT that is substantially faster than ordinary sequential al- gorithms. (Unless NC=P which is considered very un- likely.) Comparison In this section, we show that Horn DLRs subsumes sev- eral other methods for temporal constraint reasoning. Let Z, y be real-valued variables, c, d constants and A Allen’s algebra (Allen 1983) in the definitions below. Definition 20 (Nebel & Biirckert 1995) An ORD clause is a disjunction of relations of the form zry where r E I<,=, #}. The ORD-Horn subclass Z is the relations in A that can be written as ORD clauses containing only disjunctions with at most one relation of the form z = y or x 5 y and an arbitrary number of relations of the form x # y. Note that the ORD-Horn class subsumes both the con- tinuous endpoint algebra (Vilain, Kautz, & van Beek 1989) and the pointisable endpoint algebra (van Beek & Cohen 1990). Definition 21 (Koubarakis 1992) Let T E {I,>, #}. A Koubarakis formula is a formula on one of the fol- lowing forms (1) (x - y)rc, (2) xrc or (3) a disjunction of formulae of the form (x - y) # c or x # c. Definition 22 (Dechter, Meiri, & Pearl 1991) A sim- ple temporal constraint is a formula on the form c 5 (x - y) < d. - Simple temporal constraints are equivalent with the simple metric constraints (Kautz & Ladkin 1991). Definition 23 (Meiri 1991) A CPA/single interval formula is a formula on one of the following forms: (1) cq (x - y) r2 d; or (2) x ry where r E {<, 5, =, # ,>, >} and rl, r2 E {<, I}. Definition 24 (Gerevini, Schubert, & Schaeffer 1993) A TG-II formula is a formula on one of the following forms: (1) c 5 x 5 d, (2) c 5 x - y 5 d or (3) x r y where r E {<, 5, =, f, 2, >}. We can now state the main theorem of this section. Theorem 25 The formalisms defined in Definitions 20 to 24 can trivially be expressed as Horn DLRs. Note that Meiri (1991) considers two further tractable classes that cannot (in any obvious way) be trans- formed into Horn DLRs. The finding that the ORD- Horn algebra can be expressed as Horn DLRs is espe- cially important in the light of the following theorem. Theorem 26 (Nebel & Biirckert 1995) Let S be any subclass of A that contains all basic relations. Then either 1. S C Z and the satisfiability problem for S is poly- nomial, or 2. Satisfiability for S is NP-complete. By the previous theorem, we cannot expect to find tractable classes that are able to handle all basic re- lations in A and, at the same time, are able to han- dle any single relation that cannot be expressed as a Horn DLR. In other words, the qualitative fragment of HORNDLRSAT inherits the maximality of the ORD- Horn algebra. iscussion Several researchers in the field of temporal constraint reasoning have expressed a feeling that their proposed methods should be extended so they can express rela- tions between more than two time points. As a first example, in (Dechter, Meiri, & Pearl 1991) one can read “The natural extension of this work is to ex- plore TCSPs with higher-order expressions (e.g. “John drives to work at least 30 minutes more than Fred does” ; X2 - X1 + 30 5 X4 - Xs)...” Even though they do not define the exact meaning of “higher-order expressions” we can notice that their example is a sim- ple Horn DLR. Something similar can be found in (Koubarakis 1992) who wants to express “the dura- tion of interval I exceeds the duration of interval J”. Once again, this can easily be expressed as a Horn DLR. These claims seem to indicate that the use of Horn DLRs is a significant contribution to temporal reasoning. We have shown that the satisfiability problem for Horn DLRs can be carried out in polynomial time. However, the method builds on solving linear programs and it is well-known that such calculations can be com- putationally heavy. It is important to remember the reasons for introducing Horn DLRs. The main reason was not to provide an extremely efficient method, but to find a method unifying most of the other tractable classes reported. It is fairly obvious that the pro- posed method cannot outperform highly specialized al- gorithms for severely restricted classes. It should be likewise obvious that the specialized methods cannot compete with Horn DLRs in terms of expressivity. We are, as always in tractable reasoning, facing the trade- off between expressivity and computational complex- ity. We believe, though, that the complexity of decid- ing satisfiability can be drastically improved by devis- ing better algorithms than SAT. The algorithm SAT is constructed in a way that facilitates its correctness proofs and it is not optimized with respect to execu- tion time in any way. The question whether improved versions can compete with algorithms such as TIME- GRAPH II or not remains open. Throughout this paper we have assumed that time is linear, dense and unbounded but this may not be the case in real applications. For example, in a sam- pled system we cannot assume time to be dense. One Temporal Reasoning 1239 question to answer in the future is what the effects of changing the assumptions of time are. Switching to discrete time will probably make reasoning computa- tionally harder. There are some positive results con- cerning discrete time, however. Meiri (1991) presents a class of temporal constraint reasoning problems where integer time satisfiability is polynomial. Conclusion We have introduced the Horn DLR as a means for tem- poral constraint reasoning. We have proven that de- ciding satisfiability of sets of Horn DLRs is polynomial by exhibiting an algorithm based upon linear program- ming. Furthermore, we have shown that several other approaches to tractable temporal constraint reasoning can be encoded as Horn DLRs, including the ORD- Horn algebra and most methods for purely quantita- tive reasoning. Acknowledgements We would like to thank Marcus Bjareland and Thomas Drakengren for discussions and comments. We are also indebted to William C. Waterhouse who improved our original proof of Lemma 13. References American Association for Artificial Intelligence. 1991. Proceedings of the 9th (US) National Conference on Artificial Intelligence (AAAI-91), Anaheim, CA, USA: AAAI Press/MIT Press. Allen, J. F. 1983. Maintaining knowledge about temporal intervals. Communications of the ACM 26(11):832-843. Dechter, R.; Meiri, I.; and Pearl, J. 1991. Temporal constraint networks. Artificial Intelligence 49:61-95. Gerevini, A.; Schubert, L.; and Schaeffer, S. 1993. Temporal reasoning in Timegraph I-II. SIGART Bul- letin 4(3):21-25. Golumbic, M. C., and Shamir, R. 1993. Complexity and algorithms for reasoning about time: A graph- theoretic approach. Journal of the ACM 40(5):1108- 1133. Greenlaw, R.; Hoover, H. J.; and Russo, W. L. 1993. A Compendium of Problems Complete for P. Oxford: Oxford University Press. Kanellakis, P. C.; Kuper, G. M.; and Revesz, P. Z. 1995. Constraint query languages. Journal of Com- puter and System Sciences 51( 1):26-52. Karmarkar, N. 1984. A new polynomial time algo- rithm for linear programming. Combinatorics 4:373- 395. Kautz, H., and Ladkin, P. 1991. Integrating met- ric and temporal qualitatvie temporal reasoning. In AAAI-91 (1991), 241-246. Khachiyan, L. G. 1979. A polynomial algorithm in linear programming. Soviet Mathematics Doklady 20:191-194. Klee, V., and Minty, G. J. 1972. How good is the simplex algorithm ? In Shisha, O., ed., Inequalities III, 159-175. Koubarakis, M. 1992. Dense time and temporal con- straints with #. In Swartout, B., and Nebel, B., eds., Proceedings of the 3rd International Conference on Principles on Knowledge Representation and Reason- ing (KR-9,?), 24-35. Cambridge, MA, USA: Morgan Kaufmann. Meiri, I. 1991. Combining qualitative and quantita- tive constraints in temporal reasoning. In AAAI-91 (1991), 260-267. Nebel, B., and Biirckert, H.-J. 1995. Reasoning about temporal relations: A maximal tractable sub- class of Allen’s interval algebra. Journal of the ACM 42( 1):43-66. Smale, S. 1983. On the average speed of the sim- plex method of linear programming. Mathematical Programming 271241-262. van Beek, P., and Cohen, R. 1990. Exact and ap- proximate reasoning about temporal relations. Com- putational Intelligence 6(3):132-144. Vilain, M. B.; Kautz, H. A.; and van Beek, P. G. 1989. Constraint propagation algorithms for tempo- ral reasoning: A revised report. In Weld, D. S., and de Kleer, J., eds., Readings in Qualitative Reason- ing about Physical Systems. San Mateo, Ca: Morgan Kaufmann. 373-381. 1240 Planning | 1996 | 183 |
1,825 | A Connectionist Framework for Reasoning: Reasoning with Examples Dan Roth* Dept. of Appl. Math. & CS, Weizmann Institute of Science, Israel danr@wisdom.weizmann.ac.il Abstract We present a connectionist architecture that supports almost instantaneous deductive and abductive reason- ing. The deduction algorithm responds in few steps for single rule queries and in general, takes time that is linear with the number of rules in the query. The abduction algorithm produces an explanation in few steps and the best explanation in time linear with the size of the assumption set. The size of the network is polynomially related to the size of other representa- tions of the domain, and may even be smaller. We base our connectionist model on Valiant’s Neu- roidal model (Va194) and thus make minimal assump- tions about the computing elements, which are as- sumed to be classical threshold elements with states. Within this model we develop a reasoning framework that utilizes a model-based approach to reasoning (KKS93; KR94b). In particular, we suggest to inter- pret the connectionist architecture as encoding exam- ples of the domain we reason about and show how to perform various reasoning tasks with this interpre- tation. We then show that the representations used can be acquired efficiently from interactions with the environment and discuss how this learning process in- fluences the reasoning performance of the network. Introduction Any theory aiming at understanding commonsense rea- soning, the process that humans use to cope with the mundane but complex aspects of the world in evaluat- ing everyday situations, should account for the flexibil- ity, adaptability and speed of commonsense reasoning. Consider, for example, the task of language under- standing, which humans perform effortlessly and ef- fectively. It depends upon our ability to disambiguate word meanings, recognize speaker’s plans, perform pre- dictions and generate explanations. These, and other “high level” cognitive tasks such as high level vision and planning have been widely interpreted as inference tasks and collectively comprise what we call common- sense reasoning. *Research supported by the Feldman Foundation and a Grant from the Israeli Ministry of Science and the Arts. 1256 Rule-Based Reasoning h Connectionism Deductive and abductive reasoning are the basic in- ference tasks considered in the context of high level cognitive tasks. In this paper we suggest an alterna- tive to the current connectionist account of these tasks. Connectionist networks have been argued to be bet- ter suited than traditional knowledge representations for studying everyday common sense reasoning. Some of the arguments used are that these models have the ability to simultaneously satisfy multiple constraints, dynamically adapt to changes, achieve robustness and provide a useful way to cope with conflicting and uncer- tain information (Sun95; Pin95; Der90). This should be contrasted with the view that connectionist model are incapable of performing high level cognitive tasks because of their difficulties with representing and ap- plying general knowledge rules (FP88). The latter opinion, we believe, may reflect on the fact that a lot of the research on understanding high level cognition using connectionist models is actually trying to represent and apply general knowledge rules. Indeed, a lot of the research in this direction is influenced by a research program launched in the fifties, the “knowledge-base+inference engine” ap- proach (McC58), which is still the generally accepted framework for reasoning in intelligent systems. The idea is to store the knowledge, expressed in some rep- resentation language with a well defined meaning as- signed to its sentences, in a Knowledge Base (li’B). The I<B is combined with a reasoning mechanism (“in- ference engine”) that is used to determine what can be inferred from the sentences in the K B. The effort to develop a logical inference engine within a connection- ist architecture is represented by works such as (BH93; IIK91; SA90; SA93; Sun95; LD91; Pin95; Der90). Given the intractability of the general purpose knowledge base+inference engine approach to reason- ing, a significant amount of recent work in reasoning concentrates on (1) identifying classes of limited ex- pressiveness, with which one can still perform reason- ing efficiently or (2) resorting to an approximate in- ference engine. These directions have been pursued both in the knowledge representation and reasoning (II’R&R) community and in the connectionism com- From: AAAI-96 Proceedings. Copyright © 1996, AAAI (www.aaai.org). All rights reserved. munity. The former line of research is represented in KR&R by many works such as (BL84; Lev92; Rot93; SK90; Cad95) and in the connectionism community by (SA90; BH93; HK91). The latter usually builds on using Hopfield’s networks (HT82) or Boltzmann ma- chines (HS86), in an effort to solve optimization prob- lems that are relaxations of propositional satisfiabil- ity. This approach is used, for example, in (Pin95; Der90) and is related to approaches suggested in the KR&R community (SLM92; MJPSO). None of these works, however, meets the strong tractability requirements required for common-sense reasoning as argued e.g., in (Sha93). Moreover, many of these works have carried out the “knowledge baseSinference engine” research program also by ne- glecting to consider the question of how this knowledge might be acquired’ and by measuring performance of the reasoning process in absolute terms rather than with respect to the preceding learning process. We utilize a model-based approach to reasoning (KKS93; KR94b) to yield a network that is not a “logical inference engine” but, under some (formally phrased) restrictions, behaves “logically” with respect to a world it interacts with. Our model-based algo- rithms support instantaneous deduction and abduc- tion, in cases that are intractable using other knowl- edge representations. The interpretation of the con- nectionist architecture as encoding examples acquired via interaction with the environment, allows for the integration of the inference and learning processes (KR94a) and yields reasoning performance that nat- urally depends on the process of learning the network. We develop the reasoning framework within Valiant’s Neuroidal paradigm (Va194), a computational model that is intended to be consistent with the gross biological constraints we currently understand. In par- ticular, this is a programmable model which makes minimal assumptions about the computing elements, assumed to be classical threshold elements with states. In this abstract we focus on presenting the reason- ing framework: the architecture, its interpretation as a set of examples and the reasoning algorithms. The learning issues are discussed only briefly. The Reasoning Framework This paper considers two inference tasks, Deduction2 and Abduction. Deduction, the basic inference task considered in the context of high level cognitive tasks is usually modeled as follows: given a Boolean function W, represented as a conjunction of rules and assumed to capture our knowledge of the world, and a Boolean function CY, a query that is supposed to capture the ’ (Pin95) is an exception. 2We emphasize that these terms are used only to give semantics to the network’s behavior. The network is not a “logical inference engine” but, under some restrictions on the queries presented, behaves “logically” with respect to a world it had interactions with. situation at hand, decide whether W logically implies o (denoted W b CY). Abduction is a term coined by Peirce (Pei55) to describe the inference rule that con- cludes A from an observation B and the rule A + B, given that there is no “better” rule explaining B. The importance of studying abduction became clear in the past few years when some general approaches to Natu- ral Language interpretation have been advanced within the abduction framework (HSME93). We adopt an alternative, model-based approach to the study of commonsense reasoning, in which the knowledge base is represented as a set of models (sat- isfying assignments) of the domain of interest (the “world”) rather than a logical formula describing it. It is not hard to motivate a model-based approach to reasoning from a cognitive point of view and indeed, most of the proponents of this approach to reason- ing have been cognitive psychologists (JL83; JLB91; Kos83), who have alluded to the notion of “reason- ing from examples” on a qualitative basis. Building on the work of (KKS93; KR94b) we show that model- based reasoning can be implemented in a connectionist network to yield an efficient reasoning network. In our framework, when reasoning with respect to the “world” W, information about the W is stored in a network N and is interpreted as a collections of exam- ples observed in W. 3 We present both deduction and the abductive task of verifying that an explanation is consistent as a series of forward evaluation tasks. Each takes 5 computational steps. The task of producing an explanation utilizes the backwards connections in the networks, and is also instantaneous. In both cases, if the content of the network is a good representation of W, in a well defined sense, for a wide class of queries the network response is provably correct. Interaction with the network for queries presentation and learn- ing the representation is done in a unified manner, via observations and the performance of the reasoning is shown to depend naturally on this interaction. Reasoning Tasks We briefly present the reasoning tasks and some rel- evant results. See (KR94b; KR94a) for details. We consider reasoning over a propositional domain. The reasoning queries are with respect to a “world” (do- main of interest) that is modeled as a Boolean func- tion (a propositional expression) f : (0, l}n + (0, 1). Let X = (~1,. . ., xfl} be a set of variables, each of which is associated with a world’s attribute and can take the value 1 or 0 to indicate whether the associ- ated attribute is true or false in the world. (n is our complexity parameter.) An assignment x E (0, l}n satisfies f if f(x) = 1. (x is also called a model of f.) By “f entails (implies) g” , denoted f b g, we mean that every model of f is also a model of g. 3We restrict our discussion to this fragment of the net- work; in general, this will be part of a larger network and will overlap with network representations of other “worlds”. Rule-Based Reasoning & Connectionism 1257 In deduction (entailment), given Boolean functions f (assumed to capture our knowledge of the world) and cy (a query that is supposed to capture the situation at hand) we need to decide whether f implies o (de- noted f i= a). For abduction, we refer here to one of the propositional formalisms in which abduction is defined as the task of finding a minimal explanation, given a knowledge base f (the background theory), a set of propositional letters A (the assumption set), and a query letter q. An explanation of q is a minimal sub- set E 5 A such that (1) f A (/&Ex) b q and (2) f A (&v:EE x) # 8. Thus, abduction involves tests for entailment (1) and consistency (2), but also a search for a minimal4 explanation that passes both tests. Reasoning with Models The model based strategy for the deduction problem f b a is to try and verify the implication relation using model evaluation. In doing so, the knowledge base consists of a set I? of models of f rather than a Boolean function. When presented with a query CE the algorithm evaluates Q on all the models in I’. If a counterexample x such that a(x) = 0 is found, then the algorithm returns “No”. Otherwise it returns “Yes”. Clearly, the model based approach solves the infer- ence problem if I is the set of allmodels off. However, the set of all models might be too large, making this procedure infeasible computationally. A model-based approach becomes useful if one can show that it is pos- sible to use a fairly small set of models as the test set I’, and still perform reasonably good inference. Exact Reasoning using models is based on a the- ory developed in a series of papers (KKS93; KR94b; KR94a; KR95) where a characterization of when a model based approach to reasoning is feasible is de- veloped. An important feature of the theory is that the correctness of reasoning depends on the type of queries presented and not so much on the world we rea- son about (provided that the reasoner holds a “good” description of the world). The class of queries which al- lows efficient model-based reasoning is called the class of common queries (Qc). It contains a rich class of theories and, in particular, all Horn and all 1ognCNF functions. Proving the feasibility of model-based rea- soning involves showing that for the purpose of reason- ing with respect to Q,, a Boolean function f can be represented using a polynomial size set of models, I’f . Theorem 1 ((KR94b)) For any knowledge base f there exists a set rf of models whose size is poEynomially5 related to the DNF size of f. Deduc- 4Here minimal means that no subset of it is a valid ex- planation. In general this is not, by itself, adequate for choosing among explanations and more general schema can be discussed in our framework. ‘Thus, l?f is in general exponentially smaller than the number of satisfying assignments of f, and sometimes even exponentially smaller than the DNF representation. tion (with respect to Q,) and Abduction (given a query q and assumption set A) can be performed correctly in polynomial time, using rf . Approximate Reasoning is related to the notion of pat learning (Va184) and was developed in (KR94a). We assume that the occurrences of observations in the world is governed by a fixed but arbitrary and unknown probability distribution D defined on (0, 1)“. A query (Y is called (f, c)-fair if either f C a or Prob [f \ @I > E. An algorithm for approximate de- duction will is to err on non-fair queries. (Intuitively, it is allowed to err in case f p o, but the weight (un- der D) of f outside a is very small.) Along with the accuracy parameter E, we use a confidence parameter 5 which stands for the small probability that the rea- soning algorithm errs on fair queries. Theorem 2 Let & be a class of queries of interest, and let 0 < S, E be given confidence and accuracy pa- rameters. Suppose that we select m = $(ln IQ1 + In $) independent examples according to D and store in I’ all those samples that satisfy f. Then the probability that the model-based deduction procedure errs on an (f, e)-fair query in Q is less than 6. Since the queries in Q are Boolean functions of polyno- mial size, the number m of samples required is polyno- mial. Moreover, given a set of possible explanations as input, this approach efficiently supports the entailment and consistency stages of abductive reasoning. The Connectionist Framework The architecture investigated is based on Valiant’s Neuroidal model (see (Va194) for details). We present just the few aspects we need to describe the knowledge representation that supports the reasoning tasks. Valiant’s Neuroidal model is a programmable model which makes minimal assumptions on the computing elements, assumed to be classical threshold elements with states. We make a few minor abstractions for methodological purposes. Most importantly, we ab- stract away the important notion that in the localist representation assumed, every item is represented as a “cloud” of nodes rather than a single node. A 5-tuple (G(G) E), W, M, S, X) defines a network N. Here G = G(G, E) is a directed graph describing the topology of the network, W is the set of possible weights on edges of G, IM is the set of modes a node can be in at any instant, S is the update function of the mode and X is the update function of the weights. We view the nodes of the net as a set G of proposi- tions. The set E is a set of directed edges between the nodes. The set of weights W is a set of numbers. eij denotes the edge directed from i to j, and its weight 1s wij. Sometimes both eij, eji E E. The mode, (s, T), of the node describes every aspect of its instantaneous condition other than the weights on its incoming edges. s E S is a finite set of states and T is a threshold. In particular, S consists of two kinds of states F and &, 1258 Rule-Based Reasoning 81 Connectionism which stand for firing (that is, the node is active at this time), and quiescent (a non-active state). The mode transition function S specifies the updates that occur to the mode of the node from time t to t + 1. S depends on the current state of the node and the sum of weights zui = Ck{wlm E E,k E F}, of its active parents. Similarly, the weight transition function X defines for each weight WQ at time t the weight to which it will transit at time t + 1. The new value may depend on the values of the weights on the edges between node i and its parents, their firing state and the mode of i, all at time t. Two default transitions are assumed. First, a threshold transition by default occurs whenever w; > Ti at the ith node, provided that no explicit condition that overrides the default is stated. The second default assumed is that a node in a firing state ceases firing at the end of the time unit. To further specify a network we need to define the initial conditions IC, (i.e., initial weights and modes of the nodes) and input sequence IS. The interaction of the network with the outside world is modeled by assuming the existence of peripherals. They have the power to cause various sets of nodes in the network to fire simultaneously at various times. Every interaction like that we call here an observation. It specifies the set of nodes that the peripherals activate at an instant, i.e., the set of propositions that are observed to be active in the environment. The actual choices of the sets and the times in which the observations are presented to the network determine the input sequences IS. Timing is crucially important to the model. After the peripherals prompt the network and cause some subset of nodes to fire simultaneously, a cascade of computation follows, and the algorithm has to ensure that it terminates in a stable situation, before the time unit has elapsed. Typically, the peripherals will prompt low level nodes and the algorithm being exe- cuted may need to modify nodes representing higher level concepts, that are separated in the network from the prompted ones by several intermediate nodes. Knowledge Representation To emphasize the correspondence between the network and propositional reasoning, we consider a subset of the nodes in N which are controlled by the peripherals and view it as a set X = (21, . . . , x~} of propositions. For simplicity, in order to describe both the presence and the absence of an attribute xi, it is duplicated in the representation: one node describes zi and an- other describes z. We represent each interaction with the network as an observation ZI = (xi1 = zlil, xi2 = viz, * * *, q-j = vid), with d < n, vi E (0, l}, and this is translated to a corresponding node activation by the peripherals. For example, when the observation is (xi = 1,x2 = 1, x3 = 0), the peripherals activate nodes corresponding to xi, 22 and 83. An observation v can be interpreted also as a query presented to the network in the reasoning stage. The presentation of v is interpreted as the Boolean query (Y = Zi, A . . . A Zi, , where Zj = Xj if Vj = 1, Zi = 6 if Vj = 0. Definition 1 Let y be a node in N, EY = {zlezy E E) its set of parents. A node z E EY is called a model of y and e, = {i E Ezlwil = 1) its set of components. The model-based representation of y, iMY = {(z, e,)lz E E,), is the set of models and their components. We assume also that the positive and negative literals of each proposition are connected via a relay node. Fig- ure 1 depicts a model-based representation of y. The edges are assumed to be bidirectional (i.e., each line represents two edges) and all the weights on the edges drawn are assumed to be 1. Every model is connected to all 2n literals, and the n not drawn are assumed to have weight 0. Initially, all the thresholds in the repre- sentation are set to a high value, denoted by 00. The algorithms also assume a specific set of initial modes of the nodes in the representation. “1 “1 “3 “ 3 “4 “ 4 Figure 1: A Connectionist Model-Based Representation A model z can be represented as a Boolean vector. If ez = (4, d2, . . .L) is a representation of z as a set of its components (Zi E {xi, q}), than e, = [bl, bar . . . b,] is its Boolean representation, where bi E (0, 1) is defined by: bi = 1 if Ii = xi, bi = 0 if Zi = c. It can be verified that the model-based representation presented in Figure 1 is the representation of the function f = {%A= + x3, -AC --+ x2, xi A x2 A x4 -+ x3) with respect to all Horn queries. (See (KR94b).) In general, a network N will be a collection of such model-based representations. These can share nodes and any input to the network may influence many of them. Thus, although we discuss “logical” behavior, no global consistency is required. Note that while n is our complexity parameter, it is not related to the size of the whole network, but only to the number of propositions “related” to y in its local network. Rule-Based Reasoning & Connectionism 1259 Reasoning in the Network We briefly describe the reasoning algorithms, for lack of space. A complete description appears in (Rot96a). We note that within this framework, there are quite a few other ways to achieve the same goal. In particu- lar, we could define other modes and use other ways to evaluate the queries on the models stored in the model- based representation. We emphasize two design deci- sions that we view as important to the approach. First, queries are presented to the network as conjunctions. Thus, consistently with the natural interface consid- ered when learning the network, queries are viewed as observations - a list of propositions that are active (or non-active) in the environment. Second, in our algo- rithms, the top node, where the decision is being made, need not know the size of its input domain (the number of propositional letters). This is essential also to the extension to reasoning with incomplete information. Let N be a network which contains a model-based representation for f. That is, there exists a network structure as in Definition 1 and Figure 1. We im- ply nothing on the models stored in the network (i.e., which are the components of the models). We also as- sume that various nodes are in suitable initial states. Algorithms are described in the format of a sequence of steps (following (Va194)). First, we describe the initial (pre-)conditions assumed in the network. The input (“prompt”) is orchestrated by the peripherals, which also “collect” the algorithm’s response, repre- sented as a pattern of firings of one or more nodes. At each step, “prompt” describes the input at this stage - the set of nodes that the peripherals force to fire this time. Then, we define the transitions that are invoked during the following time unit at the relevant nodes. All other aspects of the algorithm are fully distributed. The effect of the algorithm on any node not directly prompted is completely determined by the transition rules and by the conditions at this node and at its par- ents. The overall algorithm can be invoked at any time, by having the preconditions of the first step satisfied as a result of an appropriate prompt. Deduction Consider the deduction problem f /= a. Queries are presented to the network as conjunctions of rules, a = Cl A . . . A Ck. Every rule has the form C = A + B, where A and B are conjunctions of literals. Since f b Cl A . . . A cr, iff f b Ci ‘d’i E (1, k), it is sufficient to consider the single6 rule case. We respond to f j= (A -+ B) using the following version of the reasoning with models algorithm. Given the set I? of models, filter out the models that do not satisfy A. Respond no (y inactive) iff one of the re- maining models (which satisfied A) does not satisfy B. 61t is easy to extend the algorithm to handle sequentially the presented rules, timed by the peripherals, and respond only after seeing the last rule. The thing to note is that it takes constant time to respond to a single rule, and the total time is linear in the number of rules. Only the top node y and the example nodes take part in the deduction algorithm AIgD. It takes five steps: in the first two steps, the A part of the query is presented by the peripherals and is evaluated on all the models; in the next two steps, the B part of the query is presented by the peripherals and is evaluated on all the models that satisfied the A part; finally, the top node fires if all the models that satisfied A satisfy also B. In the first step, an example node that receives ac- tivity wakes up and stores the total incoming weight for later comparison. A weight flip is used to evaluate the query presented to the network on the examples stored in it. In the second step, the same propositional nodes are prompted. This time, due to the weight flip, an example satisfies the observation (query) presented iff the input it sees doubles. In this case it fires and changes its mode to wait for the second part of the query. The same mechanisms works for the second part of the query, but applies only to examples which sat- isfied A. Therefore, it is sufficient for the top node to record (by setting its threshold) the number of these examples and make sure they all satisfy B also. Fi- nally, the peripherals also prompt the target node y and this is used for the case where no model satisfies A, in which the response should also be “yes”. The algorithm also makes sure that all the nodes return to their original states. Depending on the content of the representation we can prove: Theorem 3 Let y be a node in the network N, and let MY be its model-based representation. (1) If My consists of the set of models I’Fc then AIgD performs correct deduction whenever presented with a common !wJ-Y* (q If q/ consists of a set of models off ac- quired by sampling the environment according to dis- tribution D then, with high probability, AlgD performs correct deduction whenever presented with an (f, c)-fair query with respect to D. Abduction The algorithms for abductive reasoning, are not presented here. They perform the following tasks: (i) Given a candidate explanation f and a query, verify that E is a valid explanation. (ii) Provided that candidate explanations are represented as dedicated nodes in the network, given a query, the algorithm fires a valid explanation 2?. All these tasks can be performed in constant time. In addition, the peripherals can use (i) to greedily present (subsets of) the collected output of (ii), in search for a minimal explanation. The algorithm is similar to the deduction algorithm with the main distinction being that in this case we uti- lize the relay nodes and the backwards connections in order to communicate information down the network. Learning to Reason An essential part of the developed framework is that reasoning is performed by a network that has been learned from interaction with the environment (KR94a). For this purpose we have defined the interac- 1260 Rule-Based Reasoning & Connectionism tion with the network via queries that are represented as observations. This allows for combining the inter- faces to the world used by known learning models with the reasoning task. For example, the main avenue of interaction with the world used in the formal study of learning is an Example Oracle. When accessed, this oracle returns v E (0, l}“, drawn at random according to a distribution D; v can be viewed as an observa- tions v = (vii, . . . , vid) and interpreted also as a query. Examples presented in this way can be “memorized” into our network (Va194), and in combination with AlgD this provides a Learning to Reason algorithm that interacts with the environment, learns a model- based representation and supports correct entailment. Furthermore, using (2) of Theorem 3, the dependence of the reasoning performance on the learning process can be stated qualitatively. This type of interaction is supported also by the on-line L2R models (KR94a; Rot95) and can be shown to support other reasoning tasks, when augmented with membership and reason- ing queries. Conclusion This paper develops a new approach to reasoning in connectionist networks. We suggest to interpret the connectionist architecture as encoding examples and show how to perform various reasoning tasks with this interpretation. Assuming the network encodes a (reasonably small) set of representative examples of a “world”, we proved that our algorithms perform cor- rect deduction and abduction, tasks that were con- sidered intractable under other knowledge represen- tations. Moreover, our framework naturally supports Learning to Reason and the representations used can be efficiently acquired by interaction with the world. We believe that these results make this model suit- able for studying reflexive reasoning (Sha93; Va194). This work is part of a project in which we are trying to understand how networks of simple and slow neuron- like elements can encode a large body of knowledge and perform a wide range of interesting inferences almost instantaneously. It provides the theoretical framework for a system that learns knowledge representations for natural language understanding tasks. References A. Beringer and S. Holldobler. On the adequateness of the connection method. In AAAI-93, pages 9-14. R. Brachman and H. Levesque. The tractability of sub- sumption in framebased description languages. AAAI-84. M. Cadoli. Tractable Reasoning in Artificial Intelligence. Springer-verlag, 1995. Let. notes in AI, vol 941. M Dertthick. Mundane reasoning by settling on a plausi- ble model. Artificial Intelligence, 46:107-157, 1990. J . A. Fodor and Z. W. Pyilshyn. Connectionism and cogni- tive architecture: a critical analysis. Cognition, 28, 1988. S. Holldobler and F. Kurfeb. CHCL Paradlelization in Inference Systems. Springer-Verlag, 1991. G. E. Hinton and T. J. Sejnowski. Learning and re- learning in Botzmann machines. PDP (Volume I: Foun- dations), pages 282-317. MIT Press, 1986. R. Hobbs, J, M. Stickel, P. Martin, and D. Edwards. In- terpretation as abduction. Art. Intell., 63:69-142, 1993. J. J. Hopfield and D. W. Tank. Neural computation of decisions in optimization problems. Biol. Cyber-., 1982. P. N. Johnson-Laird. Mental Models. Harvard Press, 1983. P. N. Johnson-Laird and R. M. J. Byrne. Deduction. Lawrence Erlbaum Associates, 1991. H. Kautz, M. Kearns, and B. Selman. Reasoning with characteristic models. In AAAI-999, pages 34-39. S. M. Kosslyn. Image cand Mind. Harvard Press, 1983. R. Khardon and D. Roth. Learning to reason. In AAAI- 94, pages 682-687. R. Khardon and D. Roth. Reasoning with models. In AAAI-94, pages 1148-1153. R. Khardon and D. Roth. Default-reasoning with models. In IJCAI-95, pages 319-325. T. E. Lange and M. G. Dyer. High-level inferencing in a connectionist network. Connection Science, 1991. H. Levesque. Is reasoning too hard ? In Proceeding of the 3rd NEC research Symposium. 1992. J. McCarthy. Programs with common sense. In R. Brach- man and H. Levesque, Readings in IiR, 1985. S. Minton, M. D. Johnson, and A. B. Phillips. Solving large scale constraint satisfaction and scheduling problems using a heuristic repair method. In AAAI-90. C. S. Peirce. Abduction and Induction. Dover, NY, 1955. G. Pinkas. Reasoning, nonmonotonicity and learning in connectionist network that capture propositional knowl- edge. Artijiciad Intelligence, 77:203-247, 1995. D. Roth. On the hardness of approximate reasoning. In IJCAI-99, pages 613-618. D. Roth. Learning to reason: The non-monotonic case. In IJCAI-95, pages 1178-1184. D. Roth. A connectionist framework for reasoning: Rea- soning with examples. Tech. report, Dept. of App. Math. and CS, Weizmann Inst. of Science, 1996. L. Shastri and V. Ajjanagadde. An optimally efficient limited inference system. In AAAI-90, pages 563-570. L. Shastri and V. Ajjanagadde. From simple associations to systematic reasoning: A connectionist representation of rules, variables and dynamic binding using temporal synchrony. Behaviorad and Brain Sciences, 1993. L. Shastri. A computational model of tractable reasoning - taking inspiration from cognition. In IJCAI-99. B. Selman and H. Kautz. Model-preference default theo- ries. Artificial Intelligence, 45:287-322, 1990. B. Selman, H. Levesque, and D. Mitchell. A new method for solving hard satisfiability problems. In AAAI-992. R. Sun. Robust reasoning: Integrating rule-based similarity-based reasoning. Artificial Intelligence, 1995. L. G. Valiant. A theory of the learnable. Communications of the ACM, 27(11):1134-1142, November 1984. L. G. Valiant. Circuits of the Mind. Oxford University Press, November 1994. Rule-Based Reasoning & Connectionism 1261 | 1996 | 184 |
1,826 | Production Systems Nee ure an Minh Dung Computer Science rogram, Asian Institute of Technology PO Box 2754, Bangkok 10501, Thailand dung@cs.ait.ac.th aolo Mancarella Dipartimento di Informatica, University of Pisa Corso Italia 40, 56125 Pisa, Italy paolo@di.unipi.it Abstract We study action rule based systems with two forms of negation, namely classical negation and “negation as failure to find a course of actions”. We show by several examples that adding negation as failure to such systems increase their expressiveness, in the sense that real life problems can be represented in a natu- ral and simple way. Then, we address the problem of providing a formal declarative semantics to these ex- tended systems, by adopting an argumentation based approach, which has been shown to be a simple uni- fying framework for understanding the declarative se- mantics of various nonmonotonic formalisms. In this way, we naturally define the grounded (well-founded), stable and preferred semantics for production systems with negation as failure. Next, we characterize the class of stratified production systems, which enjoy the properties that the above mentioned semantics coin- cide and that negation as failure can be computed by a simple bottom-up operator. Introduction and Motivations In this section we first give examples to motivate the extension of the production systems paradigm (Wayes- Roth 1985) by the introduction of negation as failure (to find a course of actions). We then discuss its role as a specification mechanism for reactive systems. On the need for negation as failure in production systems Example 1 Imagine the situation of a person doing his household work. Clothes have to be washed and the person has two options, either hand washing or machine washing. If there is machine powder in house, then machine washing can take place. This is repre- sented by the production rule r1 : if Powder then machine-wash. If no machine powder is in house, then it can be ac- quired by either buying it in the shop (provided the shops are open) or by borrowing it from the neighbor (if he is in). The rules for acquiring powder can be represented by the following two classical production rules f-2 : if 1 Powder, Shop-Open then buy r3 : if 1 Powder, Neighbor-In then borrow. 1242 Rule-Based Reasoning & Connectionism Of course, hand washing is undesirable and will be taken up if there is no way to acquire machine pow- der. The naive representation of this rule using classi- cal negation if 1 Powder then hand-wash is clearly not correct, since the meaning of such a rule is that if there is no machine powder in house at the current state, then the clothes should be hand washed, while the intuitive meaning of “there is no way to ac- quire machine powder” is that there is no course of actions starting from the current state leading to ac- quiring machine powder. Hence in a state where there is no machine powder in house and the neighbor is in, the above naive representation would allow hand wash- ing though there is a way to acquire machine powder by borrowing it from the neighbor. Hence it fails to capture the intuitive understanding of the problem. Here we need to use a different kind of negation, called negation as failure (to find a course of actions) and denoted by the operator not. The previous naive representation is now replaced by f4 : if not Powder then hand-wash. It is not difficult to find other real life situations gov- erned by rules with negation as failure. Example 2 Consider the rules for reviewing the work of faculties at the end of each academic year in a uni- versity. The first rule specifies the conditions for of- fering tenure to assistant professors. It states that as- sistant professors with good publications and with a working experience of at least five years should be of- fered tenure. This rule could be formalized by: if Assistant-Prof(X), Good-Pub(X), Work-at-least-5-years(X) then offer-tenure(X). The second rule states that if an assistant professor has no prospect of getting a tenure then fire him. Though the intuitive meaning of this rule is clear, it is not pos- sible to represent it as a classical production rule since the premises of a classical production rule represent conditions which must be satisfied in the current state of the world while the premises of the second rule repre- sent a projection into the future. It says that if there is no possibility for an assistant professor to get a tenure in the future then sack him now. In other words, the rule says that if an assistant will fail in all possible From: AAAI-96 Proceedings. Copyright © 1996, AAAI (www.aaai.org). All rights reserved. course of actions in the future to get a tenure then fire him. To represent this rule, we use again negation as failure to find a course of actions. The second rule can then be represented as follows: if Assistant-Prof(X), not Getting-Tenure(X) then f’ire(X) In real life, we often find ourselves in situations where we have to deal with risky or undesirable actions. For example, a doctor may have to take the decision of cut- ting the foot of his patient due to some severe frostbite. This is a very risky, undesirable decision and the com- monsense rule specifying the conditions for taking this action is that the doctor is allowed to cut if there is no other way to save the foot of the patient. This can be represented, using negation as failure, as follows: if not Save then cut. Finally, we can expect that in real life, intelligent systems could be employed to satisfy multiple goals. These goals can have different priorities, and negation as failure (to find a course of actions) can be used to represent these priorities as in the following example. Example 3 Consider a robot fire fighter that should be sent into a fire to save lives and properties. The priority here is certainly saving lives first. Imagine now that the robot is standing before a valuable artifact. Should it, take it and get out of the fire? The answer should be yes only if the robot is certain that there is notthing it can do to save any life. The rule can be represented as follows if Artifact(X), In-Danger(X), not Human-Found then save(X) Note that the not Human-Found here means that no human being could be found in the current and all other possible states of the world reachable by firing a sequence of actions the robot is enabled to perform. Negation as failure as a specification mechanism for reactive systems Let us consider again the example 1. Checking whether the conditions of the rule * 1’4 . if not Powder then hand-wash are satisfied in the current state involves checking whether there is any way to acquire machine powder from the current state, a process which could be time consuming and expensive. In a concrete application, as in our example where there are only two ways to get powder: buying it in the shop or borrowing it from the neighbor, negation as failure can be “compiled” into classical negation to produce a more efficient rule: if 1 Powder, 1 Shop-Open, 1 Neighbor-In then hand-wash. However, the environment in which a production system with the above rule is applied can change. For example, you may get a new neighbor who may not have any interest for good relations to other peoples, and so you will not be able to borrow anything from him. Hence the rule for borrowing must be dropped. Consequently the above production rule must be re- vised to if 1 Powder, 1 Shop-Open then hand-wash It is clear that the rule r4 with negation as failure is still correct and serves as a specification for checking the correctness of the new rule. The point we want to make here is that in many cases, though negation as failure is not employed di- rectly, it could be used as a specification mechanism for a classical production system. This situation can be encountered quite often in many real life situations. Imagine the work of a physician in an emergency case dealing with a patient who is severely injured in a road accident. In such cases, where time is crucial, what a doctor would do is to follow certain treatments he has been taught to apply in such situations. He more or less simply react depending on the physical conditions of the patient. The treatment may even suggest a fate- ful decision to operate the patient to cut some of his organs. Now it is clear that such treatment changes according to the progress of the medical science. One treatment which was correct yesterday may be wrong today. So what decides the correctness of such treat- ments? We can think of such treatments in a simpli- fied way as a set of production rules telling the doctors what to do in a concrete state of a patient. The cor- rectness of such rules are determined by such common- sense principles like: Operate and cut an organ only if there is no other way to save the patient. And such a principle can be expressed using negation as failure. Aim of this work We have seen in the examples that using negation as failure in production systems allows one to naturally and correctly represent many real life problems. The main aim of this paper is to provide a declarative semantics to production systems where two kinds of negation are used, classical negation and negation as failure (to find a course of actions). In this respect, we show that the argumentation based approach (Dung 1995), which has been successfully adopted to under- stand logic programming with negation as failure as well as many other nonmonotonic formalisms, can also be adopted to provide a natural and simple declara- tive semantics to production systems with two kind of negations. The basic idea is that negation as fail- ure literals, such as “not Powder” in example 1, repre- sent assumptions underlying potential computations of a production system. The intuitive meaning of such an assumption is that the computation goes on by assum- ing that there is no coukse of actions (i.e. computation) from the current state of the world leading to a state which defeats the assumption itself. Referring back to example 1, assuming not Powder corresponds to as- suming that from the current state there is no course of actions leading to a state where machine powder is in house. A computation which is supported by a sequence of assumptions is plausible (acceptable) if its Rule-Based Reasoning & Connectionism 1243 underlying assumptions cannot be defeated by actually find a course of actions which defeats them. These informal, intuitive notions can be formalized by viewing a production system as an argumentation system along the lines of (Dung 1995). This provides us with many natural semantics, such as the grounded (well-founded), the preferred and the stable semantics (Dung 1995). Th ese semantics are arguably the most popular and widely accepted semantics for nonmono- tonic and commonsense reasoning in the literature (Gelfond & Lifschitz 1988; Bondarenko et al. 1995; McDermott & Doyle 1980; VanGelder, Ross, & Schlipf 1988). Moreover, we address the problem of actually com- puting negation as failure. In this respect, we intro- duce the class of stratified production systems, where negation as failure can be computed using a simple bottom-up operator. As for the case of general strati- fied argumentation systems, stratified production sys- tems enjoy the property that all the previously men- tioned semantics (grounded, preferred and stable) co- incide. Classical production systems We introduce here the notations and basic terminolo- gies we are going to use in the following. The produc- tion system language we use is similar to classical ones (see, e.g. (Forgy 1981)). We assume a first order lan- guage l representing the ontology used to describe the domain of interest. A state of the world is interpreted as a snapshot of this world, hence is represented as a Herbrand interpretation of ,C, i.e. as a set of ground atoms of ,C. The set of states is denoted by Stat. Fur- ther we assume that a set of primitive actions A is given. The semantics (effect) of actions is described by the function eJQrect : A x Stat --+ Stat. A production rule is a rule of the form if /I,. . . , 1, then a where II, . 1 *-, n are ground literals of L and a is an action in A. The conditions (resp. action) of a rule r will be referred to by cond(r) (resp. action(r)). A production system P is a set of production rules. A production rule if dl, . . . , I, then a is said to be upplicabEe in a state S iff the conditions dl, . . . , d, are true in S, i.e. S k Zr A . . . A I,. Definition 4 A partial computation C of a produc- tion system P is a sequence so -5 Sl . . . = s, n 2 0, such that Si’s are states, ri’s are production rules in P and for each i > 1, ri is applicable in Si-1, and Si = eflect(uction(ri), $-I). So will be referred to as initial(C) and S, as final(C). A partial computation C so 2 s1 . . . -2- s, is called a complete computation if no production rule in P is applicable in S,. 0 Note that, if n = 0, then the partial computation is an empty computation. The behavior of a production system P can be de- fined as the set of pairs of states (S, S’) such that S (resp. S’) is the initial (resp. final) state of some com- plete computation of P. This is formalized in the next definition. Definition 5 For a production system P, the input- output semantics of P is defined by TO(P) = ((initiul(C),f;nu/(C)) 1 C is a complete computation of P}. 0 Production rules with NAF We introduce a new form of negation into the language L, denoted by not. A general literal is now either a (classical) literal 1 or a nuf-literal not d, where 1 is a classical literal. For each classical literal d, the intuition of not I is that it is not possible to find a course of actions to achieve d. Definition 6 A general production rule has the form if II,.. . , I, then ai where each Zi is a ground general literal. 0 Given a general production rule if II,. . . , d, then ai the set of classical literals in r will be referred to as cl-cond(r), and the set of naf-literals will be referred to as hyp(r). A general production system (GPS) P is a set of general production rules. A general production rule is said to be possibly applicable in a state S if S + cl-cond(r). Notice that a rule satisfying the condition that S /= cl-cond(r) in a state S is not necessarily applicable in S since it is not clear whether its naf- conditions are satisfied in S. Definition 7 Given a GPS P, a possible partial com- putation in P is a sequence where Si’s are states, ri’s are general production rules in P, for each i 2 1, ri is possibly applicable in Si- 1 and Sa = eflect(uction(ri), Si-1). Given a GPS P, the set of all possible partial com- putations of P is denoted by C(P). 0 When a possible partial computation is acceptable: a motivating discussion The basic idea in understanding the meaning of naf- conditions not I’s in general production rules is to view them as hypotheses which can be assumed if there is no possible course of actions to achieve d. So, intuitively we can say that a rule is applicable in a state S if it is possibly applicable and each of its hypotheses could be assumed. A partial (pre)computation is then an acceptable partial computation if each of its rules is applicable. The whole problem here is to understand formally what does it mean that there is no possible course of actions starting from a state S to achieve some result 1. Let us consider again the example 1. 1244 Rule-Based Reasoning & Connectionism Example 8 Let us consider again the washing exam- ple. The effects of the actions are specified below: e@ect(hand-wash, S) = S U {Clean, Tired} eflect (machine-wash, S) = (S\ {Powder}) U {Clean} if Powder E S e*ffect (machine-wash, S) = S if Powder @ S eflect(buy, S) = S U {Powder, Less-Money} eflect (borrow, S) = S U {Powder} Assume that in the initial state we have no powder, shops are closed and the neighbor is in. This state is represented by the interpretation So = {Neighbor- In}. From this state there are three possible nonempty partial computations starting from So, namely Cl : so 2 {Neighbor-In, Clean, Tired} ca: so -2 {Neighbor-In, Powder} 2 {Neighbor-In, Clean}. c3 : so -5. {Neighbor-In, Powder} First, notice that both Cz and C’s are not based on any assumption. Our commonsense dictate that C2 and C’s represent acceptable course of actions from the initial state which lead to the commonsense result that clothes a.re machine washed. Hence they both must be accepted as possible courses of actions. On the other hand Cl is based on the assumption “not Powder”, meaning that (2’1 assumes that there is no possible way to acquire the machine powder. However, C’s repre- sents just one such possible way. Hence, C’s represents an attack against the assumption “not Powder”. So C’s can also be viewed as an attack against the acceptabil- ity of Cl as a legitimate computation. On the other hand, both C:! and C’s are not based on any assump- tion, hence there is no way they can be attacked. This example points out that the semantics of GPS’s is a form of argumentation reasoning, where arguments are represented by possible partial computations. In t,he following, we first recall the general notion of ar- gumentation systems from (Dung 1995) and then we show that the natural semantics of GPS can be defined using the theory of argumentation. Argumentation systems We review here the basic notions and definitions of argumentation systems (the reader can refer to (Dung 1995) for more details and for a discussion of the role of argumentation systems in many fields of Artificial Intelligence). An argumentation system is a pair (AR, uttucks) where AR is the set of all possible arguments and attacks C_ AR x AR, representing the attack relation- ship between arguments. If the pair (A, B) E attacks, then we say that A attacks B or B is attacked by A. Moreover, A attacks a set of .arguments H if A attacks an argument B E H. We also say that H attacks A if an argument B E H attacks A. A set H of arguments is conflict-free if no argument in H attacks H. An argument A is defended by a set of arguments H if H attacks any attack against A, i.e. for each argument B E AR, if B attacks A then H attacks B. We also say that H defends A if A is defended by H. The basic notion which underlies all the semantics for argumentation systems that we are going to review in the rest of this section, is the following, intuitive notion of acceptability of a set of arguments. A set H of arguments is acceptable if it is conflict-free and it can defend each argument in it. Let H be a set of arguments and let Def (H) be the set of all arguments which are defended by H. It is not difficult to see that H is acceptable iff H & Def (H) and H is conflict-free. Further it is easy to see that oef : P(AR) - P(AR) is monotonic. Hence the equation H = Def (H) has a least solution which is also acceptable (following from the fact that Def (0) is acceptable and if H is acceptable then also H UDef (H) is also accept able). The various semantics for argumentation systems are basically solutions of the above equation H = Def (H). In particular, the grounded (well-founded) semantics of an argumentation system is the least so- lution of the equation H = Def (H). Another semantics for argumentation systems, called the preferred semantics, is defined by the maxi- mal acceptable sets of arguments. It is not difficult to see that these sets are the conflict-free maximal solu- tions of the equation H = Def (H). In general, pre- ferred sets contain the grounded semantics, but do not coincide with it. In the next section, we will give an example for this. Finally, a popular semantics of nonmonotonic rea- soning and argumentation systems is the stable seman- tics, defined as follows. A conflict-free set of arguments H is said to be stable if it attacks each argument not belonging to it. It is not difficult to see that each sta- ble set of arguments is acceptable. Furthermore, it is also easy to see that each stable set is preferred, hence it is a maximal, conflict-free solution of the equation H = Def (H), but not vice versa. It has been shown that argumentation systems pro- vide a simple and unifying semantical framework for commonsense reasoning. For example, logic program- ming and different logics for nonmonotonic reasoning and n-person games are showed to be different rep- resentations of the argumentation systems presented above (see (Dung 1995) for further details)- where for instance, the grounded semantics of argumentation corresponds to the well-founded semantics in logic pro- gramming (Van Gelder, Ross, & Schlipf 1988) and sta- ble semantics of argumentation correspond to stable semantics of logic programming (Gelfond & Lifschitz 1988) and other prominent nonmonotonic logics like Reiter’s default logic (Reiter 1980) or Moore’s au- toepistemic logic (Moore 1985). In the following we show that general production systems with two kinds of negation are also a form of argumentation systems. Rule-Based Reasoning h Connectionism 1245 Computations as arguments The semantics of a GPS P is defined by viewing it as an argumentation framework (AR(P), attacks), where AR(P) is the set of all possible partial computations of P and the relation attacks is defined as follows. Definition 9 Let C be a possible partial computation so -Lsl I.,. ----+si %s;+1 -+...-Ys,. An attack against C is a possible partial computation C’ such that initiul(C’) = Si, for some i, and there exists an underlying assumption not I in hyp(ri+l) such that I holds in finul(C’). 0 Remark: Empty computations cannot be attacked. Hence, empty computations are contained in any se- mantics. Notice that, in the above definition, the initial state of the attack C’, which defeats the assumption not lun- derlying C, has to be the actual state Si in which such an assumption was made. In other words, whether an assumption not I can be defeated or not, depends on the state in which this assumption is made and on whether or not this state can be lead by a computation to a state in which I holds. The view of a GPS as an argumentation system, al- lows us to provide it with three different semantics: grounded (well-founded), preferred and stable seman- tics. It is easy to see that the following proposition hold. Proposition 10 (i) If a computation C attacks a computation C’ and C’ is a prefix of C”, then C also attacks C”. (ii) Let H be a set of computations which is either the grounded, or a preferred, or a stable set. For any C E H, any prefix of C also belongs to H. 0 We can now define the set of complete acceptable computations, with respect to a selected semantics. Definition 11 Let P be a GPS and R be a selected semantics of P, i.e. R is either the grounded, or a pre- ferred, or a stable, set of possible partial computations. A partial computation C E R is called an R-complete computation if there exists no other partial computa- tion C’ E R such that C is a prefix of C’. 0 If all the semantics of a GPS coincide, we simply talk about complete computations instead of R-complete computations. Referring back to the example 8, the only complete computation starting from So is C2. The input-output semantics of classical production systems can be extended to general production systems with respect to a selected (grounded, preferred, stable) semantics. Definition 12 Let P be a GPS. (i) The grounded input-output semantics of P is de- fined by ZOc(P) = {(initiuZ(C),finul(C)) 1 C is a grounded complete computation}. (ii) Let R be a set of arguments which is preferred or stable. Then Z&(P) = {(initiu/(C),finud(C)) 1 C is an R-complete computation of P }. 0 Stratified Production Systems In this section we consider only special kinds of GPS, where the actions are of two types, assert(p) and retract(p), where p is an atom. The effect of assert(p) (resp. retract(p)) on a state S is adding (resp. remov- ing) p to S (resp. from S). Moreover, the rules have the following structure: lp, II,. . . , dk, not Zk+l, . . . , not I, + assert(p) p, dl, . . . ,11, not Ik+l, . . . , not & --+ retract(p) where Zi’s are classical literals. Rules of the first kind are called assert rules, and rules of the second kind are called retract rules. If it is not important to distinguish between a rule as assert and retract rules, we will simply write d, dl, . . . , lk, not ik+l, . . . , not 1, - a. Taking inspiration from the notion of stratification in logic programming (Apt, Blair, & Walker 1988)) we define stratified GPS’s in such a way that negation as failure can be computed bottom-up. In the following, given a classical literal I, we refer to the atom of I as 1 if it is a positive atom, and as p if d is lp. Definition 13 A GPS P is stratified if there exists a partition PolJ.. .UP, of its rules such that the following conditions are satisfied. Let b)11,. . . ,!k,not /k+l,..., not dh - a be a rule in Pj. Then (i) for each li, i = 1,. . . , k, each rule containing the atom of Zi in the head must belong to U P, m<j (ii) for each Zi, i = k + 1, . . . , h, each rule containing the atom of li in the head must belong to U Pm. o m<j For stratified GPS’s, the grounded, preferred and stable semantics coincide (see theorem 15). Moreover, this semantics can be computed in a bottom-up way, by a simple operator S, that we define next. Let C be a possible partial computation s++s~...=s,. Then for any h, k such that 0 <_ h < k 5 n, the se- quence sh ‘= $+I . . . = Sk is called a subcomputation of C. Definition 14 Let P = PO U . . . U Pn be a stratified GPS. Let S be the operator defined as follows. gdu= Wo) 0 . . . u Pi+1) = {C E C(P0 u . . . u Pi+1) J * for each subcomputation C’ of C if C’ E C(Pf-J u . . . U Pi) then C’ E S(Po U . . . U Pi) * for each subcomputation of C of the form Sj-1 -X Sj such that rj E Pi+1 then for each not d E hyp(rj), there is no computation Rule-Based Reasoning & Connectionism C’ E S(P0l.J.. .UPi) such that iniiial(C’) = Sj _ 1 and final /= a} 0 Roughly speaking, the operator S formalizes the in- tuition that the acceptability of a possible partial com- putation using rules in PO U . . . U Pi+1 depends only on computations in PO U . . . U Pi. Thus, the semantics of a stratified GPS P can be computed bottom-up by iterating the operator S on the strata of P. Theorem 15 Let P be a stratified GPS. Then: (i) S(P) is grounded (ii) S(P) is the unique preferred set of computations (iii) S(P) is the unique stable set of computations o Conclusions and future work Production systems with negation as failure to find a course of actions are a natural extension of classical production systems, which increases their expressive- ness in the sense that they allow a natural and simple representation (specification) of many real life prob- lems. This extension can be given a simple seman- tics based on an argumentation theoretic framework. There are still several issues which deserve a deeper study and understanding. We have seen that our semantics reflects the inher- ent nondeterminism of production systems. In fact, in our semantics different complete computations start- ing from the same initial state can yield different fi- nal states, even for stratified GPS’s. This contrasts with many efforts in the literature aiming at find- ing a method to select one of the complete compu- tations as the expected semantics (Froidevaux 1992; Raschid 1994). Even though we believe that in many cases these efforts contrast with the inherent nonde- terministic nature of the problems represented by the production rules, there are situations in which select- ing only one out of (possibly) many complete compu- tations may not harm at all. In these cases, it is worth studying computational strategies which basically pro- vide us with a deterministic operational semantics for production systems. Still, the declarative semantics serves as a basis for reasoning about the correctness of these methods. We are investigating the application of our approach in the active databases area. Active databases-(Ceri, Dayal, & Widom 1995) is an important research topic in the database community due to the fact that they find many applications in practice. Typical active daftabase rule is an event-condition-act& form. We are currently extending our argumentation based ap- proach to active rules which also contain negation as failure in the condition part of the rules. Finally, a few words about the relationship between “negation as failure sented in this paper in logic programming. -I, ^___^ A AL-C ^-.^-_- . ..--... to find a course of actions” pre- and “negation as failure to prove” In (Dung 1995), it has been * ---L-L.-- I?-‘-^- -^------ l- ^-- L - bllUWCU Lrlab every arguIIlerllJ ci.blU11 1ra1 rre WV1 K cm1 oe represented by a simple logic program using negation as failure to prove. This means that negation as fail- ure to find a course of actions can be represented using naf to prove. In a following paper we have also showed that naf to prove is a special kind of negation as fail- ure to find a course of actions. Hence the conclusion is that the two kinds of naf have the same expressive power. Which one should be used in a concrete ap- plication depends on which one allows a more natural specification of the problem at hand. Acknowledgments This research was supported in part by EEC Keep in Touch Activity KITOll-LPKRR. eferences Apt, K.; Blair, H.; and Walker, A. 1988. Towards a theory of declarative knowledge. In Minker, J., ed., Foundations of Deductive Databases and Logic Pro- gramming. Morgan Kaufmann. Bondarenko, A.; Dung, P.; Kowalski, R.; and Toni, F. 1995. An abstract, argumentation-based theoretic ap- proach to default reasoning. Technical report, Dept. of Computing, Imperial College of Science, Technol- ogy and Medicine, London. Ceri, S.; Dayal, U.; and Widom, J. 1995. Active Database Systems. Morgan-Kauffman. Dung, P. 1995. The acceptability of arguments and its fundamental role in logic programming, nonmono- tonic reasoning and n-person games. ArtificiaE Intel- ligence 77(2). Forgy, C. 1981. OPS5 user’s manual. Technical Re- port CMU-CS-81-135, Carnegie Mellon University. Froidevaux, C. 1992. Default logic for action rule- based systems. In Neumann, B., ed., Proc. 10th Euro- pean Conference on Artificial Intelligence (ECAI 92), 413-417. John Wiley & Sons. Gelfond, M., and Lifschitz, V. 1988. The stable model semantics for logic programming. In Kowalski, R., and Bowen, K., eds., Proceedings ofthe Fifth Interna- tional Conference on Logic Programming, 1070-1080. The MIT Press. Hayes-Roth, F. 1985. Rule based systems. Commu- nications of the ACM 28(9). McDermott, D., and Doyle, J. 1980. Non-Monotonic Logic I. Artificial Intelligence 13( l-2):41-72. Moore, R. 1985. Semantical considerations on non- monotonic logics. Artificial Intelligence 25( 1):75-94. Raschid, L. 1994. A semantics for a class of strat- ified production system programs. Journal of Logic programming. Reiter, R. 1980. A logic for default reasoning. Arti- ficial Intelligence 13:81-132. VanGelder, A.; Ross, K.; and Schlipf, J. 1988. Un- founded sets and well-founded semantics for general logic programs. In Proceedings of the Seventh Sym- posium on Principles of Database Systems, ACM- SIGACT-SIGCOM, 221-230. The MIT Press. Rule-Based Reasoning & Connectionism 1247 | 1996 | 185 |
1,827 | si ase so Bing Liu and Joxan Jaffar Department of Information Systems and Computer Science National University of Singapore Lower Kent Ridge Road, Singapore 119260, Republic of Singapore (hub, joxan)@iscs.nus.sg Abstract Rule-based systems have long been widely used for building expert systems to perform practical knowledge intensive tasks. One important issue that has not been addressed satisfactorily is the disjunction, and this significantly limits their problem solving power. In this paper, we show that some important types of disjunction can be modeled with Constraint Satisfaction Problem (CSP) techniques, employing their simple representation schemes and efticient algorithms. A key idea is that disjunctions are represented as constraint variables, relations among disjunctions are represented as constraints, and rule chaining is integrated with constraint solving. In this integration, a constraint variable or a constraint is regarded as a special fact, and rules can be written with constraints and information about constraints. Chaining of rules may trigger constraint propagation, and constraint propagation may cause fn-ing of rules. A prototype system (called CFR) based on this idea has been implemented. 1. Introduction Rule-based systems are one of the great successes of AI (e.g., Newell 1973; Lucas & Van Der Gag). They are widely used to build knowledge-based systems to perform tasks that normally require human knowledge and intelligence. However, there are still some important issues that have not been addressed satisfactorily in the current rules-based systems. One of them is the disjunction. This limits their problem solving power. In the Constraint Satisfaction Problem (CSP) research, many efficient constraint propagation algorithms have been produced (Ma&worth 1977; Hentenryck et al 1992). A number of languages or systems based on the model have also been developed and used for solving real-life problems (JalYar & Maher 1994; Ilog Solver 1992). In this paper, we show that some types of important disjunctions can be modeled with CSP. Thus, it is possible to use the simple representation scheme and efficient problem solving methods in CSP to handle these types of disjunctions. Specifically, the disjunctions can be represented as constraint variables and their domains. The relations among disjunctions can be represented as constraints. In this paradigm, constraint propagation and 1248 Rule-Based Reasoning & Connectionism rule chaining are integrated. A constraint can be added as a special fact, and rules can be written with constraints and information about constraints. Chaining of rules may trigger constraint propagation, and constraint propagation may cause firing of rules. With the incorporation of CSP techniques, the power and the expressiveness of rule-based systems will be greatly increased. Based on this idea, a prototype system, called CFR, has also been implemented. The idea of incorporating CSP into a logic-based system is not new. Constraint solving has long been integrated with logic programming languages such as Prolog. This integration has resulted in a number of Constraint Logic Programming (CLP) languages (J&&r & Maher 1994), such as CLP(R) (Jaff’ar & Lassez, 1987) and Chip (Hentemyck 1989). These languages are primarily used for modeling and solving real-life optimization problems, such as scheduling and resource allocations. However, this work is different from that in CLP in a number of ways. The main difference is that CLP languages are all based on Horn clauses and backward chaining, while the proposed integration is based on forward chaining, which is suitable for solving a different class of reasoning problems. Integration of constraint solving and forward chaining has some specific problems that do not exist in CLP languages. The proposed integration is also mainly for improving reasoning capability of existing rule-based systems rather than for solving combinatorial search problems. Thus the types of constraints and their representations in the proposed approach are quite different from those in CLP languages. We regard this work as the first step to a full integration of the CSP model with forward chaining rule- based systems. The current integration presented in this paper is still restrictive in the sense that it is mainly to help model and handle the problems with some disjunctions. A full integration could potentially change the way that people use rule-based systems and change the way that people solve practical reasoning problems, which are the main applications of the rule-based systems today. It may be just like the way that CLP languages have changed the way that people model and solve practical combinatorial search problems. From: AAAI-96 Proceedings. Copyright © 1996, AAAI (www.aaai.org). All rights reserved. 2, &-Based Systems and Constraint Satisfaction Problems This section reviews rule-based systems and CSP. The coverage is by no means complete; rather the focus is on highlighting the problems with disjunctions in current rule-based systems. 2.1. Rule-Based Systems A rule-based system consists of three main components. 1. A working memory (WM): a set of facts representing the current state of the system. 2. A rule memory (RM): a set of IF-THElIz’ rules to test and to alter the WM. 3. A rule interpreter (RI): it applies the rules to the WM. The rule interpreter repeatedly looks for rules whose conditions match facts in the WM. On each cycle, it picks a rule, and performs its actions. A rule is of the form: lF <conditions> THEN <actions> There are three common connectives in a rule-based system, i.e., and, or and not. We will only discuss or here as we are mainly interested in disjunctions. or in logic can be defined as inclusive (v) or excltrsive (G3). Let us first look at the inclusive OY. For example, “if something is a block or a pyramid, then it is a pointy-object” (adapted from (Char&k et al 1987)) can be expressed as follows: IF isa(?x2 block) v isa(?x, pyramid) THEN add(isa(?x, pointy-object)) where 3x is a variable, and add adds a fact to the WM. This rule, however, cannot be used in a typical rule-based system. Instead, it is usually replaced by two rules: IF &a(?~, block) THEN add(isa(?x, pain@-object)), and IF isa(?x, pyramid) THEN add(isa(?x, pointy_object)). However, this does not say exactly the same thing as the v version does, since there might be situations where we know that either ?x is a block or ?x is a pyramid, but do not know which. In this case, neither of these rules applies, but the original one that uses v does. Now, let us look at the exclusive OK For example, the following formula says that “either NYC or albany is the capital of NY> but not both”. capital(l XYC) 03 capital(i%Yy albany) This can be rephrased as two rules: ‘?WC is the capital of IVY3 if albany is not”, and “albany is the capital of NY, if NYC is not” IF not(capitai(hY, albany)) THEN add(cupital(IVY, NYC)), and IF not(capitai(hY; AYC)) THEN add(capital(.?VY, alban)). Unfortunately, not used in current rule-based systems is different from l in logic. In a typical rule-based system, not(P) is satisfied if there is no fact in WM matching P. In general, disjunctions are difllcult to handle in reasoning. In Section 3, we will show that CSP provides a convenient model to represent these situations. 2.2. Constraint Satisfaction Problem A Constraint SatisEaction Problem (CSP) is characterized as finding values for variables subject to a set of constraints. The standard CSP has three components: e Variables: A finite set Y = (VI, vz, . . . . v,} of n variables vi, which are also referred to as constraint variables. Values: Each variable lpi is associated with a finite domain Di, which contains all the possible alternative values for vi. e Constraints: A set C = (Ci, C2, . . . . C,> ofp constraints or relations on the variables. The main approach used for solving CSPs is to embed constraint propagation (also known as consistency check) techniques in a backtrack search environment, where backtrack search performs the search for a solution and consistency check techniques prune the search space. Consistency techniques are characterized by using constraints to remove inconsistent values from the domains of variables. Past research has produced many techniques for such a purpose. The main methods used in practice are arc consistency techniques, e.g., AC-3 (Ma&worth 1977), AC-5 (Hentenryck et al, 1992), and AC-7 (Bessiere et nl 1995), and their generalizations and specializations (Hentenryck 1989; Hentenryck et al 1992; Liu 1996). For a complete treatment of these methods, please refer to (Ma&worth 1977; Mohr & Henderson 1986; Hentemyck 1989; Hentemyck et al 1992; Bessiere et al 1995; Liu 1995; Liu 1996). 3. Modeling isjunctions with CS This section shows how CSP can be used to model certain types of disjunction in a rule-based system. In this new paradigm with rules and constraints, the underlying techniques for reasoning are forward rule-chaining, constraint propagation and backtrack search. 3.1. The New Paradigm In the new paradigm, constraints are integrated into rule- based reasoning. It is described by: 1. A working memory (WM): a set of facts representing the current state of the system. There are three types .of facts: e Simple facts: these are the traditional facts used in the existing rule-based systems. 9 csp-disjunctions (inclusive and exclusive): these are special types of disjunctions (defined below) represented by the CSP model. e Constraints: these are relations on the csp- disjunctions. 2. A rule memory: a set of IF-THEAr rules. Rule-Based Reasoning & Connectionism 1249 3. A rule interpreter: this applies the rules to WM by using the traditional forward rule-chaining mechanism, and it is integrated with the constraint solver below. 4, A constraint solver: this uses consistency check and backtrack search for constraint satisfaction. It is integrated with the rule interpreter above. Thus, the key advance of this new paradigm lies in its use of the CSP model and a constraint solver, resulting in an integration of forward chaining and constraint solving. 3.2. Using Constraint Variables and Domains to Represent Disjunctions This sub-section describes how constraint variables and their domains can be used to represent disjunctions. We assume the basic definitions of term and atom, which are ground when they contain no variables. We now define the two kinds of disjunctions that we will handle, the inclusive cspdisjunctions and exclusive csp-disjunctions. In what follows, we shall, for simplicity with respect to our examples later, restrict the terms in disjunctions to differ only in the last argument. Definition 1: An exclusive csp-disjunction has the following form WV,, * b -9 Ll , GJl), Wl , * * -, GA, tn21, . . . ml, * - .f b-1, Ln)) where P(tl, . . . . t n-l) t,J is a ground atom, n 2 1, tni is a constant, and i f j implies tnj f tnj. The expression is TRUE iflexactly one of the m ground atoms is TRUE. Note that for all the atoms, the predicate symbols are the same, i.e.: P, and so are the first 11 -1 ground terms. Note also that tnj may appear in any position as long as they are at the same position in each atom. We arbitrarily choose to put them at the end. This exclusive csp-disjunction can be represented by an expression CBP(tl , . . .: tnel, D), where D is a set with the initial value (t,l, fn2, . . . . tJm>. During the reasoning process, some of the atoms (e.g., P(tl) . .., &-I, t,&) may be proven to be FALSE, then D will be modified to reflect the effect. Thus D changes during the reasoning process, but it is always a subset of (t,l, tn2, . . . . t,& When 101 = 1: we say D (= (tni}) is decided, which means that P(tl, . ..) tnml, tni) is TRUE. When D = 0, it means that the exclusive cs-disjunction is proven to be FALSE. An important point is that CDP(t,, . .., &-I, D) can be represented by a constraint variable, written as eP(tlz . .., t,,ml, 3: whose initial domain is D. For example, @Isa(john, (soldier, teacher)) can represent the fact that John is a soldier or a teacher, and that John is only in one of the professions. The corresponding constraint variable CBlsa(john, J can be used in constraints which hopefully eventually determine JoMs real profession. The second type of csp-disjunction is defined below. efhition 2: An inclusive csp-disjunction v(P(t1,. * .J”-1, t”l): P(t1, * * *, tn-1, bl2), - * -7 fvl , * * *, Ll , 4Im)) is like an exclusive csp-disjunction, except that this formula is TRUE z#3P(tl, . . . . fn-l, tni) is TRUE. This inclusive csp-disjunction can be represented by an expression vP(tl f . . . , tnml, S), where the initial value of S is the power set of (&I, tnzs . . . . tm} excluding the empty set. It is convenient to think of S in two parts (R, Q): 0 A set of required elements R: the elements that have been proven to be true, i.e., whose associated atoms have been proven to be TRUE. 0 A set of possible elements Q: the elements that belong to at least one possible value of S. Then, R and Q satis@ these. conditions: R n Q = 0 and R u Q c (tnl, ~2, . ..> t,&. The initial value of S may be (( ), U”l, 622: “‘, &}), and R will grow and Q will shrink in the reasoning process. When Q = 0 and PI = 0, we say the inclusive csg-disjunction is FALSE. When Q = 0 and PI f 0, we say S is decided, which means the following atoms are all TRUE: m, *-a, fn-1, rl), WI, . . . . h-1, r2), . ..r and P(tl , . ..) Ll, b) where R = (q p r2, . . ., rk) E (&.,I, tn2, . . .:, t,,>. We can see that vP(tl, . . . . t,l-l, (R, Q)) (or vP(tl: . . . . &-I, 5’)) can be represented by a constraint variable vP(t, , . . ., tnS1 : J whose initial domain is the pair (R, Q). Note that we now have constraint variables with a set as a domain, and with a pair of sets as a domain. Call the latter set constraint variables. For example, vIsFd@i@dke, (( >, (iohn, james, mar$>)) can represent the fact that john or janles or rnqJ is a friend of trike (or is inclusive) with R = (} and Q = fiohn, jumes: mary } . The corresponding constraint variable vlsFcC@nike, J can be used in constraints which hopefully eventually determine who are really mike’s friends. If it is decided that john is definitely a friend of mike, then R = (iohn> and Q = uames, maqf>. 3.3. Using Constraints to Represent Relations After introducing the two types of constraint variables to represent the two types of disjunctions, we now in the position to describe some of the constraints that can be used for representing relations among the disjunctions. Constraint: cs@@P1(tl I, . . ., h+l), J, @Mf21 F . . . z t2(d ), J) where t11, . . . . tl+l~, t21: . . . . and r2(m-1) are ground terms. Let D1 and 02 be the domains of the constraint variables BP1 (tll) . . .> tltn-l 1: J and @P$zl, . . .> t~(,,,-~ ), 2) respectively. This constraint ensures that the sets D1 and 02 are equal at all time. Its operational semantics is the following (which is an abstraction of the real algorithm implemented): 1250 Rule-Based Reasoning 81 Connectionism e D=D1nD2; ifD#0then if D = (v] then add Pl(tll, . . . . tl(,,+, v) to WM, add P2(t2,: . . . . f2(,,,-l), v) to WM endif D1=D;D2=D; return(TRUE); else return(FALSE) For example, we have Wsa(john, (soldier, teacher, professor, doctor)), and CMsa(james, (teacher, doctor, student)). If we know that john and jnnles have the same profession, we can e;tpress this with the constraint cst-eq(Wsa(/ohn, J, Wsa(james, J). The system will automatically propagate the constraint by using the built-in consistency algorithms to reduce both sets so that the following are obtained: @lsa(john, (teacher, doctor)), and Qlsn( james, ( teacher, doctor) ) If due to some other constraint (or information) it is decided that john is a teacher, then the following two elements will be added to WM: Isa(john, teacher), and Isa(james, teacher) If we have the following rule in the rule memory: IF Isa(?x, teacher) THEN add(has(?x, many_students)) This rule will be fired to obtain two more facts: hasuohn, many-students), and has(james, many-students) This example shows that constraint propagation and rule chaining are integrated. construint: csLnoteqFW(tl1: . . ., ~I(,-I ), .J, @P2(h? . ..) t2(4 ), 2) where h, . ..? h(,-l), hy . . . . and tz(m-1) are ground terms. Let D1 and D2 be the domains of @Pl(fll, . ..? tl(,_l), _) and QP2(t212 . . . . tz(,-l), J) respectively. Then the constraint’s operational semantics is given by: e if ID11 = 1 and 1D2/ > 1 then D2= D2-D,; if D2 =(v> then add P&,..., &(m-l;,:. v) to WM endif return(TRUE5); elseif l&l = 1 and p1I > 1 then this case is similar to the above one; elseif IDlf = 1 and lD4= 1 then if Dl f 02 then retum(TFtUE) else return(FALSE) endif else return(Tl3.m) For example, we have Wsa(john, (soldier, teacher, professor, doctor)), and Wsa(james, (teacher, doctor, student>>. The following constraint says that john and junres have different professions: cst-not-eq(@Lsa(john, _), Wsa(james, J) Constraint: cst-not_in(v, @P(tl, . . . . &-I, J) where tl, . ..? and &-I are ground terms, and v is a constant. Let D be the domain of @P(tl, . . . . &-I, J. This constraint constrains that v is not a possible element in D, w&h also means that P(tl, . . . . t,+l), v) is FALSE. We have: e D = D - (v>; if D = 0 then return(FALSE) else if D = (u> (or IDI = 1) then add P(tl,..., &,-I, u) to WM; endif return(TNJE) endif Constraint: cs~-=t_eq(vPl @I 1, . . . , h (4 1, J, vPdf21, . . ., t2(d 1, J> where tll, . . . . tl(,-l>, t21, . . . . and t7(m-lr are ground terms. Let (RI, Ql) and (R2, 92) be the domains of vPl(tll, . . . . tl (n-l),_) and vf’dh, . . . , t2(,,+ _) respectively. Then this constraint is handled by: 0 R=R1uR2;Q={rIr~Ql~QZ,r~RR); ifRcR1uQl andR&uQ2and(R#00rQ#0) then R1=R;R2=R;Ql=Q;Q2=Q; foreachr E Randr 6E Rl do add Pl(h, . ..? &A)? f9 to w-w for each r E R and r 4 R2 do add P2(f21, . . . . t2+,ml): r) to WM; retum(TRUE); else return(FALSE) For example, we have vlsFdOfimike, ((iohn), (james, Steve, david))), and vlsFdOflandrew, ((Steve >, uohn, kate, david))) If we set the constraint cst-set-eq(vIsFdOf(mike, J, VlsFdOflandrew, J)$ which says that mike and andrew have the same set of friends, we will obtain: vlsFdO@dke, ( oohns Steve ), (david))), and VlsFdOfiandrew, (oohn, Steve), (david])). Two more facts will be added in WM: i.e., IsFdOJ(irtike, Steve), and IsFdOf(andrew, john). Constraint: cst-set-not-in@? vP(tl ) . . .) r,-l, J) where tl, . . . . and tnn-l are all ground terms. Let (R: Q) be the domain of vP(tlT . .., tnml, J, &is constraint constrains that v is not a possible element in Q, which means that P(tl, . . . . &-I, v) cannot be TRUE. Its operational semantics is obvious, and omitted. 3.4. Introducing Choice Making and The consistency techniques used above for constraint solving are all based on arc consistency (Hentemyck e6 a2 1992; Liu 1995). Arc consistency alone may not be Rule-Based Reasoning & Connectionism 1251 sufficient to solve a CSP because arc consistency does not guarantee global consistency (Mackworth 1977). Then, a combination of backtrack search and consistency check is required. This approach can be described as an iterative procedure of two steps: consistency check and choice making. If a choice is proved to be wrong (when the consistency check returns FALSE), backtracking will be initiated. In the process, the previous state is restored, and an alternative is selected (Hentenryck 1989). Let us define some choice making functions. Each of them sets up a choice point for later backtracking. The choice functions are also constraints because each value selection will trigger consistency check. Choice function: cst_select@P(t~, . .., tn-l, J, func) where tl, . . . . and tnel are all ground terms, and fulzc is a user defined procedure. Let D be the domain of @3&t,, .,., tnml, J, this hnction selects a value v from D using the procedure func. func allows the user to control the selection process in order to find the solution quickly. This choice function behaves as follows: e if there is no more vaIue to be selected in D then return(FALSE) else v is selected from D usingfix; D = (VI; addP&, . ..., tnml, V) in WM; return(TRUE) endif For example, we have: OCapital(.?VY, flVYCz albaqy]), which says that the capital of New York (NQ is either AK’ or albany, but not both. We can apply the selection by using cst-select(@Capital(IU, J: func). Suppose that func chooses the first possible value first, i.e., iVYC. After it is selected, CapitaI(hiy, NYC) will be automatically added in WM, and then constraint propagation will be carried out, etc. When backtracking occurs, the second value will be tried and so on. Choice function: cst-set-select(vP(tl? . . ., tnml : _), func) where tl, . . . . and tnml are all ground terms, and fulzc is a user defined procedure. Let (R, Q) be the domain of v&t, !, . . ., fnml 2 J. This friction selects a value Y (a set) from Q (I/’ E Q) using the procedurefllnc. It behaves as follows: 0 if there is no more value to be selected from Q then return(FALSE) else A set Y is selected from & using&x; Q=0;R=RuK for each r E V do add I’&...: tnml, r) to WM; return(TRUE) endif For instance, we have vlsFdOf(mike, ((iohn >, (iames, mary, Steve))) and we know that mike has only two friends. We can try the following: cst-set-select(v.IsFdOf(mike, J? func) Suppose that fine chooses the first possible value first, i.e., james, which effectively rules out the other values. Then, mike’s Mends are john and james. We obtain vlsFdOf(nrike, (uohn, james], ())). After that, other necessary operations are performed, e.g., adding IsFdOJlntike, james) to WM and constraint propagation, etc. When a selection is proved to be wrong, backtracking will be performed. The second element, the third element, etc.: will be tried and so on. 3.5. Some Test Functions on Constraint Variables Here, we present some test functions on constraint variables. They are used to exploit the partial information provided by disjunctions for various purposes. Test function: test-in(T, @P(tl, . . . . t,,-, , ,)) where tl, . . . . and tnel are all ground terms, and T is a set of constants. Let D be the domain of 0&t,, . . . . tnml: ,). This test fin&on behaves as follows: 0 if D E T then retum(TRUE) else return(FALSE) For example, we have @CapitaZ(lVY, (MT, albany)), which says that the capital of New York (NY) is either M’C or albany, but not both, and the following rule: DF incllude(?tour, CapitaZOAiVY)) and test-in((,WC, albany), @Capiral(IVY, J) THEN add(join(l, ?tortr))) This rule allows the system to act on the partial information, i.e., test_in does not have to find the fact Capita&W, IVYC) or Capita&W, albany) in WM before firing. Instead, it only needs to check whether any one of these two cities or both are the only possible values for the capital of XY. It does not matter which. If WM has the following two facts: include(tourl6, capitaiOfi.VY)), and 03Capitai(J?Y, (AYC, albany)) the rule will fire to add join(l; tourl6)) to WM. Test function: test-set-in(T, vP(tl, . . . . &-I, J) where tl, . . . . and tn-l are all ground terms, and T is a set of constants. Let (R, Q) be the domain of VP&, . . . . tnml, J, This test function behaves as follows: e if(T~R)#&Ior(R=0andQ~T)then return(TRUE) else return(FALSE) For example, we wish to express that “if something is a block or a pyramid, then it is a pointy-&ject” (or is inclusive). We cau write: IF test-set-in(( block, pyramid>, isa(?x, 2) ‘THEN add(isa(?x, pain@-object)) 1252 Rule-Based Reasoning & Connectionism 3.6. Complications With the Idegration of Choice Making and Rule Chaining Combining backtrack search and forward chaining creates some complications. The problem lies in the handling of inconsistency. For our discussion, we class@ two types of inconsistency. The first type is the normal inconsistency in logic (IL): e.g., both A and 4 are deduced, and the other is the inconsistency of constraints (IC). IC is easy to detect and to handle because when the domain of a constraint variable is empty, it is known that there is a inconsistency, and backtracking can be used to deal with it. However, IL is hard to detect as most rule-based systems are informal systems that have no mechanism for this purpose. This has some implications for our proposed integration. 0 If a rule-based system is unable to detect IL, then (1) constraints cannot be conditions in a rule, (2) choice making and backtracking should not be allowed. The reason is that both (1) and (2) could introduce IL. Due to space limitation, we are unable to discuss this further. Interested readers, refer to (Liu & JafGr 1996). In general: if a rule-based system is unable to detect IL, (1) and (2) should not be allowed. Then, constraints can only appear as consequents of rules, and there will be no backtrack search but only consistency check. However: if an inconsistency checker is implemented for detecting IL, then both (1) and (2) can be allowed, and both IC and IL will trigger backtracking. Apart from the above two situations, a third one is also reasonable. We assume that only ICs may occur in an application, then we can also allow both (1) and (2) because IC is easily detected. Our prototype system makes this assumption. This assumption is realistic because that is the case in most existing rule-based systems. They do not have mechanisms for detecting IL. It is the user’s responsibility not to introduce any or to check it. 4. An Implementation We have implemented a prototype system (called CFR) in Common Lisp. Below are some implementation issues. 0 Apart from WM and rule memory in a rule-based system, a constraint variable memory is introduced to store constraint variables. 0 For consistency check of constraints involving normal constraint variables, we used those algorithms in (Hentenryck et al 1992; Liu 1995) as they are the most efficient algorithms. For set constraints, we designed OUT own algorithms as there is little reported work on this type of constraints. Consistency check of cst-eq, cst-not-eq, and cst-set-eq can all be done in linear time to the size of the domain D or /R u Ql. cst-not-in and cst-set-not-in can be done in constant time. 0 A choice stack is used to keep track of the choices that have been made and to remember the information necessary for restoring state upon backtracking. This is similar to CLP languages such as CHIP (Hentemyck 1989). The difference is that each choice here has to remember the facts that have been added to WM after a choice is made. When backtracking comes to the choice, these facts must be removed. 0 Finally, the pattern matching dlgorithm for rule- chaining needs to be modified to accommodate the constraint satisfaction facility. Due to the space limitation, we are unable to discuss this and many other issues. Below: we briefly describe the syntax of rules, constraint variables, and constraints in CFR. IF-THEN rules: A rule is defined using the construct: (define-rule <name <conditions> -> <actions>) For example, the rule: (define-rule is_food (edible ?x) -> (add ‘(is-food ,x))) says that if’ there is a fact in WM that matches (edible ?x), this rule will fire and add the evaluation result ‘(isfood ,x) to WM. ‘(isfood ,x) is in Lisp syntax (““‘, ““‘, and “,“ are used according to their meanings in Lisp), and x here will be substituted to whatever value ?x has after matching with the fact in WM. Constraint variable declarations: 1). ep(t,, . . . . t,,-1, D) => (corresponding to) (cst-in ‘(P tl . . . tn-l D)) e.g., 03capitaZ(AT, (NYC, aibany)) => (cst-in ‘(capital NY (NYC albany))) 2). vP(t,, . . . . b-1: CR, Q)) =’ (cst-set-in ‘(P tl . . . ht-1 CR Q>>, e.g., vlsFdOfioe, ((steve), uohn, kate))) => (cst-set-in ‘(IsFdOf joe ((Steve) (john kate)))) Constraints: 1). csQqWdh1, -., tl(ff.4 j, 3, @P2(t21: . . ., f2(m-1 j, 3) =’ (cst_eq ‘(Pl t11 . . . tl(n-1 j _) I(P2 t21 . . . t2(m-l j _)) e.g., cst-eq@Isa(/ohn, J, @Isa(james, J) => (cst-eq ‘(Isa john J ‘(Isa james -)) 2). cst~no~_eqWdhl,..., &+l),J,@P2(t21, .-., t22(m-l:f, -1) => (cst_not_eq ‘(PI tl I . . . k-1) 3 YP2 f21 . . . t22(m-1;f 3) e.g., cst-not-eq(@Isa(john, ,), @Isa(james, J) => (cst-not-eq ‘(Isa john J ‘(Isa james J) 3). cst_se~_eqWlUll,. . A (4 j, J, vW21, . . .) t2+1), _)) =’ w_set_eq ‘Vl t11 ** * h(n-1) ,) ‘(P2 f21 -*- t2(,1) J) e.g., cst-set-eq(vlsFdOf(mikez J,vlsFnOJ(ioe, J) => (cst-set-eq ‘(IsFdOf mike _) ‘(IsFdOf joe J) Due to lack of space, we will not describe the corresponding constructs in CFR for the other constraints and choice and test functions. They are quite similar to the ones above. Rule-Based Reasoning & Connectionism 1253 5. An Example We now present a simple example to illustrate how rules and constraints interact with each other in the reasoning process. The rule definitions here are self-explanatory. (define-rule professor (isa ?x scienceqrofessor) -> (add ‘(works-in-a :x university)) (cst-in ‘(teaches ,x (computer math physics chemistry biology)))) (define-rule computer (isa ?x scienRJrqfessor) (has-no ?x computer) -> (cst-not-in ‘computer ’ (,x teaches J)) (define-rule math (is_good_in 2x math) (isa ?x scienceqrofessor) -> (cst-in ‘(teaches ,x (computer math physics)))) (define-rule csp-test (test-in (physics math) (teaches ?x _)) -> (add ‘(gives-lecture-in ,x science-building))) (define-rule lab (does-not-do ?x lab-work) -> (cst-not-in ‘chemistry ’ (teaches ,x _)) (cst-not-in ‘biology ’ (teaches ,x >>) (define-rule degree (teaches ?x ?y) -> (add ‘(likes ,x ,y)) (cst-set-in ‘(has :x ‘(((ND in ,y)) ‘((MSc in ,y)))))) Let us run the system with the following facts: (add ‘(isa fred sciencegrofessor)) (add ‘(has-no fred computer)) (add ‘(isa john sciencegrofessor)) (add ‘(does-not-do john lab-work)) (cst-eq ‘(teaches john -) ‘(teaches fred -)) After all the rule chaining and constraint propagation, the working memory becomes: 1: (isa fred sciencegrofessor) 2: (works-in-a fied university) 3: (cst-fact (teaches fred _) (math physics)) 4: (has-no fred computer) 5: (isa john scienceqrofessor) 6: (works-in-a john university) 7: (cst-fact (teaches john _) (math physics)) 8: (does-not-do john lab-work) 9: (gives-lecture-in john science-building) 10: (gives-lecture-in fied science-building) Fact 3 and 7 are special facts representing two constraint variables and their remaining domains. From them, we know that botllfred and j&n teach either math or phpics, but we still do not know which. Let us say that we are not satisfied with the result. We would like to make a guess about what they teach. We can use the following selection function: (&-select ‘(teaches f&d _) #‘car) This selects math as the subject that j?ed teaches. After constraint propagation and rule chaining, we obtain the fact that john also teaches math. The following facts are deduced: 11: (teaches fred math) 12: (teaches john math) 13: (likes fied math) 14: (likes john math) 15: (hasfiXzd(PhDinmath)) 16: (has john (PhD in math)) 17: (cst-set-fact (has fred _) ((PhD in math)) (@EC in math))) 18: (cst_set_fact (has john J ((PhD in math)) ((MSc in math))) The last two facts (17 and 18) say that jkd and john have a PhD in math and may or mz~y not have a MS’c in math. If later we have some more information saying that fred does not have a PMI degree in math, this can be expressed like this: (cst-set-not-in ‘(PhD in math) ‘(has f&l -)) It immediately causes a conflict with fact 17 because fact 17 says thatped has a PhD in math. Then, backtracking is performed. The facts from 11 to 18 are removed to restore the previous state. physics is selected this time as the subject thatfred teaches, which in turn causes a nufnber of facts to be produced: 11: 12: 13: 14: 15: 16: 17: 18: (teaches fied physics) (teaches john physics) (likes fred physics) (likes john physics) (has fred (PhD in physics)) (has john (PhD in physics)) (cst-set-fact (has fred J ((PhD in physics)) ((MSc in physics))) (cst-set-fact (has john J ((PhD in physics)) ((MSc in physics))) Since math is eliminated as the possible course that fred and john teach. Fact 3 and 7 in WM become: 3: (cst-fact (teaches fred J (physics)) 7: (cst-fact (teaches john _) (physics)) The kind of reasoning illustrated here cannof be carried out in an existing rule-based system. 6. Related Work The most closely related work to our research is constraint logic programming (CLP) (Jtiar & Maher 1994) where a considerable amount of research has been done to integrate constraint satisfaction with logic programming. A number of systems have been built, and many successful 1254 Rule-Based Reasoning & Connectionism applications have also been reported (Jaf%r & Maher 1994). Two representative CLP languages are CLP(R) (Jaffar & Lassez 1987) and CHIP (Hentemyck 1989). These languages are based on Horn clauses and backward chaining. Our work is different from CLP in a number of ways. The main differences are as follows. 1. Our proposed technique is based on forward chaining rather than backward chaining as in CLP languages. Forward chaining and backward chaining reason from different directions and are suitable for solving different types of problems. Forward chaining are mainly used for building expert systems for solving real-life knowledge intensive tasks. Since the CLP languages based on backward chaining have been very successful in practice for solving practical combinatorial search problems, it is only natural that forward chaining should also be integrated with constraint solving to provide a more powerful reasoning technique for solving practical reasoning problems. 2. In CLP languages, backtracking and choice making are provided by the host language Prolog. While in forward chaining, backtracking and choice making facilities have to be added, which creates some complications as discussed in Section 3.6. To the best our knowledge, limited work has been done on combining constraint solving with forward chaining rule- based system. BABYLON (Christaller et al 1992) is one of the hybrid environments for developing expert systems that has attempted to include constraint solving in its rule- based system. BABYLON provides representation formalisms of objects, rules, Prolog and constraints. CONSAT is the constraint system of BABYLON, which is separated from others and cannot access rules. Although in the condition part of the rules, it is possible to verify whether a constraint is satisfied, the action part of a rule cannot access constraints. This is quite different from our system, within which constraint solving and rule-chaining are integrated. Rules can post and test constraints, and constraint satisfaction can also trigger chaining of rules. 7. Conclusion This paper shows how CSP can be used to model two types of important disjunctions in rule-based reasoning. These disjunctions have not been handled satisfactorily in the current rule-based systems. In the proposed scheme, the simple representation and efficient algorithms in CSP are used to deal with these types of disjunction. This results in the integration of two important types of reasoning techniques, i.e., constraint solving and (forward) rule-chaining. Hence. the power of rule-based systems is increased. The current integration of CSP with rule-based reasoning is still restricted, i.e., mainly for modeling the two types of disjunction. Our next step is to deal general constraints in a forward chaining framework. with S: Bing Liu thanks Peter Lucas from r his advice on some expert system and Yap for his help. Finally, we are grateful to the AAAI reviewers for their insightful comments. eferences Bessiere, C., Freuder, E. C. and Regin, J-C. 1995. “Using inference to reduce arc consistency computation,” IJCAI-95, 592-598. Charniak, E., Riesbeck, C., McDermott, D. and Meehan, J. 1987. Arti$cinl Intelligence Programming, Lawrence Erlbaum Associates Inc. Christaller, T., di Primio, F., Schnepf, U. and Voss, A. 1992. The AI Workbench BABYLOX. Academic Press. Hentenryck, P.V. 1989. Constraint Satisfaction in Logic Programming, MIT Press. Hentemyck’ P.V., Deville, Y. and Teng, C-M. 1992. “A generic arc consistency algorithm and its specializations,” Artificial Intelligence 27, 291-322. Ilog Solver. 1992. Reference Manual, ILOG, France. JafFar, J. and Lassez, J. 1987. “Constraint logic programming,” Proceedings of the Fourteenth Annual A CM Symposium on Principle of Programming Language. Jafhar, J. and Maher, M. 1994. “Constraint logic programming: a survey.” J. Logic Programming 19, 503-58 1. Liu, B. 1995. “Increasing fimctional constraints need to be checked only once”’ IJCAI-95, 586-59 1. Liu, B. and Jtiar, J. 1996. Using Constraints to Model Disjunction in Rule-Based Reasoning. DISCS Technical Report. Liu, B. 1996. “An improved generic arc consistency algorithm and its specializations.” To Appear in Proceedings of Fourth Pacijk Rim International Conference On ArtiJicial Intelligence (PRICU-96). Lucas, P. and Van Der Gaag, L. 1991. Principles of Expert @stems, Addison-Wesley. Ma&worth, AK. 1977. ‘Consistency in networks of relations,” Artificial Intelligence 8, 99-l 18. Mackvvorth, AK 1992. “The logic of co~Mmir~t satisfaction,” ArtiJiciai Intelligence 58, 3-20. Mohr, R. and Henderson, T. 1986. “Arc and path consistency revisited,” Artificial Intelligence 28, 225- 233. Newell, A. 1973. “Production systems: models for control structure,” In Visual Information Processing, W.G. Chase (Eds)’ Academic Press, 1973. Rule-Based Reasoning & Connectionism 1255 | 1996 | 186 |
1,828 | oall olic ropagation i Enrique Castillo*, Josh Manuel Guti&rez* and Ali S. * Department of Applied Mathematics and Computational Sciences, University of Cantabria, SPAIN CastieQccaix3.unican.es and GutierjmQccaix3.unican.es ** Department of Social Statistics, Cornell University, USA Ali-hadi@cornell.edu Abstract The paper presents an efficient goal oriented algo- rithm for symbolic propagation in Bayesian net- works. The proposed algorithm performs sym- bolic propagation using numerical methods. It first takes advantage of the independence rela- tionships among the variables and produce a re- duced graph which contains only the relevant nodes and parameters required to compute the desired propagation. Then, the symbolic expres- sion of the solution is obtained by performing numerical propagations associated with specific values of the symbolic parameters. These spe- cific values are called the canonical components. Substantial savings are obtained with this new algorithm. Furthermore, the canonical compo- nents allow us to obtain lower and upper bounds for the symbolic expressions resulting from the propagation. An example is used to illustrate the proposed methodology. Introduction Bayesian networks are powerful tools both for graphi- cally representing the relationships among a set of vari- ables and for dealing with uncertainties in expert sys- tems. A key problem in Bayesian networks is evidence propagation, that is, obtaining the posterior distribu- tions of the variables when some evidence is observed. Several efficient exact and approximate methods for propagation of evidence in Bayesian networks have been proposed in recent years (see, for example, Pearl 1988, Lauritzen and Spiegelhalter 1988, Henrion 1988, Shachter and Peot 1990, Fung and Chang 1990, Poole 1993, Bouckaert, Castillo and Gutierrez 1995). How- ever, these methods require that the joint probabilities of the nodes be specified numerically, that is, all the pa- rameters must be assigned numeric values. In practice, when exact numeric specification of these parameters may not be available, or when sensitivity analysis is de- sired, there is a need for symbolic methods which are able to deal with the parameters themselves, without assigning them numeric values. Symbolic propagation leads to solutions which are expressed as functions of the parameters in symbolic form. Recently, two main approaches have been pro- posed for symbolic inference in Bayesian networks. The symbolic probabilistic inference algorithm (SPI) (Shachter, D’Ambrosio and DelFabero 1990 and Li and D’Ambrosio 1994) is a goal oriented method which performs only those calculations that are required to respond to queries. Symbolic expressions can be ob- tained by postponing evaluation of expressions, main- taining them in symbolic form. On the other hand, Castillo, Gutierrez and Hadi 1995, 1996a, 199613, ex- ploit the polynomial structure of the marginal and conditional probabilities in Bayesian networks to ef- ficiently perform symbolic propagation by calculating the associated numerical coefficients using standard numeric network inference algorithms (such as those in Lauritzen and Spiegelhalter). As opposed to the SPI algorithm, this method is not goal oriented, but allows us to obtain symbolic expressions for all the nodes in the network. In this paper we show that this algorithm is also suitable for goal oriented prob- lems. In this case, the performance of the method can be improved by taking advantage of the independence relationships among the variables and produce a re- duced graph which contains only the nodes relevant to the desired propagation. Thus, only those opera- tions required to obtain the desired computations are performed. We start by introducing the necessary notation. Then, an algorithm for efficient computation of the desired conditional probabilities is presented and illus- trated by an example. Finally, we show how to obtain lower and upper bounds for the symbolic expressions solution of the given problem. Notation Let X = {X1,X2,... , X,} be a set of n discrete vari- ables, each can take values in the set (0, 1, . . . , ri}, the possible states of the variable Xi. A Bayesian net- work over X is a pair (D, P), where the graph D is a directed acyclic graph (DAG) with one node for each variable in X and P = {JI~(z~[T~), . . . ,pn(z,~~,)} is a set of n conditional probabilities, one for each variable, where I& is the set of parents of node Xi. Using the Bayesian Networks 1263 From: AAAI-96 Proceedings. Copyright © 1996, AAAI (www.aaai.org). All rights reserved. chain rule, the joint probability distribution of X can be written as: PhX2, * * * ,x7-J = fiPi(zih). (1) i=l Some of the conditional probability distributions (CDP) in (1) can be specified numerically and oth- ers symbolically, that is, pi(xi(ri) can be a parametric family. When pi(xi Ini) is a parametric family, we re- fer to the node Xi as a symbolic node. A convenient notation for the parameters in this case is given by eij, = pi(xi = jp, = 7-+ j E (0,. . . ,Q), (2) where 7r is any possible instantiation of the parents of Xi. Thus, the first subscript in B,j, refers to the node number, the second subscript refers to the state of the node, and the remaining subscripts refer to the parents’ instantiations. Since C,‘& Bijn = 1, for all i and r, any one of the parameters can be written as one minus the sum of all others. For example, Oirin is ri-1 eirin = i - E &jr- (3) j=O If Xi has no parents, we use 0ij to denote pi(Xi = j>, j E {o, * - * , ri}, for simplicity. Goal Oriented Algorithm Suppose that we are interested in a given goal node Xi, and that we want to obtain the CDP p(Xi = j/E = e), where E is a set of evidential nodes with known values E = e. Using the algebraic characterization of the probabilities given by Castillo, Gutierrez and Hadi 1995, the unnormalized probabilities Ij(Xi = jlE = e) are polynomials of the form: .P(Xi = jJE = e) = C Cjrmr = pj(O)y (4) m,EMj where mj are monomials in the symbolic parameters, 0, contained in the probability distribution of the Bayesian network. For example, suppose we have a discrete Bayesian network consisting of five binary vari- ables {Xi,...,Xs}, with values in the set (0, 1). The associated DAG is given in Figure 1. Table 1 gives the corresponding parameters, some in numeric and oth- ers in symbolic form. Node X4 is numeric because it contains only numeric parameters and the other four nodes are symbolic because some of their parameters are specified only symbolically. For illustrative purposes, suppose that the target node is Xa and that we have the evidence X2 = 1. We wish to compute the conditional probabilities p(Xa = j(X2 = l),j = 0,l. We shall show that p(X3 = 01x2 = 1) o.4e10e210 + o.3e301 - o.3e10e301 (5) = 0.3 - o.3elo + e10e210 , 1264 Uncertainty Node 1 Parameters xi rIi xi = 0 Xl None 810 =p(X, = 0) x2 Xl &?oo=px2=ox1=o I3201 = p(X, = 01x1 = 1) = 0.7 x3 Xl 0300 = p(X3 = 01x1 = 0) = 0.4 e301 = p(x3 = 01x1 = I) x4 x2,x3 o&-Joo = p x4 = 0 x2 = 0,x3 = 0 = 0.2 64001 = p(X4 = 01x2 = 0,x3 = 1) = 0.4 I94010 = p(X4 = 01x2 = 1, x3 = 0) = 0.7 04011 = p(X4 = 01x2 = 1, X3 = 1) = 0.8 X5 x3 f3500 = p(X5 = 0(X3 = 0) 0501 =p(X5 = 01x3 = 1) I Node I Parameters I I xi rIi xi = 1 Xl None 811 = p(X, = 1) x2 Xl &?1o=px2=1x~=o 6211 = p(X2 = 11x1 = 1) = 0.3 x3 Xl 0310 = p(X, = 11x1 = 0) = 0.6 0311 = p(X3 = 11x1 = 1) x4 x2,x3 t9mo = p(X4 = 11x2 = 0, X3 = 0) = 0.8 041~1 = p(X4 = 11x2 = 0, X3 = 1) = 0.6 e411l-J = p(X4 = 11x2 = 1,x3 = 0) = 0.3 f&111 = p(X4 = 11x2 = 1,x3 = 1) = 0.2 x5 x3 e510 = p(X5 = 11x3 = 0) I9511 = p(X5 = 1(X3 = 1) Table 1: Numeric and symbolic conditional probabilities. and p(X3 = 11x2 = 1) 0.3 - o.3elo + o.6e10e210 - o.3e301 + o.3e10e301 = 0.3 - o.3elo + e10e210 7 (6) where the denominator in (5) and (6) is a normalizing constant. Algorithm 1 gives the solution for this goal oriented problem by calculating the coefficients cjT in (4) of these polynomials. It is organized in four main parts: o PART I : Identify all Relevant Nodes. The CDP p(Xi = j] E = e) does not necessarily Figure 1: An example of a five-node Bayesian Network. involve parameters associated with all nodes. Thus, we identify the set of nodes which are relevant to the calculation of p(Xi = jl E = e), using either one of the two algorithms given in Geiger, Verma, and Pearl 1990 and Shachter 1990. Once this has been done we can remove the remaining nodes from the graph and identify the associated set of relevant parameters 0. PART II : Identify Sufficient Parameters. By considering the values of the evidence variables, the set of parameters 0 can be further reduced by identifying and eliminating the set of parameters which are in contradiction with the evidence. These parameters are eliminated using the following two rules: - Rule 1: Eliminate the parameters ejkr if xj # k for every Xj E E. - Rule 2: Eliminate the parameters Bjkr if par- ents’ instantiations 7r are incompatible with the evidence. PART III : Identify Feasible Monomials. Once the minimal sufficient subsets of parameters have been identified, they are combined in monomi- als by taking the Cartesian product of the minimal sufficient subsets of parameters and eliminating the set of all infeasible combinations of the parameters using: - Rule 3: Parameters associated with contradic- tory conditioning instantiations cannot appear in the same monomial. PART IV : Calculate Coefficients of all Poly- nomials. This part calculates the coefficients applying nu- meric network inference methods to the reduced graph obtained in Part I. If the parameters 0 are as- signed numerical values, say 8, then pj (0) can be ob- tained using any numeric network inference method to compute p(Xi = jl E = e, 0 = 0). Similarly, the monomials m, take a numerical value, the product of the parameters involved in m,. Thus, we have P(Xi = j(E = e,O = e) = x cjrm, = pj(l3). m,.EM, (7) Note that in (7) all the monomials m, , and the unnormalized probability pj (0) are known numbers, and the only unknowns are the coefficients cjr. To compute these coefficients, we need to construct a set of independent equations each of the form in (7). These equations can be obtained using sets of dis- tinct instantiations 0. To illustrate the algorithm we use, in parallel, the previous example. Algorithm 1 Computes p(Xi = jl E = e). Input: A Bayesian network (D, P), a target node Xi (4 (b) Figure 2: (a) Augmented graph D* after adding a dummy node Vi for every symbolic node Xi, and (b) the reduced graph D’ sufficient to compute p(Xi = j IE = e). and an evidential set E (possibly empty) with eviden- tial values E = e. Output: The CPD p(Xi = jlE = e). PART I: Step 1: Construct a DAG D* by augmenting D with a dummy node Vj and adding a link Vj + Xj for every node Xj in D. The node Vj represents the parameters, Oj, of node Xj. Example: We add to the initial graph in Figure 1, the nodes VI, V2, V3, Vi, and Vs The resulting graph in shown in Figure 2(a). Step 2: Identify the set V of dummy nodes in D* not d-separated from the goal node Xi by E. Ob- tain a new graph D’ by removing from D those nodes whose corresponding dummy nodes are not contained in V with the exception of the target and evidential nodes. Let 0 be the set of all the param- eters associated with the symbolic nodes included in the new graph and V. Example: The set V of dummy nodes not d- separated from the goal node X3 by the evidence node E = {X2} is found to be V = {VI, V2, V3). Therefore, we remove X4 and X5 from the graph ob- taining the graph shown in Figure 2(b). Thus, the set of all the parameters associated with symbolic nodes of the new graph is PART II: e Step 3: If there is evidence, remove from 0 the parameters Ojkr if xj # k for Xj E E (Rule 1). o Example: The set 0 contains the symbolic param- eters 8200 and 0201 that do not match the evidence X2 = 1. Then, applying Rule 1 we eliminate these parameters from 0. Bayesian Networks 1265 Step 4: If there is evidence, remove from 0 the pa- rameters O~,C* if the set of values of parents’ instan- tiations 7r are incompatible with the evidence (Rule 2). Example: Since the only evidential node X2 has no children in the new graph, no further reduction is possible. Thus, we get the minimum set of sufficient parameters: PART III: Step 5: Obtain the set of monomials M by taking the Cartesian product of the subsets of parameters in 0. Example: The initial set of candidate monomials is given by taking the Cartesian product of the minimal sufficient subsets, that is, M = (ho, ell) x (0 210, e211j x {e300, e310, e301, e311). Thus, we obtain 16 different candidate monomials. Step 6: Using Rule 3, remove from M those mono- mials which contain a set of incompatible parame- ters. Example: Some of the monomials in M contain parameters with contradictory instantiations of the parents. For example, the monomial 0ia02ia&ai con- tains contradictory instantiations of the parents be- cause @ia indicates that Xi = 0 whereas @soi in- dicates that Xi = 1. Thus, applying Rule 3, we get the following set of feasible monomials M = ~~10e210~300r ~10~210~310, ~11~211~301, ~11~211~311~. Step 7: If some of the parameters associated with the symbolic nodes are specified numerically, then remove these parameters from the resulting feasible monomials because they are part of the numerical coefficients. Example: Some symbolic nodes involve both nu- meric and symbolic parameters. Then, we remove from the monomials in M the numerical parame- ters &aa, 0310 and 0211 obtaining the set of feasi- ble monomials M = {hoezlo, fhe301, h~311). Note that, when removing these numeric parameters from 0, the monomials &002100300 and &002100310 be- come eio&ia. Thus, finally, we only have three dif- ferent monomials associated with the probabilities p(X3 = jlX2 = l),j = 0,l. PART IV: e Step 8: For each possible state j of node Xi, j = 0 Y”‘, ri, build the subset Mj by considering those monomials in M which do not contain any parameter of the form Oiqr, with q # j. o Example: The sets of monomials needed to calculate p(Xs = 01x2 = 1) and p(Xs = 11x2 = 1) are MO = {~1&10,~118sai} and Ml = (Bia&ia, &i&ir}, respectively. Then, using (4), we have: PO(Q) = @x3 = 01x2 = 1) = co1mo1+ co277Jo2 = ~ol~lo~210 + c02~11~301. (8) p1(0) = P(X3 = 11x2 = 1) = wwl + c12m12 (9) = d40~210 + c12he311. e Step 9: For each possible state j of node Xi, calcu- late the coefficients cjr of the conditional probabili- ties in (4)) r = 0, . . . , nj , as follows: Calculate nj different instantiations of 0, C = {b. . , elt3 ) such that the canonical nj x nj ma trix Tj, whose rs-th element is the value of the monomial m, obtained by replacing 0 by 8,, is a non-singular matrix. Use any numeric network inference method to compute the vector of numerical probabilities Pj = (lpj(h>,... , pj ( Bnj )) by propagating the evi- dence E = e in the reduced graph D’ obtained in Step 2. Calculate the vector of coefficients cj = (Cjl, - - . , cjnj) by solving the system of equations Tjcj = pj, (10) which implies (3 = Tr’pj. (11) Example: Thus, taking appropriate combina- tions of extreme values for the symbolic parame- ters (canonical components), we can obtain the nu- meric coefficients by propagating the evidence not in the original graph D (Castillo, Gutierrez and Hadi 1996), but in the reduced graph D’, saving a lot of computation time. We have the symbolic parameters 0 = (ho, h, e200, e210, e301, e3i1) con- tained in D’, We take the canonical components 81 = (l,O, l,O, 1,O) and e2 = (0, l,O, 1, 1,O) and using any (exact or approximate) numeric network inference methods to calculate the coefficients of PO(@). We obtain, p. (0,) = 0.4 and po(&) = 0.3. Note that, in the above equation, the vector (po(81),po(&)) can be calculated using any of the standard exact or approx- imate numeric network inference methods, because all the symbolic parameters have been assigned a numerical value: po(el) = p(x, = 01x2 = I, 0 = el) po(e2) = p(x3 = 01x2 = 1,o = e2). Then, no symbolic operations are performed to ob- tain the symbolic solution. Thus, (11) becomes 1266 Uncertainty Similarly, taking the canonical components 8i = (1, 0, 1, 0, 1,O) and 02 = (0, 1, 0, 1, 0, l), for the prob- ability pl(0) we obtain Then, by substituting in (8) and (9), we obtain the unnormalized probabilities: @x3 = 0(x2 = 1) = o.4e10e210 + o.3e11e301, (14) w3 = 11x2 = 1) = o.6e10e210 + o.3e11e311. (15) Step 10: Calculate the unnormalized probabilities pj(Q), j = 0,. . . , ri and the conditional probabilities p(Xi = j(E = e) = pj(O)/N, where N = gP,(Q) j=O is the normalizing constant. Example: Finally, normalizing (14) and (15) we get the final polynomial expressions: p(X,=OIX2= l)= o.4e10e210 + o.3e11e301 e1oe21o + o.3e11e301 + o.3e11e311 (16) and p(X3 = 11x1 = 1) = o.6~10e210 + o.3e11e311 e1oe21o + o.3e11e301 + o.3e11e311 (17) Step 11: Use (3) to eliminate dependent parameters and obtain the final expression for the conditional probabilities. Example: Now, we apply the relationships among the parameters in (3) to simplify the above expres- sions . In this case, we have: t93ri = 1 - 0301 and 011 = 1 - 810. Thus, we get Expressions (5) and (6). Equations (5) and (6) g ive the posterior distribution of the goal node X3 given the evidence X2 = 1 in symbolic form. Thus, p(X3 = jlX2 = l), j = 0,l can be evaluated directly by plugging in (5) and (6) any specific combination of values for the symbolic parameters without the need to redo the propagation from scratch for every given combination of values. In our case, u is the set of symbolic parameters and the fractional functions (18) are the symbolic ex- pressions associated with the probabilities, (5) and (6). In this case, the convex polyhedron is defined by u 5 1, u 2 0, that is, A is the identity matrix. Then, using Theorem 1, the lower and upper bounds of the symbolic expressions associated with the probabilities are attained at the vertices of this polyhedron. In our case, the vertices of the polyhedron are given by all possible combinations of values 0 or 1 of the symbolic parameters, that is, by the complete set of canonical components associated with the set of free symbolic pa- rameters appearing in the final symbolic expressions. Remark: In some cases, it is possible to obtain a set of As an example, Table 2 shows the canonical prob- canonical instantiations for the above algorithm that abilities associated with the symbolic expressions (5) leads to an identity matrix Tj. In those cases, the and (6) obtained for the CDP p(X3 = jlX2 = 1). coefficients of the symbolic expressions are directly ob- The minimum and maximum of these probabilities are tained from numeric network inferences, without the 0 and 1, respectively. Therefore, the lower and up- extra effort of solving a system of linear equations. per bounds are trivial bounds in this case. The same ok p(x3 = jlx2 = I,&) ho e210 e301 j=O j=l 0 010 0.0 1.0 Table 2: Conditional probabilities for the canonical cases associated with 1540,&o, and 8301. Sensitivity Analysis The lower and upper bound of the resulting symbolic expressions are a useful information for performing sen- sitivity analysis (Castillo, Gutierrez and Hadi 1996a). In this section we show how to obtain an interval, (1,~) c [0, l], that contains all the solutions of the given problem, for any combination of numerical val- ues for the symbolic parameters. The bounds of the obtained ratios of polynomials as, for example (5) and (6), are attained at one of the canonical components (vertices of the feasible convex parameter set). We use the following theorem given by Martos 1964. Theorem 1 If the linear fractional functional of u, c*u-co d*u-do’ where u is a vector, c and d are vector coefficients and co and do are real constants, is defined in the convex polyhedron Au 2 a~, u 2 0, where A is a constant matrix and aa is a constant vector, and the denomina- tor in (18) does not vanish in the polyhedron, then the functional reaches the maximum at least in one of the vertices of the polyhedron. Bayesian Networks 1267 ok p(x3 = jlx2 = 1, ek) alo e210 j=O j=l 0 0 0.5 0.5 Table 3: Conditional probabilities for the canonical cases associated with 810 and 0210 for 0301 = 0.5. bounds are obtained when fixing the symbolic param- eters elo or e210 to a given numeric value. However, if we consider a numeric value for the symbolic parameter 0301, for example 8301 = 0.5, we obtain the canonical probabilities shown in Table 3. Therefore, the lower and upper bounds for the prob- ability p(X3 = 01x2 = 1) become (0.4,0.5), and for p(X3 = 11x2 = 1) are (0.5,0.6), i.e., a range of 0.1. If we instantiate another symbolic parameter, for ex- ample 810 = 0.1, the new range decreases. We obtain the lower and upper bounds (0.473,0.5) for p(X3 = 01x2 = l), and (0.5,0.537) for p(X3 = 11x2 = 1). Conclusions and Recommendations The paper presents an efficient goal oriented algo- rithm for symbolic propagation in Bayesian networks, which allows dealing with symbolic or mixed cases of symbolic-numeric parameters. The main advantage of this algorithm is that uses numeric network inference methods, which make it superior than pure symbolic methods. First, the initial graph is reduced to produce a new graph which contains only the relevant nodes and parameters required to compute the desired prop- agation. Next, the relevant monomials in the symbolic parameters appearing in the target probabilities are identified. Then, the symbolic expression of the solu- tion is obtained by performing numerical propagations associated with specific numerical values of the sym- bolic parameters. An additional advantage is that the canonical components allow us to obtain lower and up- per bounds for the symbolic marginal or conditional probabilities. Acknowledgments We thank the Direction General de Investigation Cientifica y Tecnica (DGICYT) (project PB94-1056), Iberdrola and NATO Research Office for partial sup- port of this work. References Bouckaert, R. R., Castillo, E. and Gutierrez, J. M. 1995. A Modified Simulation Scheme for Inference in Bayesian Networks. International Journal of Ap- proximate Reasoning, 14:55-80. Castillo, E., Gutierrez, J. M., and Hadi, A. S. 1995. Parametric Structure of Probabilities in Bayesian Networks. Lectures Notes in Artificial Intelligence, Springer-Verlag, 946:89-98. Castillo, E., Gutierrez, J. M., and Hadi, A. S. 1996a. A New Method for Efficient Symbolic Propagation in Discrete Bayesian Networks. Networks. To appear. Castillo, E., Gutierrez, J. M., and Hadi, A. S. 199613. Expert Systems and Probabilistic Network Models. Springer-Verlag, New York. Fung, R. and Chang, K. C. 1990. Weighing and Integrating Evidence for Stochastic Simulation in Bayesian Networks, in Uncertainty in Artificial Intelligence 5, Machine Intelligence and Pattern Recognition Series, 10, (Henrion et al. Eds.), North Holland, Amsterdam, 209-219. Geiger, D., Verma, T., and Pearl, J. 1990. Identify- ing Independence in Bayesian Networks. Networks, 20:507-534. Henrion, M. 1988. Propagating Uncertainty in Bayesian Networks by Probabilistic Logic Sampling, in Uncertainty in Artificial Intelligence, (J.F. Lem- mer and L. N. Kanal, Eds.), North Holland, Ams- terdam, 2:317-324. Lauritzen, S. L. and Spiegelhalter, D. J. 1988. Lo- cal Computations with Probabilities on Graphical Structures and Their Application to Expert Sys- tems. Journal of the Royal Statistical Society (B), 50:157-224. Li, Z., and D’Ambrosio, B. 1994. Efficient Infer- ence in Bayes Nets as a Combinatorial Optimiza- tion Problem. International Journal of Approximate Reasoning, 11(1):55-81. Martos, B. 1964. Hyperbolic Programming. Naval Research Logistic Quarterly, 32:135-156. Pearl, J. 1988. Probabilistic Reasoning in Intelligent Systems: Networks of Plausible Inference. Morgan Kaufmann, San Mateo, CA. Poole, D. 1993. Average-case Analysis of a Search Algorithm for Estimating Prior and Poste- rior Probabilities in Bayesian Networks with Ex- treme Probabilities, in Proceedings of the 13th Inter- national Joint Conference on Artificial Intelligence, 13( 1):606-612. Shachter, R. D. 1990. An Ordered Examination of Influence Diagrams. Networks, 20:535-563. Shachter, R. D., D’Ambrosio, B., and DelFabero, B. 1990. Symbolic Probabilistic Inference in Belief Net- works, in Proceedings Eighth National Conference on AI, 126-131. Shachter, R. D. and Peot, M. A. 1990. Simulation Approaches to General Probabilistic Inference on Belief Networks, in Uncertainty in Artificial Intel- ligence, Machine Intelligence and Pattern Recogni- tion Series, 10 (Henrion et al. Eds.), North Holland, Amsterdam, 5:221-231. 1268 Uncertainty | 1996 | 187 |
1,829 | A Clinician’s Tool for Analyzing Non-compliance David Maxwell Chickering and Judea Pearl Cognitive Systems Laboratory Computer Science Department University of California, Los Angeles, CA 90024 dmax@cs.ucla.edu judea@cs. ucda. edu Abstract We describe a computer program to assist a clinician with assessing the efficacy of treatments in experimen- tal studies for which treatment assignment is random but subject compliance is imperfect. The major diffi- culty in such studies is that treatment efficacy is not “identifiable”, that is, it cannot be estimated from the data, even when the number of subjects is infinite, unless additional knowledge is provided. Our system combines Bayesian learning with Gibbs sampling us- ing two inputs: (1) the investigator’s prior probabili- ties of the relative sizes of subpopulations and (2) the observed data from the experiment. The system out- puts a histogram depicting the posterior distribution of the average treatment effect, that is, the proba- bility that the average outcome (e.g., survival) would attain a given level, had the treatment been taken uniformly by the entire population. This paper de- scribes the theoretical basis for the proposed approach and presents experimental results on both simulated and real data, showing agreement with the theoretical asymptotic bounds. Introduction Standard clinical studies in the biological and medi- cal sciences invariably invoke the instrument of ran- domized control, that is, subjects are assigned at ran- dom to various groups (or treatments or programs) and the mean differences between participants in different groups are regarded as measures of the efficacies of the associated programs. For example, to determine if a new drug is useful for treating some disease, subjects will be divided (at random) into a control group and a treatment group. The members of the control group are given a placebo and the members of the treatment group are given the drug in question. For each group, the clinician records the fraction of subjects that re- cover from the disease. By comparing these fractions the clinician can derive a quantitative measure of ef- fectiveness of the drug for treating the disease. In par- ticular, if fc and ft are the fractions of subjects that recovered from the control group and treatment group respectively, then the difference E = fc - ft is an indi- cation of the effectiveness of the drug. The major source of difficulty in managing and an- alyzing such experiments has been subject noncompli- ance. For example, a subject in the treatment group may experience negative side effects and will stop tak- ing the drug. Alternatively, if the experiment is testing a drug for a terminal disease, a subject suspecting that he is in the control group may obtain the drug from other sources. Imperfect compliance poses a problem because simply comparing the fractions as above may provide a misleading estimate for how effective the drug would be if applied uniformly to the population. For example, if those subjects who refused to take the drug are precisely those who would have responded ad- versely, the experiment might conclude that the drug is more effective than it actually is. It can be shown, in fact, that treatment effectiveness in such studies is non-identifiable. That is, in the absence of additional modeling assumptions, treatment effectiveness cannot be estimated from the data without bias, even as the number of subjects in the experiment approaches in- finity, and even when a record is available of the action and response of each subject (Pearl 1995a). In a popular compromising approach to the prob- lem of imperfect compliance, researchers perform an intent-to-treat analysis, in which the control and treat- ment group are compared without regard to whether the treatment was actually receivedi. The result of such an analysis is a measure of how well the treat- ment assignment effects the disease, as opposed to the desired measure of how well the treatment itself effects the disease. Estimates based on intent-to-treat analy- sis are valid only as long as the experimental conditions perfectly mimic the conditions prevailing in the even- tual usage of the treatment. In particular, the exper- iment should mimic subjects’ incentives for receiving each treatment. In situations where field incentives are more compelling than experimental incentives, as is usually the case when drugs receive the approval of a government agency, treatment effectiveness may vary significantly from assignment effectiveness. For example, imagine a study in which (a) the drug has an ‘This approach is currently used by the FDA to approve new drugs. Bayesian Networks 1269 From: AAAI-96 Proceedings. Copyright © 1996, AAAI (www.aaai.org). All rights reserved. adverse effect on a large segment of the population and (b) only those members of the segment who drop from the treatment arm recover. The intent-to-treat anal- ysis will attribute these cases of recovery to the drug since they are part of the intent-to-treat arm, while in reality these cases have recovered by avoiding the treatment (Pearl 1995b). Another approach to the problem is to use a correc- tion factor based on an “instrumental variables” for- mula (Angrist, Imbens, & Rubin 1993), according to which the intent-to-treat measure should be divided by the fraction of subjects who comply with the treat- ment assigned to them. Angrist et al. (1993) have shown that, under certain conditions, the corrected formula is valid for the subpopulation of “responsive” subjects, that is, subjects who would have changed treatment status if given a different assignment. Unfor- tunately, this subpopulation cannot be identified and, more seriously, it cannot serve as a basis for policies involving the entire population because it is instru- ment dependent-individuals who are responsive in the study may not remain responsive in the field, where the incentives for obtaining treatment differ from those used in the study. Using a graphical model with latent variables, Balke and Pearl (1994) d erive bounds, rather than point estimates, for the treatment effect, while making no assumptions about the relationship between subjects’ compliance and subjects’ physical response to treat- ment. However, the derived bounds are “asymptotic”, i.e., they ignore sampling variations by assuming that the proportions measured in the experiment are rep- resentative of the population as a whole, a condition which is valid only when the number of subjects is large. This large-sample assumption may be problem- atic when the study includes a relatively small number of subjects. In this paper we describe a system that provides an assessment of the actual treatment effect and is not limited to studies with large samples. The system uses the graphical model of Balke and Pearl (1994) to learn the treatment effect using Bayesian updating combined with Gibbs sampling. The system takes as input (1) the investigator’s prior knowledge about subject com- pliance and response behaviors and (2) the observed data from the experiment, and outputs the posterior distribution of the treatment effect. The use of graph- ical models and Gibbs’ methods for deriving posterior distributions in such models are both well known. The main contribution of this paper is a description of how these techniques can be applied to the causal analy- sis of clinical trials, and a presentation of experimental results of a practical system applied to various simu- lated and real data. While the basic idea of estimating causal effects using Bayesian analysis goes back to Ru- bin (1978), and was further used by Imbens and Rubin (1994) to estimate the correctional formula advocated by Angrist et al. (1993), we believe that this is the first time an assumption-free assessment of the aver- age treatment eflect is made available to the clinical research community. The paper is organized as follows. First, we intro- duce a graphical, causal model that represents a proto- typical clinical trial with partial compliance, and define treatment e#ect in terms of the model. Next, we de- scribe an equivalent graphical model, using potential- response variables (Balke & Pearl 1994)) that allows the compliance and response behavior to be repre- sented more efficiently. Next, we describe the general Bayesian-learning and Gibbs-sampling methods that were used to derive the posterior parameter densities in the graphical model. Finally, we describe experi- mental results obtained when our system is applied to various simulated and real data sets. We include re- sults obtained when the system is modified to answer counterfactual queries about specific individuals, e.g., “what if Joe (who died with no treatment) were to have taken the treatment?” The Graphical Model Graphical models are convenient tools for representing causal and statistical assumptions about variables in a domain (Pearl 1995a). In this section, we describe the graphical model of Figure 1, which is used to represent a prototypical clinical trial with partial compliance. We use 2, D and Y to denote observed binary variables from the experiment, where 2 represents the treatment assignment, D represents the treatment received, and Y represents the observed outcome. To facilitate the notation, we let Z, d, and y represent, respectively, the values taken by the variables 2, D, and Y, with the following interpretation: z E { zc, ~1 }, ~1 asserts that the treatment has been assigned (zc its negation); d E {do,dl}, dl asserts that the treatment has been administered (do its negation); and y E { yc, yl }, yr asserts a positive observed response (ye its negation). We use U to denote all characteristics, both observed and unobserved, that influence the value of D and Y for the subjects. The domain of U is left unspecified, and in general will combine the spaces of several random variables, both discrete and continuous. Treatien>o f 1 Received ’ Figure 1: Graphical model for a prototypical clinical trial with partial compliance 1270 Uncertainty Let ve denote the physical probability of the event E= e, or equivalently, the fraction of subjects in the population for which E = e. The graphical model explicitly represents two independence assump- tions about the joint physical probability distribution ~~,d,~,~. First, the model asserts that the treatment as- signment 2 can influence: Y only through the actual treatment D. That is, 2 and Y are conditionally in- dependent given D and U. Second, the model asserts that 2 and U are marginally independent. This second independence is ensured through the randomization of 2, which rules out both (1) the existence of a common cause for both 2 and U, and (2) the possibility that U has causal influence on 2. The two independence assumptions together induce the following decomposi- tion of the joint distribution: u~,4Y,u = %hd’dla,uVyId,u In addition to the independence assumptions, the graphical model also encodes causal assumptions (e.g., that 2 does not effect Y directly) which permit one to predict how the joint probability will change in light of exogenous local interventions (Pearl 1995a). In par- ticular, the absence of any direct link (or any spurious path) from 2 to Y implies that z.$ld+ is the same re- gardless if d is measured in an observational study, or dictated by some (exogenous) public policy. Conse- quently, if we wish to predict the distribution vlld of Y, under a new condition where the treatment D = d is applied uniformly to the population, we should cal- culate Vgr*ld = Euhd,ul where E, denotes the expectation with respect to vu. Likewise, if we are interested in the average change in Y due to treatment, we use the average causal eflect, denoted ACE(D + Y) , as defined by Holland (1988): ACE(D --$ y) = Eu[Vyljdl,u - vy,jd,,u] (1) Let D denote the observed collection of triples {x, d, y}, one for each subject, that we obtain from the experiment. Given ZJ, the objective of our system is to derive the posterior Bayesian probability distribution p(ACE(D + Y) ID). The Potential-Response Model The graphical model presented in the previous section is attractive for representing the assumptions that un- derlie a given experimental design, but may not be convenient for computation. For example, the graph of Figure 1 represents explicitly the assumptions that 2 is randomized and that 2 does not affect Y di- rectly, while making no assumption about the relation- ship between compliance and the way subjects would respond to the treatment. However, leaving the do- main of the unobserved variable U unspecified makes it difficult to derive the distribution of interest, namely, p(ACE(D + Y) ID,>. As is done by Balke and Pearl (1994), we exploit the observation of Pearl (1994) that U can always be replaced by a single discrete and finite variable such that the resulting model is equivalent with respect to all observations and manipulations of 2, D, and Y. In particular, because 2, D, and Y are all binary vari- ables, the state space of U divides into 16 equivalence classes: each equivalence class dictates two functional mappings; one from 2 to D, and the other from D to Y. To describe these equivalence classes, it is conve- nient to regard each of them as a point in the joint space of two four-valued variables C and R. The vari- able C determines the compliance behavior of a subject - through the mapping: I do if c = CO I d 0 if c=cl and % = %o 1 dl if c = cl and z = ~1 d = F&z, c) = (2) dl if c = c2 and z = zo do if c = c’; and z = ~1 ( dl if c = cg Imbens and Rubin (1994) call a subject with compli- ance behavior cc, cl, c2 and ca, respectively, a never- taker, a complier, a defier and an always-taker. Simi- larly, the variable R determines the of a subject through the mapping: response behavior Yo if r = rg Yo if r = r1 and d = do Yl if 7== rl and d = dl y = &(d,r) = Yl if r = r2 and d= do Yo if r = r2 and d = dl Yl if r = r3 (3) Following Heckerman and Shachter (1995), we call the response behavior ~0, rl, r2 and r3, respectively, never- recover, helped, hurt and always-recover. Let CR denote the variable-whose state space is the cross-product of the states of C and R. We use crij, with 0 5 i, j 5 3 to denote the state of CR correspond- ing to compliance behavior ci and response behavior rj . Figure 2 shows the graphical model that results from replacing U from Figure 1 by the 16-state variable CR. A state-minimal variable like CR is called a response variable by Balke and Pearl (1994) and a mapping vari- able by Heckerman and Shachter (1995), and its states correspond to the potential response vectors in Rubin’s model (Rubin 1978). Applying the definition of ACE(D + Y) given in Equation 1, it follows that using the model of Figure 2 we have: ACE(D+Y) = x [ i ucrtl] - [F”crs2] C4) Bayesian Networks 1271 Figure 2: Potential-response model invoking a 16 state variable CR Figure 3: Model used to represent the independences in P(W U { VCR} U {ACE(D --) y) }) Equivalently, ACE( D --+ Y) is the difference between the fraction of subjects who are helped by the treat- ment (R = rr) and the fraction of subjects who are hurt by the treatment (R = rz). Learning the Causal Effect Given the observed data 2) from the experiment, as well as a prior distribution over the unknown fractions r&R, our system uses the potential-response model de- fined in the previous section to derive the posterior distribution for ACE(D 4 Y) . In this section, we de- scribe how this computation can be done. To simplify discussion, we introduce the following notation. As- sume there are m subjects in the experiment. We use zi, di and yi to denote the observed value of 2, D and Y, respectively, for subject i. Similarly, we use cri to denote the (unobserved) compliance and response be- havior for subject i. We use Vi to denote the triple (z’, di, yi}. The posterior distribution of the causal effect can be derived using the graphical model shown in Figure 3, which explicitly represents the independences that hold in the joint (Bayesian) probability distribution defined over the variables {V, VCR, ACE(D + Y) ] . The model can be understood as m realizations of the potential-response model, one for each triple in D, connected together using the node representing the unknown fractions z&R. The model explicitly represents the assumption that, given the fractions Z&R, the probability of a subject belonging to any of the compliance-response subpopulations does not de- pend on the compliance and response behavior of the other subjects in the experiment. From Equation 4, ACE(D --+ Y) is a deterministic function of VCR and consequently ACE( D 3 Y) is independent of all other variables in the domain once these fractions are known. Determining the posterior probability for a node us- ing a graphical model is known as performing infer- ence in that model. In many cases, the independences of the model can be exploited to make the process of inference efficient. Unfortunately, because the cri are never observed, deriving the posterior distribution for ACE(D --) Y) is not tractable even with the given in- dependences. To obtain the posterior distribution, our system applies an approximation technique known as Gibbs sampling, which we describe in the following sec- tion. Gibbs Sampling Gibbs sampling is a well-known Markov chain sam- pling method that can be used to approximate the expected value of a function. The method can eas- ily be applied to approximate the posterior density of ACE(D + Y) by exploiting the independences in the model from Figure 3. Suppose we are interested in the expected value of some function f(X) with respect to the distribution p(XJY): EXIY [f] = J, ~(X)I'(XIY)~X In many cases, it may not be easy to solve the above integral analytically. However, we can approximate Exly [f] by repeatedly sampling values for X from the distribution p(X IY), and then taking an average. As- suming that N samples are taken and letting Xi denote the value for X on the it h sample we have: (5) In practice, sampling points directly from p(XIY) may be difficult. The Gibbs sampling method draws points from the distribution by repeatedly sampling from the conditiona distributions p(XilX \ Xi, Y), which are often very easy to derive in closed from. After ini- tially instantiating all the values of X, the algorithm repeatedly uninstantiates a single component Xi, and re-samples that component according to the condi- tional distribution p(Xi IX \ Xi, Y). It can be shown that as the number of iterations of the Gibbs sampler grows large, the sampled values for X are distributed as P(XIY)~. 2The resulting Markov chain must be ergodic for this result to hold, a property that can be easily established for our application. 1272 Uncertainty We can use a Gibbs sampler to approximate the the fractions VCR, and then observing the compliance posterior distribution of ACE(D + Y) as follows. Let indicator function that is 1 if and response behavior of NAR subjects, N,!r,, of which ~&CR) denote th e have behavior crij. Using this simplifying assumption, a 5 ACE(D --) Y) 5 b and 0 otherwise. Then we we update VCR by sampling from the following Dirich- have: let distribution: p(vCRlCd, . . . , Cm) = y@ N vCrij cr,j +N:r,j -1 = s fa,a(w~> * P(I/cRID) - dVcR After expanding the integral to include the unobserved compliance and response behavior for each of the sub- jects we have: i=O j=O For accurate results, the Gibbs sampler is typically run in two distinct phases. In the first phase, enough samples are drawn until it is reasonable to assume that the resulting Markov chain has converged to the cor- rect distribution. These initial samples are commonly referred to as the burn-in samples, and the correspond- ing values of the function being estimated are ignored. In the second phase, the values of the function are recorded and are used in the approximation of Equa- tion 5. There are countless techniques for determining when a series has converged, and no single method has become universally accepted among researchers. An- other complication of the Gibbs sampler is that suc- cessive samples in the second phase are inherently de- pendent, yet we use these samples to approximate in- dependent samples from the distribution. As a conse- quence of the many different methods to address these problems, tuning a Gibbs sampler for the best results tends to be more of an art than a science. p(u 5 ACE@ + Y) 5 blD) .dvcR . dcrl . . . . . dcrm Thus we can use the approximation of Equation 5 in conjunction with the Gibbs sampler to estimate the probability that ACE( D ---) Y) falls within any inter- val [a, b]. The conditional distributions from which we sample are easily derived in light of the indepen- dences depicted in Figure 3. In particular, letting X = {VCR, crl, . . . , crm}, we have: p(cri IX \ cri, lD) . . . . = ~~.p(d’,y~l2’,cr~).~~~i where a is the normalization constant. p(d” , yi j.zi, cri) is either one or zero, depending on whether the ob- served values of xi, di and yi agree with the given compliance and response behavior. Note that we have used the fact that if the fractions VCR are known, then the probability of cri is simply v,,i. To update VCR we sample from the posterior distri- bution: &‘CRjx \ WR$) = @ VCrij Ncraj * p( VCR) i=O j=O where ,B is the normalization constant and NcrSj is the number of times crij occurs in X. One choice of the functional form for p(VCR) is par- ticularly convenient for our application. In partic- ular, if the prior p(vCR) is a Dirichlet distribution, then both efficiently computing the posterior distri- bution in closed form and sampling from that distri- bution are easy. Assuming that the prior distribu- tion for VCR is Dirichlet implies there exists exponents N’ cr00 J - - * j N&, such that where y is the normalization constant. Let N& = ~~==, c,“,, NLpij. Having the given Dirichlet prior can be thought of as at some point being ignorant about The approach we took for the results presented in the next section can be explained as follows. We ran the Gibbs sampler for enough iterations to ensure a relatively smooth estimate of the distribution, always discarding a large number of the initial points sam- pled. We then repeated the same schedule, starting with a different random seed, and compared the re- sulting outputs. If the distributions were reasonably distinct, we repeated the process using more samples. We emphasize that the any one of the many methods of data analysis can readily be applied to the output of our system. Experimental We have applied the Gibbs sampling algorithm to the model of Figure 3 for various real and simulated data sets. Our system takes as input (1) the observed data ZJ, expressed as the number of cases observed for each of the 8 possible instantiations of {z, d, y}, and (2) a Dirichlet prior over the unknown fractions VCR, ex- pressed as the 16 exponents N&R. The system outputs the posterior distribution of ACE(D -+ Y) , expressed in a histogram. To investigate the effect of the prior distribution on the output, we have run all our experiments us- ing two different priors as input. The first is a flat (uniform) distribution over the 16-vector VCR, and is commonly used to express ignorance about the do- main. The second prior is skewed to represent a de- pendency between the compliance and response be- Bayesian Networks 1273 Table 1: Population fractions resulting in an identifi- able ACE(D 3 Y) z d Y 0 0 0 0 0 1 0 1 0 0 1 1 1 0 0 1 0 1 1 1 0 1 1 1 havior of the subjects. Figure 4 shows the distribu- vz ,d,Y 0.275 0.0 0.225 0.0 0.225 0.0 0.0 0.275 tion of ACE(D + Y) induced by these two prior dis- tributions. Note that the skewed prior of Figure 4b assigns almost all the weight to negative values of ACE(D + Y) . 0 (4 1 -1 0 1 (b) Figure 4: (a) The prior distribution of ACE(D ---) Y) induced by flat priors over the parameters VCR, and (b) the dis- tribution for ACE(D -+ Y) induced by skewed priors over the parameters. In the following sections, we present the output of our system using (1) a simulated data set for which the causal effect is identifiable, (2) a real data set from an experiment designed to determine the effect of cholestyramine on reduced cholesterol level, and (3) a real data set from a study to determine the effect of vitamin A supplementation on childhood mortality. Simulated Data Example: Identifiable Causal Effect As we noted in the introduction, Balke and Pearl (1994) h ave derived the tightest bounds for ACE(D + Y) under the large-sample assumption. They show that for some distributions of 2, D and Y, the resulting upper and lower bounds collapse to a single point. We say that ACE(D -+ Y) is identi- fiable-in this case. In -this section,‘ we show the out- put of our system when run on data sets derived from a distribution for which ACE(D + Y) is identifiable. One such distribution is shown in Table 1, yielding ACE(D --) Y) = 0.55. We transformed the (continuous) data from the Lipid study to the binary variables D and Y using the same method as Balke and Pearl (1994). The resulting data set is shown in Table 2. Using the large-sample assumption, Balke and Pearl (1994) use the given data to derive the bounds 0.39 5 ACE(D -+ Y) 5 0.78. Figure 5 shows the the output of our system when Figure 6 shows posterior applied to data sets of various sizes drawn from the dis- densities for ACE(D --) Y) given the data. The den- tribution shown in Table 1, using both the flat and the sity of Figure 6a corresponds to flat priors (over the skewed prior. As expected, as the number of cases in- creases, the posterior distributions become increasingly concentrated near the value 0.55. In general, because the skewed prior for ACE(D --) Y) is concentrated fur- ther from 0.55 than the uniform prior, more cases are needed before the posterior distribution converges to the value 0.55. (4 fb) (c) (4 (4 03 w (h) Figure 5: Output histograms for identifiable treatment effect using two priors. (a), (b), (c) and (d) show the posteriors for ACE( D -+ Y) using the flat prior and a data set consisting of 10, 100, 1000 and 10000 subjects, respectively. (e), (f), (g) and (h) show the posteriors for ACE(D + Y) using the skewed prior with the same respective data sets. Real Data Example: Effect of Cholestyramine on Reduced Cholesterol Consider the Lipid Research Clinics Coronary Primary Prevention data described in . A portion of this data consisting of 337 subjects was analyzed by Efron and Feldman (1991) using a model that incorporates sub- ject compliance as an explanatory variable; this same data set is the focus of this section. A population of subjects was assembled and two pre- liminary cholesterol measurements were obtained: one prior to a suggested low-cholesterol diet and one fol- lowing the diet period. The initial cholesterol level was taken as a weighted average of these two measures. The subjects were randomized into two groups: in the first group all subjects were prescribed cholestyramine (zl), while the subjects in the other group were prescribed a placebo (~0). During several years of treatment, each subject’s cholesterol level was measured multiple times, and the average of these measurements was used as the post-treatment cholesterol level. The compliance of each subject was determined by tracking the quan- tity of prescribed dosage consumed. 1274 Uncertainty Table 2: Observed data for the Lipid study and the Vitamin A study z cl Y 0 0 0 0 0 1 0 1 0 0 1 1 1 0 0 1 0 1 1 1 0 1 1 1 Lipid Study Vitamin A Study 0 bservations Observations 158 11514 14 74 0 0 0 0 52 2385 12 34 23 9663 78 12 I parameters) and the density of Figure 6b corresponds to skewed priors. Rather remarkably, even with only 337 cases in the data, both posterior distributions are highly concentrated within the large-sample bounds. Figure 7: Output histograms for the Vitamin A Sup- plementation data. (a) Using flat priors and (b) using skewed priors. (4 fb) Figure 6: Output histograms for the Lipid data. (a) Using flat priors and (b) using skewed priors. Real Data Example: Effect of Vitamin A Supplements on Child Mortality In this section, we consider an experiment described by Sommer et al. (1986) designed to determine the im- pact of vitamin A supplementation on childhood mor- tality. In the study, 450 villages in northern Sumatra were randomly assigned to participate in a vitamin A supplementation scheme or serve as a control group for one year. Children in the treatment group received two large doses of vitamin A (dl), while those in the con- trol group received no treatment (do). After the year had expired, the number of deaths yo were counted for both groups. The results of the study are shown in Table 2. Under the large-sample assumption, the method of Balke and Pearl (1994) yields the bounds: -0.01 5 ACE(D + Y) 5 0.19. Figure 7 shows posterior den- sities for ACE(D + Y) given the data. The density of Figure 7a corresponds to flat priors over the parame- ters and the density of Figure 7b corresponds to skewed priors over the parameters. It is interesting to note that for this study, the choice of the prior distribution has a significant effect on the posterior. This suggests that if the clinician is not very confident in the prior, a sensitivity analysis should be performed. A Counterfactual Query In addition to assessing the average treatment effect, the system is also capable (with only minor modifica- tion) of answering a variety of counterfactual queries concerning individuals with specific characteristics. In this section, we show the result of our system when modified to answer the following query: What is the probability that Joe would have had an improved cholesterol reading had he taken cholestyramine, given that (1) Joe was in the control group of the Lipid study, (2) Joe took the placebo as prescribed, and (3) Joe’s cholesterol level did not improve. We can answer the above query by running the Gibbs’ sampler on a model identical to that shown in Figure 3, except that the function ACE(D + Y) (Equation 4) is replaced by another function of VCR, one that represents our query. If Joe was in the control group and took the placebo, that means that he is either a complier or a never- taker. Furthermore, because Joe’s cholesterol level did not improve, Joe’s response behavior is either never- recover or helped. Consequently, Joe must be a mem- ber of one of the following four compliance-response populations: { crol, cro2, cr11, cr12). Joe would have improved had he taken cholestyramine if his response behavior is either helped (rl) or always-recover (r3). It follows that the query of interest is captured by the function Figure 8a and Figure 8b show the prior distribution over f(r&R) that follows from the flat prior and the skewed prior, respectively. Figure 8c and Figure 8d show the posterior distribution p(f(z&RID)) obtained by our system when run on the Lipid data, using the flat prior and the skewed prior, respectively. From the bounds of Balke and Pearl (1994), it follows that under the large-sample assumption, 0.51 5 f(YCRIZ)) 5 0.86. Bayesian Networks 1275 (4 1 0 1 fb) f c) (4 Figure 8: Prior (a, b) and posterior (c,d) distributions for a subpopulation f(vc~IZ)) specified by the counter- factual query “Would Joe have improved had he taken the drug, given that he did not improve without it”. (a) corresponds to the flat prior, (b) to the skewed prior. Thus, despite 39% non-compliance in the treatment group, and despite having just 337 subjects, the study strongly supports the conclusion that, given Joe’s spe- cific history, he would have been better off taking the drug. Moreover, the conclusion holds for both priors. Conclusion This paper identifies and demonstrates a new appli- cation area for network-based inference techniques - the management of causal analysis in clinical exper- imentation. These techniques, which were originally developed for medical diagnosis, are shown capable of circumventing one of the major problems in clinical experiments - the assessment of treatment efficacy in the face of imperfect compliance. While standard di- agnosis involves purely probabilistic inference in fully specified networks, causal analysis involves partially specified networks in which the links are given causal interpretation and where the domain of some variables are unknown. The system presented in this paper provides the clin- ical research community, we believe for the first time, an assumption-free3, unbiased assessment of the aver- age treatment effect. We offer this system as a practical tool to be used whenever full compliance cannot be en- forced and, more broadly, whenever the data available is insufficient for answering the queries of interest to the clinical investigator. Acknowledgements. The research of D. Chickering was supported by NSF grant #IRI-9119825 and a grant 3 “Assumption-transparent” may be a better term, since the two basic assumptions in our analysis (i.e., random- ized assignment and no-side-effects) are vividly displayed in the graph (e.g., Figure l), and the impact of the prior distribution is shown by histograms such as those of Figure 4. 1276 Uncertainty from Rockwell International. The research of J. Pearl was suppported by gifts from Microsoft Corporation and Hewlett-Packard Company. References Angrist, J.; Imbens, G.; and Rubin, D. 1993. Iden- tification of causal effects using instrumental vari- ables. Technical Report 136, Department of Eco- nomics, Harvard University, Cambridge, MA. Forth- coming, Journal of American Statistical Association, 1996. Balke, A., and Pearl, J. 1994. Counterfactual prob- abilities: Computational methods, bounds and appli- cations. In Proceedings of Tenth Conference on Un- certainty in Artificial Intelligence, Seattle, WA, 46- 54. Morgan Kaufman. Efron, B., and Feldman, D. 1991. Compliance as an explanatory variable in clinical trials. Journal of the American Statistical Association 86(413):9-26. Heckerman, D., and Shachter, R. 1995. Decision- theoretic foundations for causal reasoning. Journal of Artificial Intelligence Research 3:405-430. Holland, P. W. 1988. Causal inference, path analysis, and recursive structural equations models. In Clogg, C., ed., Sociological Methodology. Wachington, DC: American Socialogical Association. chapter 13, 449- 484. Imbens, G., and Rubin, D. 1994. Bayesian infer- ence for causal effects in randomized experiments with noncompliance. Technical report, Harvard University. Lipid Research Clinic Program. 1984. The lipid re- search clinics coronary primary prevention trial re- sults, parts I and II. Journal of the American Medical Association 251(3):351-374. January. Pearl, J. 1994. From Bayesian networks to causal network. In Proceedings of the UNICOM Seminar on Adaptive Computing and Information Processing, Brunel University, London, 165-194. Also in A. Gam- merman (Ed.), Bayesian Networks and Probabilistic Reasoning, Alfred Walter Ltd., London, l-31, 1995. Pearl, J. 1995a. Causal diagrams for experimental research. Biometrika 82(4):669-710. Pearl, J. 199513. Causal inference from indirect ex- periments. Artificial Intelligence in Medicine Journal 7(6):561-582. Rubin, D. 1978. Bayesian inference for causal effects: The role of randomization. Annals of Statistics 7:34- 58. Sommer, A.; Tarwotjo, I.; Djunaedi, E.; West, K. P.; Loeden, A. A.; Tilden, R.; and Mele, L. 1986. Impact of vitamin A supplementation on childhood mortality: A randomized controlled community trial. The Lancet i:1169-1173. | 1996 | 188 |
1,830 | Building Classifiers using ayesian Networks Nir Friedman Stanford University Dept. of Computer Science Gates Building 1A Stanford, CA 943059010 nir@cs.stanford.edu Abstract Recent work in supervised learning has shown that a surpris- ingly simple Bayesian classifier with strong assumptions of independence among features, called naive Bayes, is com- petitive with state of the art classifiers such as C4.5. This fact raises the question of whether a classifier with less re- strictive assumptions can perform even better. In this paper we examine and evaluate approaches for inducing classifiers from data, based on recent results in the theory of learn- ing Bayesian networks. Bayesian networks are factored representations of probability distributions that generalize the naive Bayes classifier and explicitly represent statements about independence. Among these approaches we single out a method we call Tree AugmentedNaive Bayes (TAN), which outperforms naive Bayes, yet at the same time maintains the computational simplicity (no search involved) and robustness which are characteristic of naive Bayes. We experimentally tested these approaches using benchmark problems from the U. C. Irvine repository, and compared them against C4.5, naive Bayes, and wrapper-based feature selection methods. 1 Introduction A somewhat simplified statement of the problem of super- vised learning is as follows. Given a training set of labeled instances of the form (al, . . . , a,), c, construct a classi- 7; f capable of predicting the value of c, given instances al,-*-, a,) as Input. The variables Ai, . . . , A, are called features or attributes, and the variable C is usually referred to as the class variable or label. This is a basic problem in many applications of machine learning, and there are nu- merous approaches to solve it based on various functional representations such as decision trees, decision lists, neural networks, decision-graphs, rules, and many others. One of the most effective classifiers, in the sense that its predic- tive performance is competitive with state of the art clas- sifiers such as C4.5 (Quinlan 1993), is the so-called naive Bayesian classifier (or simply naive Bayes) (Langley, Iba, & Thompson 1992). This classifier learns the conditional probability of each attribute Ai given the label C in the training data. Classification is then done by applying Bayes rule to compute the probability of C given the particular *Current address: SRI International, 333 Ravenswood Way, Menlo Park, CA 94025, moises@erg.sri.com Moises Goldszmidt* Rockwell Science Center 444 High St., Suite 400 Palo Alto, CA 94301 moises@rpal.rockwell.com instantiation of Al, . . . , A,. This computation is rendered feasible by making a strong independence assumption: all the attributes Ai are conditionally independent given the value of the label C.’ The performance of naive Bayes is somewhat surprising given that this is clearly an unrealistic assumption. Consider for example a classifier for assessing the risk in loan applications. It would be erroneous to ignore the correlations between age, education level, and income. This fact naturally begs the question of whether we can improve the performance of Bayesian classifiers by avoid- ing unrealistic assumptions about independence. In order to effectively tackle this problem we need an appropriate language and effective machinery to represent and manip- ulate independences. Bayesian networks (Pearl 1988) pro- vide both. Bayesian networks are directed acyclic graphs that allow for efficient and effective representation of the joint probability distributions over a set of random vari- ables. Each vertex in the graph represents a random vari- able, and edges represent direct correlations between the variables. More precisely, the network encodes the follow- ing statements about each random variable: each variable is independent of its non-descendants in the graph given the state of its parents. Other independences follow from these ones. These can be efficiently read from the net- work structure by means of a simple graph-theoretic crite- ria. Independences are then exploited to reduce the number of parameters needed to characterize a probability distribu- tion, and to efficiently compute posterior probabilities given evidence. Probabilistic parameters are encoded in a set of tables, one for each variable, in the form of local conditional distributions of a variable given its parents. Given the in- dependences encoded in the network, the joint distribution can be reconstructed by simply multiplying these tables. When represented as a Bayesian network, a naive Bayesian classifier has the simple structure depicted in Fig- ure 1. This network captures the main assumption behind the naive Bayes classifier, namely that every attribute (ev- ery leaf in the network) is independent from the rest of the attributes given the state of the class variable (the root in the network). Now that we have the means to represent and ‘By independence we mean probabilistic independence, that is A is independentof B given C wheneverPr(AIB, C) = Pr(AIC) for all possible values of A, B and C. Bayesian Networks 1277 From: AAAI-96 Proceedings. Copyright © 1996, AAAI (www.aaai.org). All rights reserved. CC-7 / d J”\, 0 0 00 0 Al A2 A, Figure 1: The structure of the naive Bayes network. manipulate independences, the obvious question follows: can we learn an unrestricted Bayesian network from the data that when used as a classifier maximizes the prediction rate? Learning Bayesian networks from data is a rapidly grow- ing field of research that has seen a great deal of activ- ity in recent years, see for example (Heckerman 1995; Heckerman, Geiger, & Chickering 1995; Lam & Bacchus 1994). This is a form unsupervised learning in the sense that the learner is not guided by a set of informative examples. The objective is to induce a network (or a set of networks) that “best describes” the probability distribution over the training data. This optimization process is implemented in practice using heuristic search techniques to find the best candidate over the space of possible networks. The search process relies on a scoring metric that asses the merits of each candidate network. We start by examining a straightforward application of current Bayesian networks techniques. We learn networks using the MDL metric (Lam & Bacchus 1994) and use them for classification. The results, which are analyzed in Sec- tion 3, are mixed: although the learned networks perform significantly better than naive Bayes on some datasets, they perform worse on others. We trace the reasons for these re- sults to the definition of the MDL scoring metric. Roughly speaking, the problem is that the MDL scoring metric mea- sures the “error” of the learned Bayesian network over all the variables in the domain. Minimizing this error, however, does not necessarily minimize the local “error” in predict- ing the class variable given the attributes. We argue that similar problems will occur with other scoring metrics in the literature. In light of these results we limit our attention to a class of network structures that are based on the structure of naive Bayes, requiring that the class variable be a parent of ev- ery attribute. This ensures that in the learned network, the probability Pr( Cl Al , . . . , An) will take every attribute into account. Unlike the naive Bayes structure, however, we allow additional edges between attributes. These additional edges capture correlations among the attributes. Note that the process of adding these edges may involve an heuris- tic search on a subnetwork. However, there is a restricted sub-class of these structures, for which we can find the best candidate in polynomial time. This result, shown by 1278 Uncertainty Geiger (1992), is a consequence of a well-known result by Chow and Liu (1968) (see also (Pearl 1988)). This ap- proach, which we call Tree Augmented Naive Bayes (TAN), approximates the interactions between attributes using a tree-structure imposed on the naive Bayes structure. We note that while this method has been proposed in the liter- ature, it has never been rigorously tested in practice. We show that TAN maintains the robustness and computational complexity (the search process is bounded) of naive Bayes, and at the same time displays better performance in our experiments. We compare this approach with C4.5, naive Bayes, and selective naive Bayes, a wrapper-based feature subset selection method combined with naive Bayes, on a set of benchmark problems from the U. C. Irvine repository (see Section 4). This experiments show that TAN leads to significant improvement over all of these three approaches. 2 Learning Bayesian Networks Consider a finite set U = {Xi, . . . , X,} of discrete random variables where each variable Xi may take on values from a finite domain. We use capital letters, such as X, Y, 2, for variable names and lowercase letters 2, y, z to denote specific values taken by those variables. The set of val- ues X can attain is denoted as KzZ(X), the cardinality of this set is denoted as 11 X 1 I = I kZ(X) I. Sets of variables are denoted by boldface capital letters X, U, Z, and assign- ments of values to the variables in these sets will be denoted by boldface lowercase letters x, y, z (we use VaZ(X) and 11x1 I in the obvious way). Finally, let P be a joint proba- bility distribution over the variables in U, and let X, ‘!I?, Z be subsets of U. X and U are conditionally independent given Z if for all x E l&Z(X), y E KzZ(‘lr’), 2 E KzZ(Z), P(x I z, y) = P(x I z) whenever P(y, z) > 0. A Bayesian network is an annotated directed acyclic graph that encodes a joint probability distribution of a do- main composed of a set of random variables. Formally, a Bayesian network for U is the pair B = (G, 0). G is a directed acyclic graph whose nodes correspond to the random variables X1, . . . , X,, and whose edges represent direct dependencies between the variables. The graph struc- ture G encodes the following set of independence assump- tions: each node Xi is independent of its non-descendants given its parents in G (Pearl 1988).2 The second compo- nent of the pair, namely 0, represents the set of param- eters that quantifies the network. It contains a parameter e z* I&* = P(QI&*) for each possible value zi of Xi, and III,, of IIx, (the set of parents of Xi in G). B defines a unique joint probability distribution over U given by: n n f33(Xlj..-,Xn)= h(Xi Irrx,) = ~x*p-Ix* (1) i=l i=l Example 2.1: Let U* = {Ai, . . . , An, C}, where the vari- ables Al,..., A, are the attributes and C is the class variable. In the naive Bayes structure the class variable *Formally there is a notion of minimality associated with this definition, but we will ignore it in this paper (see (Pearl 1988)). is the root, i.e., IIc = 8, and the only parent for each attribute is the class variable, i.e., IIA, = {C}, for all 1 5 i 2 n. Using (1) we have that Pr(Ai , . . . , An, C) = Pr(C) . ny’, Pr(AaIC). F rom the definition of conditional probability we get that Pr(CIAi , . . . , An) = a . Pr(C) . ny’, Pr(Ai IC), where Q is a normalization constant. This is the definition of naive Bayes commonly found in the literature (Langley, Iba, & Thompson 1992). The problem of learning a Bayesian network can be stated as follows. Given a trair&zg set D = (~1, . . . , UN} of in- stances of U, find a network B that best matches D. The common approach to this problem is to introduce a scor- ing function that evaluates each network with respect to the training data, and then to search for the best network. In gen- eral, this optimization problem is intractable, yet for certain restricted classes of networks there are efficient algorithms requiring polynomial time (in the number of variables in the network). We will indeed take advantage of these effi- cient algorithms in Section 4 where we propose a particular extension to naive Bayes. We start by examining the com- ponents of the scoring function that we will use throughout the paper. Let B = (G, 0) be a Bayesian network, and let D = {Ul,.., , us} (where each ui assigns values to all the vari- ables in U) be a training set. The log-likelihood of B given D is defined as N LL(BID) = >: log(l?+)). i=l This term measures the likelihood that the data D was gen- erated from the model B (namely the candidate Bayesian network) when we assume that the instances were indepen- dently sampled. The higher this value is, the closer B is to modeling the probability distribution in the data D. Let & (.) be the measure defined by frequencies of events in D. Using (1) we can decompose the the log-likelihood accord- ing to the structure of the network. After some algebraic manipulations we can easily derive: Now assume that the structure of the network is fixed. Stan- dard arguments show that LL( BI D) is maximized when 0% P-L* = b(G In[x,). Lemma 2.2: Let B = (G, 0) and B’ = (G, 0’) such that %,pI = &(xi(IIxi). Then LL(B’(D) >_ LL(BID). Thusrwe have a closed form solution for the parameters that maximize the log-likelihood for a given network structure. This is crucial since instead of searching in the space of Bayesian networks, we only need to search in the smaller space of network structures, and then fill in the parameters by computing the appropriate frequencies from the data. The log-likelihood score, while very simple, is not suit- able for learning the structure of the network, since it tends to favor complete graph structures (in which every vari- able is connected to every other variable). This is highly undesirable since such networks do not provide any useful representation of the independences in the learned distribu- tions. Moreover, the number of parameters in the complete model is exponential. Most of these parameters will have extremely high variance and will lead to poor predictions. This phenomena is called overjitting, since the learned pa- rameters match the training data, but have poor performance on test data. The two main scoring functions commonly used to learn Bayesian networks complement the log-likelihood score with additional terms to address this problem. These are the Bayesian scoring function (Heckerman, Geiger, & Chicker- ing 1995), and the one based on minimal description length (MDL) (Lam & Bacchus 1994). In this paper we con- centrate on MDL deferring the discussion on the Bayesian scoring function for the full paper.3 The motivation underlying the MDL method is to find a compact encoding of the training set D. We do not repro- duce the derivation of the the MDL scoring function here, but merely state it. The interested reader should consult (Friedman & Goldszmidt 1996; Lam & Bacchus 1994). The MDL score of a network B given D, written MDL( B I D) is MDL(B(D) = ; log NIBI - LL(BID) (4) where I B I is the number of parameters in the network. The first term simply counts how many bits we need to encode the specific network B, where we store l/2 . log N bits for each parameter in 0. The second term measures how many bits are needed for the encoded representation of D. Mini- mizing the MDL score involves tradeoffs between these two factors. Thus, the MDL score of a larger network might be worse (larger) than that of a smaller network, even though the former might match the data better. In practice, the MDL score regulates the number of parameters learned and helps avoid overfitting of the training data. Note that the first term does not depend on the actual parameters in B, but only on the graph structure. Thus, for a fixed the network structure, we minimize the MDL score by maximizing the LL score using Lemma 2.2. It is important to note that learning based on the MDL score is asymptotically correct: with probability 1 the learned distribution converges to the underlying distribu- tion as the number of samples increases (Hecketman 1995). Regarding the search process, in this paper we will rely on a greedy strategy for the obvious computational reasons. This procedure starts with the empty network and succes- sively applies local operations that maximally improve the score and until a local minima is found. The operations ap- plied by the search procedure are: arc addition, arc deletion and arc reversal. In the full paper we describe results using other search methods (although the methods we examined so far did not lead to significant improvements.) 3 ayesian Networks as Classifiers 3There are some well-known proposals (Heckerman 1995). connections between these two Bayesian Networks 1279 Error 19162213 9 4 61514 2111121018717208213 5 Figure 2: Error curves comparing unsupervised Bayesian networks (solid line) to naive Bayes (dashed line). The hor- izontal axis lists the datasets, which are sorted so that the curves cross only once. The vertical axis measures fraction of test instances that were misclassified (i.e., prediction er- rors), Thus, the smaller the value, the better the accuracy. Each data point is annotated by a 90% confidence interval. Using the methods just described we can induce a Bayesian network from the data and then use the resulting model as a classifier. The learned network represents an approximation to the probability distribution governing the domain (given the assumption that the instances are independently sampled form a single distribution). Given enough samples, this will be a close approximation Thus, we can use this network to compute the probability of C given the values of the attributes. The predicted class c, given a set of attributes al,...,% is simply the class that attains the maximum posterior PB(cjal, . . . , a,), where PB is the probability distribution represented by the Bayesian network B. It is important to note that this procedure is unsupervised in the sense that the learning procedure does not distinguish the class variable from other attributes. Thus, we do not inform the procedure that the evaluation of the learned net- work will be done by measuring the predictive accuracy with respect to the class variable. From the outset there is an obvious problem. Learning unrestricted Bayesian networks is an intractable problem. Even though in practice we resort to greedy heuristic search, this procedure is often expensive. In particular, it is more expensive than learning the naive Bayesian classifier which can be done in linear time. Still, we may be willing to invest the extra effort re- quired in learning a (unrestricted) Bayesian network if the prediction accuracy of the resulting classifier outperforms that of the naive Bayesian classifier. As our first experi- ment shows, Figure 2, this is not always the case. In this experiment we compared the predictive accuracy of classifi- cation using Bayesian networks learned in an unsupervised fashion, versus that of the naive Bayesian classifier. We run this experiment on 22 datasets, 20 of which are from the U. C. Irvine repository (Murphy & Aha 1995). Appendix A describes in detail the experimental setup, evaluation meth- ods, and results (Table 1). As can be seen from Figure 2 the classifier based on unsu- pervised networks performed significantly better than naive Bayes on 6 datasets, and performed significantly worse on 6 datasets. A quick examination of the datasets revels that all the datasets where unsupervised networks performed poorly contain more than 15 attributes. It is interesting to examine the two datasets where the un- supervised networks performed worst (compared to naive Bayes): “soybean-large” (with 35 attributes) and “satim- age” (with 36 attributes). For both these datasets, the size of the class’ Markov blanket in the networks is rather small- less than 5 attributes. The relevance of this is that pre- diction using a Bayesian network examines only the values of attributes in the class variable’s Markov blanket4 The fact that for both these datasets the Markov blanket of the class variable is so small indicates that for the MDL metric, the “cost” (in terms of additional parameters) of enlarging the Markov blanket is not worth the tradeoff in overall ac- curacy. Moreover, we note that in all of these experiments the networks found by the unsupervised learning routine had better MDL score than the naive Bayes network. This suggest that the root of the problem is the scoring metric- a network with a better score is not necessarily a better classifier. To understand this problem in detail we re-examine the MDL score. Recall that the likelihood term in (4) is the one that measures the quality of the learned model. Also recall thatD= {ur,..., UN) is the training set. In a classification task each ui is a tuple of the form (a:, . . . , ah, c”) that assigns values to the attributes Al, . . . , A, and to the class variable C. Using the chain rule we can rewrite the log- likelihood function (2) as: U(BID) = glogPB(c”lal,. . .,a;) + i= 1 N ClogP&zf,...,ak) i=l (5) The first term in this equation measures how well B esti- mates the probability of the class given the attributes. The second term measures how well B estimates the joint dis- tribution of the attributes. Since the classification is deter- mined by PB(CIAI, . . . , A,) only the first term is related to the score of the network as a classifier (i.e., its prediction accuracy). Unfortunately, this term is dominated by the sec- ond term when there are many attributes. As n grows larger, the probability of each particular assignment to Al, . . . , A, becomes smaller, since the number of possible assignments grows exponentially in 72. Thus, we expect the terms of the form l+(Al, . . . , An) to yield smaller values which in turn will increase the value of the log function. However, 4More precisely, for a fixed network structure the Markov blan- ket of a variable X (Pearl 1988) consists of X’s parents, X’s chil- dren, and parents of X’s children in G. This set has the property that conditioned on X’s Markov blanket, X is independent of all other variables in the network. 1280 Uncertainty Glucose Figure 3: A TAN model learned for the dataset “pima”. at the same time, the conditional probability of the class will remain more of less the same. This implies that a rela- tively big error in the first term will not reflect in the MDL score. Thus, as indicated by our experimental results, using a non-specialized scoring metric for learning an unrestricted Bayesian network may result in a poor classifier when there are many attributes.5 A straightforward approach to dealing with this problem would be to specialize the scoring metric (MDL in this case) for the classification task. We can easily do so by restricting the log-likelihood to the first term of (5). Formally, let the conditional log-likelihood of a Bayesian network B given dataset D be CLL(BID) = CL, logPg(CiIAt,. . . , Ah). A similar modification to Bayesian scoring metric in (Heck- erman, Geiger, & Chickering 1995) is equally easy to define. The problem with applying these conditional scoring met- rics in practice is that they do not decompose over the struc- ture ofthe network, i.e., we do not have an analogue of (3). As a consequence it is no longer true that setting the param- eters 8,i~~,i = &(G I&,> maximizes the score for a fixed network str&ture.6 We do not know, at this stage, whether there is a computational effective procedure to find the pa- rameter values that maximize this type of conditional score. In fact, as reported by Geiger (1992), previous attempts to defining such conditional scores resulted in unrealistic and sometimes contradictory assumptions. 4 Learning Restricted Networks for Classifiers In light of this discussion we examine a different ap- proach. We limit our attention to a class of network struc- tures that are based on the naive Bayes structure. As in naive Bayes, we require that the class variable be a parent of every attribute. This ensures that, in the learned network, the probability P(CIAI, . . . , An) will take every attribute into account, rather than a shallow set of neighboring nodes. 51n the full paper we show that the same problem occur in the Bayesian scoring metric. 6We remark t h a t decomposition still holds for a restricted class of structures-essentially these where C does not have any chil- dren. However, these structures are usually not useful for classifi- cation. In order to improve the performance of a classifier based on naive Bayes we propose to augment the naive Bayes struc- ture with “correlation” edges among the attributes. We call these structures augmented naive Bayes. In an the augmented structure an edge from Ai to Aj im- plies that the two attributes are no longer independent given the class variable. In this case the influence of Ai on the as- sessment of class variable also depends on the value of Aj . Thus, in Figure 3, the influence of the attribute “Glucose” on the class C depends on the value of “Insulin”, while in naive Bayes the influence of each on the class variable is independent of other’s. Thus, a value of “Glucose” that is surprising (i.e., P(gj ) 1 ) c is ow , may be unsurprising if the value of its correlated attribute, “Insulin,” is also unlikely (i.e., P(gIc, i) is high). In this situation, the naive Bayesian classifier will over-penalize the probability of the class vari- able (by considering two unlikely observations), while the network in Figure 3 will not. Adding the best augment naive Bayes structure is an in- tractable problem. To see this, note this essentially involves learning a network structure over the attribute set. How- ever, by imposing restrictions on the form of the allowed interactions, we can learn the correlations quite efficiently. A tree-augmented naive Bayes (TAN) model, is a Bayesian network where IIc = 0, and IIA, contains C and at most one other attribute. Thus, each attribute can have one correlation edge pointing to it. As we now show, we can exploit this restriction on the number of correla- tion edges to learn TAN models efficiently. This class of models was previously proposed by Geiger (1992), using a well-known method by Chow and Liu (1968), for learn- ing tree-like Bayesian networks (see also (Pearl 1988, pp. 387-390)).7 We start by reviewing Chow and Liu’s result on learning trees. A directed acyclic graph is a tree if IIxl contains exactly one parent for all Xi, except for one variable that has no parents (this variable is referred to as the root). Chow and Liu show that there is a simple procedure that constructs the maximal log-probability tree. Let n be the number of random variables and N be the number of training instances. Then Theorem 4.1: (Chow & Lui 1968) There is a procedure of time complexity O( n2. N), that constructs the tree structure BT that maximizes LL( BT 1 D). The procedure of Chow and Liu can be summarized as follows. Compute the mutual information I(Xi ; Xj ) = of variables, i # j. Build a complete undirected graph in which the vertices are the variables in U. Annotate the weight of an edge connecting Xi to Xj by I( Xi ; Xi ) . Build a maximum weighted spanning tree of this graph (Cormen, Leiserson, & Rivest 1990). 7These structures are called “Bayesian conditional trees” by Geiger. Bayesian Networks 1281 Error 0.7' I.....,..,.X.,,..,..I.. 1 9 4 6 211152219 71810 8 211714 31316 51220 Figure 4: Error curves comparing smoothed TAN (solid line) to naive Bayes (dashed line). Error 0.35; T 3215 1 7 6 8 20191815 917 210221611141312 4 Figure 5: Error curves comparing smoothed TAN(solid line) to C4.5 (dashed line). 4. Transform the resulting undirected tree to a directed one by choosing a root variable and setting the direction of all edges to be outward from it. (The choice of root variable does not change the log-likelihood of the network.) The first step has complexity of O(n2 . N) and the third step has complexity of O(n2 logn). Since we usually have that N > log n, we get the resulting complexity. This result can be adapted to learn the maximum likeli- hood TAN structure, Theorem 4.2: (Geiger 1992) There is a procedure of time complexity O(n2 . N) that constructs the TAN structure BT that maximize LL( BT ID). The procedure is very similar to the procedure de- scribed above when applied to the attributes Al, . . . , A,. The only change is in the choice of weights. In- stead of taking I( Ai ; Aj ), we take the condi- tional mutual information given C, I(Ai; Aj IC) = &(wJjlC) Cai,uj,~ +oCaay ajj cl log pD(a.lc)pD(a,~c)* Roughly I speaking, this measures the gain in log-likelihood of adding Ai as a parent of Aj when C is already a parent. There is one more consideration. To learn the parameters in the network we estimate conditional frequencies of the form &(X ]IIx). This is done by partitioning the train- ing data according to the possible values of IIx and then computing the frequency of X in each partition. A prob- lem surfaces when some of these partitions contain very few instances. In these small partitions our estimate of the conditional probability is unreliable. This problem is not serious for the naive Bayes classifier since it partitions the data according to the class variable, and usually all values of the class variables are adequately represented in the training data. In TAN models however, for each attribute we asses the conditional probability given the class variable and an- other attribute. This means that the number of partitions is at least twice as large. Thus, it is not surprising to encounter unreliable estimates, especially in small datasets. To deal with this problem we introduce a smoothing op- eration on the parameters learned in TAN models. This operation takes every estimated parameter B,ln, and bi- ases it toward the observed marginal frequency of X, & (2). Formally, we let the new parameter 6” (z III,) = a * hl(~,rr,) + (1 - a!)P~(;c), taking d! = N:PD(nz) N.h(&)+s ’ where s is the smoothing parameter, which is usually quite small (in all the experiments we choose s = 5).8 It is easy to see that this operation biases the learned parameters in a manner that depends on our confidence level as expressed by s: the more instances we have in the partition from which we compute the parameter, the less bias is applied. If the number of instances that with a particular parents’ value as- signment is significant, than the bias essentially disappears. On the other hand, if the number of instances with the a particular parents’ value assignment is small, then the bias dominates. We note that this operation is performed after the structure of the TAN model are determined. Thus, the smoothed model has exactly the same qualitative structure as the original model, but with different numerical param- eters. In our experiments comparing the prediction error of smoothed TAN to that of unsmoothed TAN, we observed that smoothed TAN performs at least as well as TAN and oc- casionally significantly outperforms TAN (see for example the results for “soybean-large”, “segment”, and “lymphog- raphy” in Table 1). From now on we will assume that the version of TAN uses the smoothing operator unless noted otherwise. Figure 4 compares the prediction error of the TAN classi- fier to that of naive Bayes. As can be seen, the performance of the TAN classifier dominates that of naive Bayes.9 This result supports our hypothesis that by relaxing the strong independence assumptions made by naive Bayes we can indeed learn better classifiers. *In statistical terms, we are essentially applying a Dirichlet prior on Bx inx with mean expected value i)D (X) and equivalent sample size s. We note that this use of Dirichlet priors is related to the class of Dirichlet priors described in (Heckerman, Geiger, & Chickering 1995). ‘In our experiments we also tried smoothed version of naive Bayes. This did not lead to significant improvement over the unsmoothed naive Bayes. 1282 Uncertainty Finally, we also compared TAN to C4.5 (Quinlan 1993), a state of the art decision-tree learning system, and to the selective naive Bayesian classifier (Langley & Sage 1994; John, Kohavi, & Pfleger 1995). The later approach searches for the subset of attributes over which naive Bayes has the best performance. The results displayed in Figure 5 and Table 1, show that TAN is quite competitive with both approaches and can lead to big improvements in many cases. 5 Concluding Remarks This paper makes two important contributions. The first one is the analysis of unsupervised learning of Bayesian networks for classification tasks. We show that the scoring metrics used in learning unsupervised Bayesian networks do not necessarily optimize the performance of the learned networks in classification. Our analysis suggests a possible class of scoring metrics that are suited for this task. These metrics appear to be computationally intractable. We plan to explore effective approaches to learning with approxi- mations of these scores. The second contribution is the experimental validation of tree augmented naive Bayesian classifiers, TAN. This approach was introduced by Geiger (1992), yet was not extensively tested and as a consequence has received little recognition in the machine learning com- munity. This classification method has attractive computa- tional properties, while at the same time, as our experimental results show, it performs competitively with reported state of the art classifiers. In spite of these advantages, it is clear that in some sit- uations, it would be useful to model correlations among attributes that cannot be captured by a tree structure. This will be significant when there is a sufficient number of train- ing instances to robustly estimate higher-order conditional probabilities. Thus, it is interesting to examine the problem of learning (unrestricted) augmented naive Bayes networks. In an initial experiment we attempted to learn such networks using the MDL score, where we restricted the search pro- cedure to examine only networks that contained the naive Bayes backbone. The results were somewhat disappoint- ing, since the MDL score was reluctant to add more than a few correlation arcs to the naive Bayes backbone. This is, again, a consequence of the fact that the scoring metric is not geared for classification. An alternative approach might use a cross-validation scheme to evaluate each candidate while searching for the best correlation edges. Such a procedure, however, is computationally expensive. We are certainly not the first to try and improve naive Bayes by adding correlations among attributes. For ex- ample, Pazzani (1995) suggests a procedure that replaces, in a greedy manner, pairs of attributes Ai, Aj, with a new attribute that represents the cross-product of Ai and Aj. This processes ensures that paired attributes influence the class variable in a correlated manner. It is easy to see that the resulting classifier is equivalent to an augmented naive Bayes network where the attributes in each “cluster” are fully interconnected. Note that including many attributes in a single cluster may result in overfitting problems. On the other hand, attributes in different clusters remain (condition- ally) independent of each other. This shortcoming does not occur in TAN classifiers. Another example is the work by Provan and Singh (1995), in which a wrapper-based feature subset selection is applied to an unsupervised Bayesian net- work learning routine. This procedure is computationally intensive (it involves repeated calls to a Bayesian network learning procedure) and the reported results indicate only a slight improvement over the selective naive Bayesian clas- sifier. The attractiveness of the tree-augmented naive Bayesian classifier is that it embodies a good tradeoff between the quality of the approximation of correlations among at- tributes, and the computational complexity in the learning stage. Moreover, the learning procedure is guaranteed to find the optimal TAN structure. As our experimental results show this procedure performs well in practice. Therefore we propose TAN as a worthwhile tool for the machine learn- ing community. Acknowledgements The authors are grateful to Denise Draper, Ken Fertig, Dan Geiger, Joe Halpern, Ronny Kohavi, Pat Langley and Judea Pearl for comments on a previous draft of this paper and useful discussions relating to this work. We thank Ronny Kohavi for technical help with the MLC++ library. Parts of this work were done while the first author was at Rockwell Science Center. The first author was also supported in part by an IBM Graduate fellowship and NSF Grant IRI-95- 03109. erirnental Methodology an esults We run our experiments on the 22 datasets listed in Table 1. All of the datasets are from the U. C. Irvine repository (Murphy & Aha 1995), with the exception of “mofn-3-7- 10” and “corral”. These two artificial datasets were used for the evaluation of feature subset selection methods by (John, Kohavi, & Pfleger 1995). All these datasets are accessible at the MLC++ ftp site. The accuracy of each classifier is based on the percentage of successful predictions on the test sets of each dataset. We estimate the prediction accuracy for each classifier as well as the variance of this accuracy using the MLC++ system (Kohavi et al. 1994). Accuracy was evaluated using the holdout method for the larger datasets, and using 5-fold truss validation (using the methods described in (Kohavi 1995)) for the smaller ones. Since we do not deal, at the current time, with missing data we had removed instances with missing values from the datasets. Currently we also do not handle continuous attributes. Instead, in each invoca- tion of the learning routine, the dataset was pre-discretized using a variant of the method of (Fayyad & Irani 1993) using only the training data, in the manner described in (Dougherty, Kohavi, & Sahami 1995). These preprocess- ing stages where carried out by the MLC++ system. We note that experiments with the various learning procedures were carried out on exactly the same training sets and evalu- ated on the same test sets. In particular, the cross-validation folds where the same for all the experiments on each dataset. Bayesian Networks 1283 Table 1: Experimental results 1 2 6 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 Dataset australian breast chess cleve corral crx diabetes flare german heart hepatitis letter lymphography mofn-3-7-10 pima satimage segment shuttle-smaI1 soybean-large vehicle vote waveform-2 1 # Attributes 14 10 36 13 6 15 8 IO 20 13 19 16 18 10 8 36 19 9 35 18 16 21 # Instances Train Test1 690 683 C T -5 c -5 2130 1066 296 cv-5 128 cv-5 653 cv-5 768 cv-5 1066 cv-5 1000 cv-5 270 cv-5 80 cv-5 15000 5000 I48 cv-5 300 1024 768 cv-5 4435 2000 1540 770 3866 1934 562 cv-5 846 cv-5 435 cv-5 300 4700 Accuracy NBC &sup TAN TAN* c4.5 SNBC 86.23+- 1.10 86.23+-l .76 8 1.30+- 1.06 84.20+- 1.24 85.65~1.82 86.67+-1.81 97.36+-0.50 96.92+-0.63 95.75+-1.25 96.92t0.67 94.73+-0.59 96.19+-0.63 87.15+-1.03 95.5940.63 92.40+-0.81 92.3 I+-0.82 99.53+-0.21 94.28t0.71 82.76+-1.27 81.39c1.82 79.06+-0.65 81.76t0.33 73.31+-0.63 78.06t2.41 85.88+-3.25 97.6Oh2.40 95.32+-2.26 96.06t2.5 1 97.69+-2.31 83.57+-3.15 86.22+-1.14 85.60+-0.17 83.77+-l .34 85.76+-1.16 86.22+-0.58 85.92t 1.08 74.48t0.89 75.39-t-0.29 75.13+-0.98 75.52+-1.11 76.04+-0.85 76.04+-0.83 79.46+-1.11 82.74+-l .90 82.74+-l .60 82.27+-l .86 82.55+- 1.75 83.40+-1.67 74.70+-1.33 72.30-k- 1.57 72.20+-l .54 73.10t1.54 72.20-t-1.23 73.7ot2.02 81.4843.26 82.22+-2.46 82.96t2.5 1 83.33+-2.48 81.1 l-t-3.77 81.85t2.83 91.25+-1.53 91.25+-4.68 85.00+-2.50 91.25+-2.50 86.25h4.15 90.0044.24 74.96+-0.61 75.02+-0.61 83.44+-0.53 85.86+-0.49 77.70-l-0.59 75.36t0.61 79.72+- 1.10 75.03+- 1.58 66.87+-3.37 85.03+-3.09 77.03t1.21 77.72t2.46 86.43+-l .07 85.94-k-l .09 91.70+-0.86 91.11-t-0.89 85.55t1.10 87.5Ot 1.03 75.51+-1.63 75.00+-l .22 75.13+-1.36 75.52+-1.27 75.13-i--1.52 74.86+-2.61 81.75+-0.86 59.2oc1.10 77.55to.93 87.20+-0.75 83.15+-0.84 82.05t0.86 91.17+-1.02 93.51-t-0.89 85.32+-l .28 95.58+-0.74 93.64+0.88 93.25to.90 98.34+-0.29 99.17+-0.21 98.8640.24 99.53+-0.15 99.17+-0.21 99.28t0.19 91.29-t-0.98 58.54+-4.84 58.17t1.43 92.17+-1.02 92.ooc1.11 92.89+-1.01 58.28+-l .79 61 .OO+-2.02 67.86t2.92 69.63+-2.11 69.74+-1.52 61.36+-2.33 90.34+-0.86 94.94+-0.46 89.20t1.61 93.56+-0.28 95.63+-0.43 94.71+-0.59 77.89+-0.61 69.45+-0.67 75.38+-0.63 78.38+-0.60 74.70~0.63 76.53~0.62 Finally, in Table 1 we summarize the accuracies of the six learning procedures we discussed in this paper: NBC-the naive Bayesian classifier; Unsup-unsupervised Bayesian networks learned using the MDL score; TAN-TAN net- works learned according to Theorem 4.2; TAN”-smoothed TAN networks; C4.5-the decision-tree classifier of (Quin- lan 1993); SNBC-the selective naive Bayesian classifier, a wrapper-based feature selection applied to naive Bayes, us- ing the implementation of (John, Kohavi, & Pfleger 1995). References Chow, C. K., and Lui, C. N. 1968. Approximating dis- crete probability distributions with dependence trees. IEEE Trans. on Info, Theory 141462-467. Cormen, T. H.; Leiserson, C. E.; and Rivest, R. L. 1990. Introduction to Algorithms. MIT Press. Dougherty, J.; Kohavi, R.; and Sahami, M. 1995. Super- vised and unsupervised discretization of continuous fea- tures. In ML ‘95. Fayyad, U. M., and Irani, K. B. 1993. Multi-interval discretization of continuous-valued attributes for classifi- cation learning. In IJCAZ ‘93, 1022-1027. Friedman, N., and Goldszmidt, M. 1996. Discretization of continuous attributes while learning Bayesian networks. In ML ‘96. Geiger, D. 1992. An entropy-based learning algorithm of Bayesian conditional trees. In UAZ ‘92. 92-97. Heckerman, D.; Geiger, D.; and Chickering, D. M. 1995. Learning Bayesian networks: The combination of knowl- ege and statistical data. Machine Learning 20: 197-243. Heckerman, D. 1995. A tutorial on learning Bayesian networks. Technical Report MSR-TR-95-06, Microsoft Research. 1284 Uncertainty John, G.; Kohavi, R.; and Pfleger, K. 1995. Irrelevant features and the subset selection problem. In ML ‘94. 121-129. Kohavi, R.; John, G.; Long, R.; Manley, D.; and Pfleger, K. 1994. MLC++: A machine learning library in C++. In Tools with Artifcial Intelligence. 740-743. Kohavi, R. 1995. A study of cross-validation and bootstrap for accuracy estimation and model selection. In IJCAI ‘95. 1137-l 143. Lam, W., and Bacchus, F. 1994. Learning Bayesian be- lief networks. An approach based on the MDL principle. Computational Intelligence 101269-293. Langley, P., and Sage, S. 1994. Induction of selective Bayesian classifiers. In UAI ‘94. 399-406. Langley, P.; Iba, W.; and Thompson, K. 1992. An analysis of bayesian classifiers. In AAAZ ‘90. 223-228. Murphy, P. M., and Aha, D. W. 1995. UC1 repository of machine learning databases. http : / /www . its . uci . edu/-mlearn/MLRepository.html. Pazzani, M. J. 1995. Searching for dependencies in Bayesian classifiers. In Proc. of the 5’th Int. Workshop on Artificial Intelligence and Statistics. Pearl, J. 1988. Probabilistic Reasoning in Intelligent Systems. Morgan Kaufmann. Quinlan, J. R. 1993. C4.5: Programs for Machine Learn- ing. Morgan Kaufmann. Singh, M., and Provan, G. M. 1995. A comparison of induction algorithms for selective and non-selective bayesian classifiers. In ML ‘9.5. | 1996 | 189 |
1,831 | 1 Advantages of a Leveled Commitment Contracting Tuomas VV. Sandholm and Victor {sandholm, lesser}@cs.umass.edu University of Massachusetts at Amherst Department of Computer Science Amherst, MA 01003 Abstract In automated negotiation systems consisting of self-interested agents, contracts have tradition- ally been binding. Such contracts do not al- low agents to efficiently accommodate future events. Game theory has proposed contingency contracts to solve this problem. Among com- putational agents, contingency contracts are of- ten impractical due to large numbers of inter- dependent and unanticipated future events to be conditioned on, and because some events are not mutually observable. This paper proposes a leveled commitment contracting protocol that allows self-interested agents to efficiently ac- commodate future events by having the possi- bility of unilaterally decommitting from a con- tract based on local reasoning. A decommit- ment penalty is assigned to both agents in a contract: to be freed from the contract, an agent only pays this penalty to the other party. It is shown through formal analysis of several contracting settings that this leveled commit- ment feature in a contracting protocol increases Pareto efficiency of deals and can make con- tracts individually rational when no full com- mitment contract can. This advantage holds even if the agents decommit manipulatively. Introduction The importance of automated negotiation systems is likely to increase as a result of three developments. One is the growth of a standardized communication infrastructure-EDI, NII, KQML, Telescript etc-over which separately designed agents belonging to different organizations can interact in an open environment and safely carry out transactions [8; 161. The second is the advent of small transaction commerce on the Internet for purchasing goods, information, and communication bandwidth. The third is an industrial trend toward vir- tual enterprises: dynamic alliances of small enterprises which together can take advantage of economies of scale. In such multiagent systems consisting of self-interested agents, contracts have traditionally been binding [12; 13; 4; 61. Once an agent agrees to a contract, it has to follow through with it no matter how future events unravel. Although a contract may be profitable to an agent when viewed ez ante, it need not be profitable when viewed after some future events have occurred, i.e. ez post. Similarly, a contract may have too low expected *Supported by NSF under Grant No. IRI-9523419. payoff ex ante, but in some realizations of the future events, the same contract may be desirable when viewed ex post. Normal full commitment contracts are unable to efficiently take advantage of the possibilities that such- probabilistically known-future events provide. On the other hand, many multiagent systems consist- ing of cooperative agents incorporate some form of de- commitment possibility in order to allow the agents to accommodate new events. For example, in the origi- nal Contract Net Protocol [19], the agent that had con- tracted out a task could send a termination message to cancel the contract even when the contractee had already partially fulfilled the contract. This was possible be- cause the agents were not self-interested: the contractee did not mind losing part of its effort without a mone- tary compensation. Similarly, the role of decommitment among cooperative agents has been studied in meeting scheduling [ 181 and in cooperative coordination [2]. Game theory has suggested utilizing the potential provided by probabilistically known future events via contingency contracts among self-interested agents [II]. The contract obligations are made contingent on future events. There are games in which this method increases the expected payoff to both parties of the contract com- pared to any full commitment contract. Also, some deals are enabled by contingency contracts in the sense that there is no full commitment contract that both agents prefer over their fall-back positions, but there is a contin- gency contract that each agent prefers over its fall-back. There are at least three problems regarding the use of contingency contracts in automated negotiation among self-interested agents. First, it is often impossible to enu- merate all possible relevant future events in advance. Second, contingency contracts get cumbersome as the number of relevant events to monitor increases. In the limit, all domain events (e.g. new tasks arriving or re- sources breaking down) and all negotiation events (other contracts) can affect the value of the obligations of the original contract, and should therefore be conditioned on. Furthermore, these future events may not only af- fect the value of the original contract independently: the value may depend on combinations of the future events 117; 13; 121. The third problem is that of verifying the unraveling of the events. Sometimes an event is only observable by one of the agents. This agent may have an incentive to lie to the other party of the contract about the event in case the event is associated with an unad- vantageous contingency to the directly observing agent. Thus, to be viable, contingency contracts would require an event verification mechanism that is not manipulable and not prohibitively complicated. 126 Agents From: AAAI-96 Proceedings. Copyright © 1996, AAAI (www.aaai.org). All rights reserved. We propose another method for taking advantage of the possibilities provided by probabilistically known fu- ture events. Instead of conditioning the contract on future events, a mechanism is built into the contract that allows unilateral decommitting at any point in time. This is achieved by specifying in the contract decommit- ment penalties, one for each agent. If an agent wants to decommit-i.e. to be freed from the obligations of the contract-it can do so simply by paying the decom- mitment penalty to the other party. We will call such contracts leveled commitment contracts because the de- commitment penalties can be used to choose a level of commitment. The method requires no explicit condi- tioning on future events: each agent can do its own con- ditioning dynamically. Therefore no event verification mechanism is required either. This paper presents for- mal justifications for adding this decommitment feature into a contracting protocol. Principles for assessing decommitment penalties have been studied in law [l; lo], but the purpose has been to assess a penalty on the agent that has breached the contract after the breach has occurred. Similarly, penalty clauses for partial failure-such as not meeting a deadline-are commonly used in contracts, but the purpose is usually to motivate the agents to follow the contract. To our knowledge, the possibility of explicitly allowing decommitment from the whole contract for a predetermined price has not been studied as an active method for utilizing the potential provided by an uncer- tain future.’ Somewhat unintuitively, it turns out that the mere existence of a decommitment possibility in a contract can increase each agent’s expected payoff. Key microeconomic concepts are now introduced. So- cial welfare is the sum of the payoffs of the agents un- der consideration [7; 51. It does not address distribu- tion. Pareto eficiency measures both societal good and distribution [7; 51. A vector of payoffs to the agents Pareto dominates another vector if each agent’s payoff in the first vector is no less than in the second, and there exists an agent whose payoff in the first vector is greater than in the second. Social welfare and Pareto efficiency can be measured either ex ante as expected values or ex post as realizations. Strategies (mappings from observed history of the game to actions) S, of the contractor and Sb of the contractee are in Nash equi- librium if Sa is a best-expected payoff maximizing- response to Sb, and Sb is a best response to Sa [9; 7; 51. Finally, a strategy is a dominant strategy if it is a best response to any strategy of the other agent [7]. We analyze contracting situations from the perspec- tive of two agents: the contractor who pays to get a task done, and the contractee who gets paid for han- dling the task. Handling a task can mean taking on any types of constraints. The method is not specific to clas- sical task allocation. The contractor tries to minimize the contract price p that it has to pay. The contractee tries to maximize the payoff p that it receives from the contractor. Outside offers from third parties will be ex- plicitly discussed. The contracting setting consists of two games. First, the contracting game involves the agents choosing a contract-or no contract, i.e. the null deal-before any future events have unraveled. Secondly, the decommitting gam .e involves the agents deciding in whether to decommit or to carry out the obligations of the contract-after the future events have unraveled. The decommitment game is a subgame of the contracting game: affect the the - expected outcomes of the decommitting game agents’ preferences over contracts in the con- tracting game. The decommitting game will be analyzed using the Nash equilibrium and the dominant strategy concepts. The contracting game will be analyzed with respect to individual rationality (IR): is the contract bet- ter for an agent than the null deal? Contractor’s decommitment penalty. Contractee’s decommitment penalty. Price of contractor’s best (full commitment) outside offer. Ex ante probability density function of a.. Ex ante probability density function of 6. Probability that the contractee decommits. Table 1: symbols used in the paper. We TestTict our analysis to contracts where a 2 0 and b 2 0, i.e. we rule out contracts that specify that the decommitting agent receives a payment from the victim of the decommitment. In our contracting settings, the future of both agents involves uncertainty. Specifically, the agents might re- ceive outside offers. The contractor’s best outside offer 6 is only probabilistically known ex ante by both agents, and is characterized by a probability density function f(Z). If the contractor does not receive an outside offer, 6 corresponds to its best outstanding outside offer or its fall-back payoff, i.e. payoff that it receives if no contract is made. The contractee’s best outside offer g is also only probabilistically known ex ante, and is characterized by a probability density function g(vb).2 If the contractee does not receive an outside offer, -“b corresponds to its best outstanding outside offer or its fall-back payoff. The variables ?i and g are assumed statistically independent. The contractor’s options are either to make a contract with the contractee or to wait for ti Similarly, the con- tractee’s options are either to make a contract with the contractor or to wait for vb. The two agents have many mutual contracts to choose from. A leveled commitment contract is specified by the contract price p, the con- tractor’s decommitment penalty a, and the contractee’s decommitment penalty b. The agents also have the pos- sibility to make a full commitment contract. The con- tractor has to decide on decommitting when it knows its outside offer 6 but does not know the contractee’s out- side offer I;. Similarly, the contractee has to decide on decommitting when it knows its outside offer “b but does not know the contractor’s. This seems realistic from a ‘Decommit i g t n has been studied in other settings, e.g. where there is a constant inflow of agents, and they have a time cost for searching partners of two types: good or bad [3]. 2Games where at least one agent’s future is certain, are a subset of these games. In such games all of the probability mass of f(5) and/or g(g) is on one point. Negotiation & Coalition 127 practical automated contracting perspective. We do not assume that the agents decommit truth- fully. An agent may not decommit although its out- side offer accompanied by a penalty is better than the contract because the agent believes that there is a high probability that the opponent will decommit. This would save the agent its decommitment penalty and give the agent a decommitment penalty from the opponent. Games of this type differ significantly based on whether the agents have to decommit sequentially or simultane- ously. The next two sections analyze these cases. Finally, Section 4 gives some practical prescriptions for building automated negotiation systems, and Section 5 concludes. 2 Sequential decommitting (SEQD) In our sequential decommitting (SEQD) game, one agent has to declare decommitment before the other. We study the case where the contractee has to decommit first. The case where the contractor has to go first is analogous. Figure 1 presents the game tree. There are two alter- native types of leveled commitment contracts that differ on what happens if both agents decommit. In the first, both agents have to pay the decommitment penalties to each other. In the second, neither agent has to pay. -------------__--------------------------------------------------------------------------------. contracting __---------__________________________L__-----------------------------------------,, game j Decommitting cmllrnder : pame drnlnrn4b / -&+b,i .IHLI jj (or .%Y, ) ;I !! game Wee of the fgure represents two alternative pTotocol8, i.e. two d@eTent games. In the first, both agents have to pay the decommit- ment penalties to each other if both decommit. In the second, neither agent has to pay if both decommit. The payoffs of the latter pToto- co1 are in parentheses when they difler fTom the fosmer. The dotted lines TepTesent information sets: the contractor does not know the contTactee’s outside offer and vice versa. The contractor’s payo& are usually negative because it has to pay for having the task handled. We now analyze the decommitting game using domi- nance in subgames as the solution concept. Reasoning about the agents’ actions starts at the leaves of the tree and proceeds backwards to the beginning of the game. In the subgame where the contractee has decommitted, the contractor’s best move is not to decommit because -Z-a+ b 5 -ii+b (because a >_ 0). This also holds for a contract where neither agent has to pay a decommitment penalty if both decommit-because -6 5 -ii + b. In the subgame where the contractee has not decommitted, the contractor’s best move is to decommit if -8 - a > -p. This happens with probability JTia f(ii)dii. Put to- gether, the contractee gets “b - b if it decommits, “b + a if it does not but the contractor does, and p if neither decommits. Thus the contractee decommits if P--o i-b> s f (a)di@ + a] + f (~W[pl -00 s- p--o If &Ya f(Z)& = 0, this is equivalent to -b > a which is false because a and b are nonnegative. In other words, if the contractee surely decommits, the contractor does not. On the other hand, the above is equivalent to (1) when &ya f(Z)& > 0. N ow the contractee’s IR con- straint states that the expected payoff from the contract is no less than the expected payoff from the outside offer: Similarly, the contractor’s IR constraint states that the expected payoff from the contract is no less than that from the outside offer: 2 E[-it] = r f (~)[--4dh: (3) --oo Because the contractor can want to decommit only if -5 - a > -p, its decommitment penalty can be chosen so high that it will surely not decommit (assuming that ii is bounded from below). In this case the contractee will decommit whenever p < g - b. If $ is bounded from above, the contractee’s decommitment penalty can be chosen so high that it will surely not decommit. Thus, full commitment contracts are a subset of leveled com- mitment ones. This reasoning holds for contracts where both agents have to pay the penalties if both decom- mit, and for contracts where neither agent has to pay a penalty if both decommit. Because full commitment contracts are a subset of leveled commitment contracts, the former can be no better in the sense of Pareto effi- ciency or social welfare than the latter. It follows that if there exists an IR full commitment contract, then there also exist IR leveled commitment contracts. However, leveled commitment contracts can enable deals that are impossible via full commitment contracts: Theorem 2.1 Encsbling in a §EQD game. There are SEQD games (defined by f(Z) and g(i)) where no full commitment contract satisfies the IR constraints but a leveled commitment contract does. 128 Agents Proof. Let f(ti) = Ok ftEe:wts$ loo C and g(g) = ing vb’(p,a,b) = p+ b+JIi= f(awbl s,:, f (XW = p + b, i.e. the 1 ii5 0 ft&:w!sz ‘lo A full commitment contract F . contractee decommits truthfully. The contractor’s IR constraint (Eq. 3) becomes cannot satisfy both IR constraints because that would require E[L] 5 PF 5 E[ii] which is impossible because 55 = E[i;] > E[ti] = 50. Ch oose a leveled commitment contract where p = 52.5, a = 30, and b = 20. By substi- tuting these in Equations 1, 2, and 3, it turns out that both agents’ IR constraints are strictly satisfied. The substitutions are straightforward but tedious [14]. •I In the game of the above proof, both IR constraints are satisfied by a wide range of leveled commitment contracts-and for no full commitment contract. Which leveled commitment contracts defined by p, a, and b sat- isfy the constraints ? There are many values of p for which some a and b exist such that the constraints are satisfied. We analyze contracts where p = 52.5 as an example. Now which values of a and b satisfy both IR constraints? There are three qualitatively different cases. Case 1. Either agent might decommit e In the case where a < p there is some chance that the contrac- tor will decommit (it may happen that -6 > -p + a). If Vb*(p,a, b) < 110 (’ 1.e. iess than maximum possible 6), there is some chance that the contractee will decommit. This occurs if \6 > p + b. We programmed a model of the IR constraints (Equations 3 and 2) for this case. To make the algebra tractable (constant f(6) and s(g)), versions of these IR constraint equations were used that assumed 0 5 a < p, and 0 < i* < 110, without loss of general- ity. The corresponding decommitment penalties a and b that satisfy the IR constraints are plotted in Figure 2 left. Furthermore, the boundaries of the programmed model need to be checked. The boundaries a = 0, a = p, and vb* = 110 are plotted in Figure 2 left. The constraint &* > 0 is always satisfied in this case and is not plotted. Figure 2: Decommitment penalties a and b that satisfy both agenta IR constraints in the example SEQD game. Right: either agent might decommit (a < p, and g’(p, a, b) < 110). Middle: only con- tractor might decommit (a < p, and &‘(p, a, b) 2 110). Left: a 2 p, i.e. only the contructee might decommit. Case 2, Contractor will surely not decommit. When a 2 p, the contractor will surely not decom- mit because its best possible outside offer is 8 = 0. Note that a can be arbitrarily high. The correspond- 1,. 110 100 dQ 1 0 100 f(X)[-~+b]dXdh- s s 0 p-tb 0 96) f(~)h+f~d~ 2 W-4 (4) If p + b > 110, this is equivalent to -p 2 E[-ii] which is false. If 0 < p + b < 110, this is equivalent to -=-[(‘IO-(p+b)).(F +lOOb)+(p+b).(-lOOp)] 1 E[-X] (31 llo &-[(57.5 - b) - (-5000 1OOb) (52.5 + + + b) . (-5250)] 2 -50 t+ 2.5 5 b < 52.5 by the quadratic equation solution formula. Similaily the co&ractee’s IR constraint (Eq. 2) be- comes 110 l+b 1 100 sm f(&)[i;-b]d&dt;+ 0 6”” sm 1100 f(WWd~ 2 Etfd (5) If p + b 1 110, this is equivalent to p _> E[g] which is false. If 0 < p + b < 110, this is equivalent to * &&j(~-l’Ob-(T- (i + b)’ (p+b)b)).lOO+(p+b)p.loo] > 55 e b 5 approximately 34.05 or b >_ approximately 80.95 by the quadratic equation solution formula. The latter violates p + b < 110. Put together, in the open region 2.5 5 b 5 34.05, a 2 p (Fig. 2 right) this type of contracts are IR for both agents even though the agents decommit insincerely. Case 3, Contractee will surely not decommit. If b is so high that 6* (p, a, b) 2 110, the contractee will surely not decommit. The contractor will decom- mit whenever -5 - a > -p e ii < p - a, i.e. the decommitting threshold ii* = p - a. The contractor’s IR constraint becomes e lllo -&L’-= f(X)[--X - a]d& + !I; f(s)[-p]d&]dg 2 -50 s P-a s 100 e> f(X)[-X - a]dX + f(~)[--plda 2 -50 0 p--o If a 2 p, this is equivalent to -p 2 -50 which is false. If 0 5 a < p, this is equivalent to iid J p--o [-X - a]d& + 0 s 100 [-p]dX] 2 -50 P-” * &[( + + (p - a)(-a)) + ((100 - (P - a)). (-p)) 2 -50 e a s approximately 30.14 or a 1 approximately 74.86 by the quadratic equation solution formula. The latter violates a < p. Negotiation & Coalition 129 Similarly, the contractee’s IR constraint becomes [~p’a’bk,,[[~~)[~ + a]dX + ~~~)(plda]dh 1 E[g] (7) J 110 s P--O s 100 * $j[P+al f(aW + [PI f(X)dX]di; > 55 0 0 p-0 p--o 100 # $$g + llOa] s f(a)da + 110~ s f(a)4 1 55 0 p--o p--o 100 e [55 + =I s f(aW + P s_ f(&)dtX > 55 If a 2 p, this :s equivalent t: j > 55 which is false. If 0 < a < p, this is equivalent to [55+a](p-a)~+p[lOO-(p-a)1~ 255 e 2.5 5 a < 47.5 Thus the open region 2.5 5 a 5 30.14, 6* > 110 (Fig. 2 middle) is where this type of contracts are IR for both agents even though the agents decommit insincerely. In addition to enabling deals that are impossible via full commitment contracts, leveled commitment con- tracts can increase the efficiency of deals which are pos- sible via full commitment contracts (the reverse cannot occur because the former can emulate the latter) if there is enough ez ante variance in the outside offers: Theorem 2.2 Pareto irnprovernent. If a SEQD game has at least one IR full commitment contract F and 1. i is bounded from above, f is bounded, and ml J-, f(Z)& > 0, or 2. ii is bounded from below, g is bounded, and then that game has a leveled commitment contract that increases both agents’ expected payoffs over any full com- mitment contract. Therefore, the leveled commitment contract is Pareto superior and IR. Proof. We prove this under condition 1. The proof under 2 is analogous. With F, the contractor’s payoff is -pF, and the contractee’s PF. We construct a leveled commitment contract where the contractee will surely not decommit because its penalty is chosen high and “b is bounded from above. Choose p = PF, and a = PF - E[g] + E. The contractor decommits if -ii - a > -P * 6 < p - a = E[g] - E. This has nonzero probability because bounded f and J:zl f(lc)da > o imply 2~ > 0 s.t. J-:2-= f(a)da > o. The contractee’s expected payoff increased: it is PF if the contractor does not decommit, and E[gJ + a = PF + E > PF if the contractor does. The contractor’s expected payoff also increased: s P---cr -PF < f(X)[--X - aW+ f(a)[--&a -00 s* p--p s P-” *O< f(a)[-s - = + PF]dg -00 which is implied by Jrf’-‘f(il)da > o (Because f is bounded, all of this probability mass cannot be on a single point ii = p - a (= E[g] - E)). III 130 Agents 3 Simultaneous decommitting In our simultaneous decommitting games, both agents have to declare decommitment simultaneously. There are two alternative leveled commitment protocols that differ on what happens if both agents decommit, Fig. 3. In the first, both agents have to pay the decommitment penalties to each other. In the second, neither agent has to pay. The next two sections analyze them. .------- ----_ :ontracting pIlIe I.C”.lcd -_--------_----_-___------------------------------------------------------------, Ikcommitting :ame P commit” (SIMUDBP) game. The parentheaired payoff8 Tepresent the "SIMUltaneous Decommit - Neither Pay8 if both decommit” (SIMUDNP) game. The dashed lines represent the agenta’ infor- mation sets. When decommitting, the contractor doe8 not know the contructee’s outside offer and vice veT8a. Furthermore, the contrac- tor ha8 to decide on decommitting before it ha8 observed the con- tTactee ‘8 decommitting decision, and vice verda. 3.1 Both pay if both decommit Simultaneous decommitting games where both agents have to pay the penalties if both decommit will be called SIMUDBP games, Fig. 3. Let pb be the probability that the contractee decommits, which depends on f(S), g(g), p, a, and b. The contractor decommits if pb. (--a + b - a) + (1 - pb)(-6 - a) > pb * (-a + b) + (1 - pb)(-P) If pb = 1, this equates to a < 0, but we already ruled out contracts where an agent gets paid for decommitting. On the other hand, the above inequality is equivalent to X<p-a 1 - pb “&’ &*(p, u,b,&*) When pa < 1 (8) If the contractor’s outside offer is below the threshold (6 < 6*), the contractor is best off by decommitting. The contractee decommits if SW s a* (p,%b,b*) f (a)d@ - b] + f (X)dX[i; - b + a] a*(p,a,b,g*) -co r s a*(p,d,6*) > f (~)Wd + f (+a[6 + al a+ (P,a,b,6*) -CCl If c.fXy(,o,a,b,i;*) f (ti)d’ = 0, this equates to b < 0, but we ruled out contracts where an agent gets paid for decom- mitting. However, the above inequality equates to If the contractee’s outside offer exceeds the threshold (& > &*), the contractee is best off by decommitting. The probability that the contractee will decommit is pb = I= s(W (10) g+ (P ,a,b,a+) Condition 8 states the contractor’s best response (de- fined by n*) to the contractee’s strategy that is defined by 2. Condition 9 states the contractee’s best response ‘i* to the contractor’s strategy that is defined by ii*. Con- dition 8 uses the variable pa which is defined by Equa- tion 10. So together, Equations 8, 9, and 10 define the Nash equilibria of the decommitting game. Now the contractor’s IR constraint becomes The first row corresponds to the contractee decommit- ting, while the second corresponds to the contractee not decommitting. The second integral in each row corre- sponds to the contractor decommitting, while the third integral corresponds to the contractor not decommitting. Similarly, the contractee’s IR constraint becomes If 5 is bounded from below, the contractor’s decom- mitment penalty a can be chosen so high that the con- tractor’s decommitment threshold ii* (p, a, b, 8”) becomes lower than any & In that case the contractor will surely not decommit. Similarly, if “b is bounded from above, the contractee’s decommitment penalty b can be chosen so high that the contractee’s decommitment threshold g*(p) a, b, ;i*) is greater than any “b. In that case the con- tractee will surely not decommit. Thus, full commit- ment contracts are a subset of leveled commitment ones. Therefore, the former can be no better in the sense of Pareto efficiency or social welfare than the latter. In addition to leveled commitment contracts never be- ing worse than full commitment ones, the following the- orem shows that in SIMUDBP games they can enable a deal that is impossible via full commitment contracts. Theorem 3.1 Enabling in a SIMUDBP game. There are SIMUDBP games (defined by f(Z) and g(g)) where no full commitment contract satisfies the IR con- straints but a leveled commitment contract does. Proof. Let f(6) = $ ftie:wts$ loo and g(g) = & if 0 2 i; 5 110 0 otherwise. No full commitment contract F satisfies both IR constraints because that would re- quire .+?!$I < PF 5 E[ti] which is impossible because 55 = E[Q > E[ii] = 50. We build a leveled commitment contract with p = 52.5 as an example. Four cases result: Case I. Either agent might decommit. If 0 < &* < 100, and 0 < i* < 110, there is a nonzero prob- ability for each agent to decommit. The unique Nash equilibrium is plotted out for different values of a and b in Figure 4. The equilibrium decommitment thresholds ti” and vb* differ from the truthful ones. Yet there exist equilibria in the proper range of ii* and vb*. l:::i:fi ~~~ qri .I 1 s / II I Figure 4: The Nash equilibriim decommitment thresholds I’ and 6* of OUT example SIMUDBP game for different values of the de- commitment penalties a and b. The Nash equilibrium deviates fTom truthful decommitting. If 0 < 1’ < 100, and 0 < g’ < 110, there is dome chance that eitheT agent will decommit. It is not guaranteed that all of these Nash equilib- ria satisfy the agents’ IR constraints however. We pro- grammed a model of Equations 8, 9, and 10 and the IR constraints. To make the algebra tractable (constant f(z) and d)), versions of these equations were used that assumed 0 < ai” < 100, and 0 < vb* < 110, without loss of generality. Therefore the first task was to check the boundaries of the validity of the model. The boundaries ii* = 0 and vb” = 110 are plotted in Figure 5. The bound- ary 6* = 100 turns out to be the line b = 0. There exists no boundary g* = 0 because g* was always positive. b . Contractee not decomnit -100 contracke lR \ ) ,-,-' con[rac.(,,. ,R Figure 5: P Three different regions of contracts that aTe IR for both agenta and allow an equilibrium in the SIMUDBP decommit- ting game. In the dark gray aTea eitheT agent might decommit but in the Zight gray aTeas only one of the agents might. Curves represent the IR con&Taints and validity constraints of the programmed model that Tequires 0 < ii* < 100, and 0 < 6* < 110. Each agent’s IR constraint induces three curves (Fig. 5)) two of which actually bound the IR region. The third is also a root, but at both sides of that curve, the IR constraint is satisfied. The dark gray area of Fig- ure 5 represents the values of the decommitment penal- ties a and b for which the validity constraints of the programmed model and the IR constraints are satisfied. In other words, for any such a and b, there exists de- commitment thresholds G* and g* such that these form a Nash equilibrium, and there is a nonzero probability for either agent to decommit, and each agent has higher expected payoff with the contract than without it. Negotiation & Coalition 131 Case 2, Contractor will surely not decommit. If 8* 5 0, the contractor will surely not decommit. Now b(p, a, b, &*) = p+ JZ ;(&,,, = p+ b, i.e. the contractee decommits truthfully. The contractor’s IR constraint becomes the same as in case 2 of the example SEQD game (Eq. 4). Th is constraint was proven equivalent to 2.5 < b s 52.5. The contractee’s IR constraint also equates to that in the SEQD game (Eq. 5). It was proven equivalent to b 5 approximately 34.05. Thus these con- tracts are IR for both agents and in equilibrium in the open region 2.5 < b 5 34.05, ii’ 5 0, Fig. 5. Case 3, Contractee will surely not decommit. If &” 1 110, the contractee will surely not decommit (pb = 0). Now ii*(p,u, b,vb*) = p - e = p - a, i.e. the contractor decommits truthfully. The contractor’s IR constraint becomes the same as in case 3 of the example SEQD game (Eq. 6). Th’ 1s constraint was proven equiv- alent to a 5 approximately 30.14. The contractee’s IR constraint equates to Eq. 7 of the SEQD game. It was proven equivalent to 2.5 5 a ,< 47.5. Thus the open re- gion 2.5 5 a < 30.14, b > 110 is where these contracts are IR for both agents, and in equilibrium, Fig. 5. Case 4, Trivial case. A contract where at least one agent will surely decommit, i.e. ii* > 100 or vb* 5 0 can be IR-barely because it does not increase either agent’s payoff. For it to be IR for the decommitting agent, the decommitment penalty has to be zero: the decommitting agent gets the same payoff as without the contract. Similarly, the other agent gets the same payoff as it would get without the contract. This contract is equivalent to no contract at all: decommitment occurs and no payment is transferred. cl In addition to enabling deals that are impossible using full commitment, leveled commitment contracts can in- crease the efficiency of a deal even if a full commitment contract were possible (the reverse cannot occur): Theorem 3.2 Pareto efficiency improaement. Theorem 2.2 applies to SIMUDBP games. Proof. When one agent is known not to decommit, SIMUDBP games are equivalent to SEQD games. •I 3.2 Neither pays if both decommit Simultaneous decommitting games where a protocol is used where neither agent has to pay a decommit- ting penalty if both agents decommit (SIMUDNP games, Fig. 3) can be analyzed in the same way as SIMUDBP games, but the decommitting thresholds differ [14]. If ti is bounded from below, and g from above, a can be chosen so high that the contractor wiIl surely not decommit, and b so high that the contractee will not. So, full commitment contracts are a subset of leveled commitment ones. Thus the former cannot enable a deal whenever the latter cannot. Also, leveled commitment can enable a deal that is impossible via full commitment: Theorem 3.3 Enabling in a SIMUDNP game. There exist SIMUDNP games (defined by f (ii) and g(g)) where no full commitment contract satisfies the IR con- straints but a leveled commitment contract does. The proof is like that of Theorem 3.1 except that the formulas for decommitting differ [14]. With the same f(8), g(i), and p as in the proof of Theorem 3.1, the Nash equilibria of the SIMUDNP game are as shown in Figure 6. The decommitment thresholds Z* and vb” differ from the truthful ones. They are closer to the truthful ones than what they were with a protocol where both agents pay if both decommit, Figure 4. The shapes of the curves using these two protocols also differ significantly. I, II I Figure i: ihe Nazi equilibriim decommitment tkesiolds ii*“’ an: ub’ of OUT example SIMUDNP game foT different values of the de- commitment penaltiea a and b. The Nash equilibrium deviates from truthful decommitting. If 0 < 1’ < 100, and 0 < 6* < 110, there ia dome chance that either agent will decommit. We programmed a model to check which of the SIMUDNP equilibria allow contracts that are IR for both agents. The results are quantitatively different, but qual- itatively same as in SIMUDBP games (Fig. 5) [14]. Leveled commitment contracts can also increase the efficiency of a deal even if a full commitment contract were possible (the reverse cannot occur): Theorem 3.4 Pareto eficiency improuement. Theorem 2.2 applies to SIMUDNP games. Proof. When one agent is known not to decommit, SIMUDNP games are equivalent to SEQD games. •I 4 rescriptions for system buil The results from the above canonical games suggest that it is worthwhile from a contract enabling and a contract Pareto improving perspective to incorporate the decom- mitment mechanism into automated contracting proto- cols. The decommitment penalties are best chosen by the agents dynamically at contract time as opposed to statically in the protocol. This allows the tuning of the penalties not only to specific negotiation situations and environmental uncertainties, but also to specific belief structures of the agents. An extended paper analyzes the impact of agents’ biased beliefs on the benefits of contracts, and the distribution of these gains 1141. In the presented instance of the simultaneous de- committing game, the Nash equilibrium decommitting strategies were usually closer to truthful ones when a protocol was used where neither pays if both decommit than when a protocol was used where both pay if both decommit. Also, as an agent’s opponent’s decommit- ment penalty approaches zero, the agent becomes truth- ful in the former protocol, but starts to increasingly bias its decommitment decisions in the latter. This suggests using the former protocol in practical systems. It also minimizes the number of payment transfers because it does not require any such transfer if both decommit. 132 Agents In a web of multiple mutual contracts among several agents, classical full commitment contracts induce one negotiation focus consisting of the obligations of the con- tracts. Under the protocol proposed in this paper, there are multiple such foci, and any agent involved in a con- tract can swap from one such focus to another by de- committing from a contract. Such a swap may make it beneficial for another agent to decommit from another contract, and so on. To avoid loops of decommitting and recommitting in practise, recommitting can be disabled. This can be implemented by a protocol that specifies that if a contract offer is accepted and later either agent decommits, the original offer becomes void-as opposed to staying valid according to its original deadline which may not have been reached at decommitment time. Even if two agents cannot explicitly recommit to a con- tract, it is hard to specify and monitor in a protocol that they will not make another contract with an identical content, This gives rise to the possibility of the equiva- lent of useless decommit-recommit loops. Such loops can be avoided by a mechanism where the decommitment penalties increase with real-time or with the number of domain events or negotiation events. This allows a low commitment negotiation focus to be moved in the joint search space while still making the contracts meaningful by some level of commitment. The increasing level of commitment causes the agents to not backtrack deeply in the negotiations, which can also save computation. The initially low commitment to contracts can also be used as a mechanism to facilitate linking of deals. Of- ten, there is no contract over a single item that is bene- ficial, but a combination of contracts among two agents would be [13; 171. E ven contracts [13; 171 if explicit clustering of issues into is not used, an agent can agree to an unbeneficial contract in anticipation of synergic future contracts from the other agent that will make the first contract beneficial [17]. If no such contracts appear, the agent can decommit. Similarly, low commitment con- tracts can be used to facilitate deals among more than two agents. Even without explicit multiagent contract protocols [17], multiagent contracts can be implemented by one agent agreeing to an unbeneficial contract in an- ticipation of synergic future contracts from third parties that will make the first contract beneficial [17]. If no such contracts appear, the agent can decommit. In many practical automated contracting settings lim- ited computation resources bound the agents’ capability to solve combinatorial problems [17; 15; 133. The value of a contract may only be probabilistically known to an agent at contract time. The leveled commitment con- tracting protocol allows the agent to continue delibera- tion regarding the value of the contract after the con- tract is made. If the value turns out to be lower than expected, the agent can decommit. However, decom- mitment penalties which increase quickly in time may be appropriate with computationally limited agents so that the agents do not need to consider the combinato- rial number of possible future worlds where alternative combinations of decommitments have occurred [17]. 5 Conclusions A protocol was presented for automated contracting that allows agents to accommodate future events more profitably than traditional full commitment contracts. Each contract specifies a decommitment penalty for both agents involved. To decommit, an agent just pays that penalty to the other agent. This mechanism is bet- ter suited for complex computerized contracting settings than contingency contracts. The analysis handled the fact that agents decommit manipulatively. This analysis also serves as a normative tool for agents to decide which contracts to accept based on individual rationality. Leveled commitment contracts can emulate full com- mitment ones by setting the decommitting penalties high. Therefore, full commitment contracts cannot be better than leveled commitment ones in the sense of Pareto efficiency or social welfare to the two agents. Nei- ther can they enable a deal that is impossible-based on individual rationality-via a leveled commitment con- tract. We proved that in these games the new proto- col surprisingly enables deals that are impossible via full commitment contracts. It also increases the expected payoff to both agents in settings where a full commit- ment contract is possible. Obviously one can also con- struct game instances where the null deal is so profitable to both agents that no contract-even a leveled commit- ment one-is individually rational to both. References 111 bl 131 [41 bl b1 [71 I4 [Ql 1101 b11 Ml b31 t141 [Iti1 b61 1171 b31 1191 .I. D. Calameri and J. M. Perillo. The L8W of Contracts. West Pub- lishing Co., 2nd edition, 1977. K. Decker and V. R. Lesser. Designing 8 family of coordination 8@0- rithms. In ICMAS, pages 73-80, San Francisco, CA, June 1996. P.Diamond and E.Maskin. An equilibrium analysis of search and breach Of Contract, i: Steady States. Bell .J of Economics, 10:282-316, 1979. E. Ephrati and J. S. Rosenschein. The Clarke tax as a consensus mech- anism among automated agents. In AAAI, page5 173-178, 1991. D. Fudenberg and 3. Tirole. Game Theory. MIT Press, 1991. S. Kr8us. Agents contracting tasks in non-COll8bor8tiVe In AAAI, page5 243-248, Washington D.C., July 1993. environments. D. Kreps. A course in microeconomic theory. Princeton U Press, 1990. D. M. Kristol, S. H. Low, and N. F. Maxemchuk. Anonymous internet mercantile protocol. 1994. Submitted. J. Nash. Equilibrium points in n-person games. Proc. of the National Academy of Sciences, 36:48-49, 1960. R. Posner. Economic Analysis of Law. Little, Brown &Co, 2nd cd, 1977. H. Raiffa. The Art and Science of Negotiation. Harvard U Press, 1962. J. S. Rosenschcin and G. Zlotkin. Rules of Encounter. MIT Press, 1994. T. W. Sandholm. An implementation of the contract net protocol based on marginal cost calculations. In AAAI, pages 266-262, July 1993. T. W. Sandholm and V. R. Lesser. Advantage5 of 8 leveled commitment contracting protocol. Umass TR 96-72, 1996. T. W. Sandholm and V. R. Lesser. Coalition formation among bounded rational agents. In IJCAI, pages 662-669, Montreal, Canada, Aug. 1996. Extended version: Umass TR 96-71. T. W. Sandholm and V. R. Lesser. Equilibrium analysis of the possi- bilitieo of unenforced exchange in multiagent systems. In IJCAI, pages 694-701, Montreal, Canada, Aug. 1996. T. W. Sandholm and V. R. Lesser. Issues in automated negotiation and electronic commerce: Extending the contract net framework. In ICMAS-96, pages 328-336, San Francisco, CA, June 1996. S. Sen. Tradeoffs in Contract-Based Distributed Scheduling. PhD the- sis, Univ. of Michigan, 1993. R. G. Smith. The contract net protocol: High-level communication and control in a distributed problem solver. IEEE Transactions on Computers, C-29(12):1104-1113, Dec. 1980. Negotiation & Coalition 133 | 1996 | 19 |
1,832 | Generalized Queries on robabilistic Context- David V. Pynadath and chael I? Wellman Artificial Intelligence Laboratory University of Michigan 1101 Beal Avenue Ann Arbor, MI 48 109 USA {pynadath,wellmanj @umich.edu Abstract Probabilistic context-free grammars (PCFGs) provide a simple way to represent a particular class of dis- tributions over sentences in a context-free language. Efficient parsing algorithms for answering particular queries about a PCFG (i.e., calculating the probability of a given sentence, or finding the most likely parse) have been applied to a variety of pattern-recognition problems. We extend the class of queries that can be answered in several ways: (1) allowing missing to- kens in a sentence or sentence fragment, (2) supporting queries about intermediate structure, such as the pres- ence of particular nonterminals, and (3) flexible condi- tioning on a variety of types of evidence. Our method works by constructing a Bayesian network to repre- sent the distribution of parse trees induced by a given PCFG. The network structure mirrors that of the chart in a standard parser, and is generated using a similar dynamic-programming approach. We present an algo- rithm for constructing Bayesian networks from PCFGs, and show how queries or patterns of queries on the net- work correspond to interesting queries on PCFGs. Introduction Most pattern-recognition problems start from ob- servations generated by some structured stochas- tic process. Probabilistic context-free gram- mars (PCFGs) (Gonzalez & Thomason 1978; Charniak 1993) have provided a useful method for modeling uncertainty in a wide range of structures, including programming languages (Wetherell 1980), images (Chou 1989), speech signals (Ney 1992), and RNA sequences (Sakakibara et al. 1995). Domains like plan recognition, where non-probabilistic grammars have provided useful models (Vilain 1990), may also benefit from an explicit stochastic model. Once we have created a PCFG model of a process, we can apply existing PCFG parsing algorithms to answer a vari- ety of queries. However, these techniques are limited in the types of evidence they can exploit and the types of queries they can answer. In particular, the standard techniques gen- erally require specification of a complete observation se- quence. In many contexts, we may have only a partial se- quence available, or other kinds of contextual evidence. In addition, we may be interested in computing the probabili- ties of types of events that the extant techniques do not di- rectly support. Finally, the PCFG model itself imposes re- strictions on the probabilistic dependence structure, which we may wish to relax. To extend the forms of evidence, queries, and distribu- tions supported, we need a flexible and expressive repre- sentation for the distribution of structures generated by the grammar. We adopt Bayesian networks for this purpose, and define an algorithm to generate a network represent- ing the distribution of possible parse trees corresponding to a given PCFG. We then present algorithms for extending the class of queries to include the conditional probability of a symbol appearing anywhere within any region of the parse tree, conditioned on any evidence about symbols ap- pearing in the parse tree. The Bayesian network also pro- vides a flexible structure for future extensions to context- sensitive probabilities, similar to the probabilistic parse ta- bles of (Briscoe & Carroll 1993). robabilistic Context-Free Gra ars A probabilistic context-free grammar is a tuple (HT,HN,E~,P), where HT is the set of terminal symbols, HN the set of nonterminal symbols, El E HN the start symbol, and P the set of productions. Productions taketheformE -+ 5 (p), withE E HN,~$ E (HTUHN)+, andp = Pr( E + [), the probability that E will be ex- panded into the string [. The probability of applying a particular production to an intermediate string is condi- tionally independent of what productions were previously applied to obtain the current string, or what productions will be applied to the other symbols in the current string, given the presence of the left-hand symbol. Therefore, the probability of a given derivation is simply the product of the probabilities of the individual productions involved. The probability of a string in the language is the sum taken over all possible derivations. In the grammar (from (Charniak 1993)) shown in Figure 1, the start symbol is S. Bayesian Networks 1285 From: AAAI-96 Proceedings. Copyright © 1996, AAAI (www.aaai.org). All rights reserved. s -+ s - *P - *P - *P - “P - “P - “P - “P - *P “P “P n * PP * *P V ” *P “PP “*PPP (0.8) (0.2) (0.4) (0.4) (0.2) (0.3) (0.3) (0.2) (0.2) Figure 1: A probabilistic context-free grammar. PP - p *p (1.0) P - like (1.0) v - swat (0.2) n -+ flies (0.45) n - ants (0.5) s ~1~4~2) I “P (194.1) E-yh Pp (3,2,1) I \ nP (2,1,3) 1 v (4,1,3) verb (1,1,2) noun (2,1,2) 1 1 1 Pr P I (3,W noun (4,1,2) swat U,l,l) flies (2,l.l) I like (3,lJ ants (4,W Figure 2: Parse tree for Swat flies like ants, with (i, j, k) indices labeled. Indexing Parse Trees Calculating the probability of a particular parse tree can sometimes be useful, but we may also wish to derive the probability of some more abstract feature of a parse tree. To pose such queries, we require a scheme to specify events as the appearance of symbols at designated points in the parse tree. We use three indices to identify a node in a parse in terms of the structure of the subtree rooted at that node. Two indices delimit the leaf nodes of the subtree, defining a sub- string of the entire terminal sequence. The index i refers to the position of the substring within the entire terminal string, with i = 1 indicating the start of the string. The index j refers to the length of the substring. For example, the pp node in the parse tree of Figure 2 is the root of the subtree whose leaf nodes are like and ants, so i = 3 and j = 2. These i and j indices are commonly used in PCFG algo- rithms. However, we cannot always uniquely specify a node with these two indices alone. In the branch of the parse tree pass- ing through np, n, and flies, all three nodes have i = 2 and j= 1. To differentiate them, we introduce the k index, de- fined recursively. If a node has no child with the same i and j indices, then it has k: = 1. Otherwise, its k: index is one more than the Ic index of its child. Thus, the flies node has k = 1, the n node above it has a Ic = 2, and its parent np haslc = 3. We have labeled each node in the parse tree of Figure 2 with its (i, j, k) indices. We can think of the k index of a node as its level of ab- straction, with higher values indicating more abstract sym- bols. For instance, the flies symbol is a specialization of the n concept, which, in turn, is a specialization of the np concept. Each possible specialization corresponds to an ab- straction production of the form E --) E’. In a parse tree involving such a production, the nodes for E and E’ will have identical i and j values, but the k value for E will be one more than that of E’. We denote the set of abstraction productions as PA E P. All other productions are decomposition productions, in the set PO = P \ PA, and have two or more symbols on the right-hand side. If a node E is expanded by a decom- position production, the sum of the j values for its children will equal its own j value, since the length of the original substring derived from E must equal the total lengths of the substrings of its children. In addition, since each child must derive a string of nonzero length, no child has the same j in- dex as E, which must then have a k value of 1. Therefore, abstraction productions connect nodes whose indices match in the i and j components, while decomposition productions connect nodes whose indices differ. Dynamic Programming Algorithm We can compute the probability of a string by summing probabilities over the set of its possible parse trees, which grows exponentially with the string’s length. Fortunately, parse trees often share common subtrees, a fact exploited by the standard dynamic programming approach for both probabilistic and non-probabilistic CFGs (Jelinek, Lafferty, & Mercer 1992). The central structure is a table, or chart, storing previous results for each substring in the input sen- tence. Each entry in the chart corresponds to a substring xi * * . x:i+j - 1 (ignoring abstraction level, k) of the observa- tion string 21 . . . XL. For each symbol E, an entry contains the probability that the corresponding substring is derived from that symbol, Pr(xi e. .xd+j-1IE). At the bottom of the table are the results for substrings of length one, and the top entry holds the result for the en- tire string, Pr(xl . . . XL 1 El), which is exactly the probabil- ity of the observed string. We can compute these probabil- ities bottom-up, since we know that Pr (xi 1 E) = 1 if E is the observed symbol xi, and 0 otherwise. We can define all other probabilities recursively as the sum, over all produc- tions E + 5 (p), of the product of p and the probability Pr(xi . 0 . xi+j- 1 I[). Here, we can make use of the PCFG independence assumptions, and compute this probability as the product of the probabilities of the individual symbols, where we have to consider all possible substring lengths for these symbols. A slight alteration to this procedure also al- lows us to obtain the most probable parse tree for the ob- served string. To compute the probability of the sentence Swat flies like ants, we would use the algorithm to generate the table shown in Figure 3, after eliminating any intermediate entries Uncertainty s+vp: BOO72 s-+np(2) VP(~): .000035 s-cnp(l) VP(~): BOO256 vp+v np pp: .OO14 vp-tv np: .00216 j = 4 VP+vpp: .016 1 3 np+n pp: .036 np+ n np: VP-+ v np: 2 0.0018 0.024 np-+n: 0.02 v+swaf: 0.2 n-bswat: 0.05 a=1 rip-m:: 0.18 vhflies: 0.4 n-bflies: 0.45 2 pp+p np: 0.2 rip-m:: 0.2 p-+ like: 1 .O n+ ants: v+like: 0.4 0.5 3 4 Figure 3: Chart for Swat flies like ants. that were not referenced by higher-level entries. There are also separate entries for each production, though this is not necessary if we are only interested in the final sentence prob- ability. In the top entry, there are two listings for the produc- tion s-+np VP, with different substring lengths for the right- hand side symbols. The sum of all probabilities for produc- tions with s on the right-hand side in this entry yields the total sentence probability of 0.001011. This algorithm is capable of computing any “inside” probability, the probability of a particular terminal string ap- pearing inside the subtree rooted by a particular nontermi- ml. We can work top-down in an analogous manner to com- pute any “outside” probability (Charniak 1993), the proba- bility of a subtree rooted by a particular nonterminal appear- ing amid a particular terminal string. Given these probabil- ities we can compute the probability of any particular non- terminal symbol appearing in the parse tree as the root of a subtree covering some substring. For example, in the sen- tence Swat flies like ants, we can compute the probability that like ants is a prepositional phrase, using a combina- tion of inside and outside probabilities. The Left-to-Right Inside (LRI) algorithm (Jelinek, Lafferty, & Mercer 1992) specifies how we can manipulate certain probability matri- ces and combine the results with the inside probabilities to obtain the probability of a given initial substring, such as the probability of a sentence (of any length) beginning with the words Swat flies. Furthermore, we can use such initial sub- string probabilities to compute the conditional probability of the next observation given all previous observations. However, there are still many types of queries not covered by existing algorithms. For example, given observations of arbitrary partial observation strings, it is unclear how to ex- ploit the standard chart directly. Similarly, we are unaware of methods to handle observation of nonterminals only (e.g., the last two words form a prepositional phrase). We seek, therefore, a mechanism that would admit observational evi- dence of any form as part of a query about a PCFG, without requiring us to enumerate all consistent parse trees. ayesian Networks for PCFGs Bayesiavz networks (Pearl 1987) provide an expressive and efficient representation for probability distributions. They are expressive in that they can represent any joint distribu- tion over a finite set of discrete-valued random variables. They are efficient in that they exploit an important class of conditional independence relationships among the ran- dom variables. Moreover, Bayesian networks are conve- nient computational devices, supporting the calculation of arbitrary conditional probability expressions involving their random variables. Therefore, if we can create a Bayesian network representing the distribution of parse trees for a given probabilistic grammar, then we can incorporate par- tial observations of a sentence as well as other forms of ev- idence, and determine the resulting probabilities of various features of the parse trees. We base our Bayesian-network encoding of PCFGs on the parse tree indexing scheme presented in the previous section The random variable NQ~ denotes the symbol in the parse tree at the position indicated by the (i, j, L) in- dices. Index combinations not appearing in the tree corre- spond to N variables taking on the null value nil. To sim- plify the dependency structure, we also introduce random variables Pdjk to represent the productions that expand the corresponding symbols Naj k . However, the identity of the production is not quite sufficient to render the corresponding children in the parse tree conditionally independent, so we dictate that the P variable take on different values for each breakdown of the right-hand symbols’ substring lengths. This increases the state space of the variables, but simplifies the dependency structure. rogramming Phase To complete the specification of the network, we identify the symbols and productions making up the domains of our ran- dom variables, as well as the conditional probability tables representing their dependencies. The PCFG specifies the relative probabilities of different productions for each non- terminal, but to specify the probabilities of alternate parse trees in terms of the Najk variables we need the probabili- ties of the length breakdowns. We can calculate these with a modified version of the standard dynamic programming al- gorithm sketched in the previous section. This modified algorithm constructs a chart based on the set of all possible terminal strings, up to a bounded length n. Our resulting chart defines a function ,8( E, j, k) (analogous to the inside probability in the standard parsing algorithm), specifying the probability that symbol E is the root node of a subtree, at abstraction level L, with a terminal substring of length j. Because this probability is not relative to a partic- ular observation string, we can ignore the i index. As in the previous dynamic programming algorithms, we can define this function recursively, initializing the entries to Bayesian Networks 1287 k E 1 ,6(&T, 4, k) 11 k E 1 /3(E,3, k) 11 k E Figure 4: Final table for sample grammar. 0. Again, we start at j = 1 and work upward to j = n. For each terminal symbol 2, P(z, 1,l) = 1. For X: > 1, only ab- straction productions are possible, because, as discussed be- fore, decomposition productions are applicable only when k = 1. For each abstraction production E ---) E’ (p), we incrementP(E, j,Ic)byp.P(E’, j,k-1). Iflc = l,onlyde- composition productions are applicable, so for each decom- position production E + El Es s . . Em (p), each substring length breakdown ji, . . . , j, (such that the Et j, = j), and each abstraction level kt legal for each j, , we increment /3( E, j, k) by p . nz, p( Et, jt , kt ). The table of Figure 4 lists the nonzero ,0 values for our grammar over strings of maximum length 4. For analysis of the complexity of this algorithm, it is use- ful to define d as the maximum abstraction level, and m as the maximum number of symbols on a production’s right- hand side. For a maximum string length of n, the table re- quires space 0( n2d 1 HN I), exploiting the fact that /3 for ter- minal symbols is one in the bottom row and zero elsewhere. For a specific value of j, there are O(d) possible k values greater than 1, each requiring time 0( 1 PA I). For k = 1, the algorithm requires time 0( I PD I jmB1dm), for the eval- uation of all decomposition productions, as well as all pos- sible combinations of substring lengths and levels of ab- stractions for each symbol on the right-hand side. There- fore, the whole algorithm would take time O(n[d I PA I + IPD In “-ldm]) = O(IPln”d”). As an alternative, we can modify the standard chart pars- ing algorithm (Younger 1967; Earley 1970) to compute the required values for ,8 by recording the k values and probabil- ities associated with each edge. We would also ignore any distinctions among terminal symbols, since we are comput- ing values over all possible terminal strings. Therefore, the time required for computing /!? is equivalent to that required for parsing a terminal string of length n, which is 0( n3) ig- noring the parameters of the grammar. Network Generation Phase Upon completion of the dynamic programming phase, we - - can use the table entries to compute the domains of random variables N. e ag k and Pij k and the required conditional proba- bilities. We begin at the top of the abstraction hierarchy for strings of the maximum length n starting at position 1. The corresponding symbol variable can be either El or the spe- cial null symbol nil*, indicating that the parse tree begins at some other point below. The prior probability of the start symbol is proportional to ,0( El, n, k), while that of nil* is proportional to the sum of all other ,B values for the start symbol El. The exact probabilities are normalized so that the sum equals one. We start with this node and pass through all of the nodes in order of decreasing j and k. With each N node, we insert the possible productions into the do- main of its corresponding P node. For a production rule r that maps E to El . . . Ena with probability p and a breakdown of substring lengths and abstraction lev- els (ji , ICI), . . . , (j,, km), the conditional probability Pr(&jk = r ((h, h), - - -, (jm,km>) INijk = E) m ~-nP(EtA,kt). m e exact probability is normalized so that the sum over all rules P and breakdowns ((j, , kt)) for a particular left-hand symbol E is 1. For any symbol E’ # E in the domain of Nij k , we can set the conditional probability Pr(Pijk = r ((jl, kl), - - -, (h, km)) INijk = E’) = 0. A symbol variable which takes on the value nil has no children, so its production variable will also take on a null value (i.e., Pr(Pijk = tlillNijr, = nil) = 1). For the special symbol nil*, there are two possibilities, either the parse tree starts at the next level below, or it starts further down the tree. In the first case, the production is nil* + El, and has a conditional probability proportional to the ,f3 value of El at the j and k value immediately below the current position, given that Ndjk = nil*. In the second case, the production is nil* + nil*, and has a conditional probability proportional to the sum of the p values of El at all j and k more than one level below, given that Nijk = nil*. When all possible values for a production variable are added, we add a link from the corresponding node to the variables corresponding to each symbol on the right-hand side and insert these symbols into the domains of these child variables. A child nodes takes on the appropriate right-hand side symbol with probability 1 if the parent node has taken on the value of the given production. A child node takes on the value nil with probability 1 if none of its parent nodes assign it a symbol value. Figure 5 illustrates the network structure resulting from applying this algorithm to the table of Figure 4 with a length bound of 4. In general, the resulting network has O(n2d) nodes. The n N. ali variables have 0( I HT I) states each, while the O(n2d) other N variables have 0( I HN I) possi- ble states. The Pijk variables for k > 1 (of which there are O(n2 d)) have a domain of 0( I PA I) states. For Pij 1 vari- ables, there are states for each possible decomposition pro- duction, for each possible combination of substring lengths, and for each possible level of abstraction of the symbols on the right-hand side. Therefore, the Pij 1 variables (of which 1288 Uncertainty JN-1-421 Figure 5: Network from example grammar. there are O(n2)) have a domain of 0( IPD I jmeldm) states. Unfortunately, even though each particular P variable has only the corresponding N variable as its parent, a given Ndjk variable could have potentially O(i . (n - i - j)) P variables as parents, and the size of a node’s conditional probability table is exponential in the number of its par- ents. If we define T to be the maximum number of en- tries of any conditional probability table in the network, then the total time complexity of the algorithm is then O(n2dlPAIT + n21PDlnmw1dmTm + ndT + n2dT) = OWln m+ldmTm), which dwarfs the complexity of the dynamic programming algorithm for the ,8 function. How- ever, this network is created only once for a particular gram- mar and length bound. Inference The Bayesian network can answer any of the queries ad- dressed by the usual parsing algorithm. To find the prob- ability of a particular terminal string 21 . . . XL, we can in- stantiate the variables Nili to be xi, for i < 15, and nil, for i > L. Then, we can use any of the standard Bayesian net- work propagation algorithms to compute the probability of this evidence. The result is the conditional probability of the sentence, given that the string is bounded in length by n. We can easily acquire the unconditional probability, since the probability of a string having length no more than n is the sum of the ,8 values for El over all lengths of n and un- der. To find the most probable parse tree, we would use the standard network algorithms for finding the most probable configuration of the network. The network represents a distribution over strings of bounded length, so we cannot obtain the same probability of an initial substring ~1x2 . a . XL as (Jelinek, Lafferty, & Mer- cer 1992), which considered all completion lengths. How- ever, we can find initial substring probabilities over comple- tions of length bounded by n - L. The algorithm is identical to that for the probability of the entire sentence, except that we do not instantiate the Nili variables beyond i = L to be nil. The procedure for finding the probability that a particu- lar symbol derives a particular substring is complicated by the fact that there are multiple levels of abstraction possible for a particular substring. Therefore, after we instantiate the evidence, we must query all of the N variables for the par- ticular i and j values of interest. We can start with the a par- ticular k value and find the posterior probability of Nij k be- ing the symbol of interest. Having “counted” this event, we set the likelihood of Ndjk being that symbol to be zero, and proceed to a different k. We maintain a running total as we proceed, with the final probability being the result when all of the nodes have been counted. In general, we can answer any query about an event that can be expressed in terms of the basic N and P random vari- ables. Obviously, if we are interested in whether a symbol appeared at a particular i, j, k location in the parse tree, we only need to examine the marginal probability distribution of the corresponding N variable. Alternatively, we can find the probability of a particular symbol deriving any part of a particular substring (specified by i and j indices) by per- forming a similar procedure to that for an exact substring described above. However, in this case, we would continue computing posterior probabilities over all i and j variables within the bounds. As another example, consider the case of possible four- word sentences beginning with the phrase Swat flies. In the network of Figure 5, we instantiate Nl11 to be swat and N2i1 to be flies and then propagate this evidence. We then need only to examine the joint distributions of Nail and N411 to find that like flies is the most likely completion. This is similar to the Left-to-Right Inside algorithm of (Je- linek, Lafferty, & Mercer 1992), except that we can find the most probable joint configuration over multiple time steps, instead of over only the one immediately subsequent. A greater advantage is in the utilization of evidence. Any of the queries mentioned previously can be conditioned on any event that can be expressed in terms of N and P vari- ables. If we only have a partial observation of the string, we simply instantiate the Nili variables corresponding to the positions of whatever observations we have, and then propagate to find whatever posterior probability we require. In addition, we can exploit additional evidence about non- terminals within the parse tree. For instance, we may want to find the probability of the sentence Swat flies like ants with the additional stipulation that like ants is a preposi- tional phrase. In this case, we instantiate the Nili variables as usual, but we also instantiate N321 to be pp. Bayesian Networks 1289 Conclusion The algorithms presented here automatically generate a Bayesian network representing the distribution over all parses of strings (bounded in length by some parameter) in the language of a PCFG. The first stage uses a dynamic pro- gramming approach similar to that of standard parsing al- gorithms, while the second stage generates the network, us- ing the results of the first stage to specify the probabilities. This network is generated only once for a particular PCFG and length bound. Once created, we can use this network to answer a variety of queries about possible strings and parse trees. In general, we can use the standard inference algo- rithms to compute the conditional probability or most proba- ble configuration of any collection of our basic random vari- ables, given any other event which can be expressed in terms of these variables. These algorithms have been implemented and tested on a number of grammars, with the results verified against those of existing dynamic programming algorithms when appli- cable, and against enumeration algorithms when given non- standard queries. When answering standard queries, the time requirements for network inference were comparable to those for the dynamic programming techniques. Our net- work inference methods achieved similar response times for some other types of queries, providing a vast improvement over the much slower brute force algorithms. The network representation of the probability distribu- tion also allows possible relaxations of the independence assumptions of the PCFG framework. We could extend the context-sensitivity of these probabilities within our net- work formalism by adjusting the probability tables associ- ated with our production nodes. For instance, we may make the conditional probabilities a function of the (i, j, k) index values. Alternatively, we may introduce additional depen- dencies on other nodes in the network, or perhaps on fea- tures beyond the parse tree itself. The context-sensitivity of (Charniak & Carroll 1994), which conditions the produc- tion probabilities on the parent of the left-hand side symbol, would require only an additional link from N nodes to their potential children P nodes. Other external influences could include explicit context representation in natural language problems or influences of the current world state in plan- ning, as required by many plan recognition problems (Py- nadath & Wellman 1995). Therefore, even though the evidence propagation is ex- ponential in the worst case, our method incurs this cost in the service of greatly increased generality. Our hope is that the enhanced scope will make PCFGs a useful model for plan recognition and other domains that require more flexi- bility in query forms and in probabilistic structure. In addi- tion, these algorithms may extend the usefulness of PCFGs in natural language processing and other pattern recognition domains where they have already been successful. Acknowledgments We are grateful to the anonymous re- viewers for careful reading and helpful suggestions. This work was supported in part by Grant F49620-94-I-0027 from the Air Force Office of Scientific Research. References Briscoe, ‘I., and Carroll, J. 1993. Generalized probabilistic LR parsing of natural language (corpora) with unification- based grammars. Computational Linguistics 19( 1):25-59. Charniak, E., and Carroll, G. 1994. Context-sensitive statistics for improved grammatical language models. In Proceedings of the National Conference on AI, 728-733. Charniak, E. 1993. Statistical Language Learning. Cam- bridge, MA: MIT Press. Chou, I? 1989. Recognition of equations using a two- dimensional stochastic context-free grammar. In Proceed- ings SPIE, Visual Communications and Image Processing ZV, 852-863. Earley, J. 1970. An efficient context-free parsing algo- rithm. Communications of the Association for Computing Machinery 13(2):94-102. Gonzalez, R. C., and Thomason, M. S. 1978. Syntac- tic pattern recognition: An introduction. Reading, MA: Addison-Wesley Publishing Company. 177-2 15. Jelinek, F.; Lafferty, J. D.; and Mercer, R. 1992. Basic methods of probabilisticcontext free grammars. In Laface, P., and DeMori, R., eds., Speech Recognition and Under- standing. Berlin: Springer. 345-360. Ney, H. 1992. Stochastic grammars and pattern recogni- tion. In Laface, P., and DeMori, R., eds., Speech Recogni- tion and Understanding. Berlin: Springer. 3 19-344. Pearl, J. 1987. Probabilistic Reasoning in Intelligent Sys- tems: Networks of Plausible Inference. San Mateo, CA: Morgan Kaufmann. Pynadath, D. V., and Wellman, M. P. 1995. Accounting for context in plan recognition, with application to traffic monitoring. In Proceedings of the Conference on Uncer- tainty in AI, 472-48 1. Sakakibara, Y.; Brown, M.; Underwood, R. C.; Mian, I. S.; and Haussler, D. 1995. Stochastic context-free grammars for modeling RNA. In Proceedings of the 27th Hawaii In- ternational Conference on System Sciences, 284-293. Vilain, M. 1990. Getting serious about parsing plans: A grammatical analysis of plan recognition. In Proceedings of the National Conference on AI, 190-197. Wetherell, C. S. 1980. Probabilistic languages: a review and some open questions. Comp. Surveys 12(4):361-379. Younger, D. 1967. Recognition and parsing of context-free languages in time n 3. Info. and Control lO(2): 189-208. 1290 Uncertainty | 1996 | 190 |
1,833 | On the Foundations of Qualitative Decision Theory Ronen I. Brafman Computer Science Department University of British Columbia Vancouver, B.C., Canada V6T 124 brafman@cs.ubc.ca Abstract This paper investigates the foundation of rnaxipnin, one of the central qualitative decision criteria, using the approach taken by Savage (Savage 1972) to inves- tigate the foundation and rationality of classical de- cision theory. This approach asks “which behaviors could result from the use of a particular decision pro- cedure?” The answer to this question provides two important insights: (1) under what conditions can we employ a particular agent model, and (2) how ratio- nal is a particular decision procedure. Our main re- sult is a constructive representation theorem in the spirit of Savage’s result for expected utility maximiza- tion, which uses two choice axioms to characterize the maxapnin criterion. These axioms characterize agent behaviors that can be modeled compactly using the maxcirnin model, and, with some reservations, indicate that rnaxionin is a reasonable decision criterion. Introduction Decision theory plays an important role in fields such as statistics, economics, game-theory, and industrial engineering. More recently, the realization that deci- sion making is a central task of artificial agents has led to much interest in this area within the artificial intel- ligence research community. Some of the more recent work on decision theory in AI concentrates on qual- itative decision making tools. For example, Boutilier (Boutilier 1994) and Tan and Pearl (Tan & Pearl 1994) examine semantics and specification tools for qualita- tive decision makers, while Darwiche and Goldszmidt (Darwiche & Goldszmidt 1994) experiment with qual- itative probabilistic reasoning in diagnostics. There are two major reasons for this interest in qual- itative tools. One reason is computational efficiency: one hopes that qualitative tools, because of their sim- plicity, will lead to faster algorithms. Another rea- son is a simpler knowledge acquisition process: often, qualitative information is easier to obtain from experts and layman. However, while there is abundant work on the foundations of quantitative approaches to deci- sion making, usually based on the principle of expected Moshe Tennenholtz Faculty of Industrial Engineering and Mgmt. Technion - Israel Institute of Technology Haifa 32000, Israel moshet@ie.technion.ac.il utility maximization (e.g.,(Savage 1972; Anscombe & Aumann 1963; Blum, Brandenburger, & Dekel 1991; Kreps 1988; Hart, Modica, & Schmeidler 1994))) we are aware of very little work on the foundations of qual- itative methods.’ Work on the foundations of decision theory is moti- vated by two major applications: agent modeling and decision making. Agent modeling is often the main concern of economists and game-theorists; they ask: under what assumptions can we model an agent as if it were using a particular decision procedure? In artificial intelligence, we share this concern in vari- ous areas, most notably in multi-agent systems, where agents must represent and reason about other agents. Decision making is often the main concern of statisti- cians, decision analysts, and engineers. They ask: how should we model our state of information? And how should we choose our actions? The relevance of this question to AI researchers is obvious. The foundational approach helps answer these questions by describing the basic principles that underlie various decision pro- cedures. One of the most important foundational results in the area of classical decision theory is Savage’s theorem (Savage 1972), d escribed by Kreps (Kreps 1988) as the “crowning achievement” of choice theory. Savage pro- vides a number of conditions on an agent’s preference among actions. Under these conditions, the agent’s choices can be described as stemming from the use of probabilities to describe her state of information, utilities to describe her preferences over action out- comes, and the use of expected utility maximization to choose her actions. For example, one of Savage’s pos- tulates, the sure-thing principle, roughly states that: if ‘An interesting related work is the axiomatic approach taken by Dubois and Prade (Dubois & Prade 1995), which proves the existence of a utility function representing a pref- erence ordering among possibility distributions. Many ax- iom systems that are weaker than Savage’s appear in (Fish- burn 1988), but we are not aware of any that resemble ours. Foundations 1291 From: AAAI-96 Proceedings. Copyright © 1996, AAAI (www.aaai.org). All rights reserved. an agent prefers action a over b given that the possible worlds are sr and &, and she prefers a over b when the possible worlds are ss and ~4, then she should still prefer a over b when the possible worlds are ~1, ~2, ss and ~4. Economists use Savage’s results to understand the assumptions under which they can use probabilities and utilities as the basis for agent models; decision the- orists rely on the intuitiveness of Savage’s postulates to justify the use of the expected utility maximization principle. Our aim in this paper is to initiate similar work on the foundations of qualitative decision making. Given that we have compelling practical reasons to investi- gate such tools, we would like to have as sound an understanding of the adequacy of qualitative decision tools as we do of the classical, quantitative tools; both for the purpose of decision making and agent modeling. Our main contribution is a representation theorem for the maximin decision criterion.2 Using a setting sim- ilar to that of Savage, we provide two conditions on an agent’s choice over actions under which it can be represented as a qualitative decision maker that uses maximin to make its choices. One of these conditions is similar to Savage’s sure thing principle. It says that if an agent prefers action a over b when the possible worlds are sr and s2 and she prefers a over b when her possible worlds are s2 and sa, then she still prefers a over b when the possible worlds are si, s2 and ~3. The other condition is more technical, and we defer its presentation to Section 4. Beyond qualitative decision theory, the results pre- sented in this paper have another interesting interpre- tation: There are different ways in which we can en- code an agent’s behavior (or program). One simple manner is as an explicit mapping from the agent’s local state to actions. Th.is is often highly inefficient in terms of space. Alternative, implicit, representations are of- ten used if we desire to cut down on program storage or transmission costs. Probabilities and utilities and their qualitative counterparts can be used to obtain a compact, albeit implicit, representation of programs. Our (constructive) results characterize a class of agent programs that can be represented in O(n log n) space, where n is the number of states of the world. This is to be contrasted with a possibly exponential explicit representation. In Section 2 we define a model of a situated agent and two alternative representations for its program or behavior. One is a simple policy that maps an agent’s 2 (Hart, Modica, & Schmeidler 1994) presents an axiom- atization for maximin in the context of 2-person zero-sum games. However, their axiomatization is probabilistic, and does not fit the framework of qualitative decision theory. 1292 Uncertainty state of information to actions, while the other rep- resents the agent’s program (or behavior) implicitly using the maximin decision criterion. Our aim is to present conditions under which policies can be repre- sented implicitly using the maximin criterion. This will be carried out in two steps: In Section 3 we dis- cuss the case of an agent which has to decide among two actions in various states, while in Section 4 we con- sider the case where the agent has any finite number of actions to choose from. Proofs of these results, which are constructive, are omitted due to space constraints. Section 5 concludes with a discussion of some issues raised by our results and a short summary. The Basic Model In this section, we define a fairly standard agent model and the concept of a policy, which describes the agent’s behavior. Then, we suggest one manner for implicitly representing (some) policies using the concept of utility and the decision criterion maximin. Definition 1 States is a finite set of possible states of the world. An agent is a pair of sets, (LocalStates, Actions), which are called, respectively, the agent’s set of local states and actions. PW : LocalStates + Zstates \ 8 is the function de- scribing the set of world states consistent with every local state. PW satisfies the following conditions: (1) PW(1) = PW(1’) iff 1 = I’, and (2) For each subset S of States, there exists some 1 E LocalStates such that PW(Z) = s. Each state in the set States describes one possi- ble state of the world, or the agent’s environment. This description does not tell us about the inter- nal state of the agent (e.g., the content of its reg- isters). These internal states are described by ele- ments of the set of local states. Intuitively, local states correspond to the agent’s possible states of in- formation, or its knowledge (see (Fagin et al. 1995; Rosenschein 1985)) .’ I n addition to a set of possible lo- cal states, the agent has a set Actions of actions. One can view these actions as the basic control signals the agent can send to its actuators. With every local state I E LocalStates we associate a subset PW(Z) of States, understood as the possible states of the world consistent with the agent’s infor- mation at 1. That is, s E PW(Z) iff the agent can be in local state I when the current state of the world is s. In fact, in this paper we identify I with PW(Z) and use both interchangeably. Hence, we require that I = E’ iff p W) = PW(Z’) and that for every S C States there exists some I E LocalStates such that PW(Z) = S. Like other popular models of decision making (e.g., (Savage 1972; Anscombe & Aumann 1963)), our model considers one-shot decision making. The agent starts at some initial state of information and chooses one of its possible actions based on its current state of infor- mation (i.e., its local state); this function is called the agent’s poEicy (see also protocol (Fagin et al. 1995) and strategy (Lute & Raiffa 1957)). This policy maps each state of information of the agent into an action. Definition 2 A policy for agent (LocalStates,Actions) is a function P : LocalStates + Actions. A naive description of the policy as an explicit mapping between local states and actions is exponen- tially large in the number of possible worlds because lLocaZStatesl = 21Statesl. Requiring a designer to sup- ply this mapping explicitly is unrealistic. Hence, a method for implicitly specifying policies is desirable. In particular, we would like a specification method that helps us judge the quality of a policy. Classical deci- sion theory provides one such manner: the policy is implicitly specified using a probability assignment pr over the set States and a real valued utility function u over a set 0 of action outcomes. The action to be per- formed at local state I is obtained using the principle of expected utility maximization: argmaxaEActiotas{ c p(s) . u@(s))} sEPW(I) where a(s) is the outcome of action a when the state of the world is s. We wish to present a different, more qualitative representation. We will not use a prob- ability function, and our utility function u(v) .) takes both the state of the world and the action as its argu- ments and returns some value in a totally pre-ordered set. (Notice the use of qualitative, rather than quan- titative representation of utilities.) For convenience, we will use integers to denote the relative positions of elements within this set. In our representation, the agent’s action in a local state 1 is defined as: That is, the agent takes the action whose worst-case utility is maximal. Maximin is a qualitative decision criterion that seems-w4l;tailored to risk-averse agents. Definition 3 A policy P has a maximin represen- tation if there exists a utility function on States x Actions such that for every 1 E LocalStates p(l) = a??maxaEActioras ( sEgg4a~ 4)). That is, P has a maximin representation if for every local state I, an agent with this utility function that makes her decision by applying maximin to the utilities of actions in PW(Z), would choose the action P(Z). Given an arbitrary agent and a policy P adopted by the agent, it is unclear whether this policy has a maximin representation. It is the goal of this pa- per to characterize the class of policies that have this representation. From this result, we can learn about the conditions under which we can use the maximin representation to model agents and under- stand the rationality of using maximin as a deci- sion criterion. Unlike the exponential naive repre- sentation of policies, the maximin representation re- quires only O(ZogM - IStates . IActions!) space, where M = maXa,a’EActions;s,s’EStates IU(a, S) - u(a’, s’) 1. epresenting Binary Decisions This section presents two representation theorems for maximin for agents with two possible actions. We start by describing a basic property of maximin rep- resentable policies. Definition 4 We say that a policy P is closed under union if P(U) = P(W) implies P(U U W) = P(U), where U, W C States. That is, suppose that the agent would take the same action a when its local state is I or I’, and let i be the local state in which the agent considers possible all worlds that are possible in I and in I’. That is PW(i) = PW(Z) U PW(Z’). If the agent’s policy is closed under unions, it would choose the action a at i. For example, suppose that our agent is instructed to bring coffee when it knows that the weather is cold or warm and when it knows that the weather is warm or hot. If all the agent knows is that the weather is cold, warm, or hot, it should still bring coffee if its pol- icy is closed under unions. This sounds perfectly rea- sonable. Consider another example: Alex likes Swiss chocolate, but dislikes all other chocolates. He finds an unmarked chocolate bar and must decide whether or not he should eat it. His policy is such that, if he knows that this chocolate is Swiss or American, he will eat it; if he knows that this bar is Swiss or French, he will eat it as well. If Alex’s policy is closed under unions, he will eat this bar even if he knows it must be Swiss, French, or American. Our first representation theorem for maximin shows that policies containing two possible actions that are closed under unions are representable using a utility function defined on Actions and States. Theorem 1 Let P be a policy assigning only one of two possible actions at each local state, and assume that P is closed under union. Then, P is maximin representable. Foundations 1293 Notice that this corresponds to a completeness claim, while soundness, which implies that the above conditions hold for maximin, is easily verified. The following example illustrates our result. Example I Consider the following policy (or precon- dition for wearing a sweater) in which Y stands for “wear a sweater” and N stands for “do not wear a sweater”. (cold} ( } ok ( hot} {c,o} (c,h} (o,h} ( c,o,h Y N N Y N N N It is easy to verify that this policy is closed under unions. For example, the sweater is not worn when the weather is ok or when the weather is either hot or cold, hence it is not worn when there is no information at all, i.e., the weather is either cold, ok, or hot. Using the proof of Theorem 1 we construct the fol- lowing utility function representing the policy above: A slight generalization of this theorem allows for policies in which the agent is indifferent between the two available choices. In the two action case discussed here, we capture such indifference by assigning both actions at a local state, e.g., P(1) = {a, a’}. Hence we treat the policy as assigning sets of actions rather than actions. We refer to such policies as set-valued policies, or s-policies. Closure under union is defined in this context as follows: Definition 5 An s-policy P is closed under unions if for every pair of local states U, W c States, P(UU W) is either P(W), P(U), or P(U) UP(W). We require a number of additional definitions before we can proceed with the representation theorem for s- policies. First, we define two binary relationships on subsets of States: Definition 6 U >p W, where U, W C States, if P(U u W) = P(U) and P(U) # P(W). U =p W, where U, W c States, if P(U),P(W) and P(U U W) are all different. That is U >p W tells us that the preferred action in U is preferred in U U W. U =p W is basically equivalent to U #p W and W #p U. Next, we define a condition on these relations which closely resembles transitivity. Definition 7 We say that >p is transitive-like if whenever U1*1 . . ‘*k-l Uk, where *j E (>p, =p), and Wl) #P(h), we have that Ul * Uk. Here, * is >p if any of the *i are >p, and it is =p otherwise. 1294 Uncertainty Finally, we say that P respects domination if the action assigned to the union of a number of sets does not depend on those sets that are dominated by other sets w.r.t. >p. Definition 8 We say that P respects domination if for all W, U, V E States we have that W >p U implies that P(W U U U V) = P(W U V). We have the following representation theorem for s- policies: Theorem 2 Let P be an s-policy for an agent (LocalStates, Actions) such that (I) (Actions1 = (a, a’), (2) P is closed under unions, (2) P respects domination, (4) >p is transitive-like. Then, P is max- imin representable. A General Existence Theorem In the previous section we provided representation the- orems for a class of policies in which the agent chooses between two actions. We would like to generalize these results to represent choice among an arbitrary set of ac- tions. We will assume that, rather than a single most preferred action, the agent has a total order over the set of actions associated with each local state. This total order can be understood as telling us what the agent would do should its first choice became unavail- able. The corresponding representation using maximin will tell us not only which action is most preferred, but also, which action is preferred to which. efinition 9 A generalized policy for an agent (LocalStates,Actions) is a function P : LocalStates + TO(Actions), where TO(Actions) is the set of total orders on Actions. Generalized policy P is maximin representable if there exists a utility function u(., .) on Statesx Actions such that a is preferred to a’ in local state 1 according to P(1) ifl for every pair of actions a, a’ E Actions and for every local state 1 E LocalStates. The generalization of closure under unions to gener- alized policies is not a sufficient condition on policies for obtaining a maximin representation. The following definition introduces an additional property needed: Definition 10 Let {+ w 1 W C_ S}, be a set of total orders over Act ions. aven s s’ E States and a, a’ E Actions , we write (~7) < ( > s’, a’) if (1) a’ ss a, a Fsl a’, and a’ +{s,s~) a; or (2) s = s’ and a’ gs a. We say that < is transitive-like if whenever (~1, al) < (~2, a2) < . . . < (sk, ak) and either (1) s 27’ zkid a 1 3 a’ 3 2 Figure 1: (s, a) < (s’, a’) ak s-s1 al and al +-sk al, or (2) s1 = Sk, then (sl, al) < (Sk, ak) . The left table in Figure 1 helps us clarify this def- inition. In it, we depict the conditions under which (s, a) < (s’, a’) holds. Th ere are three columns in this table, each showing the agent’s preference relation over actions in different local states. The possible worlds in these local states are s, s’, and {s,s’}. In s the agent prefers a’ over a, in s’ it prefers a over a’, but when all the agent knows is that the world is either in state s or s’, it prefers a’ over a. Roughly, we can say that (s, a) < (s’, a’) if the agent dislikes taking action a in state s more than it dislikes taking action a’ in state S’. The following example illustrates the transitivity- like condition. Example 2 Suppose that there are three possible states of the world: snowing and cold, raining and cold, neither and warm. I prefer skiing to walking when it is snowing, but prefer walking to skiing when it is raining. However, when I am uncertain about whether it will rain or snow, I’d choose to walk. In this case (ski, rain) < (walk, snow). I prefer skiing to jogging when it is warm, and I prefer jogging to skiing when it is raining. However, I really dislike jogging when it is not cold, so I prefer skiing to jogging if I am uncertain whether it is warm or snowing. Hence (jog, warm) < (ski, rain). Suppose that, in addition, I prefer walking to jogging when it is warm, and I prefer jogging to walking when it snows. The transitivity-like condition implies that (jog, warm) < (walk, snow), and hence I’d prefer walking to jogging if I am un- certain whether it will be warm or it will snow. This seems quite plausible. Theorem 3 Let Actions be an arbitrary set of ac- tions, and let >w, for every W C S, be a total order over Actions such that 1. if a sw a’ and a >v a’ then a Fwuv a’ j and 2. < is transitive-like. Then, {>w 1 W E S} is maximin representable. Again, it is easy to see that a preference relation based on maximin will have the properties described in this theorem, and this result can be viewed as a sound and complete characterization of the maximin criterion for total orders. In addition, this theorem characterizes a class of policies that can be represented using O(n . log(n)) space in contrast with the exponentially large naive represent ation. Discussion Decision theory is clearly relevant to AI, and there is little doubt about the need for decision making tech- niques that are more designer friendly and have nice computational properties. Qualitative decision proce- dures could offer such an alternative, but the question is: how rational are they ? One method of addressing this question is experimentation, as in (Darwiche & Goldszmidt 1994). However, the prominent approach for understanding and justifying the rationality of deci- sion criteria has been the axiomatic approach. This ap- proach characterizes the properties of a decision crite- rion in a general, domain independent manner. Given a particular domain of application, we can assess the rationality of employing a particular decision criterion using its characteristic properties. Our work provides one of a few results within the axiomatic approach that deals with qualitative decision criteria and helps us un- derstand the inherent properties of maximin, assess the rationality of using maximin, and understand the con- ditions under which an arbitrary agent can be modeled as if it were a qualitative decision maker. In classical decision theory, the agent has both a util- ity function and a probability function. In our repre- sentation theorems, the emphasis has been on utilities rather than beliefs. The agent’s state of information is modeled by means of the set of worlds consistent with its current local state, PW(1). Most authors (e.g., (Fa- gin et al. 1995)) regard this set as representing the agent’s knowledge, rather than belief. However, the concept of belief can be incorporated into this model by imposing additional structure on the set States in the form of a ranking function. This model has been suggested by e.g., (Brafman & Tennenholtz 1994; Friedman & Halpern 1994; Lamarre & Shoham 1994). Given a ranking function r : States + N, we define the agent’s beliefs at local state 1 as: B(1) = {s E PW(1) Is’ E PW(Z) implies r(s) 5 r(s’)}. B(1) are often called the agent’s plausible states at the local state 1. We can modify maximin by applying it to the plausible states, instead of the possible states (see, e.g.,(Brafman & Tennenholtz 1994)). That is, at state 1 the agent chooses ar9maXaactions Q-$j) 4% 41. Foundations 1295 A similar approach is taken in (Boutilier 1994; Tan & Pearl 1994)). Clearly, any behavior that is maximin representable can be represented using the ranked maximin repre- sentation suggested above. (We would use a ranking function that maps all states to the same integer). We can show that the converse is true as well. That is, if an agent can be represented as using ranked maximin it can also be represented as using the standard max- imin approach discussed in this paper; a formal proof is deferred to the full paper. Therefore, the ranked maximin representation is no more expressive than our standard maximin representation, i.e., it can capture the same set of behaviors. Hence, ranked maximin is not, a priori, a more rational decision criterion. Our work differs from most other foundational work in decision theory in its definition of the utility func- tion. We define utilities as a function of both the agent’s action and the state of the world. Savage, and many others, define the notion of an outcome, i.e., a description of the state of the world following the per- formance of an action. In these works, utilities are a function of outcomes. Savage defines actions as map- pings between states and outcomes, and it is possible to obtain the same outcome when two different actions are performed in two different states of the world. Our approach is motivated by the fact that, in practice, an agent chooses an action, not an outcome. That is, the only physically observable aspect of the agent’s behavior is its choice of action, e.g., the control sig- nal it sends to its actuators. The outcome of these actions is not directly chosen by the agent. Our rep- resentation is identical to the standard representation if it is assumed that the outcomes of different actions on different states are different. Moreover, using utility functions that depend on both the state and the action makes practical sense in our qualitative context: it is reasonable when the manner in which the outcome was received is important, e.g., the cost of an action, and it allows us to use the utility function to encode both the desirability of the action’s outcomes and the likeli- hood of the state in which it is obtained. Nevertheless, obtaining representation theorems for maximin in the more standard framework is an interesting challenge. Acknowledgments: Comments from Joe Halpern, Daniel Lehmann, and the anonymous referees provided much help in improving the content and presentation of this paper. We thank Ehud Kalai, Dov Monderer, and Ariel Rubinstein for useful comments and pointers to related work in other disciplines. References Anscombe, F. J., and Aumann, R. J. 1963. A defini- tion of subjective probability. Annals of M’athematical Statistics 34:199-205. Blum, L.; Brandenburger, A.; and Dekel, E. 1991. Lexicographic probabilities and equilibrium refine- ments. Econometrica 59:61-79. Boutilier, C. 1994. Toward a Logic for Qualitative Decision Theory. In Proc. of the 4th Int. Conf. on Prin. of Knowledge Rep. and Rears., 75-86. Brafman, R. I., and Tennenholtz, M. 1994. Belief ascription and mental-level modelling. In Proc. of the 4th Int. Conf . on Print. of Knowledge Rep. and Reas., 87-98. Darwiche, A., and Goldszmidt, M. 1994. On the re- lation between kappa calculus and probabilistic rea- soning . In Proc. 10th Conf. on Uncertainty in A I, 145-153. Dubois, D., and Prade, H. 1995. Possibility Theory as a Basis for Qualitative Decision Theory. In Proc. 14th International Joint Conference on Artificial In- telligence, 1924-1930. Fagin, R.; Halpern, J. Y.; Moses, Y.; and Vardi, M. Y. 1995. Reasoning about Knowledge. MIT Press. Fishburn, P. C. 1988. Nonlinear Preference and Util- ity Theory. Johns Hopkins University Press. Friedman, N., and Halpern, J. Y. 1994. A knowledge- based framework for belief change. Part I: Founda- tions. In Proc. of the 5th Conf. on Theoretical Aspects of Reas. About Knowledge. Hart, S.; Modica, S.; and Schmeidler, D. 1994. A neo bayesian foundation of the maxmin value for two-parson zero-sum games. International Journal of Game Theory 23~347-358. Kreps, D. M. 1988. Notes on the Theory of Choice. Boulder: Westview Press. Lamarre, P., and Shoham, Y. 1994. Knowledge, cer- tainty, belief and conditionalization. In Proc. of 4th Intl. Conf. on Principles of Knowledge Rep. and Reas. Lute, R. D., and Raiffa, H. 1957. Games and Deci- sions. New York: John Wiley & Sons. Rosenschein, S. J. 1985. Formal Theories of Knowl- edge in AI and Robotics. New Generation Computing 3(3):345-357. Savage, L. J. 1972. The Foundations of Statistics. New York: Dover Publications. Tan, S., and Pearl, J. 1994. Specification and Eval- uation of Preferences under Uncertainty. In Proc. of the 4th Int. Conf. on Principles of Knowledge Rep. and Reas., 530-539. 1296 Uncertainty | 1996 | 191 |
1,834 | Nir Friedman Stanford University Gates Building 1A Stanford, CA 9430590 10 nir@cs.stanford.edu Abstract In recent years, a number of different semantics for de- faults have been proposed, such as preferential structures, E- semantics, possibilistic structures, and K-rankings, that have been shown to be characterized by the same set of axioms, known as the KLM properties (for Kraus, Lehmann, and Magidor). While this was viewed as a surprise, we show here that it is almost inevitable. We do this by giving yet another semantics for defaults that uses plausibility measures, a new approach to modeling uncertainty that generalize other ap- proaches, such as probability measures, belief functions, and possibility measures. We show that all the earlier approaches to default reasoning can be embedded in the framework of plausibility. We then provide a necessary and sufficient con- dition on plausibilities for the KLM properties to be sound, and an additional condition necessary and sufficient for the KLM properties to be complete. These conditions are easily seen to hold for all the earlier approaches, thus explaining why they are characterized by the KLM properties. 1 Introduction There have been many approaches to default reasoning proposed in the literature (see (Ginsberg 1987; Gabbay, Hogger, & Robinson 1993) for overviews). The recent literature has been guided by a collection of axioms that have been called the KLMproperties (since they were dis- cussed in (Kraus, Lehmann, & Magidor 1990)), and many of the recent approaches to default reasoning, including preferential structures (Kraus, Lehmann, & Magidor 1990; Shoham 1987), e-semantics (Adams 1975; Geffner 1992b; Pearl 1989), possibilistic structures (Dubois & Prade 1991), and K-rankings (Goldszmidt & Pearl 1992; Spohn 1987), have been shown to be characterized by these properties. This has been viewed as somewhat surprising, since these approaches seem to capture quite different intuitions. As Pearl (1989) said of the equivalence between e-semantics and preferential structures, “It is remarkable that two totally different interpretations of defaults yield identical sets of conclusions and identical sets of reasoning machinery.” The goal of this paper is to explain why all these ap- proaches are characterized by the KLM properties. Our key tool is the use of yet another approach for giving se- mantics to defaults, that makes use of plausibility measures oseph U. Halpern IBM Almaden Research Center 650 Harry Road San Jose, CA 95 120-6099 halpern@almaden.ibm.com (Friedman & Halpern 1995a). A plausibility measure asso- ciates with a set a plausibility, which is just an element in a partially ordered space. The only property that we require is that the plausibility of a set is at least as great as the plausibility of any of its subsets. Probability distributions, Dempster-Shafer belief functions (Shafer 1976), and possi- bility measures (Dubois & Prade 1990) are all easily seen to be special cases of plausibility measures. Of more interest to us here is that all the approaches that have been used to give semantics to defaults that can be characterized by the KLM properties can be embedded into the plausibility framework. In fact, we show much more. All of these approaches can be understood as giving semantics to defaults by considering a class P of structures (preferential structures, possibilistic structures, etc.). A default d is then said to follow from a knowledge base A of defaults if all structures in P that satisfy A also satisfy d. We define a notion of qualitative plausibility measure, and show that the IUM properties are sound in a plausibility structure if and only if it is qualitative. Moreover, as long as a class P of plausibility structures sat- isfies a minimal richness condition, we show that the KIM properties will completely characterize default reasoning in P. We then show that when we map preferential structures (or possibilistic structures or any of the other structures con- sidered in the literature on defaults) into plausibility struc- tures, we get a class of qualitative structures that is easily seen to satisfy the richness conditioning. This explains why the KLM axioms characterize default reasoning in all these frameworks. Far from being surprising that the IUM ax- ioms are complete in all these cases, it is almost inevitable. The KLM properties have been viewed as the “conser- vative core” of default reasoning (Pearl 1989), and much recent effort has been devoted to finding principled meth- ods of going beyond KLM. Our result shows that any ap- proach that gives semantics to defaults with respect to a collection P of structures will almost certainly not go be- yond KIM. This result thus provides added insight into and justification for approaches such as those of (Bacchus et al. 1993; Geffner 1992a; Goldszmidt & Pearl 1992; Gold- szmidt, Morris, & Pearl 1993; Lehmann & Magidor 1992; Pearl 1990) that, roughly speaking, say d follows from A if d is true in a particular structure Pa E P that satisfies A Foundations 1297 From: AAAI-96 Proceedings. Copyright © 1996, AAAI (www.aaai.org). All rights reserved. (not necessarily all structures in P that satisfy A). This paper is organized as follows. In Section 2, we review the relevant material from (Friedman & Halpern 1995a) on plausibility measures. In Section 3, we review the KLM properties and various approaches to default rea- soning that are characterized by these properties. In Sec- tion 4, we show how the various notions of default reasoning considered in the literature can all be viewed as instances of plausibility measures. In Section 5, we define qualita- tive plausibility structures, show that the KLM properties are sound in a structure if and only if it is qualitative, and provide a weak richness condition that is necessary and suf- ficient for them to be complete. In Section 6, we discuss how plausibility measures can be used to give semantics to a full logic of conditionals, and compare this with the more traditional approach (Lewis 1973). In the full paper (Friedman & Halpern 1995b) we also consider the relation- ship between our approach to plausibility and epistemic en- trenchment (Gardenfors & Makinson 1988). We conclude in Section 7 with a discussion of other potential applications of plausibility measures. 2 Plausibility Measures A probability space is a tuple ( W, ,T, /-1), where W is a set of worlds, F is an algebra of measurable subsets of W (that is, a set of subsets closed under union and complementation to which we assign probability) and p is aprobabilitymeasure, that is, a function mapping each set in F to a number in [0, l] satisfying the well-known Kolmogorov axioms (~(0) = 0, p(W) = 1, and ,u(A U B) = p(A) + p(B) if A and B are disjoint). A plausibility space is a direct generalization of a prob- ability space. We simply replace the probability measure p by a plausibility measure Pl, which, rather than mapping sets in F to numbers in [0, I], maps them to elements in some arbitrary partially ordered set. We read PI(A) as “the plausibility of set A”. If PI(A) 5 PI(B), then B is at least as plausible as A. Formally, a plausibility space is a tuple S = (W, F’, Pl), where W is a set of worlds, J= is an algebra of subsets of W, and PI maps the sets in F to some set D, partially ordered by a relation 50 (so that SD is reflex- ive, transitive, and anti-symmetric). We assume that D is pointed: that is, it contains two special elements TD and ID such that 1~5~ d 2~ TD for all d E D; we further assume that Pl( W) = TD and Pl(@) =J_D . The only other assumption we make is Al. If A z B, then PI(A) SD Pi(B). Thus, a set must be at least as plausible as any of its subsets. Some brief remarks on the definition: We have delib- erately suppressed the domain D from the tuple S, since the choice of D is not significant in this paper. All that matters is the ordering induced by SD on the subsets in F. The algebra F also does not play a significant role in this paper; for our purposes, it suffices to take F = 2*. We have chosen to allow the generality of having an algebra of measurable sets to make it clear that plausibility spaces generalize probability spaces. For ease of exposition, we 1298 Uncertainty omit the F from here on in, always taking it to be 2w, and just denote a plausibility space as a pair (W, Pl). As usual, we define the ordering <D by taking d 1 <D d2 if dl ID d2 and dl # d2. We omit the subscript S from 50, <D, 7-0 and ID whenever it is clear from context. Clearly plausibility spaces generalize probability spaces. We now briefly discuss a few other notions of uncertainty that they generalize: A belieffunction Belon W is a function Bel : 2* --+ [0, l] satisfying certain axioms (Shafer 1976). These axioms certainly imply property Al, so a belief function is a plausibility measure. A fuzzy measure (or a Sugeno measure) f on W (Wang & Klir 1992) is a function f : 2* H [0, 11, that sat- isfies Al and some continuity constraints. A possi- bility measure (Dubois & Prade 1990) Poss is a fuzzy measure such that Poss( W) = 1, Pass(0) = 0, and Pass(A) = sup,,A(Poss({w}). An ordinal ranking (or n-ranking) K on W (as defined by (Goldszmidt & Pearl 1992), based on ideas that go back to (Spohn 1987)) is a function mapping subsets of wtoLV* = JV U {co} such that K(W) = 0, n(8) = 00, and n(A) = min,EA(K;({u})). Intuitively, an ordinal ranking assigns a degree of surprise to each subset of worlds in W, where 0 means unsurprising and higher numbers denote greater surprise. It is easy to see that if K: is a ranking on W, then (W, K) is a plausibility space, where x LJ,J* y if and only if y 5 z under the usual ordering on the ordinals. A preference ordering on W is a partial order < over W (Kraus, Lehmann, & Magidor 1990; Shoham 1987). Intuitively, w -c: w’ holds if w is preferred to w’.l Pref- erence orders have been used to provide semantics for default (i.e., conditional) statements. In Section 4 we show how to map preference orders on W to plausibil- ity measures on W in a way that preserves the ordering of events of the form {w} as well as the truth values of defaults. A parametrized probability distribution (PPD) on W is a sequence {Pi-i : i 2 0) of probability measures over W. Such structures provide semantics for defaults in e-semantics (Pearl 1989; Goldszmidt, Morris, & Pearl 1993). In Section 4 we show how to map PPDs into plausibility structures in a way that preserves the truth- values of conditionals. Plausibility structures are motivated by much the same concerns as two other recent symbolic generalizations of probability by Darwiche (1992) and Weydert (1994). Their approaches have a great deal more structure though. They start with a domain D and several algebraic operations that have properties similar to the usual arithmetic operations (e.g., addition and multiplication) over [0, 11. The result ‘We follow the standard notation for preference here (Lewis 1973; Kraus, Lehmann, & Magidor 1990), which uses the (perhaps confusing) convention of placing the more likely world on the left of the 4 operator. is an algebraic structure over the domain D that satisfies various properties. Their structures are also general enough to capture all of the examples above except preferential orderings. These orderings cannot be captured precisely because of the additional structure. Moreover, as we shall see, by starting with very little structure and adding just what we need, we can sometimes bring to light issues that may be obscured in richer frameworks. We refer the interested reader to (Friedman & Halpern 1995a) for a more detailed comparison to (Darwiche 1992; Weydert 1994). 3 Approaches to Default Reasoning: A Review Defaults are statements of the form “if cp then typi- cally/likely/by default +“, which we denote cp+$. For example, the default “birds typically fly” is represented Bird+FZy. There has been a great deal of discussion in the literature as to what the appropriate semantics of de- faults should be, and what new defaults should by entailed by a knowledge base of defaults. For the most part, we do not get into these issues here. While there has been little consensus on what the “right” semantics for defaults should be, there has been some consensus on a reasonable “core” of inference rules for default reasoning. This core, known as the KLM properties, was suggested by (Kraus, Lehmann, & Magidor 1990), and consists of the following axiom and rules of inference (where we use + to denote material implication): LLE. From cp ($ ‘p’ and cp-+ infer cp’+$ (left logical equivalence) RW. From 1c, j +’ and cp-+ infer ‘p--)$’ (right weaken- ing) REF. ‘p--)(p (reflexivity) AND. From cp+$~l and (p-)$2 infer cp--+$l A $2 OR. From cpl+$ and (pz--‘$ infer ‘p1 V cp2-4 CM. From cp-+ monotonicity) and (p--)7& infer (cautious LLE states that the syntactic form of the antecedent is irrel- evant. Thus, if ‘p1 and (~2 are equivalent, we can deduce (p2+1c( from cpl+$~. RW describes a similar property of the consequent: If T/J (logically) entails $J’, then we can de- duce cp+’ from cp++. This allows us to can combine default and logical reasoning. REF states that ‘p is always a default conclusion of cp. AND states that we can combine two default conclusions: If we can conclude by default both $1 and $2 from ‘p, we can also conclude $1 A $2 from ‘p. OR states that we are allowed to reason by cases: If the same default conclusion follows from each of two antecedents, then it also follows from their disjunction. CM states that if $1 and $~2 are two default conclusions of cp, then discov- ering that $1 holds when ‘p holds (as would be expected, given the default) should not cause us to retract the default conclusion $2. This system of rules is called system P in (Kraus, Lehmann, & Magidor 1990). The notation A l-p cp+$ denotes that cp+lc, can be deduced from A using these inference rules. There are a number of well-known semantics for defaults that are characterized by these rules. We sketch a few of them here, referring the reader to the original references for further details and motivation. All of these semantics involve structures of the form (W, X, T), where W is a set of possible worlds, T(W) is a truth assignment to primitive propositions, and X is some “measure” on W such as a preference ordering, a K-ranking, or a possibility measure. We define a little notation that will simplify the discussion below. Given a structure (W, X, T), we take [p] s W to be the set of of worlds satisfying cp, i.e., [(p] = {w E W : 7r(w)(cp) = true}. The first semantic proposal was provided by Kraus, Lehmann and Magidor (1990), using ideas that go back to (Lewis 1973; Shoham 1987). Recall that a preference ordering on W is partial order (i.e., h-reflexive and transi- tive relation) + over W. A preferential structure is a tuple (W, 4, T), where 4 is a partial order on W.2 The intu- ition (Shoham 1987) is that a preferential structure satisfies a conditional cp+$ if all the the most preferred worlds (i.e., the minimal worlds according to 4) in [cp] satisfy T,LJ. However, there may be no minimal worlds in [(PI. This can happen if [p] contains an infinite descending se- quence . . . 4 w2 4 201. What do we do in these struc- tures? There are a number of options: the first is to as- sume that, for each formula ‘p, there are minimal worlds in [(p]; this is the assumption actually made in (Kraus, Lehmann, & Magidor 1990), where it is called the smooth- ness assumption. A yet more general definition-one that works even if _( is not smooth-is given in (Lewis 1973; Boutilier 1994). Roughly speaking, cp-+$ is true if, from a certain point on, whenever ‘p is true, so is $. More formally, (W, i, T) satisfies cp+$, if for every world wl E [(p], there is a world w2 such that (a) w2 5 201 (so that w2 is at least as normal as WI), (b) w2 E [cp A GI], and (c) for all worlds w3 -( w2, we have w3 E [p =+ $1 (so any world more normal than 202 that satisfies cp also satisfies $). It is easy to verify that this definition is equivalent to the ear- lier one if 3 is smooth. A knowledge-base A preferentially entails cp-+qb, denoted A bP cp+$, if every preferential structure that satisfies (all the defaults in) A also satisfies P--+. Lehmann and Magidor show that preferential entailment is characterized by system P. *We note that the formal definition of preferential structures in (Kraus, Lehmann, & Magidor 1990; Lehmann & Magidor 1992) is slightly more complex. Kraus, Lehmann, and Magidor distinguish between states and worlds. In their definition, a preferential struc- ture is an ordering over states together with a labeling function that maps states to worlds. They take worlds to be truth assignments to primitive propositions. Our worlds thus correspond to states in their terminology, since we allow two worlds w # 20’ such that r(w) = n(w’). Despite these minor differences, all the results that we prove for our version of preferential structures hold (with almost no change in proof) for theirs. Foundations 1299 Theorem 3.1: (Lehmann & Magidor 1992; Boutilier 1994) A bp cp+ ifand only if A l-p cp+$. Thus, reasoning with preferential structures corresponds in a precise sense to reasoning with the core properties of default reasoning. As we mentioned earlier, we usually want to add addi- tional inferences to those sanctioned by the core. Lehmann and Magidor (1992) hoped to do so by limiting attention to a special class of preferential structures. A preferential structure (VV, 4, ?r) is rational if 4 is a modular order, so that for all worlds u, ZI, w E IV, if w _( o, then either u 4 w or w _( u. It is not hard to show that modularity implies that possible worlds are clustered into equivalence classes, each class consisting of worlds that are incomparable to one another, with these classes being totally ordered. Thus, rational structures form a “well-behaved” subset of pref- erential structures. Unfortunately, Lehmann and Magidor showed that restricting to rational structures gives no ad- ditional properties (at least, as far as the limited language of defaults is concerned). We say that a knowledge base A rationally entails cp+$, denoted A br cp+$, if every rational structure that satisfies A also satisfies (P+$.~ Theorem 3.2: (Lehmann & Magidor 1992) A br cp+ if and only if A t-p cp-+. Thus, we do not gain any new patterns of default inference when we restrict our attention to rational structures. Pearl (1989) considers a semantics for defaults grounded in probability, using an approach due to Adams (1975). In this approach, a default cp-+ denotes that Pr($ 1 cp) is ex- tremely high, i.e., almost 1. Roughly speaking, a collection A of defaults implies a default (p-+ if we can ensure that Pr(cpl$) is arbitrarily close to 1, by taking the probabilities of the defaults in A to be sufficiently high. The formal definition needs a bit of machinery.4 Recall that a PPD on W is a sequence (Pri : i 2 0) of probability measures over W. A PPD structure is a tuple ( W, { Pri : i > 0}, K), where (Pri} is PPD on W. Intuitively, it satisfies a conditional p4$ if the conditional probability $ given ‘p goes to 1 in the limit. Formally, cp-+ is satisfied if limi,, Pri( 1[$] I[$]) = 1 (Goldszmidt, Morris, & Pearl 1993) (where Pri ([$I 1 [p]) is taken to be 1 if Pri ([cp]) = 0). A e-entails cp-+$, denoted A i==E cp4$, if every PPD structure that satisfies all the defaults in A also satisfies cp-+$. Surprisingly, Geffner shows that e-entailment is equivalent to preferential entailment. Theorem 3.3: (Geffner 1992b) A bE cp-$ ifand only if A l--p ‘P--~(P.~ Possibility measures and ordinal rankings provide two more semantics for defaults. A possibility structure is a 3Rational entailment should not be confused with the notion of rational cZosure, also defined by Lehmann and Magidor. 4We adopt the presentation used in (Goldszmidt, Morris, & Pearl 1993). ‘Geffner’s result is stated in terms of the original formulation of c-entailment, as described in (Pearl 1989). However, results of (Goldszmidt, Morris, & Pearl 1993) show that the formulation we describe here is equivalent to the original one. tuple PS = (W, Pass, T) such that Poss is a possibility mea- sure on W. We say PS + posS cp-$ if either Poss([cp]) = 0 or Poss([p A +I) > Poss([rp A +jj). Intuitively, cp+$ holds vacuously if ‘p is impossible; otherwise, it holds if cp A ?I, is more “possible” than cp A +. For example, Bird+FZy is satisfied when Bird A FZy is more possible than Bird A ~Fly. Similarly, an ordinal ranking structure is a tuple R = (W, K, r) if K is a an ordinal ranking on W. We say that R bK. cp+$ if either K([cp]) = cc or K( [cp A $1) < K( [‘p A +j). We say that A possibilistically entails cp-@, denoted A b pass cp-tlc, (resp., A K-entails cp+$, denoted A bK. cp+$) if all possibility structures (resp., all ordinal ranking structures) that satisfy A also satisfy cp-+$. These two approaches are again characterized by the KLM properties. Theorem 3.4: (Geffner 1992b; Dubois & Prade 1991) Ab pass (p*$ if and only if A kn cp-+$ if and only if A l--p (p+. Why do we always seem to end up with the KLM proper- ties? As we are about to show, thinking in terms of plausi- bility measures provides the key to understanding this issue. 4 efault Reasoning sing Plausibility We can give semantics to defaults using plausibility mea- sures much as we did using possibility measures. A plau- sibility structure is a tuple PL = (W, Pl, T), where (W, Pl) is a plausibility space and T maps each possible world to a truth assignment. We define PL ~PL cp+$ if either Pl([cpn) =J- or Pl([rcp A $1) > Pl([cp A +I). Notice that if Pl is a probability function Pr, then cp+G holds exactly if either Pr( [p]) = 0 or Pr( [$] 1 [p]) > l/2. How does this semantics for defaults compare to others that have been given in the literature? It is immediate from the defini tions that the semantics we give to defaults in possi- bility structures is the same as that given to them if we view these possibility structures as plausibility structures (using the obvious mapping described above, and similarly for or- dinal ranking structures. What about preferential structures and PPD structures? Can we map them into plausibility structures while still preserving the semantics of defaults? As we now show, we can. Theorem 4.1: (a) Let _( be a preference ordering on W. There is a plausibility measure Pl, on W such that (W -04 I=p v-4 ifs(W PI+ j 4 I=PL cp-4. (b) Let P P = (Pri} be a PPD on W. There is a plausibility measure Plpp on W such that (W, {Pri},n) j=E C~+T,I!J ifs(W, WP, n> I==PL 9-b. Proof: We first sketch the proof of part (a). Let 4 be a preference order on W. We define a plausibility measure PI, on W as follows. Let Do be the domain of plausibility values consisting of one element d, for every element zu E W. We use 4 to determine the order of these elements: d, < d, if w 4 V. (Recall that w 4 20’ denotes that w is preferred to 20’.) We then take D to be the smallest set containing Do closed under least upper bounds (so that 1300 tincertainty every set of elements in D has a least upper bound in D). In the ful I paper, we show that this construction results in the followi&ordering over subsets of W: PI+(A) 5 PI<(B) if and only if for all w E A - B, there is a world w’ E B such that w’ 4 w and there is no w” E A - B such that w” 4 w’. It is not hard to show that PI< satisfies the requirements of the theorem. We next sketch the proof of part (b). Let PP = {Prl, Pr2, . . . , ) be a PPD on W. We define Plpp so that Plpp(A) 2 Plpp(B) iff limi_+m Pri(BIA U B) = 1. It is easy to see that this definition uniquely determines the effect of Plpp. It is also easy to show that Plpp satisfies the requirements of the theorem. Thus, each of the semantic approaches to default reason- ing that were described above can be mapped into plausi- bility structures in a way that preserves the semantics of defaults. 5 Default Entailment in Plausibility Structures In this section we characterize default entailment in plausi- bility structures. To do so, it is useful to have a somewhat more tures. general definition of entailment in plausibility struc- Definition 5.1: If P is a class of plausibility structures, then a knowledge base A entails cp+1c, with respect to P, denoted A +=p cp+$, if every PL E P that satisfies A also satisfies cp++. The classes of structures we are interested in include PpL, the class of all plausibility structures, and Pposs, F, 7, P’, and PC, the classes that arise from mapping possibil- ity structures, ordinal ranking structures, preferential struc- tures, rational structures, and PPDs, respectively, into plau- sibility structures. (In the case of possibility structures and ordinal ranking structures, the discussed in Section 2; in the mapping is the obvious one case-f preferential and ratio- nal structures and PPDs, the mapping is the one described in Theorem 4.1.) Recall the semantics of defaults. that afi these mappings preserve It is easy to check that our semantics for defaults does not guarantee that the axioms of system P hold in all struc- tures in PpL. In particular, they do not hold in probabil- ity structures. It is easy to construct a plausibility struc- ture PL where PI is actually a probability measure Pr such that Pr@ A 40) > 0, Pr([a]Ib]) > .5,Pr(UrDIILp]l) > -5, but Pr( [q A rl] lb]) < .5 and Pr( [r] [([p A 40) < S. Re- call that if Pr( ‘p) > 0 then cp+ti holds if and only if Pr($lv) > .5. Thus, PL /==p~ (true+p) A (true-q), but PLkPLtrue+p A q and PL&tpL(true A p+q). This gives us a violation of both AND and CM. We can similarly construct a counterexample to OR. On the other hand, as the following result shows, plausibility structures do satisfy the other three axioms of system P. Let system P’ be the system consisting of LLE, RW, and REF. Theorem 5.2: If A l--p/ cp-+$, then A kpp~ cp+$. What extra conditions do we have to place on plausibility structures to ensure that AND, OR, and CM are satisfied? We focus first on the AND rule. We want an axiom that cuts out probability functions, but leaves more qualitative notions. Working at a semantic level, taking [cp] = A, [$1]1 = &, and [$2] = B2, and using x to denote the complement of X, the AND rule translates to: A2’. Forall sets A, B1, and B2, ifPl(AnB,) > PI(AnBl) and Pl(A n B2) > Pl(A fl g), then Pl(A n B, n B2) > Pl(A n (B1 n &)). It turns out that in the presence of A 1, the following some- what simpler axiom is equivalent to A2’: A2. If A, B, and C are pairwise disjoint sets, PI(A U B) > PI(C), andPl(AUC) > PI(B), thenPl(A) > PI(BUC). Proposition 5.3: A plausibility measure satisfies A2 if and only if it satis$es A2’. A2 can be viewed as a generalization of a natural re- quirement of qualitative plausibility: if A, B, and C are pairwise disjoint, PI(A) > PI(B), and PI(A) > PI(C), then PI(A) > Pl( B U C). Moreover, since A2 is equivalent to A2’, and A2’ is a direct translation of the AND rule into con- ditions on plausibility measures, any plausibility structure whose plausibility measure satisfies A2 satisfies the AND rule. Somewhat surprisingly, a plausibility measure PI that satisfies A2 satisfies CM. Moreover, Pl satisfies the non- vacuous case of the OR rule. That is, if Pl( [(PI]) > I, then from cpl-+11, and (p2+$ we can conclude (cpl V (p2)-+~4.~ To handle the vacuous case of OR we need an additional axiom: A3. If PI(A) = PI(B) = I, then Pl(A U B) = 1. Thus, A2 and A3 capture the essence of the IUM properties. To make this precise, define a plausibility space (W, Pl) to be qualitative if it satisfies A2 and A3. We say PL = (W, Pl, 7r) is a qualitative plausibility structure if (TV, PI) is a qualitative plausibility space. Let PQPL consist of all qualitative plausibility structures. Theorem 5.4: If P C PQ pL, then for all A, p and 11, if A t--p cp+$, then A bp cp+$. Thus, the KLM axioms are sound for qualitative plausi- bility structures. We remark that Theorem 5.4 provides, in a precise sense, not only a sufficient but a necessary condition for a set of preferential structures to satisfy the KLM prop- erties. As we show in the full paper, if the KLM axioms are sound with respect to P, then even if there is a structure 6We remark that if we dropped requirement Al, then we can define properties of plausibilities measures that correspond pre- cisely to CM and OR. The point is that in the presence of Al, A2-which essentially corresponds to AND-implies CM and the non-vacuous case of OR. Despite appearances, Al does not corre- spond to RW. Semantically, RW says that if A and B are disjoint sets such that Pl( A) > Pl( I?), and A C A’, B’ C B, and A’ and B’ are disjoint, then Pl(A’) > Pl( B’). While this follows from A 1, it is much weaker than A 1. Foundations 1301 P= (VV,Pl,n) EPth a is not qualitative, P is “essentially t qualitative” for a11 practical purposes. More precisely, we can show that Pl’, the restriction of Pl to sets of the form 1~1 is qualitative. This, of course, leads to the question of which plausibil- ity structures are qualitative. All the ones we have been focusing on are. Theorem 5.5: Each of Pposs, Pli, P’, PP, and Pr is a subset of PQpL. It follows from Theorems 5.4 and 5.5 that the KLM prop- erties hold in all the approaches to defaults considered in Section 3. While this fact was already known, this result gives us a deeper understanding as to why the KLM prop- erties should hold. In a precise sense, it is because A2 and A3 holds for all these approaches. In the full paper we also show that each of the classes considered in Theorem 5.5 is, in a nontrivial sense, a subset of PQpL; this remains true even if we restrict to totally ordered plausibility measures in the case of Pposs and PlF.7 We now turn to the problem of completeness. To get soundness we had to ensure that P did not contain too many structures, in particular, no structures that are not qualitative. To get completeness we have to ensure that P contains “enough” structures. In particular, if A fp (p+$, we want to ensure that there is a plausibility structure PL E P such that PL b=p~ A and PLk,,cp+$. The following weak condition on P does this. Definition 5.6: We say that P is rich if for every collec- tion ~1, . . . , yn of mutually exclusive formulas, there is a plausibility structure PL = (W, PI, n) E P such that: The requirement of richness is quite mild . It says that we do not have a priori constraints on the relati ve plausibilities of a collection of disjoint sets. Certainly every collection of plausibility measures we have considered thus far can be easily shown-to satisfy this richness condition. Theorem 5.7: Each of Pposs, PK, Pp, Pr, PC, and PQpL is rich. More importantly, richness is a necessary and sufficient condition to ensure-that the KLM properties are complete. Theorem 5.8: A set P of qualitative plausibility structures is rich if and only iffor all A and defaults cp+$+ we have that A bp cp+ implies A I-_p cp+$. Putting together Theorems 5.4,5.5, and 5.8, we get Corollary 5.9: For P E {Pposs, PK, P*, Pr, PC, PQpL], and all A, cp, and $, we have A l-p cp-$ if and only if A +p (PA@. 7Since, for example, the range of a possibility measure is [O,l], there are totally ordered plausibility measures that are not possi- bility measures, although they may put the same ordering on sets. However, for example, we can have a qualitative plausibility mea- sure on {1,2} such that Pl({l}) = Pl((2)) < P1({1,2}). This cannot correspond to a possibility measure, since Poss({ 1)) = Poss( { 2)) would imply that Poss( { 1)) = Poss( { 1,2}). Not only does this result gives us a straightforward and uniform proof that the KLM properties characterize default reasoning in each of the systems considered in Section 3, it gives us a general technique for proving completeness of the KLM properties for other semantics as well. All we have to do is to provide a mapping of the intended semantics into plausibility structures, which is usually straightforward, and then show that the resulting set of structures is qualitative and rich. Theorem 5.8 also has important implications for attempts to go beyond the KLM properties (as was the goal in in- troducing rational structures). It says that any semantics for defaults that proceeds by considering a class P of structures satisfying the richness constraint, and defining A b=p cp+$ to hold if cp+ll, is true in every structure in P that satisfies A cannot lead to new properties for entailment. Thus, to go beyond KLM, we either need to consider interesting non-rich classes of structures, or to define a no- tion of entailment that does not amount to considering what holds in all the structures of a given class. It is possible to construct classes of structures that are arguably interesting and violate the richness constraint. One way is to impose independence constraints. For example, suppose we con- sider all structures where p is independent of Q in the sense that true-+q holds if and only if p+q holds if and only if ‘p-q holds, so that discovering either p or lp does not af- fect whether or not q is believed.8 Restricting to such struc- tures clearly gives us extra properties. For example, from true-q we can infer p+q, which certainly does not follow from the KLM properties. Such structures do not satisfy the richness constraint, since we cannot have, for example, whn) > wbhd) > wl~bin~ > wmm. Much of the recent work in default reasoning (Bacchus et al. 1993; Geffner 1992a; Goldszmidt & Pearl 1992; Gold- szmidt, Morris, & Pearl 1993; Lehmann & Magidor 1992; Pearl 1990) has taken the second approach, of not looking at entailment with respect to a class of structures. Roughly speaking, these approaches can be viewed as taking the basic idea of preferential semantics-placing a preference ordering on worlds-one step further: They try to get from a knowledge base one preferred structure (where the structure itself puts a preference ordering on worlds)-for example, in (Goldszmidt, Morris, & Pearl 1993), the PPD of maxi- mum entropy is considered-and carry out all reasoning in that preferred structure. We believe that plausibility mea- sures will provide insight into techniques for choosing such preferred structures, particularly through the use of indepen- dence, but the discussion of this issue is beyond the scope of this paper. 6 A Logic of Defaults Up to now, we have just focused on whether a set of defaults imphes another default. We have not considered a full logic of defaults, with negated defaults, nested defaults, and dis- *We remark that if we define independence appropriately in plausibility structures, this property does indeed hold; see (Fried- man & Halpem 1995a). 1302 Uncertainty junctions of defaults. It is easy to extend all the approaches we have defined so far to deal with such a logic. Condi- tional logic is a logic that treats 4 as a modal operator. The syntax of the logic is simple: let ICC be the language defined by starting with primitive propositions, and close off under A, 1, and +. Formulas can describe logical combination of defaults (e.g., (p+~) V (p 47)) as well as nested defaults (e.g., (P++-+~). The semantics of conditional logic is similar to the se- mantics of defaults.’ The usual definition (Lewis 1973) associates with each world a preferential order over worlds. We now give a similar definition based on plausibility mea- sures. Given a preferential structure PL = (IV, Pl, ‘i’r), we define what it means for a formula cp to be true at a world w in PL. The definition for the propositional connectives is standard, and for 3, we use the definition already given: e (PL, 2~) k p if r(w) + p for a primitive propositionp e (PL, w> b lcp if (PL, w) p ‘P e (PL,20)~cpA1C1if(PL,ul)~=and(PL,zu)~= * (PL, W) j= (p++ if either Pl([(p]p~) =I or Pl([p A $J]JPL) > Pl( I[cpA+~]lp~), where we define [(P]PL = {w E w : (Pl, w) b ‘p}? We now want to axiomatize default reasoning in this framework. Clearly we need axioms and inference rules that generalize those of system P. Let Np be an abbreviation for lcp+faZse. (This operator is called the outer modality in (Lewis 1973).) Expanding the definition of 4, we get that NV holds at w if and only if Pl([lcp]) =_L. Thus, Np holds if lcp is considered completely implausible. Thus, it implies that cp is true “almost everywhere”. Let system C be the system consisting of LLE, RW, and the following axioms and inference rules: CO. Cl. C2. C3. C4. c5. MP. It is The All the propositional tautologies P-P ((cP-++1) A (‘p47h2)) * (cp4($1 A $2)) ((Pl4+) A (P24@)) * ((PI v (P2)4$) ((P14(02) A (w4@)) * ((~1 A ‘p2)4$) [(P-+) * N(cp-+ti)] A [+4$) a N+p4ti)] From ‘p and cp +- + infer $. easy to see that system C is very similar to system P. richer language lets us replace a rule like AND by the axiom C2. Similarly, Cl to REF, C3 to OR and C4 to CM. We need CO and MP to deal with propositional reasoning. Finally, C5 captures the fact that the plausibility function Pl is independent of the world. Thus, if a default is true (false) at some world, it is true (false) at all of them. If we had enriched plausibility structures to allow a different plausibility function Pl, for each world w (as is done in the ‘The connections between default reasoning and conditional logics are well-known; see (Boutilier 1994; Kraus, Lehmann, & Magidor 1990; Katsuno & Satoh 1991). loWe redefine [v]p,r_ since 9 can involve conditional statements. Note that if cp does not contain occurrences of+ then this definition is equivalent to the one we gave earlier. Again, we omit the subscript when it is clear from the context. general definition of conditional logic (Lewis 1973)) then we would not need this axiom. There is no KLM property analogous to C5 since a formula such as N ((p-+$) involves nested 4’s. It is well known (Lewis 1973; Burgess 1981; Friedman & Halpern 1994) that system C captures reasoning in pref- erential structures. Theorem 6.1: (Burgess 1981; Friedman & Halpern 1994: System C is a sound and complete axiomatization of ,Ccc’ with respect to PP. Since the axioms of system C are clearly valid in all the structures in PQpL and PP C PQPL, we immediately get Corollary 6.2: System C is a sound and complete axioma- tization of ICC with respect to PQpL. This result shows that, at least as far as the language ,Cc goes, plausibility structures are no more expressive than preferential structures. We return to this issue in Section 7. The language lee allows us to make distinctions that we could not make using just implication between defaults, as in Section 4. For example, consider the following axiom: C6. (p---+ A l(cp A <47,b) a cp4lc Axiom C6 corresponds to the rule of rational mono- tonicity discussed in (Kraus, Lehmann, & Magidor 1990; Lehmann & Magidor 1992). It is not hard to show that C6 is valid in systems where the plausibility ordering is modu- lar. In particular, it is valid in each of Pr , Pposs, and Pn, although it is not valid in PP. In fact, it is well-known that system C+C6 is a sound and complete axiomatization of ,Cc with respect to P’ (Burgess 1981).” 7 Conch.lsions We feel that this paper unifies earlier results regarding the KLM properties, and explains why they arise so frequently. It also points out the advantage of using plausibility mea- sures as a semantics for defaults. Do we really need plausibility measures? If all we are interested in is propositional default reasoning and the KLM properties, then the results of Section 6 show that preferen- tial structures provide us all the expressive power we need. Roughly speaking, this is so because when doing proposi- tional reasoning, we can safely restrict to finite structures. (Technically, this is because we have a jnite model prop- erty: if a formula in ,C c is satisfiable, it is satisfiable in a finite plausibility structure (Friedman & Halpern 1994).) As we show in companion paper (Friedman, Halpern, & Koller 1996), preferential structures and plausibility struc- tures are no longer equally expressive once we move to a first-order logic, precisely because infinite structures now play a more important role. The extra expressive power of plausibility structures makes them more appropriate than “To capture Prc and Pposs, we need the additional axiom 7( true+faZse) . This axiom together with system C also pro- vides a complete axiomatization for PC. These results are based on well-known results in conditional logic (Burgess 1981; Friedman & Halpem 1994) and are proved in the full paper. Foundations 1303 preferential structures for providing semantics for first-order default reasoning. Beyond their role in default reasoning, we expect that plausibility measures will prove useful whenever we want to express uncertainty and do not want to (or cannot) do so using probability. For example, we can easily define a plau- sibilistic analogue of conditioning (Friedman & Halpern 1995a). While this can also be done in many of the other approaches we have considered, we believe that the gen- erality of plausibility structures will allow us to again see what properties of independence we need for various tasks. In particular, in (Friedman & Halpern 1996), we use plau- sibilistic independence to define a plausibilistic analogue of Markov chains. We plan to further explore the properties and applications of plausibility structures in future work. Acknowledgements The authors are grateful to Ronen Brafman, Adnan Dar- wiche, Moises Goldszmidt, Adam Grove, and Daphne Koller .for useful discussions relating to this work. The authors were supported in part by the Air Force Office of Scientific Research (AFSC), under Contract F49620-9 1 -C- 0080 and by NSF grant IRI-95-03 109. The first author was also supported in part by Rockwell Science Center. References Adams, E. 1975. The Logic of Conditionals. Reidel. Bacchus, F.; Grove, A. J.; Halpern, J. Y.; and Koller, D. 1993. Statistical foundations for default reasoning. In N- CAZ ‘93,563-569. Available at http://logos.uwaterloo.ca. Boutilier, C. 1994. Conditional logics of normality: a modal approach. Artificial Intelligence 68:87-154. Burgess, J. 198 1. Quick completeness proofs for some logics of conditionals. Notre Dame J. of Formal Logic 22~76-84. Darwiche, A. 1992. A Symbolic Generalization of Proba- bility Theory. Ph.D. Dissertation, Stanford University. Dubois, D., and Prade, H. 1990. An introduction to possi- bilistic and fuzzy logics. In Shafer, G., and Pearl, J., eds., Readings in Uncertain Reasoning. Morgan Kaufmann. Dubois, D., and Prade, H. 1991. Possibilistic logic, pref- erential models, non-monotonicity and related issues. In IJCAI ‘91,4 19-424. Friedman, N., and Halpern, J. Y. 1994. On the complexity of conditional logics. In KR ‘94,202-213. Friedman, N., and Halpern, J. Y. 1995a. Plausibility measures: a user’s manual. In UAZ ‘95, 175-184. Friedman, N., and Halpern, J. Y. 1995b. Plausibility measures and default reasoning. Technical Report RJ 9959, IBM. Available at http: //robotics -Stanford. edu/users/nir. Friedman, N., and Halpern, J. Y. 1996. A qualitative Markov assumption and its implications for belief change. Submitted to UAI ‘96. Friedman, N.; Halpern, J. Y.; and Keller, D. 1996. Condi- tional first-order logic revisited. In AAAZ ‘96. Gabbay, D. M.; Hogger, C. J.; and Robinson, J. A., eds. 1993. Nonmonotonic Reasoning and Uncertain Reason- ing, volume 3 of Handbook of Logic in Artificial Intelli- gence and Logic Programming. Oxford University Press. Gardenfors, P., and Makinson, D. 1988. Revisions of knowledge systems using epistemic entrenchment. In Proc. 2nd Con. on Theoretical Aspects of Reasoning about Knowledge. 83-95. Geffner, H. 1992a. Default Reasoning. MIT Press. Geffner, H. 1992b. High probabilities, model preference and default arguments. Mind and Machines 2:5 l-70. Ginsberg, M. L., ed. 1987. Readings in Nonmonotonic Reasoning. Morgan Kaufmann . Goldszmidt, M., and Pearl, J. 1992. Rank-based systems: A simple approach to belief revision, belief update and reasoning about evidence and actions. In KR ‘92, 661- 672. Goldszmidt, M.; Morris, P.; and Pearl, J. 1993. A maximum entropy approach to nonmonotonic reasoning. IEEE Trans. of Pattern Analysis and Machine Intelligence 15(3):220-232. Katsuno, H., and Satoh, K. 1991. A unified view of consequence relation, belief revision and conditional logic. In IJCAZ ‘91,406-4 12. Kraus, S.; Lehmann, D.; and Magidor, M. 1990. Non- monotonic reasoning, preferential models and cumulative logics. ArtiJcial Intelligence 44: 167-207. Lehmann, D., and Magidor, M. 1992. What does a con- ditional knowledge base entail? ArtiJiciaZ Intelligence 55: l-60. Lewis, D. K. 1973. Counterfactuals. Harvard University Press. Pearl, J. 1989. Probabilistic semantics for nonmonotonic reasoning: a survey. In KR ‘89,505-5 16. Pearl, J. 1990. System Z: A natural ordering of defaults with tractable applications to nonmonotonic reasoning. In Theoretical Aspects of Reasoning about Knowledge: Proc. 3rd Conference, 121-135. Shafer, G. 1976. A Mathematical Theory of Evidence. Princeton University Press. Shoham, Y. 1987. A semantical approach to nonmonotonic logics. In Proc. 2nd IEEE Symp. on Logic in Computer Science, 275-279. Spohn, W. 1987. Ordinal conditional functions: a dynamic theory of epistemic states. In Harper, W., and Skyrms, B., eds., Causation in Decision, Belief Change and Statistics, volume 2. Reidel. 105-134. Wang, Z., and Klir, G. J. 1992. Fuzzy Measure Theory. Plenum. Weydert, E. 1994. General belief measures. In UAZ ‘94, 575-582. 1304 Uncertainty | 1996 | 192 |
1,835 | First-Order Conditional Logic Revisited Nir Friedman Joseph Y. Walpern Daphne Koller Dept. of Computer Science Stanford University Gates Building 1A Stanford, CA 94305-9010 IBM Almaden Research Center 650 Harry Road San Jose, CA 95 120-6099 halpern@almaden.ibm.com Dept. of Computer Science Stanford University Gates Building 1A Stanford, CA 94305-90 10 nir@cs.stanford.edu koller@cs.stanford.edu Abstract Conditional Zogics play an important role in recent attempts to investigate default reasoning. This paper investigates first- order conditional logic. We show that, as for first-order probabilistic logic, it is important not to confound statisti- cal conditionals over the domain (such as “most birds fly”), and subjective conditionals over possible worlds (such as “I believe that lweety is unlikely to fly”). We then address the issue of ascribing semantics to first-order conditional logic. As in the propositional case, there are many possi- ble semantics. To study the problem in a coherent way, we use plausibility structures. These provide us with a general framework in which many of the standard approaches can be embedded. We show that while these standard approaches are all the same at the propositional level, they are signifi- cantly different in the context of a first-order language. We show that plausibilities provide the most natural extension of conditional logic to the first-order case: We provide a sound and complete axiomatization that contains only the KLM properties and standard axioms of first-order modal logic. We show that most of the other approaches have additional properties, which result in an inappropriate treatment of an infinitary version of the lottery paradox. ordering 4. If u, + w’, then the world w is strictly more preferred/more normal than w’. The formula Sird+FZy holds if in the most preferred worlds in which Bird holds, FZy also holds. (See Section 2 for more details about this and the other approaches.) 1 Introduction In recent years, conditional logic has come to play a major role as an underlying foundation for default reasoning. Two of the more successful default reasoning systems (Geffner 1992; Goldszmidt, Morris, & Pearl 1993) are based on con- ditional logic. Unfortunately, while it has long been rec- ognized that first-order expressive power is necessary for a default reasoning system, most of the work on conditional logic has been restricted to the propositional case. In this paper, we investigate the syntax and semantics ofJirst-order conditional logic, with the ultimate goal of providing a first- order default reasoning system. The extension of these approaches to the first-order case seems deceptively easy. After all, we can simply have a preferential ordering on first-order, rather than proposi- tional, worlds. However, there is a subtlety here. As in the case of first-order probabilistic logic (Bacchus 1990; Halpern 1990), there are two distinct ways to define condi- tionals in the first-order case. In the probabilistic case, the first corresponds to (objective) statistical statements, such as “90% of birds fly”. The second corresponds to subjec- tive degree of belief statements, such as “the probability that Tweety (a particular bird) flies is 0.9”. The first is captured by putting a probability distribution over the domain (so that the probability of the set of flying birds is 0.9 that of the set of birds), while the second is captured by putting a proba- bility on the set of possible worlds (so that the probability of the set of worlds where Tweety flies is 0.9 that of the set of worlds where Tweety is a bird). The same phenomenon occurs in the case of first-order conditional logic. Here, we can have a measure (e.g., a preferential ranking) over the domain, or a measure over the set of possible worlds. The first would allow us to capture qualitative statistical statements such as “most birds fly”, while the second would allow us to capture subjective beliefs such as “I believe that the bird Tweety is likely to fly”. It is important to have a language that allows us to distinguish between these two very different statements. Having distinguished between these two types of conditionals, we can ascribe semantics to each of them using any one of the standard approaches. Many seemingly different approaches have been pro- posed for giving semantics to conditional logic, including preferential structures (Lewis 1973; Boutilier 1994; Kraus, Lehmann, & Magidor 1990), e-semantics (Adams 1975; Pearl 1989), possibility theory (Benferhat, Dubois, & Prade 1992), and K-rankings (Spohn 1987; Goldszmidt & Pearl 1992). In preferential structures, for example, a model con- sists of a set of possible worlds, ordered by a preference There have been previous attempts to formalize first- order conditional logic; some are the natural exten- sion of some propositional formalism (Delgrande 1987; Brafman 1991), while others use alternative approaches (Lehmann & Magidor 1990; Schlechta 1995). (We defer a detailed discussion of these approaches to the full paper; see also Section 5.) How do we make sense of this plethora of alternatives? Rather than investigating them separately, we use a single common framework that generalizes almost all of them. This framework uses a notion of uncertainty Foundations 1305 From: AAAI-96 Proceedings. Copyright © 1996, AAAI (www.aaai.org). All rights reserved. called a plausibility measure, introduced by Friedman and Halpern (1995). A plausibility measure associates with set of worlds its plausibility, which is just an element in a par- tially ordered space. Probability measures are a subclass of plausibility measures, in which the plausibilities lie in [0, 11, with the standard ordering. In (Friedman & Halpern 1996), it is shown that the different standard approaches to condi- tional logic can all be mapped to plausibility measures, if we interpret Bird4FZy as “the set of worlds where Bird A FZy holds has greater plausibility than that of the set of worlds where Bird A +Zy holds”. The existence of a single unifying framework has al- ready proved to be very useful in the case of propositional conditional logic. In particular, it allowed Friedman and Halpern (1996) to explain the intriguing “coincidence” that all of the different approaches to conditional logic result in an identical reasoning system, characterized by the KLM axioms (Kraus, Lehmann, & Magidor 1990). In this paper, we show that plausibility spaces can also be used to clarify the semantics of first-order conditional logic. However, we show that, unlike the propositional case, the different ap- proaches lead to different properties in the first-order case. Intuitively, these are infinitary properties that require quan- tifiers and therefore cannot be expressed in a propositional language. We show that, in some sense, plausibilities pro- vide the most natural extension of conditional logic to the first-order case. We provide a sound and complete axioma- tization for the subjective fragment of conditional logic that contains only the KLM properties and the standard axioms of first-order modal logic. i (We provide a similar axiom- atization for the statistical fragment of the language in the full paper.) Essentially the same axiomatization is shown to be sound and complete for the first-order version of C- semantics, but the other approaches are shown to satisfy additional properties. One might think that it is not so bad for a conditional logic to satisfy additional properties. After all, there are some properties- such as indifference to irrelevant information- that we would like to be able to get. Unfortunately, the ad- ditional properties that we get from using these approaches are not the ones we want. The properties we get are re- lated to the treatment of exceptional individuals. This issue is perhaps best illustrated by the lottery paradox (Kyburg 1961).2 Suppose we believe about a lottery that any partic- ular individual typically does not win the lottery. Thus we get Vx(true-+lWinner(x)). (1) ‘By way of contrast, there is no (recursively enumerable) ax- iomatization of first-order probabilistic logic (Halpern 1990). ‘We are referring t o Kyburg’s original version of the lottery paradox (Kyburg 1961), and not to the finitary version discussed by Poole (1991). As Poole showed, any logic of defaults that satisfies certain minimal properties-properties which are satisfied by all the logics we consider-is bound to suffer from his version of the lottery paradox. Unfortunately, in many of the standard approaches, such as Delgrande’s (1987) version of first-order preferential struc- tures, from (1) we can conclude true4Vx(l Winner(x)). (3) Intuitively, from (1) it follows that in the most preferred worlds, each individual d does not win the lottery. There- fore, in the most preferred worlds, no individual wins. This is exactly what (3) says. Since (2) says that in the most pre- ferred worlds, some individual wins, it follows that there are no most preferred worlds, i.e., we have true4faZse. While this may be consistent (as it is in Delgrande’s logic), it implies that all defaults hold, which is surely not what we want. Of all the approaches, only c-semantics and plausi- bility structures, both of which are fully axiomatized by the first-order extension of the KLM axioms, do not suffer from this problem. It may seem that this problem is perhaps not so serious. After all, how often do we reason about lotteries? But, in fact, this problem arises in many situations which are clearly of the type with which we would like to deal. Assume, for example, that we express the default “birds typically fly” as Delgrande does, using the statement Vx(Sird(x)4FZy(x)). (4 If we also believe that Tweety is a bird that does not fly, so that our knowledge base contains the statement true4Bird( Tweety) A +Zy( Tweety), we could similarly conclude true4faZse. Again, this is surely not what we want. Our framework allows us to deal with these problems. Using plausibilities, (1) and (2) do not imply true4faZse, since (3) does not follow from (1). That is, the lottery paradox simply does not exist if we use plausibilities. The flying bird example is somewhat more subtle. If we take Tweety to be a nonrigid designator (so that it might denote different individuals in different worlds), the two statements are consistent, and the problem disappears. If, however, Tweety is a rigid designator, the pair is inconsistent, as we would expect. This inconsistency suggests that we might not always want to use (4) to represent “birds typically fly”. After all, the former is a statement about a property believed to hold of each individual bird, while the latter is a state- ment about the class of birds. As argued in (Bacchus et al. 1994), defaults often arise from statistical facts about the domain. That is, the default “birds typically fly” is often a consequence of the empirical observation that “almost all birds fly”. By defining a logic which allows us to express statistical conditional statements, we provide the user an alternative way of representing such defaults. We would, of course, like such statements to impact our beliefs about individual birds. In (Bacchus et al. 1994), the same issue was addressed in the probabilistic context, by presenting an approach for going from statistical knowledge bases to sub- jective degrees of belief. We leave the problem of providing a similar mechanism for conditional logic to future work. The rest of this paper is organized as follows. In Sec- tion 2, we review the various approaches to conditional 1306 Uncertainty logic in the propositional case; we also review the defi- nition of plausibility measures from (Friedman & Halpern 1996) and show how they provide a common framework for these different approaches. In Section 3, we discuss the two ways in which we can extend propositional conditional logic to first-order-statistical conditionals and subjective conditionals-and ascribe semantics to both using plausi- bilities. In Section 4, we provide a sound and complete axiomatization for first-order subjective conditional asser- tions. In Section 5, we discuss the generalization of the other propositional approaches to the first-order case, by investigating their behavior with respect to the lottery para- dox. We also provide a brief comparison to some of the other approaches suggested in the literature, deferring de- tailed discussion to the full paper. We conclude in Section 6 with discussion and some directions for further work. 2 Propositional conditional logic The syntax of propositional conditional logic is simple. We start with a set @ of propositions and close off under the usual propositional connectives (1, V, A, and a) and the conditional connective 4. That is, if ‘p and $ are formulas in the language, so is cp-++. Many semantics have been proposed in the literature for conditionals. Most of them involve structures of the form (W, X, T), where W is a set of possible worlds, K(W) is a truth assignment to primitive propositions, and X is some “measure” on W such as a preference ordering (Lewis 1973; Kraus, Lehmann, & Magidor 1990).3 We now describe some of the proposals in the literature, and then show how they can be generalized. Given a structure (W, X, T), let [p] C W be th e set of worlds satisfying cp. A possibility measure (Dubois & Prade 1990) Poss is a function Poss : 2w H [0, l] such that Poss( W) = 1, Poss(cI)) = 0, and Pass(A) = sup,,A(Poss({w}). A possibility structure is a tuple (W, Pass, T), where Poss is a possibility measure on W. It satisfies a conditional (p+$ if either Poss( [(PI) = 0 or Poss([cp A $1) > Poss([p A +J) (Dubois & Prade 1991). That is, ei- ther ‘p is impossible, in which case the conditional holds vacuously, or ‘p A $ is more possible than cp A T/J. A K-ranking (or ordinal ranking) on W (as defined by (Goldszmidt & Pearl 1992), based on ideas that go back to (Spohn 1987)) is a function K : 2w --+ N*, where nv* = LV U {oo}, such that K(W) = 0, ~(0) = 00, and K(A) = minzuEA(K({w})). Intuitively, an ordinal ranking assigns a degree of surprise to each subset of worlds in W, where 0 means unsurprising and higher numbers denote greater surprise. A K-structure is a tuple ( W, K, n), where K is an ordinal ranking on W. It satisfies a conditional cp+$ if either K( [pII) = 00 or K( [(PA@]) < 4~9 A +o. 3We could also consider a more general definition, in which one associates a different “measure” with each world, as done by Lewis, for example (Lewis 1973). It is straightforward to extend our definitions to handle this. Since this issue is orthogonal to the main point of the paper, we do not discuss it further here. A preference ordering on W is a partial order + over W (Kraus, Lehmann, & Magidor 1990; Shoham 1987). Intuitively, w 4 w’ holds if w is preferred to w’. A preferential structure is a tuple (W, 4, T), where 4 is a partial order on W. The intuition (Shoham 1987) is that a preferential structure satisfies a conditional cp++ if all the most preferred worlds (i.e., the minimal worlds according to 4) in [(pn satisfy $. However, there may be no minimal worlds in [(p& This can happen if [[CpJ contains an infinite descending sequence . . . 4 202 4 wt. The simplest way to avoid this is to assume that 4 is well-founded; we do so here for simplicity. A yet more general definition-one that works even if 4 is not well- founded-is given in (Lewis 1973; Boutilier 1994). We discuss that in the full paper. A parameterized probability distribution (PPD) on W is a sequence {Pri. : i 2 0) of probability measures over W. A PPD structure is a tuple (W, {Pri : i > 0)) T), where {Pr;} is PPD over W. Intuitively, it satisfies a conditional cp+$ if the conditional probability 1c, given cp goes to 1 in the limit. Formally, cp+lc, is satisfied if lim++, Pri (Uti] I IWJ) = 1 (where Pri (IMI I UpI) is t&n to be 1 if Pri ([VI) = 0). PPD structures were introduced in (Goldszmidt, Morris, & Pearl 1993) as a reformulation of Pearl’s e-semantics (Pearl 1989). These variants are quite different from each other. However, as shown in (Friedman & Halpern 1996), we can provide a uniform framework for all of them using the notion of plau- sibility measures. In fact, plausibility measures generalize other types of measures, including probability measures (see (Friedman & Halpern 1995)). A plausibility measure PI on W is a function that maps subsets of W to elements in some arbitrary partially ordered set. We read PI(A) as “the plausibility of set A”. If PI(A) 5 PI(B), then B is at least as plausible as A. Formally, a plausibility space is a tuple S = (W, PI), where W is a set of worlds and Pl maps subsets of W to some set D, partially ordered by a relation 5 (so that 5 is reflexive, transitive, and anti-symmetric). As usual, we define the ordering < by taking dl < d2 if dl < d2 and dl # d2. We assume that D is pointed: that is, it contains two special elements T and I such that J-5 d 5 T for all d E D; we further assume that Pl( W) = T and PI(@) =1. Since we want a set to be at least as plausible as any of its subsets, we require: Al. If A C B, then PI(A) 5 PI(B). Clearly, plausibility spaces generalize probability spaces. Other approaches to dealing with uncertainty, such as pos- sibility measures, K-rankings, and belief functions (Shafer 1976), are also easily seen to be plausibility measures. We can give semantics to conditionals using plausibility in much the same way as it is done using possibility. A plausibility structure is a tuple PL = (W,Pl, sir), where Pl is a plausibility measure on W. We then define: o PL b cp+$ if either Pl( [(PI) =I or Pl([cp A $1) > pw A +n). Intuitively, cp+$ holds vacuously if cp is impossible; oth- erwise, it holds if ‘p A $ is more plausible than ‘p A v,b. It is Foundations 1307 easy to see that this semantics for conditionals generalizes the semantics of conditionals in possibility structures and K-structures. As shown in (Friedman & Halpern 1996), it also generalizes the semantics of conditionals in preferential structures and PPD structures. More precisely, a mapping is given from preferential structures to plausibility structures such that (IV, 4, n) b cp if and only if (IV, Pl< , ;~r) b cp, where PI4 is the plausibility measure that corresponds to 4. A similar mapping is also provided for PPD structures. These results show that our semantics for conditionals in plausibility structures generalizes the various approaches examined in the literature. Does it capture our intuitions about conditionals? In the AI literature, there has been dis- cussion of the right properties of default statements (which are essentially conditionals). While there has been little consensus on what the “right” properties for defaults should be, there has been some consensus on a reasonable “core” of inference rules for default reasoning. This core, is known as the KLM properties (Kraus, Lehmann, & Magidor 1990).4 Do conditionals in plausibility structures satisfy these properties? In general, they do not. To satisfy the KLM properties we must limit our attention to plausibility struc- tures that satisfy the following conditions: A2. If A, B, and C are pairwise disjoint sets, Pl(A U B) > Pl(C),andPl(AUC) > Pi(B), thenPl(A) > PI(BUC). A3. If PI(A) = Pi(B) =I, then Pl(A U B) =1. A plausibility space (W, Pl) is qualitative if it satisfies A2 and A3. A plausibility structure (IV, PI, 7r) is qualitative if (IV, Pl) is a qualitative plausibility space. In (Friedman & Halpern 1996) it is shown that, in a very general sense, qualitative plausibility structures capture default reasoning. More precisely, the KLM properties are sound with respect to a class of plausibility structures if and only if the class consists of qualitative plausibility structures. Furthermore, a very weak condition is necessary and sufficient in order for the KLM properties to be a complete axiomatization of con- ditional logic. As a consequence, once we consider a class of structures where the KLM axioms are sound, it is almost inevitable that they will also be complete with respect to that class. This explains the somewhat surprising fact that KLM properties characterize default entailment not just in preferential structures, but also in E-semantics, possibility measures, and K-rankings. Each one of these approaches corresponds, in a precise sense, to a class of qualitative plausibility structures. These results show that plausibility structures provide a unifying framework for the characteri- zation of default entailment in these different logics. 3 First-order conditional logic We now want to generalize conditional logic to the first- order case. As mentioned above, there are two distinct notions of conditionals in first-order logic, one involving statistical conditionals and one involving subjective con- ditionals. For each of these, we use a different syntax, 4Due to space limitations we do not review the KLM properties here; see (Friedman & Halpem 1996) in this proceedings. 1308 Uncertainty analogous to the syntax used in (Halpern 1990) for the prob- abilistic case. The syntax for statistical conditionals is fairly straightfor- ward. Let <p be a first-order vocabulary, consisting of predi- cate and function symbols. (As usual, constant symbols are viewed as O-at-y function symbols.) Starting with atomic formulas of first-order logic, we form more complicated formulas by closing off under truth-functional connectives (i.e., A, V ‘, 1, and +), first-order quantification, and the fam- ily of modal operators cp 0~ $,-where Z is a sequence of distinct variables. We denote the resulting language iCstat. The intuitive reading of cp ~2 $J is that almost all of the Z’s that satisfy cp also satisfy $J. Thus, the wz modality binds the variables 5’ in ‘p and $. A typical formula in this language is 3y(P(z, y) c\/fZ Q(x, y)), which can be read “there is some y such that most z’s satisfying P(x, y) also satisfy Q(z, Y)“.~ Note that we allow arbitrary nesting of first-order and modal operators. The syntax for subjective plausibilities is even sim- pler than that for statistical plausibilities. Starting with a first-order vocabulary a, we now close off under truth- functional connectives, first-order quantification, and the single modal operator +. Thus, a typical formula is Vz(P(z)+3yQ(z, y)). Let ,Csuaj be the resulting lan- guage (the “subj” stands for “subjective”, since the con- ditionals are viewed as expressing subjective degrees of belief). We can ascribe semantics to both types of conditionals using any one of the approaches described in the previous section, (In fact, we do not even have to use the same approach for both.) However, since we can embed all of the approaches within the class of plausibility structures, we use these as the basic semantics. As in the propositional case, we can then analyze the behavior of the other approaches simply by restricting attention to the appropriate subclass of plausibility structures. To give semantics to lCstat, we use (@St-order) statisti- cal plausibility structures, which generalize the semantics of statistical probabilistic structures (Halpern 1990) and statistical preferential structures (Brafman 1991). Statis- tical plausibility structures are tuples of the form PL = (Dam, K, P), where Dom is a domain, 7r is an interpretation assigning each predicate symbol and function symbol in @ a predicate or function of the right arity over Dom, and P associates with each number n a plausibility measure PI, on Damn. As usual, a valuation maps each variable to an element of Dom. Given a structure PL and a valuation V, we can associate with every formula cp a truth value in a straightforward way. The only nontrivial case is ‘p -5 $. We define &Y.,~ ,Q ( P> = {cf : (PL, z1[2/4) b cp}, where z~[Z/d is a valuation that maps each x in IE: to the corre- sponding element in d and agrees with v elsewhere. o (PL, V) b ‘p ~3 1c, if either PI, (1(p~,~ ,2)(v)) =I or PL(+L,,,&J A ti>) > ~L&Y+,z)(‘P A +>), where n is the length of Z. ‘This syntax is borrowed from Brafman (1991), which in turn is based on that of (Bacchus 1990; Halpem 1990). We remark that we need the sequence of plausibility mea- sures to deal with tuples of different arity. The analogous se- quence of probability measures was not needed in (Halpern 1990), since, given a probability measure on Dom, we can consider the product measure on Damn. In the full paper, we place some requirements on PI, to force it to have the key properties we expect of product measures. We omit further discussion of statistical plausibilities here, and focus instead on subjective plausibilities. To give semantics to C subj, we use (‘rst-order) subjec- tive plausibility structures. These are tuples of the form PL = (Dom, IV, PI, 7r), where Dom is a domain, (IV, PI) is a plausibility space and ~(20) is an interpretation assign- ing to each predicate symbol and function symbol in @ a predicate or function of the right arity over Dom. We de- fine the set of worlds that satisfy ‘p given the valuation v to be EP](PL+) = -b : (PL, UI, V) k cp}. (We omit the subscript whenever it is clear from context.) For subjective conditionals, we have We do not treat terms as rigid designators here. That is, in different worlds, a term can denote different individuals. For example, if n( zo) (c) # 7r( w’) (c), the constant c denotes different individuals in ‘w and UI’. Because terms are not rigid designators, we cannot substitute terms for universally quantified variables. (A similar phenomenon holds in other modal logics where terms are not rigid (Garson 1977).) For example, let Clap be an abbreviation for lcp+faZse. Notice that (PL, W) /= q cp if Pl( [-(p]) =I; i.e., q cp asserts that the plausibility of lcp is the same as that of the empty set, so that cp is true “almost everywhere”. We define 0~ as ~Olcp; this says that ‘p is true in some non-negligible set of worlds. Suppose c is a constant that does not appear in the formula ‘p. As we show in the full paper, VxO~(x) + 09(c) is not valid in our framework; that is, we cannot substitute constants for universally quantified variables. We could substitute if c were rigid. We can get the effect of rigidity by assuming that 3x( 0(x = c)) holds. Thus, we do not lose expressive power by not assuming rigidity. 4 Axiomatizing default reasoning in plausibility structures We now want to show that plausibility structures provide an appropriate semantics for a first-order logic of defaults. As in the propositional case, this is true only if we restrict attention to qualitative plausibility structures, i.e., those sat- isfying conditions A2 and A3 above. Let PzTf be the class of all subjective qualitative plausibility structures. We provide a sound and complete axiom system for Pz$F, and show that it is the natural extension of the KLM properties to the first-order case. The axiomatization C”” bj, specified in Figure 1, consists of three parts. The first set of axioms (CO-C5 together with the rules MP, LLE, and RW) is simply the standard axiomatization of propositional conditional logic (Hughes CO. Cl. c2. c3. c4. cs. Fl. All instances of propositional tautologies Y-Y b-44 A ((P-$2)) * (P-+1 A $2)) Ew-+Yv * ((P2-4)) * ((Pl v $92)~$9 ((w-+w) A (w-4)) * ((Pl A P2)-4) KY-+,) * %J-491 A HP-4) * q +--+~,)] Vxy + y[x/t], where t is substitutable for x in the sense . -_ - discussed below F2. ‘v’x(p G- y!+ + (Vxp 3 b’x$) F3. cp + Vx y if x does not occur free in 9 F4. x = x FS. x = y + (p + cp’), where p is a quantifier-free and +-free formula and ‘p’ is obtained from y by replacing zero or more occurrences of x in ~7 by y F6. q vxp * vxop F7. x = y + 0(x = y) F8. x # Y * “(x # Y> Ml? From cp and cp j $ infer + LLE. From cpl H ~2 infer cpl-$ e ~2-+$ RW. From $1 + $2 infer v+$l + cp--+&. Figure 1: The system CSUbj consists of all generalizations of the following axioms (where cp is a generalization of 1c, if cp is of the form Vxi . . . Vxn $) and rules; x and y denote variables, while t denotes an arbitrary term. & Cresswell 1968); the second set (axioms Fl-F5) consists of the standard axioms of first-order logic (Enderton 1972); the final set (F6-F8) contains the standard axioms relating the two (Hughes & Cresswell 1968). F6 is known as the Barcan formula; it describes the relationship between 0 and ‘v’ in structures where all the worlds have the same domain (as is the case here). F7 and F8 describe the interaction between 0 and equality, and hold because we are essentially treating variables as rigid designators. It remains to explain the notion of “substitutable” in Fl. Clearly we cannot substitute a term t for x with free variables that might be captured by some quantifiers in cp; for example, while Vx3y( x # y) is true as long as the domain has at least two elements, if we substitute y for x, we get 3y(y # y), which is surely false. In the case of first-order logic, it suffices to define “substitutable” so as to make sure this does not happen (see (Enderton 1972) for details). However, in modal logics such as this one, we have to be a little more careful. As we observed in Section 3, we cannot substitute terms for universally quantified variables in a modal context, since terms are not in general rigid. Thus, we require that if cp is a formula that has occurrences of +, then the only terms that are substitutable for x in cp are other variables. Theorem 4.1: Csubj is a sound and complete axiomatiza- tion of Csubj with respect to P2UTJF. We claim that CSubj is the weakest “natural” first-order extension of the KLM properties. The bulk of the proposi- tional fragment of this axiom system (axioms Cl-C4, LLE, and RW) corresponds precisely to the KLM properties. The Foundations 1309 remaining axiom (C5) captures the fact that the plausibility function PI is independent of the world. This property does not appear in (Kraus, Lehmann, & Magidor 1990) since they do not allow nesting of conditionals. As discussed above, the remaining axioms are standard properties of first-order modal logic. 5 Alternative Approaches In the previous section we showed that CsUbj is sound and complete with respect to PyUTJF. What happens if we use one of the approaches described in Section 2 to give seman- tics to conditionals? As noted above, we can associate with each of these approach a subset of qualitative plausibility structures. Let ?f;yj , PfU bj , P,“, b j , ~~~~~ , and piU bj be the subsets of PyUfJ! that correspond to well-founded preferen- tial orderings, preferential orderings, K-rankings, possibility measures, and PPDs, respectively. From Theorem 4.1, we immediately get Theorem 5.1: Csubj is sound in P~~~j, Ps”;“bj, P~ubj, PK subj, PftijS, and Piz,bj. Is CSubj complete with respect to these approaches? Even at the propositional level, it is well known that be- cause K rankings and possibility measures induce plausibil- ity measures that are total (rather than partial) orders, they satisfy the following additional property: c6. cp++ A +P-+~~> =j (P A++). In addition, the plausibility measures induced by K rankings, possibility measures, and E semantics are easily seen to have the property that T > 1. This leads to the following axiom: C7. 1 (true--tfaZse) . In the propositional setting, these additional axioms and the basic propositional conditional system (i.e., CO-U, MP, LLE, and RW) lead to sound and complete axiomatization of the corresponding (propositional) structures. Does the same phenomenon occur in the first-order case? For c-semantics, it does. Theorem 5.2: Csubj +C7 is a sound and complete axiom- atization of Lsubj w.rt. Piubj. But, unlike the propositional case, the remaining approaches all satisfy properties beyond CSubj, C6, and C7. And these additional properties are ones that we would argue are un- desirable, since they cause the lottery paradox. Recall that the lottery paradox can be represented with two formulas: (1) Vx(true+lWinner(x)) states that every individual is unlikely to win the lottery, while (2) true+ElxWinner(x) states that is is likely that some individual does win the lottery. We start by showing that (1) and (2) are consis- tent in PzTf. We define a first-order subjective plau- sibility structure PLt,, = (Dowof, Wet, P1lof, u) as fol- lows: Domtot is a countable domain consisting of the in- dividuals 1,2,3, . . .; IVlot consists of a countable num- ber of worlds 2oi,u12, wg, . . .; Pll,, gives the empty set plausibility 0, each non-empty finite set plausibility l/2, and each infinite set plausibility 1; finally, the denotation of Winner in world wi according to Q,~ is the singleton set {di} (that is, in world wi the lottery winner is in- $idual di). It is easy to check that [lWinner(di)] = - { wi}, so PI& [rl Winner(d = 1 > l/2 = Pl( [Winner(di)]); h ence, PLtot satisfies (1). On the other hand, [3x Winner(x)] = PI& [13x Winner(x)]); W, so Plt,,([3xWinner(x)]) > h ence PLt,, satisfies (2). It is also easy to verify that Pll,, is a qualitativemeasure, i.e., satisfies A2 and A3. A similar construction allows us to capture a situation where birds typically fly but we know that Tweety does not fly. What happens to the lottery paradox in the other ap- proaches? First consider well-founded preferential struc- tures, i.e., Pf;yj. In these structures, ~47) holds if $ holds in all the preferred worlds that satisfy ‘p. Thus, (1) implies that for any domain element d, d is not a winner in the most preferred worlds. On the other hand, (2) implies that in the most preferred worlds, some domain element wins. To- gether both imply that there are no preferred worlds. When, in general, does an argument of this type go through? As we now show, it is a consequence of A2*. If {Ai : i E I} are pairwise disjoint sets, A = UiE=Ai, 0 E I, and for all i E I - {0}, Pl(A - Ai) > Pl(Ai), then Pl(Ae) > Pl(A - Ao). Recall that A2 states that if Ao, Al, and A2 are disjoint, Pl(Ao u A) > PI(&), and Pl(Ao U AZ) > Pl(At), then Pl(Ao) > Pl(Ai U AZ). It is easy to check that for any finite number of sets, a similar property follows from Al and A2 by induction. A2* asserts that a condition of this type holds even for an infinite collection of sets. This is not implied by Al and A2. To see this, consider the plausibility model PLt,, that we used to capture the infinite lottery: Take A0 to be empty and take Ai, i > 1, to be the singleton consisting ofthe world wi. ThenPlI,,(A-Ai) = 1 > l/2 = PIi,,( but PII,, = 0 < 1 = Pl(Ui>oAi). Hence, A2* does not hold for plausibility structures in general. It does, however, hold for certain subclasses: Proposition 5.3: AZ* holds in every plausibility structure in ~‘;L’;(i - - and P,“, b j. In the full paper we show that A2* is characterized by the axiom called V3 by Delgrande: V3* Vx(cp+$) + (cp-+Vx7/~) if x does not occur free in cp. This axiom can be viewed as an infinitary version of ax- iom C2 (which is essentially IUM’s And Rule). Since A2* holds in Pf’hyj and P~ubj, it follows that V3 does as well. It is easy to see that the axiom V3 leads to the lottery para- dox: From Vx(true-+l Winner(x)), V3 would imply that true+Vx( 1 Winner(x)). As we show in the full paper, A2* does not hold in Przii and P~ubj. In fact, the infinite lottery is consistent in these classes, although a somewhat unnatural model is required to express it. For example, we can represent the lottery via a possibility structure (Domt,,, Wtot, Pass, Q,~), where all the components besides Poss are just as in the plausibil- ity structure PLt,, that represents the lottery scenario, and POSS(Wi) = i/(i + 1). Th is means that if i > j, then it 13 10 Uncertainty is more possible that individual i wins than individual j. Moreover, this possibility approaches 1 as i increases. It is not hard to show that -this possibility structure satisfies formulas (1) and (2). We can block this type of behavior by considering a crooked lottery, where there is one individual who is more likely to win than the rest, but is still unlikely to win. To formalize this in the language, we add the following formula that we call Crooked: dx(Winner(x)+faZse) A 3yVx(x # y * (( Winner( 2) V Winner(y)) -+ Winner( 9))) The first part of this formula states that each individual has some plausibility of winning; in the language of plausibility, this means that Pi(d) >I for each domain element d. The second part states that there is an individual who is more likely to win than the rest. To see this, recall that ((pV$) +$ implies that either Pl([cp V $1) =I (which cannot happen here because of the first clause of Crooked) or Pl( [(p]) < Pl( [[$I). We take the crooked lottery to be formalized by the formula Yx(true+l Winner(x)) A (true-ax Winner( 2)) A Crooked. Note, that \Jz (true+ lVGnner(z)) implies that every individual is unlikely to win. It is easy to model the crooked lottery using plausibil- ity. Consider the structure PLiOl = (Donq,,, VVIOt, Pl$,, Q~), which is identical to PLlot except for the plausibility mea- sure Pl$,. We define Pl$,(wt ) = 3/4; PI;&&) = l/2 for i > 1; P&,(A) of a finite set A is 3/4 if w1 E A, and l/2 if wt @ A; and PII,, = 1 for infinite A. It is easy to verify that PL$, satisfies Crooked, taking dt to be the special individual who is most likely to win (since Pl( [Winner(dt)]) = 3/4 > l/2 = Pl( [winn+&)]) for i > 1). It is also easy to verify that PLiOt j= Vx(true-+~Winner(x:)) A (true+!lxWinner(x)). As we show in the full paper, the crooked lottery cannot be captured in P,“,“ig and P~~bi. This shows that, once we move to first-order logic, possibility structures and preferen- tial structures satisfy-extra properties over and above those characterized by CSubj. Although our focus thus far has been on subjective con- ditionals, the situation for statistical conditionals is similar. We have already remarked that we can construct “statisti- cal” first-order analogues of all the approaches considered in the propositional case. As in the subjective case, all of them suffer From problems except for the one based on e-semantics. We illustrate this using by considering the ex- tension of well-founded preferential structures to first-order conditionals over the domain, as defined by Brafman (199 1). Consider the statement Vy(true -2 %Varried( x, y)) (5) This states that for any individual y, most individuals are not married to y. This seems reasonable since each y is married to at most one individual, which clearly constitutes a small fraction of the population. The analogue of V3 holds in Brafman’s logic, for the same reason that it does in Pf;‘l(j. As a consequence, (5) implies true cvfI bfy+farried( x, y) . That is, most people are not married! This certainly does not seem to be a reasonable conclusion. It is straightforward to construct similar examples for the statistical variants of the other approaches, again, with the exception of plausibility structures and c-semantics. We note that these problems occur for precisely the same reasons they occur in the sub- jective case. In particular, property A2*, when stated for the plausibility over domain elements, is the necessary property for the statistical analogue of ‘~‘3. We observe that problems similar to the lottery paradox occur in the approach of Lehmann and Magidor (1990), which can be viewed as a hybrid of subjective and statistical conditionals based on on preferential structures. Finally, we observe that the approach of (Schlechta 1995), which is based on a novel representation of “large” subsets, is in the spirit of our notion of statistical defaults (although his language is somewhat less expressive than ours). We defer a detailed discussion of these approaches to the full paper. We have shown how to ascribe semantics to a first-order logic of conditionals in a number of ways. Our analysis shows that, once we move to the first-order case, significant differences arise between approaches that were shown to be equivalent in the propositional case. This vindicates the intuition that there are significant differences between these approaches, which the propositional language is simply too weak to capture. Our analysis also supports our choice of plausibility structures as the semantics for first-order de- faults: it shows that, with the exception of c-semantics, all the previous approaches have significant shortcomings, which manifest themselves in lottery-paradox type situa- tions. What does all this say about default reasoning? As we have argued, statements like “birds typically fly” should perhaps be thought of as statistical statements, and should thus be represented as Bird(x) -Z Fly(z). Such a repre- sentation gives us a logic of defaults, in which statements such as “birds typically fly” and “birds typically do not fly” are inconsistent, as we would expect. Of course, what we really want to do with such typicality statements is to draw default conclusions about individuals. Suppose we believe such a typicality statement. What other beliefs should follow? In general, Vx(Bird( x) +FZy( x)) does not follow; we should not necessarily believe that all birds are likely to fly. We may well know that Tacky the penguin does not fly. As long as Tacky is a rigid des- ignator, this is simply inconsistent with believing that all birds are likely to fly. In the absence of information about any particularly bird, ‘v’x(Bird(x) +FZy(x)) may well be a reasonable belief to hold. Moreover, no matter what we know about exceptional birds, it seems reasonable to believe true -E (Bird(x)+FZy(x)): almost all birds are likely to fly (assuming we have a logic that allows the obvious com- bination of statistical and subjective plausibility). Unfortunately, we do not have a general approach that will let us go from believing that birds typically fly to believing that almost all birds are likely to fly. Nor do we have an Foundations 1311 approach that allows us to conclude that Tweety is likely to fly given that birds typically fly and Tweety is a bird (and that we know nothing else about Tweety). These issues were addressed in the first-order setting by both Lehmann and Magidor (1990) and Delgrande (1988). The key feature of their approaches, as well as other propositional approaches rests upon getting a suitable notion of irrelevance. While we also do not have a general solution to the problem of irrelevance, we believe that plausibility structures give us the tools to study it in an abstract setting. We suspect that many of the intuitions behind probabilistic approaches that allow us to cope with irrelevance (Bacchus et al. 1994) can also be brought to bear here. We hope to return to this issue in future work. Acknowledgements We would like to thank Ronen Brafman, Ron Fagin, and Adam Grove for their comments on an earlier version of the paper. Some of this work was done while all three authors were at the IBM Almaden Research Center, supported by the Air Force Office of Scientific Research (AFSC) under Contract F49620-9 1 -C-0080. Some was done while Daphne Koller was at U.C. Berkeley, supported by a University of California President’s Postdoctoral Fellowship. This work is also partially supported by NSF grant IRI-95-03 109. References Adams, E. 1975. The Logic of Conditionals. Reidel. Bacchus, F.; Grove, A. J.; Halpern, J. Y.; and Koller, D. 1994. From statistical knowledge bases to de- grees of belief. Technical Report RJ 9855, IBM. Available at http://robotics.stanford.edu/ users / koller. A preliminary version of this work ap- peared in IJCAI ‘93,1993, pp. 563-569. Bacchus, F. 1990. Representing and Reasoning with Prob- abilistic Knowledge. MIT Press. Benferhat, S.; Dubois, D.; and Prade, H. 1992. Rep- resenting default rules in possibilistic logic. In KR ‘92. pp. 673-684. Boutilier, C. 1994. Conditional logics of normality: a modal approach. Artificial Intelligence 68:87-154. Brafman, R. I. 1991. A logic of normality: Predicate calculus incorporating assertions. Master’s thesis, Hebrew University, Jesrusalem. Delgrande, J. P. 1987. A first-order conditional logic for prototypical properties. Artificial Intelligence 33: 105-l 30. Delgrande, J. P. 1988. An approach to default reasoning based on a first-order conditional logic: revised report. Artificial Intelligence 36:63-90. Dubois, D., and Prade, H. 1990. An introduction to possi- bilistic and fuzzy logics. In Shafer, G., and Pearl, J., eds., Readings in Uncertain Reasoning. Morgan Kaufmann. Dubois, D., and Prade, H. 1991. Possibilistic logic, pref- erential models, non-monotonicity and related issues. In IJCAZ ‘91. pp. 419-424. Enderton, H. B. 1972. A Mathematical Introduction to Logic. Academic Press. Friedman, N., and Halpern, J. Y. 1995. Plausibility mea- sures: a user’s manual. In UAZ ‘9.5. pp. 175-184. Friedman, N., and Halpern, J. Y. 1996. Plausibility measures and default reasoning. In AAAZ ‘96. An ex- tended version appeared as IBM Research Report RJ 9959, available at http://robotics.stanford.edu/ users/nir. Garson, J. W. 1977. Quantification in modal logic. In Gabbay, D., and Guenthner, F., eds., Handbook of Philo- sophical Logic, Vol. II. Reidel. pp. 249-307. Geffner, H. 1992. Default reasoning: causal and condi- tional theories. MIT Press. Goldszmidt, M., and Pearl, J. 1992. Rank-based systems: A simple approach to belief revision, belief update and reasoning about evidence and actions. In KR ‘92. pp. 661- 672. Goldszmidt, M.; Morris, I?; and Pearl, J. 1993. A maximum entropy approach to nonmonotonic reasoning. IEEE Trans. of Pattern Analysis and Machine Intelligence 15(3):220-232. Halpern, J. Y. 1990. An analysis of first-order logics of probability. Artificial Intelligence 46:3 1 l-350. Hughes, G. E., and Cresswell, M. J. 1968. An Introduction to Modal Logic. Methuen. Klaus, S.; Lehmann, D.; and Magidor, M. 1990. Non- monotonic reasoning, preferential models and cumulative logics. Artificial Intelligence 44: 167-207. Kyburg, Jr., H. E. 1961. Probability and the Logic of Rational Belief. Wesleyan University Press. Lehmann, D., and Magidor, M. 1990. Preferential log- its: the predicate calculus case. In Theoretical Aspects of Reasoning about Knowledge: Proc. 3rd Con.. pp. 57-72. Lewis, D. K. 1973. Counteeactuals. Harvard University Press. Pearl, J. 1989. Probabilistic semantics for nonmonotonic reasoning: a survey. In KR ‘89, pp. 505-5 16. Poole, D. 199 1. The effect of knowledge on belief: con- ditioning, specificity and the lottery paradox in default reasoning. ArtiJiciaZ Intelligence 49( l-3):282-307. Schlechta, K. 1995. Defaults as generalized quantifiers. J. Logic and Computation 5(4):473-494. Shafer, G. 1976. A Mathematical Theory of Evidence. Princeton University Press. Shoham, Y. 1987. A semantical approach to nonmonotonic logics. In Proc. 2nd IEEE Symp. on Logic in Computer Science, pp. 275-279. Spohn, W. 1987. Ordinal conditional functions: a dynamic theory of epistemic states. In Harper, W., and Skyrms, B., eds., Causation in Decision, Belief Change and Statistics, volume 2. Reidel. pp. 105-134. 1312 Uncertainty | 1996 | 193 |
1,836 | IBM Research Division Almaden Research Center, Dept. K53-B2 650 Harry Road San Jose, CA 95 120-6099 halpern@almaden.ibm.com Abstract Cox’s well-known theorem justifying the use of probability is shown not to hold in finite domains. The counterexample also suggests that Cox’s assumptions are insufficient to prove the result even in infinite domains. The same counterexample is used to disprove a result of Fine on comparative conditional probability. 1 Introduction One of the best-known and seemingly most compelling ar- guments in favor of the use of probability is given by Cox (1946). Suppose we have a function Be1 that associates a real number with each pair (U, V) of subsets of a domain W such that U # 8. We write Bel( V 1 U) rather than Bel( U, V), since we think of Bel( V ] U) as the credibility or likelihood of V given U. ’ Cox further assumes that Bel(V]U) is a function of Bel( V ] U) (where 7 denotes the complement of V in W), that is, there is a function S such that Al. Bel(v]U) = S(Bel(V]U)) if U # 8, and that Bel( V n V' I U) is a function of Bel( V' IV n U) and Bel( VIU), that is, there is a function F such that A2Vel&; V'IU) = F(Bel(V’]V n U), Bel(V]U)) if . Notice that if Be1 is a probability function, then we can take S(Z) = 1 - x and F(x, y) = xy. Cox makes much weaker assumptions: he assumes that F is twice differen- tiable, with a continuous second derivative, and that S is twice differentiable. Under these assumptions, he shows that Be1 is isomorphic to a probability distribution in the sense that there is a continuous one-to-one onto function g : IR --+ lR such that g o Be1 is a probability distribution on W, and g(Bel(V]U))xg(Bel(U)) = g(Bel(VflU)) if U # 8, (1) where Bel( U) is an abbreviation for Bel( U 1 W). Not surprisingly, Cox’s result has attracted a great deal of interest in the AI literature. For example ’ Cox writes VI U rather than Bel( VI V), and takes U and V to be propositions in some language rather than events, i.e., subsets of a given set. This difference is minor-there are well-known mappings from propositions to events, and vice versa. I use events here since they are more standard in the probability literature. e Cheeseman (1988) has called it the “strongest argument for use of standard (Bayesian) probability theory”. 8 Horvitz, Heckerman, and Langlotz (1986) used it as a basis for comparison of probability and other nonproba- bilistic approaches to reasoning about uncertainty. o Heckerman (1988) uses it as a basis for providing an axiomatization for belief update. The main contribution of this paper is to show (by means of an explicit counterexample), that Cox’s result does not hold in finite domains, even under strong assumptions on S and F (stronger than those made by Cox and those made in all papers proving variants of Cox’s results). Since finite domains are arguably those of most interest in AI appli- cations, this suggests that arguments for using probability based on Cox’s result-and other justifications similar in spirit-must be taken with a grain of salt, and their proofs carefully reviewed. Moreover, the counterexample suggests that Cox’s assumptions are insufficient to prove the result even in infinite domains. It is known that some assumptions regarding F and S must be made to prove Cox’s result. Dubois and Prade (1990) give an example of a function Bel, defined on a finite domain, that is not isomorphic to a probability distribution. For this choice of Bel, we can take F(x, y) = min(z, y) and S(x) = 1 - x. Since min is not twice differentiable, Cox’s assumptions block the Dubois-Prade example. Aczel (1966, Section 7 (Theorem 1)) does not make any assumptions about F, but he does make two other assump- tions, each of which block the Dubois-Prade example. The first is that the Bel( VI U) takes on every value in some range [e, E], with e < E. In the Dubois-Prade example, the do- main is finite, so this certainly cannot hold. The second is that if V and V' are disjoint, then there is a continuous function G : JR2 + R, strictly increasing in each argument, such that A3. Bel(V U V'IU) = G(Bel(V]U), Bel(V’]U)). Dubois and Prade point out that, in their example, there is no function G satisfying A3 (even if we drop the require- ment that G be continuous and strictly increasing in each argument).2 With these assumptions, he gives a proof much 21n fact Acztl allows there to be a different function Gv for each set U on the right-hand side of the conditional. However, the Foundations 1313 From: AAAI-96 Proceedings. Copyright © 1996, AAAI (www.aaai.org). All rights reserved. in the spirit of that of Cox to show that Be1 is essentially a probability distribution. Reichenbach (1949) earlier proved a result similar to Aczel’s, under somewhat stronger assumptions. In par- ticular, he assumed A3, with G being +. Other variants of Cox’s result have also been consid- ered in the literature. For example, Heckerman (1988) and Horvitz, Heckerman, and Langlotz (1986) assume that F is continuous and strictly increasing in each argument and S is continuous and strictly decreasing. Since min is not strictly continuous in each argument, it fails this restriction too.3 Aleliunas (1988) gives yet another collection of as- sumptions and claims that they suffice to guarantee that Be1 is essentially a probability distribution. The first to observe potential problems with Cox’s result is Paris (1994). As he puts it, “Cox’s proof is not, perhaps, as rigorous as some pedants might prefer and when an attempt is made to fill in all the details some of the attractiveness of the original is lost.” Paris provides a rigorous proof of the result, assuming that the range of Be1 is contained in [0, l] and using assumptions similar to those of Horvitz, Heckerman, and Langlotz. In particular, he assumes that F is continuous and strictly increasing in (0, 112 and that S is decreasing. However, he makes use of one additional assumption that, as he himself says, is not very appealing: A4. For any 0 5 Q, ,0, y 5 1 and E > 0, there are sets U1, U2, U3, and U4 such that U3 n U2 fl U1 # 8, and each of ]Bel(U& 0 U2 n VI) - CX], )Bel(Us]U2 n VI) --PI, and IBel(U2lUi) - y] is less than E. Notice that this assumption forces the range of Be1 to be dense in [0, 11. This means that, in particular, the domain W on which Be1 is defined cannot be finite. Is this assumption really necessary? Paris suggests that Aczel needs something like it. (This issue is discussed in further detail below.) The counterexample of this paper gives further evidence. It shows that Cox’s result fails in finite domains, even if we assume that the range of Be1 is in [O, 11, S(x) = 1 - x (so that, in particular, S is twice differ- entiable and monotonically decreasing), G(x, y) = x + y, and F is infinitely differentiable and strictly increasing on (0, 112. We can further assume that F is commutative, F(0, x) = F(z, 0) = 0, and that F(x, 1) = F(l, x) = x. The example emphasizes the point that the applicability of Cox’s result is far narrower than was previously believed. It remains an open question as to whether there is an ap- propriate strengthening of the assumptions that does give us Cox’s result in finite settings. In fact, the example shows even more. In the course of his proof, Cox claims to show that F must be an associa- tive function, that is, that F(x, F(y, z)) = F(F(x, y), z). For the Be1 of the counterexample, there can be no associa- tive function F satisfying A2. It is this observation that is the key to showing that there is no probability distribution isomorphic to Bel. What is going on here? Actually, Cox’s proof just shows that F(x, F(y, z)) = F(F(x, y), z) only for those triples (x, y, z) such that, for some sets 171, U2, U3, and U4, we have x = Bel(U4lUs n U2 n U,), y = Bel(UsIU2 n VI), and z = Bel( 772 I VI). If the set of such triples (x, y, z) is dense in [0, l]“, then we conclude by continuity that F is associative. The content of A4 is precisely that the set of such triples is dense in [0, 113. Of course, if W is finite, we cannot have density. As my counterexample shows, we do not in general have associativity in finite domains. Moreover, this lack of associativity can result in the failure of Cox’s theorem. A similar problem seems to exist in Aczel’s proof (as already observed by Paris (1994)). While Aczel’s proof does not involve showing that F is associative, it does involve showing that G is associative. Again, it is not hard to show that G is associative for appropriate triples, just as is the case for F. But it seems that Aczel also needs an assumption that guarantees that the appropriate set of triples is dense, and it is not clear that his assumptions do in fact guarantee this.4 As shown in Section 2, the problem also arises in Reichenbach’s proof. This observation also shows that another well-known re- sult in the literature is not completely correct. In his semi- nal book on probability and qualitative probability (1973), Fine considers a non-numeric notion of comparative (con- ditional)probability, which allows us to say “U given V is at least as probable as U’ given V”‘, denoted U I V t: U’ 1 V’. Conditions on t are given that are claimed to force the ex- istence of (among other things) a function Be1 such that UIV k U’IV’ iff Bel(U]V) > Bel(U’IV’) and an asso- ciative function F satisfying A2. (This is Theorem 8 of Chapter II in (Fine 1973).) However, the Be1 defined in my counterexample to Cox’s theorem can be used to give a counterexample to this result as well, The remainder of this paper is organized as follows. In the next section there is a more detailed discussion of the problem in Cox’s proof. The counterexample to Cox’s the- orem is given in Section 3. The following section shows that it is also a counterexample to Fine’s theorem. Section 5 concludes with some discussion. 2 To understand the problems with Cox’s proof, I actually con- sider Reichenbach’s proof, which is similar in spirit Cox’s proof (it is actually even close to AczCl’s proof), but uses some additional assumptions, which makes it easier to ex- plain in detail. AczCl, Cox, and Reichenbach all make critical use of functional equations in their proof, and they make the same (seemingly unjustified) leap at correspond- ing points in their proofs. Dubois-Prade example does not even satisfy this weaker condition. 3Actually, the restriction that F be strictly increasing in each argument is a little too strong. If e = BeI( then it can be shown that F(e,o) = F(z, e) = e for all z, so that F is not strictly increasing if one of its arguments is e. 41 should stress that my counterexample is not a counterexam- ple to Aczel’s theorem, since he explicitly assumes that the range of Be1 is infinite. However, it does point out potential problems with his proof, and certainly shows that his argument does not apply to finite domains. 13 14 Uncertainty In the notation of this paper, Reichenbach (1949, pp. 65- 67) assumes (1) that the range of Bel(.].) is a subset of [O, 11, (2) Bel(VIU) = 1 if U C V, (3) that if V and V’ are disjoint, then Bel(VUV’]U) = Bel(V]U) +Bel(V’]U) (thus, he assumes that A3 holds, with G being +), and (4) that A2 holds with a function F that is differentiable. (He remarks that the result holds even without assumption (4), although the proof is more complicated; AczCl in fact does not make an assumption like (4).) Reichenbach’s proof proceeds as follows: Replacing V’ in A2 by VI U Vi, where VI and V2 are disjoint, we get that Bel(Vn(Vi uV2)lU) = F(Bel(Vi uV$frlU), Bel(V]U)). (2) Using the fact that G is +, we immediately get Bel(V n (VI u V2)]U) = Bel(V n V$J) + Bel(V n V$) (3) and F(Bel(Vi U V# fl U), Bel(V]U)) = F(Bel(Vi IV n U) + Bel(V2IV n U), Bel(V]U)) (4) Moreover, by A2, we also have, for i = 1,2, Bel(V n V&Y) = F(Bel(V n V$f n U), Bel(V]U)). (5) Putting together (2), (3), (4), and (5), we get that F(Bel(V n Vl IV n U), Bel(V]U)) + F(Bel(V n V# n U), Bel(V]U)) = F(Bel(V n Vl IV n U) + Bel(V n V2]V n U, Bel(V]U)) (6) Takingx = Bel(VnVI]VflU),y = Bel(VnV2]VnU), and z = Bel(V]U) in (6), we get the functional equation F(x, z) + F(y, z) = F(x + y, 2). (7) Suppose we assume (as Reichenbach implicitly does) that this functional equation holds for all (x, y, z) E P = uxc, Y, 4 E LO, II3 : x + y < 1). The rest of the proof now follows easily. First, taking x = 0 in (7), it follows that F(O, z) + F(Y, z> = F(Y, z>, from which we get that F(0, z) = 0. Next, fix z and let gz (2) = F(x, z). Since F is, by assump- tion, differentiable, from (7) we have that g:(x) = ;Fo(F(x + Y, 2) - F(x, z)/Y) = ;eoF(y, z)/Y. It thus follows that g: (x) is a constant, independent of x. Since the constant may depend on z, there is some function h such that gi (x) = h(z). Using the fact that F(0, z) = 0, elementary calculus tells us that g,(x) = F(x, z) = h(z)x. Using the assumption that for all U, V, we have Bel( VI U) = 1 if U C_ V, we get that Bel(V]U) = Bel(V fl VlU) = F(Bel(V]V n U), Bel(V]U)) = F( 1, Bel(V]U)). Thus, we have that F(l, z) = h(z) = z. We conclude that F(x, z) = xz. Note, however, that this conclusion depends in a crucial way on the assumption that the functional equation (7) holds for all (x, y, z) E P. 5 In fact, all that we can conclude from (6) is that it holds for all (x, y, z) such that there exist U, V, VI, and V2, with VI and V2 disjoint, such that x = Bel(V n Vl]V n U), y = Bel(V n V$7 n U), and z = Bel(V]U). Let us say that a triple that satisfies this condition is ac- ceptable. As I mentioned earlier, Aczel also assumes that Bel(V]U) takes on all values in [e, E], where e = Bel(@]U) and E = Bel( U] U). (In Reichenbach’s formulation, e = 0 and E = 1.) There are two ways to interpret this assump- tion. The weak interpretation is that for each x E [0, 11, there exist U, V such that Bel( V]U) = x. The strong in- terpretation is that for each U and x, there exists V such that Bel(V]U) = x. It is not clear which interpretation is intended by Aczel. Neither one obviously suffices to prove that every triple in P is acceptable, although it does seem plausible that it might follow from the second assumption. In any case, both Aczel and Reichenbach (as well as Cox, in his analogous functional equation) see no need to check that Equation (7) holds throughout P. However, it turns out to be quite necessary to do this. Moreover, it is clear that if W is finite, there are only finitely tuples in P which are acceptable, and it is not the case that all of P is. As we shall see in the next section, this observation has serious consequences as far as all these proofs are concerned. 3 The Counterexample to Cox’s Theorem The goal of this section is to prove Theorem 3.1: There is a function Belo, a finite domain W, and functions S, F, and G satisfying Al, A2, and A3 respectively such that Belo(V]U) E [0, l]forU # 8, S(x) = 1 - x (so that S is strictly decreasing and in- finitely diferentiable), G( x, y) = x + y (so that G is strictly increasing in each argument and is infinitely difSerentiable), F is infinitely di 2fs erentiable, nondecreasing in each ar- gument in [0, l] , and strictly increasing in each argu- ment in (0, 112. Moreovel; F is commutative, F(x, 0) = F(0, x) = 0, and F(x, 1) = F(l, x) = x. Howevel; there is no one-to-one onto function g : [0, I] --+ [0, l] satisfying (1). Note that the hypotheses on Belo, S, G, and F are at least as strong as those made in all the other variants of Cox’s re- sult, while the assumptions on g are weaker than those made in the variants. For example, there is no requirement that g be continuous or increasing nor that g o Belo is a probabil- ity distribution (although Paris and Aczel both prove that, ‘Actually, using the continuity of F, it suffices that the func- tional equation holds for a set of triples which is dense in P. Foundations 1315 under their assumptions, g can be taken to satisfy all these requirements). This serves to make the counterexample quite a strong one. Proof: Consider a domain W with 12 points: WI, . . . . ~12. We associate with each point w E W a weight f(w), as follows. f(w) = 3 f(w4) = 5 x lo4 f(W2) = 2 f(w5) = 6 x lo4 f(w3) = 6 f(w6) = 8 x lo4 f(W7) = 3 x lo* f(WiO) = 3 x 10 18 f(w8) = 8 x lo8 f(wii) = 2 x 10 18 f(w9) = 8 x lo* f(w12) = 14 x 10 18 For a subset U of W, we define f(U) = zwEU f(w). Thus, we can define a probability distribution Pr on W by taking Pr( U) = f( U)/f( W). Let f’ be identical to f, except that f’(wio) = (3 - 6) x lo’* and f’(wii) = (2 + S) x 101*, where 6 is defined below. Again, we extend f’ to subsets of W by defining f’(U) = xwEU f’(w). Let W’ = {w0,~11,~12). If U # 0, define Belo(V]U) = f’(V n U)/f(U) if W’ C U f( V n U)/f( U) otherw:se. Belo is clearly very close to Pr. If U # 8, then it is easy to see that ]Belo(V]U) - Pr(V]U)] = If’(V n U) - f(V fi U)l/f(U) 5 6. We choose 6 > 0 so that if Pr(V]U) > Pr(V’]U’), then Belo(V]U) > Belo(V’]U’). (8) Since the range of Pr is finite, all sufficiently small S sat- isfy (8). The exact choice of weights above is not particularly im- portant. One thing that is important though is the following collection of equalities: Pr(wl I(wI, ~2)) = Pr(wlol(wlo, ~113) = 3/5 pr((wl, w2j-l(% ~2, w33) = Pr(w4](w4, ~53) = 5/11 pr((W4, wS)l{w4, w5, w6)) = pr((w7, w8))( W7,W8,W93> = 1 l/19 Pr(w4I(w4, w5, w6)) = pr({wlo, w)@10, ~11, ~12)) = s/19 pr(wl I(wl, w2, w3)) = Pr(w7l(w7, wS>> = 3/l 1. (9) It is easy to check that exactly the same equalities hold if we replace Pr by Belo. Although, as is shown below, the function F satisfying A2 can be taken to be infinitely differentiable and increasing in each argument, the equalities in (9) suffice to guarantee that it cannot be taken to be associative, that is, we do not in general have F(X) F(Y) 4) = F(F(% Y), z>- Indeed, there is no associative function F satisfying A2, even if we drop the requirements that F be differentiable or increasing. Lemma 3.2: For Belo as dejined above, there is no associa- tive function F satisfying A2. Proof: Suppose there were such a function F. From (9), we must have that F(5/11,11/19) = F(Bek&l( w4, w53), Be10(+‘4, w531(w4, w5, w63)) = Belo(w4](w4, ws, W6)) = s/19 and that F(3/5,V 1) = F(Beb(wll(wl, w)), Beb({wl, w)I(w~, w, ~3))) = Belo(wi [(wi, ~2, ~33) = 3/l 1. It follows that F(3/5, F(5/11,1 l/19)) = F(3/5,5/19) and that F(F(3/5,5/11), 1 l/19) = F(3/11,1 l/19). Thus, if F were associative, we would have F(3/5,5/19) = F(3/11,11/19). On the other hand, from (9) again, we see that F(3/5, WJ) = F(Belo(wd(wlo, wd), Beb((wlo, wll>l-blo, ~1, ~12))) = B&(wlol{wo, wll, w)) = (3 - 6)/l% while F(3/11,1 l/19) = F(BelO(w7b7, w8>>, BelO((w7, w8)l(w7, w8, w9))) = Belo(w71(w7, ws, ~93) = 3/l% It follows that F cannot be associative. The next lemma shows that Belo cannot be isomorphic to a probability function. Lemma 3.3: For Belo as defined above, there is no one-to- one onto function g : [0, l] -+ [0, l] satisfying (I). Proof: Suppose there were such a function g. First note that g(Belo(U)) # 0 if U # 8. For if g(Bele(U)) = 0, then it follows from (I) that for all V C U, we have g(Belo(V)) = g(Bel~(V]U))xg(Bel~(U)) = g(Belo(V]U))xO = 0. Thus, g(Bele(V)) = g(Belo(U)) for all subsets V of U. Since the definition of Belo guarantees that Bele(V) # Bele(U) if V is a strict subset of U, this contradicts the assumption that g is one-to-one. Thus, g(Belo(U)) # 0 if U # 8. It now follows from (1) that if U # 8, then s(Belo(Vlu)) = g(B&(V n U))/g(B&(U)). (10) Now define F(x) y) = g-‘(g(x) x g(y)). Notice that, by applying the observation above repeatedly, if V fl U # 8, we get F(Belo(V’]V n U), Belo(V]U)) = g-‘((g(B&(V’lV n u>> x g(B&(VlU)) = g-‘(g(Belo(V’ n V n U))/@&(U))) = g-‘(g(Belo(V’ n VIU))) = Bele(V’ n V(U). 1316 Uncertainty Thus, F satisfies A2. Moreover, notice that F is associative, since But this contradicts Lemma 3.2. Despite the fact that Bela is not isomorphic to a probability function, functions S, F, and G can be defined that satisfy Al, A2, and A3, respectively, and all the other requirements stated in Theorem 3.1. The argument for S and G is easy; all the work goes into proving that an appropriate F exists. Lemma 3.4 : There exists an infinitely diflerentiable, strictly decreasing function S : [0, l] -+ [0, l] such that BeZo(vlU) = S(BeZo(VlU)) for all sets U, V 5 W with 27 # 0. In fact, we can take S(x) = 1 - x. Proof: This is immediate from the obs Bel@]U) = 1 - Belo(V]U) for U, V 5 W. Lemma 3.5 There exists an infinitely diflerentiablefunction G : [0, 112 + [0, 11, increasing in each argument, such that if U, V, V’ & W, V fl V’ = 8, and U # 8, then BeZo(V U V’IU) = G(BeEo(VIU), BeZo(V’, U)). In fact, we can take G(x, y) = x + y. Proof: This is immediate from the definition of Belo. Thus, all that remains is to show that an appropriate F exists. The key step is provided by the following lemma, which essentially shows that there is a well defined F that is increasing. Lemma 3.6: If U2 fl U1 # 8 and V2 fl iJ # 8, then if BeZo(V3jV2 n VI) 5 BeZo(U3JU2 n Ul) and Bek@#i) 5 Belo(U2lUl), then B&(V3 n v2lVi) < BeloW n U2lU1), (b) (4 ifBelo(V3IJJin K) -c Belo(U3IU2n VI), Bel0(V#i) 5 Belo(U2[U1), BeZo(U$J2 n VI) > 0, andBeZo(U2jU1) > 0, then BeZo( V3 n V2 I VI ) < Be&-J U3 fl U2 I U1 ), ifBelo(fiIV2 17 K) L Belo(U3IU2n VI), Bel0(V$$) < Belo(U$~), BeZ@$J2 n U,) > 0, andBeZo(U:!IUl) > 0, then BeZo(V3 n V$VI) < BeZo(U3 n U4U1), Proof: First observe that if Belo(VsIV2 n VI) < Belo(UsIU2 n VI) and Belo(V2IVt) 5 Belo(U2(Ut), then from (8), it follows that Pr( V3 IV-2 n VI) 5 Pr( U3 I U2 n U,) and Pr(V2lVi) 5 Pr(U2lUt). If we have either Pr(VsIV2 n Vi) -c Pr(U3(U2 f-7 VI) or Pr(V2(Vi) < Pr(U2(Ut), then we have either Pr( V3 n I4 IV1 ) < Pr( U3 n U2 I U, ) or Pr( U3 IV2 n VI) = 0 or Pr( U2 IV,) = 0. It follows that either Belo(V3 n V2lVl) < Belo(Us fl U21U1) (this uses (8) again) or that Belo(Vs il V2IVl) = Belo(Us rl U2jUl) = 0. In either case, the lemma holds. Thus, it remains to deal with the case that Pr(Vs IV2 n V’) = Pr(Us(U2 r\ Ut) and Pr(&(Vr) = Pr(U2(Ur), and hence Pr( V3 n &IV1 ) = Pr( U3 n U2 I VI). The details of this analysis are left to the full paper. Lemma 3.7: There exists a function F : [0, 112 + [0, l] satisfying all the assumptions of the theorem. Proof: Define a partial function F’ on [0, 112 whose domain consists of all pairs (x, y) such that for some subsets U, V, V’ of W, we have x = For such (x, y), B&(V’]VnU) andy = Bele(V]U). we define F’( x , y) = Bela(V’ fl V/U). A priori, it is possible that there exist sets U1, U2, Us, VI, V2, $5 such that x = Belo(UsIU2 il U,) = Bela(VsIV2 n VI) and y = Belo(U2IUl) = B&(E$4), yet Belo(U3 n u216) # Belo(V3 fl V21V1). If this were the case, then F/(x, y) would not be well defined. However, Lemma 3.6 says that this cannot happen. Moreover, the lemma assures us that F’ is increasing on its domain, and strictly increasing as long as one of its arguments is not 0. Notice that if Belo(V]U) = x # 0 for some V, U, then (0, x), (x, 1) and (1, x) are in the domain of F’, and F/(x, 1) = F’( 1, x) = x, while F’(0, x) = 0. It is easy to see that there are no pairs (x, 0) in the domain of F’. Finally, there are no pairs (x, y) and (y, x) that are both in the domain of F’ unless one of x or y is 1. The domain of F’ is finite. It is straightforward to extend F’ to a commutative, infinitely differentiable, and increas- ing function F defined on all of [0, 112, which is strictly increasing on (0, 112, and satisfies F(x, 1) = F(l, x) = x and F(x,O) = F(0, x) = 0. (Note that to make F com- mutative, we first define it on pairs (x, y) such that x 2 y, and then if x < y, we define F((x, y) = F(y, x). Since F’ is commutative on its domain of definition, this approach does not run into problems.) Clearly F satisfies A2, since (by construction) F’ does, and A2 puts constraints only on the domain of F’. orem 3.1 now follows from Lemmas 3.3,3.4,3.5, and 4 The Counterexample to Fine’s T Fine is interested in what he calls comparative conditional probability. Thus, rather than associating a real number with each “conditional object” VlU, he puts an ordering > on such objects. As usual, V IU > V’IU’ is taken to be an abbreviation for VlU t V’IU and not(V’]U’ t VIU). Fine is interested in when such an ordering is induced by a real-valued belief function with reasonable properties. He says that a real-valued function P on such objects agrees with k if P(VIU) 2 P(V’IU’) iff VIU k V’IU’. Fine then considers a number of axioms that > might satisfy. For our purposes, the most relevant are the ones Fine denotes QCC 1, QCC2, QCCS, and QCC7. QCCl just says that k is a linear order: QCCl. VJU 2 V’IU’ or V’IU k VIU. QCC2 says that h is transitive: QCCZ If VI I UI k V2 I U2 and V2 IV2 k V3 I U3, then VlUl ?z v3IU3. QCC5 is a technical condition involving notions of order topology. The relevant definitions are omitted here (see (Fine 1973) for details), since QCC5, as Fine observes, Foundations 1317 holds vacuously in finite domains (the only ones of interest here). QCC5. The set (VlU> h as a countable basis in the order topology induced by t . Finally, QCC7 essentially says that > is increasing, in the sense of Lemma 3.6. QCC7. (a) If VjIV2 n VI > UsI& n UI and V2lV1 > U2jU1 then vi nvilvi k u3 n u21ul. (b) If &IV2 tl VI t U2IUr and V2IVt t U&U2 fl U1 then v3 n vilvi k u3 n u21ul. CC> If tip!2 n v + u3)u2 n ul, &I& k u21h and &IV1 5 0lW, then V3 n V21Vl s U3 n U2jU1. Fine then claims the following theorem: Fine’s Theorem: (Fine 1973, Chapter II, Theorem 8) Zf k satisfies QCCl, QCC2, QCCS, then there exists some agreeing function P. There exists a function F of two variables such that 1. P(V n V’IU) = F(P(V’JV n U), P(VIU)),6 2. F(x, Y> = F(Y, xc>, 3. F(x, y) is increasing in x for y > P(@IW), 4. F(x, F(Y, 4) = F(F(x, Y), 4, 5. F(P(WIU), Y) = Y, 6. F(P(@(U), y) = P(@IU). irk also satis$es QCC7. The only relevant clauses for our purposes are Clause (l), which is just A2, and Clause (4), which says that F is associative. As Lemma 3.2 shows, there is no associative function satisfying A2 for Belo. As I now show, this means that Fine’s theorem does not quite hold either. Before doing so, let me briefly touch on a subtle issue regarding the domain of t. In the counterexample of the previous section, Belo(V] U) is defined as long as U # 8. Fine does not assume that the t relation is necessarily defined on all objects VI U such that U, V c W and U # 0. He assumes that there is an algebra F of subsets of W (that is, a set of subsets closed under finite intersections and complementation) and a subset F’ of F closed under finite intersections and not containing the empty set such that k is defined on conditional objects VlU such that V E F and U E F’. Since J=’ is closed under intersection and does not contain the empty set, F’ cannot contain disjoint sets. If W is finite, then the only way a collection F’ can meet Fine’s restriction is if there is some nonempty set UO such that all elements in F’ contain UO. This restriction is clearly too strong to the extent that comparative conditional probability is intended to generalize probability. If Pr is a probability function, then it certainly makes sense to compare Pr( V I U) and Pr( V’ I U’) even if U and U’ are disjoint sets. Fine [private communication, 19951 suggested that it might be 6Fine assumes that P(V rl V’lU) = F(P(V/IU), P(V’jV rl U)). I have reordered the arguments here for consistency with Cox’s theorem, 13 18 Uncertainty better to constrain QCC7 so that we do not condition on events U that are equivalent to 8 (where U is equivalent to 8 if 8 k U and U > 8). Since the only event equivalent to 8 in the counterexample of the previous section is 8 itself, this means that the counterexample can be used without change. This is what is done in the proof below. In the full paper, I indicate how to modify the counterexample so that it satisfies Fine’s original restrictions. Theorem 4.1: There exists an ordering k satisfying QCCI, QCC2, QCCS, and QCC7, such that for every function P agreeing with h, there is no associative function F of two variables such that P(V n V’)lU) = F(P(V’lV n u>, VIW Proof: Let W and Belo be as in the counterexample in the previous section. Define t so that Belo agrees with k. Thus, VlU L: V’IU’ iff Belo(VlU) 2 Belo(V’]U’). Clearly t satisfies QCCl and QCC2. As was mentioned earlier, since W is finite, t vacuously satisfies QCC5. Lemma 3.6 shows that k satisfies parts (a) and (c) of QCC7. To show that k also satisfies part (b) of QCC7, we must prove that if Belo(V3(V2 n VI) 2 Belo(U2IUl) and Belo(V21Vt) 1 Belo(UsIU2 fl VI), then Belo(Vs fl V~VI) 2 Belo(UsfW21Ul). Theproofofthisisalmost identical to that of Lemma 3.6; we simply exchange the roles of Pr(V2 [VI) and Pr( V3 (Vi n VI) in that proof. I leave the details to the reader. Lemma 3.2 shows that there is no associative func- tion F satisfying A2 for Belo. All that was used in the proof was the fact that Belo satisfied the inequalities of (9). But these equalities must hold for any function agreeing with 2. Thus, exactly the same proof shows that if P is any function agreeing with k, then there is no associative function F satisfying P(V n V’IU) = F(P(V’IV n U), P(VIU)). 5 Discussion Let me summarize the status of various results in the light of the counterexample of this paper: Cox’s theorem as originally stated does not hold in finite domains. Moreover, even in infinite domains, the coun- terexample and the discussion in Section 2 suggest that more assumptions are required for its correctness. In par- ticular, the claim in his proof that F is associative does not follow. Although the counterexample given here is not a coun- terexample to Aczel’s theorem, his assumptions do not seem strong enough to guarantee that the function G is associative, as he claims it is. The variants of Cox’s theorem stated by Heckerman (1988), Horvitz, Heckerman, and Langlotz (1986), and Aleliunas (1988) all succumb to the counterexample. The claim that the function F must be associative in Fine’s theorem is incorrect. Fine has an analogous result (Fine 1973, Chapter II, Theorem 4) for unconditional compar- ative probability involving a function G as in Aczel’s theorem. This function too is claimed to be associative, and again, this does not seem to follow (although my counterexample does not apply to that theorem). Of course, the interesting question now is what it would take to recover Cox’s theorem. Paris’s assumption A4 suf- fices. As we have observed, A4 forces the domain of Be1 to be infinite, as does the assumption that the range of Be1 is all of [0, 11. We can always extend a domain to an infinite- indeed, uncountable-domain by assuming that we have an infinite collection of independent fair coins, and that we can talk about outcomes of coin tosses as well as the original events in the domain. (This type of “extendibil- ity” assumption is fairly standard; for example, it is made by Savage (1954) in quite a different context.) In such an extended domain, it seems reasonable to also assume that Be1 varies uniformly between 0 (certain falsehood) and 1 (certain truth). If we also assume A4 (or something like it), we can then recover Cox’s theorem. Notice, however, that this viewpoint disallows a notion of belief that takes on only finitely many or even countably many gradations. Suppose we really are interested in a particular finite do- main, and we do not want to extend it. What assumptions do we then need to get Cox’s theorem? The counterex- ample given here could be circumvented by requiring that F be associative on all tuples (rather than just on the tu- ples (x, y,z) that arise as x = Belo(Ud]Us rl U2 fl Ul), y = Belo(U3IU2 n VI), and z = Belo(U However, if we really are interested in a single domain, the motiva- tion for making requirements on the behavior of F on belief values that do not arise is not so clear. Moreover, it is far from clear that assuming that F is associative suffices to prove the theorem. For example, Cox’s proof makes use of various functional equations involving F and S, analo- gous to the equation (7) that appears in Section 2. These functional equations are easily seen to hold for certain tu- ples. However, as we saw in Section 2, the proof really requires that theq’ hold for all tuples. Just assuming that F is associative does not appear to suffice to guarantee that the functional equations involving S hold for all tuples. Futher assumptions appear necessary. One condition (suggested by Nir Friedman) that does seem to suffice (although I have not checked details) is that of assuming that essentially all beliefs are distinct. More precisely, we could assume 0 if 8 c U c V, 8 c U’ c V’, and (U, V) # (U’, V’), then Bel(U]V) # Bel(U’]V’). Even if this condition suffices, note that it precludes, for example, a uniform probability distribution, and thus again seems unduly restrictive. So what does all this say regarding the use of probability? Not much. Although I have tried to argue here that Cox’s justification of probability is not quite as strong as previously believed, and the assumptions underlying the variants of it need clarification, I am not trying to suggest that probability should be abandoned. There are many other justifications for its use. and an anonymous referee for useful comments on an earlier draft of the paper. I’d also like to thank Judea Pearl for pointing out Reichenbach’s work to me. This work was supported in part by NSF grant IRI-95-03 109. eferences Aczel, J. 1966. Lectures on Functional Equations and Their Applications. New York: Academic Press. Aleliunas, R. 1988. A summary of a new normative the- ory of probabilistic logic. In Proceedings of the Fourth Workshop on Uncertainty in Artificial Intelligence, Min- neapolis, MN, 8-14. Also in R. Shachter, T. Levitt, L. Kanal, and J. Lemmer, editors, Uncertainty in Artificial Intelligence 4, pages 199-206. North-Holland, New York, 1990. Cheeseman, l? 1988. An inquiry into computer under- standing. Computational Intelligence 4( 1):58-66. Cox, R. 1946. Probability, frequency, and reasonable expectation. American Journal of Physics 14( 1): 1-13. Dubois, D., and Prade, H. 1990. The logical view of conditioning and its application to possibility and evidence theories. International Journal of Approximate Reasoning 4( 1):23-46. Fine, T. L. 1973. Theories of Probability. New York: Academic Press. Heckerman, D. 1988. An aximoatic framework for belief updates. In Lemmer, J. F., and Kanal, L. N., eds., Un- certainty in Artificial Intelligence 2. Amsterdam: North- Holland. 1 l-22. Horvitz, E. J.; Heckerman, D.; and Langlotz, C. P. 1986. A framework for comparing alternative formalisms for plau- sible reasoning. In Proc. National Conference on Artificial Intelligence (AAAZ ‘86), 210-214. Paris, J. B. 1994. The Uncertain Reasoner’s Companion. Cambridge, U.K.: Cambridge University Press. Reichenbach, H. 1949. The Theory of Probability. Univer- sity of California Press, Berkeley. This is a translation and revision of the German edition, published as Wahrschein- lichkeitslehre, in 1935. Savage, L. J. 1954. Foundations of Statistics. John Wiley & Sons. Acknowledgments I’d like to thank Peter Cheeseman, Terry Fine, Ron Fagin, Nir Friedman, David Heckerman, Eric Horvitz, Jeff Paris, Foundations 1319 | 1996 | 194 |
1,837 | Pete Bonasso* and To *Met&a, Inc., Johnson Space Center, NASA Houston, TX 77059 bonass@aio.jsc,nasa.gov **Department of Computer Science, Brown University, Providence, RI 02912 tld@cs.brown.edu There have been five years of robot competitions and exhibitions since the inception of this annual event in 1992. Since that first show we have seen 30 different teams compete and almost that many more exhibit their robots. These teams ranged from universities to industry and government research labs to one or two inventors working out of garages. Their composition ranged from seasoned AI researchers to eager undergraduates, and they hailed from the United States, Canada, Europe and the Far East. Despite the concerns of some about the relevance and even the appropriateness of such an event, the robots have become a key attraction of the national and international conferences. In this talk, we look back on the form and function of the five years of exhibitions and competitions and attempt to draw some lessons in retrospect as well as future implications for the AI community and our society at large. A cornerstone of this event has always been the emphasis on fully autonomous robots, and hence the apparent need for AI. We will survey the role that the hallmarks of AI -- planning, learning, machine vision and spoken language understanding -- have played in the competitions, particularly with those teams ranked high in the standings. We will touch on the use of single and multiple agents, reactive and deliberative control schemes, use of active perception, and the basic problem-solving approaches brought to bear each year by the teams in the competition. The past five years have also seen an increase in the need for and even the use of autonomous mobile robots in the service industries -- those industries requiring the use of robots in natural environments among humans. We are beginning to see the almost routine use of autonomous robots vacuuming large warehouse and hotel areas, ferrying x-rays and medicines in hospitals, and even filling pharmaceutical prescriptions. We will sample from the competition results and some of the exhibitions to suggest which AI technologies can or cannot be made relevant for service industry needs. Finally, we want to convey some of the atmosphere of the competitions both in front of and behind the scenes: the camaraderie, the sleepless nights, the sharing of ideas, the last minute requisitions for hardware and the up to the minute software hacks. There is every much a thrill of victory and an agony of defeat in these events as there are in sports contests in other settings. Through the liberal use of video tape and anecdotes, we hope we can make these aspects of the competition realizable so that listeners can glimpse the intangible benefits of this important coming together of people and ideas to produce intelligent -- and useful -- robots. Invited Talks 1321 From: AAAI-96 Proceedings. Copyright © 1996, AAAI (www.aaai.org). All rights reserved. | 1996 | 195 |
1,838 | Moving Up the Information Food Chain: Deploying Softbots on the World Wide Web Qren Etzioni Department of Computer Science and Engineering University of Washington Seattle, WA 98195 http://www.cs.washington.edu/homes/etzioni Abstract I view the World Wide Web as an information food chain (figure 1). The maze of pages and hyperlinks that comprise the Web are at the very bottom of the chain. The WebCrawlers and Alta Vistas of the world are information herbivores; they graze on Web pages and regurgitate them as searchable indices. Today, most Web users feed near the bottom of the infor- mation food chain, but the time is ripe to move up. Since 1991, we have been building information carni- vores, which intelligently hunt and feast on herbivores in Unix (Etzioni, Lesh, & Segal 1993), on the Inter- net (Etzioni & Weld 1994)) and on the Web (Dooren- bos, Etzioni, & Weld 1996; Selberg & Etzioni 1995; Shakes, Langheinrich, & Etzioni 1996). Motivation Today’s Web is populated by a panoply of primitive but popular information services. Consider, for exam- ple, an information cow such as Alta Vista. Alta Vista requires massive memory resources (to store an index of the Web) and tremendous network bandwidth (to create and continually refresh the index). The cost of these resources is amortized over millions of queries per day. As a result, the CPU cycles devoted to satisfying each individual query are sharply curtailed. There is no time for intelligence. Furthermore, each query is independent of the previous one. No attempt is made to customize Alta Vista’s responses to a particular in- dividual. The result is homogenized, least-common- denominator service. In contrast, visionaries such as Alan Kay and Nicholas Negroponte have been advocating agents - personal assistants that act on your behalf in cy- berspace. While the notion of agents has been popular for more than a decade, we have yet to build agents that are both widely used and intelligent. The Web presents a golden opportunity and an implicit chal- lenge for the AI community. As the old adage goes “If not us, then who? And if not now, v-hen?” The challenge of deploying web agents will help re- vitalize AI and forge closer links with other areas of 1322 AAAI-96 computer science. But be warned, the Web commu- nity is hungry, impatient, and skeptical. They expect: Robustness: a working system, accessible seven days a week, twenty-four hours a day. Speed: virtually all widely-used Web resources be- gin transmitting useful (or at least entertaining) in- formation within seconds. Added Value: any increase in sophistication had better yield a tangible benefit to users. Is the Web challenge a distraction from our long- term goal of understanding intelligence and building intelligent agents? I believe that the field benefits from a mixture of long-term and short-term goals and from both empirical and theoretical work. Work toward the goal of deploying intelligent agents on the Web is a valuable addition to the current mix for two rea- sons. First, the Web suggests new problems and new constraints on existing techniques. Second, intelligent Web agents will provide tangible evidence of the power and utility of AI techniques. Next time you encounter AI bashing, wouldn’t it be satisfying to counter with a few well-chosen URLs? Personally, I find the Web irre- sistible. To borrow Herb Simon’s phrase, it is today’s “Main Chance.” Simon describes his move from the “academic backwater” of public administration to AI and cognitive psychology as “gravitating toward the sun” (Simon 1991, pages 113-114). While AI is not an academic backwater, the Web is today’s sun. Turning towards the sun and responding to the Web challenge, my collaborators and have begun to deploy a species of information carnivores (called softbots) on the Web. Softbots Softbots (software robots) are intelligent agents that use software tools and services on a person’s behalf (see figure 2 for a softbot family tree). Tool use is one of the hallmarks of intelligence. In many cases, softbots rely on the same tools and utilities available to human computer users - tools for sending mail, printing files, and so on. Mobile robots have yet to achieve the physical analog - using vacuum cleaners, From: AAAI-96 Proceedings. Copyright © 1996, AAAI (www.aaai.org). All rights reserved. Personal Assistants Mass Services Figure 1: The Information Food Chain lawn mowers, etc.l Much of our work has focused on the Internet soft- bot (also known as Rodney) (Etzioni & Weld 1994). Rodney enables a person to state what he or she wants accomplished. Rodney disambiguates the request and dynamically determines how and where to satisfy it, utilizing a wide range of Internet services and Unix commands. Rodney relies on a declarative represen- tation of the different software tools at its disposal, enabling it to chain together multiple tools in response to a user’s request. Rodney uses automatic plan- ning technology to dynamically generate the appro- priate action sequence. The Internet softbots project has led to a steady stream of technical results (e.g., (Etzioni et al. 1992; Etzioni, Golden, & Weld 1994; Golden, Etzioni, & Weld 1994; Kwok & Weld 1996; Perkowitz & Etzioni 1995)). Closely related projects include (Kirk et al. 1995; Arens et al. 1993). Unfortunately, we have yet to produce a planner- based softbot that meets the stringent demands of the Web community. While continuing our ambitious long-term project to develop planner-based softbots, we have embraced a new strategy for the creation of intelligent agents which I call “useful first.” Instead of starting with grand ideas about intelligence and issu- ‘softbots are an attractive substrate for intelligent- agent research for the following reasons (Etzioni 1993; 1994). First, the cost, effort, and expertise necessary to develop and systematically experiment with software arti- facts are relatively low. Second, software environments cir- cumvent many of the thorny but peripheral problems that are inescapable in physical environments. Finally, in con- trast to simulated physical worlds, software environments are readily available (sophisticated simulations can take years to perfect), intrinsically interesting, and real. How- ever, Softbots are not intended to replace robots; Robots and softbots are complimentary. ing a promissory note that they will eventually yield useful intelligent agents, we take the opposite tack; we begin with useful softbots deployed on the Web, and issue a promissory note that they will evolve into more intelligent agents. We are still committed to the goal of producing agents that are both intelligent and useful. However, I submit that we are more likely to achieve this conjunctive goal if we reverse the traditional sub- goal ordering and focus on building useful systems first. The argument for “useful first” is analogous to the argument made by Rod Brooks (Brooks 1991) and oth- ers (Etzioni 1993; Mitchell et al. 1990) for building complete agents and testing them in a real world. As Brooks put it, “with a simplified world. . . it is very easy to accidentally build a submodule of the sys- tems which happens to rely on some of those simplified properties. . . the disease spreads and the complete sys- tem depends in a subtle way on the simplified world.” This argument applies equally well to user demands and real-time constraints on Web agents. There is a huge gulf between an AI prototype and an agent ready for deployment on the Web. One might argue that this gulf is of no interest to AI researchers. However, the demands of the Web community con- strain the AI techniques we use, and lead us to new AI problems. We need to recognize that intelligent agents are ninety-nine percent computer science and one per- cent AI. The AI is critical but we cannot ignore the context into which it is embedded. Patrick Winston has called this the “raisin bread” model of AI. If we want to bake raisin bread, we cannot focus exclusively on the raisins.2 Operating on a shoestring budget, we have been able 2See (Brachman 1992) for an account of the massive re-engineering necessary to transform an “intelligent first” knowledge representation system into a usable one. Invited Talks 1323 Figure 2: The Softbot Family Tree. The black boxes represent softbots developed at the University of Washington. MetaCrawler, Ahoy!, and ShopBot have been deployed on the Web. to deploy several softbots on the Web within one year. I review our fielded softbots and then consider both the benefits and pitfalls of the “useful first” approach. MetaCrawler The MetaCrawler softbot3 provides a single, unified in- terface for Web document searching (Selberg & Etzioni 1995). MetaCrawler supports an expressive query lan- guage that allows searching for documents that con- tain certain phrases and excluding documents contain- ing other phrases. MetaCrawler queries nine of the most popular information herbivores in parallel. Thus, MetaCrawler eliminates the need for users to try and re- try queries across different herbivores. Furthermore, users need not remember the address, interface and capabilities of each one. Consider searching for doc- uments containing the phrase “four score and seven years ago.” Some herbivores support phrase search- ing whereas others do not. MetaCrawler frees the user from having to remember such details. If a herbi- vore supports phrase searching, MetaCrawler automat- ically invokes this feature. If a herbivore does not support phrase searching, MetaCrawler automatically downloads the pages returned by that herbivore and performs its own phrase search locally. In a recent article, Forbes Magazine asked Lycos’s Michael Maudlin “why aren’t the other spiders as smart as MetaCrawler ?” Maudlin replied “with our vol- ume I have to turn down the smarts...MetaCrawler will too if it gets much bigger.” Maudlin’s reply misses an important point: because MetaCrawler relies on infor- mation herbivores to do the resource-intensive grazing of the Web, it is sufficiently lightweight to run on an average PC and serve as a personal assistant. Indeed, MetaCrawler-inspired PC applications are now on the market. MetaCrawler demonstrates that Web services and their interfaces may be de-coupled. MetaCrawler is a meta-interface with three main benefits. First, the same interface can be used to access multiple services simultaneously. Second, since the meta-interface has relatively modest resource requirements it can reside on an individual user’s machine, which facilitates cus- tomization to that individual. Finally, if a meta- interface resides on the user’s machine, there is no need to “turn down the smarts.” In a Web-mediated client/server architecture, where intelligence resides in the client, “volume” is no longer a limiting factor on the “smarts” of the overall system. While MetaCrawler does not currently use AI tech- niques, it is evolving rapidly. For example, we are in- vestigating the use of document clustering to enable users to rapidly focus on relevant subsets of the refer- ences returned by MetaCrawler. In addition, we are in- vestigating mixed-initiative dialog to help users focus their search. Most important, MetaCrawler is an en- abling technology for softbots that are perched above it in the information food chain. Ahoy! The Home Page Finder The Ahoy! softbot4 specializes in locating people’s home pages on the Web by filtering MetaCrawler out- put (Shakes, Langheinrich, & Etzioni 1996). Ahoy! takes as input a person’s name and affiliation, and attempts to find the person’s home page. Ahoy! queries MetaCrawler and uses knowledge of Web geog- raphy (e.g., the URLs of home pages at the University of Washington end with Washington. edu) and home page appearance (a home page title is likely to contain a person’s last name) to filter MetaCrawler’s output. Typically, Ahoy! is able to cut the number of refer- ences returned by a factor of forty but still maintain 1324 AAAI-96 very high accuracy. Since Ahoy!% filtering algorithm is heuristic, it asks its users to label its answers as correct or not. Ahoy! uses the feedback it receives from its users to continually improve its performance. It rapidly collects a set of home pages and near misses (la- beled as such by users) to use as training data for an algorithm that attempts to learn the conven- tions underlying home page placement. For exam- ple, home pages at the University of Washington’s Computer Science Department typically have the form http://www.cs.washington.edu/homes/clastname>. After learning, Ahoy! is able to locate home pages of in- dividuals even before they are indexed by MetaCrawler’s herd of information herbivores. In the context of Ahoy!, the “useful first” constraint led us to tackle an important impediment to the use of machine learning on the Web. Data is abundant on the Web, but it is unlabeled. Most concept learning techniques require training data labeled as positive (or negative) examples of some concept. Techniques such as uncertainty sampling (Lewis & Gale 1994) reduce the amount of labeled data needed, but do not elimi- nate the problem. Instead, Ahoy! attempts to harness the Web’s interactive nature to solve the labeling prob- lem. Ahoy! relies on its initial power to draw numerous users to it and to solicit their feedback; it then uses this feedback to solve the labeling problem, make general- izations about the Web, and improve its performance. Note that by relying on feedback from multiple users, Ahoy! rapidly collects the data it needs to learn; sys- tems that are focused on learning an individual users taste do not have this luxury. Ahoy!‘s boot-strapping architecture is not restricted to learning about home pages; user feedback may be harnessed to learn in a variety of Web domains. ShopBot ShopBot5 is a softbot that carries out comparison shop- ping at Web vendors on a person’s behalf (Doorenbos, Etzioni, & Weld 1996). Whereas virtually all previ- ous Web agents rely on hard-coded interfaces to the Web sites they access, ShopBot autonomously learns to extract product information from Web vendors given their URL and general information about their product domain (e.g., software). Specifically, S hopBot learns how to query a store’s searchable product catalog, learns the format in which product descriptions are presented, and learns to extract product attributes such as price from these descriptions. ShopBot’s learning algorithm is based in part on that of the Internet Learning Agent (I LA) (Perkowitz & Et- zioni 1995). I LA learns to extract information from un- familiar sites by querying with familiar objects and an- alyzing the relationship of output tokens to the query object. ShopBot borrows this idea from /LA; ShopBot learns by querying stores for information on popular products, and analyzing the stores’ responses. How- ever, ShopBot tackles a more ambitious learning prob- lem than I LA because Web vendors are far more com- plex and varied than the Internet directories that I LA was tested on. In the software shopping domain, ShopBot has been given the home pages for 12 on-line software vendors. After its learning is complete, ShopBot is able to speed- ily visit the vendors, extract product information such as availability and price, and summarize the results for the user. In a preliminary user study, ShopBot users were able to shop four times faster (and find better prices!) than users relying only on ‘a Web browser (Doorenbos, Etzioni, & Weld 1996). Discussion Every methodology has both benefits and pitfalls; the softbot paradigm is no exception. Perhaps the most important benefit has been the discovery of new re- search challenges, the imposition of tractability con- straints on AI algorithms, and the resulting innova- tions. In recent years, planner-based softbots have led us to the challenge of incorporating information goals, sensory actions, and closed world reasoning into plan- ners in a tractable manner. Our focus on tractabil- ity led us to formulate UWL (Etzioni et al. 1992) and Local Closed World Reasoning (Etzioni, Golden, & Weld 1994; 1995). We expect “useful first” to be equally productive over the next few years. For ex- ample, MetaCrawler has led us to investigate on-line, real-time document clustering. Previous approaches to document clustering typically assume that the entire document collection is available ahead of time, which permits analysis of the collection and extensive pre- processing. In the context of MetaCrawler, document snippets arrive in batches and the delay due to docu- ment clustering has to be minimal. As a result, clus- tering must take place as the snippets are rolling in. I acknowledge that our approach has numerous pit- falls. Here are a couple, phrased as questions: will we fail to incorporate substantial intelligence into our softbots? Does the cost of deploying softbots on the Web outweigh the benefit? Our preliminary success in incorporating AI techniques into our deployed softbots makes me optimistic, but time will tell. Each of the softbots described above uses multiple Web tools or services on a person’s behalf. Each softbot enforces a powerful abstraction: a person is able to state what they want, the softbot is responsible for deciding which Web services to invoke in response and how to do so. Each softbot has been deployed on the Web, meeting the requirements of robustness, speed, and added value. Currently, MetaCrawler receives close to 100,000 hits a day. Ahoy! and ShopBot have yet to be announced publicly. However, shortly after its Invited Talks 1325 release on the Web, Ahoy! was discovered by Yahoo and mentioned in its directory. Immediately, it began receiving hundreds of queries per day. Having satisfied the “useful first” constraint, our challenge is to make our current softbots more intel- ligent, inventing new AI techniques and extending fa- miliar ones. We are committed to doing so while keep- ing our softbots both usable and useful. If we succeed, we will help to rid AI of the stereotype “if it works, it ain’t AI.” To check on our progress, visit the IJRLs mentioned earlier. Softbots are standing by.. . Acknowledgments I would like to thank Dan Weld, my close collaborator on many of the softbots described above, for his numer- ous contributions to the softbots project and its vision; he cannot be held responsible for the polemic tone and sub- versive methodological ideas of this piece. I would also like to thank my co-softbotists David Christianson, Bob Doorenbos, Marc Friedman, Keith Golden, Nick Kushm- crick, Cody Kwok, Neal Lesh, Mark Langheinrich, Su- jay Parekh, Mike Perkowitz, Richard Segal, and Jonathan Shakes for making softbots real. Thanks are due to Steve Hanks and other members of the UW AI group for help- ful discussions and collaboration. I am indebted to Ema Nemes for her assistance in writing this paper and creat- ing its figures. This research was funded in part by Of- fice of Naval Research grant 92-J-1946, by ARPA / Rome Labs grant F30602-95-1-0024, by a gift from Rockwell In- ternational Palo Alto Research, and by National Science Foundation grant IRI-9357772. References Arens, Y.; Chee, C. Y.; Hsu, C.-N.; and Knoblock, C. A. 1993. Retrieving and integrating data from multiple infor- mation sources. International Journal on Intelligent and Cooperative Information Systems 2(2):127-158. Brachman, R. 1992. “Reducing” CLASSIC to Practice: Knowledge Representation Theory Meets Reality. In Proc. 3rd Int. Conf. on Principles of Knowledge Repre- sentation and Reasoning. Brooks, R. 1991. Intelligence without representation. Ar- tificial Intelligence 47:3.39-159. Doorenbos, B.; Etzioni, 0.; and Weld, D. 1996. A scal- able comparison-shopping agent for the world-wide web. Technical Report 96-01-03, University of Washington, De- partment of Computer Science and Engineering. Available via FTP from pub/ai/ at ftp. cs . Washington. edu. Etzioni, O., and Weld, D. 1994. A Softbot-Based Interface to the Internet. CACM 37(7):72-76. See http://www.cs.washington.edu/research/softbots. Etzioni, 0.; Hanks, S.; Weld, D.; Draper, D.; Lesh, N.; and Williamson, M. 1992. An Approach to Planning with Incomplete Information. In Proc. 3rd Int. Conf. on Principles of Knowledge Representation and Reasoning. San Francisco, CA: Morgan Kaufmann. Available via FTP from pub/ai/ at f tp . cs . Washington. edu. Etzioni, 0.; Golden, K.; and Weld, D. 1994. Tractable closed-world reasoning with updates. In Proc. 4th Int. Conf. on Principles of Knowledge Representation and Reasoning, 178-189. San Francisco, CA: Morgan Kauf- mann. Etzioni, 0.; Golden, K.; and Weld, D. 1995. Sound and efficient closed-world reasoning for planning. Technical Report 95-02-02, University of Washington. Available via FTP from pub/ai/ at f tp . cs . Washington. edu. Etzioni, 0.; Lesh, N.; and Segal, R. 1993. Build- ing softbots for UNIX (preliminary report). Techni- cal Report 93-09-01, University of Washington. Avail- able via anonymous FTP from pub/etzioni/softbots/ at cs.washington.edu. Etzioni, 0. 1993. Intelligence without robots (a reply to brooks). AI Magazine 14(4). Available via anonymous FTP from pub/etzioni/softbots/ at cs.washington.edu. Etzioni, 0. 1994. Etzioni Responds. AI Magazine. Re- sponse to commentary on “Intelligence without Robots (A Reply to Brooks)“. Golden, K.; Etzioni, 0.; and Weld, D. 1994. Omnipotence without omniscience: Sensor management in planning. In Proc. 12th Nat. Conf. on A-I., 1048-1054. Menlo Park, CA: AAAI Press. Kirk, T.; Levy, A. Y.; Sagiv, Y.; and Srivastava, D. 1995. The information manifold. In Working Notes of the A A A I Spring Symposium: Information Gathering from Heterogeneous, Distributed Environments, 85-91. Stan- ford University: AAAI Press. To order a copy, contact sss@aaai.org. Kwok, C., and Weld, D. 1996. Planning to gather information. Technical Report 96-01-04, University of Washington, Department of Computer Science and Engineering. Available via FTP from pub/ai/ at ftp.cs.washington.edu. Lewis, D., and Gale, W. 1994. Training text classifiers by uncertainty sampling. In 17th Annual Int’l ACM SIGIR Conference on Research and Development in Information Retrieval. Mitchell, T. M.; Allen, J.; Chalasani, P.; Cheng, J.; Et- zioni, 0.; Ringuette, M.; and Schlimmer, J. C. 1990. Theo: A framework for self-improving systems. In VanLehn, K., ed., Architectures for Intelligence. Hillsdale, NJ.: Erl- baum. Perkowitz, M., and Etzioni, 0. 1995. Category transla- tion: Learning to understand information on the internet. In Proc. 15th Int. Joint Conf. on A.I. Selberg, E., and Etzioni, 0. 1995. Multi-Service Search and Comparison Using the MetaCrawler. In Proc. 4th World Wide Web Conf., 195-208. See http://www.cs.washington.edu/research/metacrawler. Shakes, J.; Langheinrich, M.; and Etzioni, 0. 1996. Ahoy! the home page finder. Technical re- port, University of Washington. To appear, see http://www.cs.washington.edu/research/ahoy. Simon, H. 1991. Models of My Life. Basic Books. 1326 AAAI-96 | 1996 | 196 |
1,839 | ynamics in the genesis of trust as the basis for communication by representations. Walter J Freeman Department of Molecular & Cell Biology, LSA 129 University of California at Berkeley CA 94720 wfreeman@garnet.berkeley.edu Abstract A theory of brain dynamics is proposed according to which brains construct external representations by actions into the world for communication. The prior brain patterns constitute meanings, not representations of meanings. The representations have no meaning in themselves. They are shaped in accordance with meaning inside transmitting brains, and they can elicit the construction of meaning inside receiving brains, provided that trust has been established between the transmitters and the receivers through appropriate neurochemical changes. The Nature of IIdS There are three classes of ory about the nature of minds, each with its remarkable successes, and also its intractable problems. A. Material, Empirical - minds are “nothing but . ..‘I the activity of neurons according to most neurobiologists; hierarchies of reflexes for behaviorists; a chemical stew for geneticists, clinicians, and pharmacologists; or quantum coherences for physicists. These approaches have given powerful tools for investigating and treating disorders of both brain and behavior, but have conceived minds either as epiphenomena or as mysteries, leaving unexplained how the meaningless firing of neurons can lead to meaningful subjective experiences (Searle, 1995). Cognitive, Idealist - minds are sets of representations, such as thoughts and ideas that are processed according to rules discovered by psychologists, or images and symbols that are manipulated according to syntactical rules. The intractable problems are those of introducing motivations, drives http://sulcus.berkeley.edu and instincts, and of devising rules on how to attach meanings and values to signs. robots are built in conformance to look-up tables and difference equations, can they ever be conscious, or have free will? ntentional, Existentia actions into the world, from John Dewey (1914) (“mind is action into the stimulus”) and Merleau-Ponty (1945) (“La Structure du Comportement”). Though discussed in extenso by pragmatists, Cestaltists including JJ Gibson (1963), Piagetians and others, the intractable problems have been how to account for the inner construction of intentional behavior and perception through self-organizing brain dynamics, and for the genesis of knowledge in the face of the problem of solipsism (Freeman, 1995). A biological approach to the brain-mind problem is to study the evolution of minds and brains, on the premiss that animals have minds and brains that are prototypic of our own, and that their brains and behaviors can tell us much about our own minds. servations of t Experimental observations of the brain activity that follows sensory stimulation of animals show that sensory cortices engage in construction of activity patterns in response to stimuli. The operation is not that of filter, storage, retrieval, addressing or correlation mechanisms. It is a state transition by which a cortex switches abruptly from one basin of attraction to another, thereby to change one spatial pattern to another like frames in a cinema (Freeman, 1975, 1992). The transitions in the primary sensory cortices are shaped by interactions with the limbic system, which formulate the intentional Invited Talks 1327 From: AAAI-96 Proceedings. Copyright © 1996, AAAI (www.aaai.org). All rights reserved. nature of percepts. They result from goal- directed actions in time and space. Each transition involves learning, so that cumulatively a trajectory is formed by each brain over its lifetime. Each spatial pattern as it occurs reflects the entire content of individual experience. It is a meaning and not the representation of a meaning. It is the basis for consciousness. Inferences made from EEG studies about the nature of meaning are as follows. Brains are open with respect to energy and information, but closed systems with respect to meaning. Brains create their own frames of reference, and can have no direct communication, such as by ESP. Each consciousness is isolated from all others. Brains have no direct access to the physical world. All perceptions are constructs from raw sensory input. Intentionality is texture and context in the dynamical structure of space-time memory. It is based in a neural net by neurochemical modulations of synapses and trigger zones. Meanings are places in this structure. A Theory of Representation Four findings led to these conclusions and the demise of a theory of representations in the experiments designed to test (Freeman, 1983; Skarda and Freeman, 1987): 1, The EEG spatial amplitude patterns observed during training lacked invariance with respect to the conditioned stimuli over time and learning. 2. The EEG spatial patterns in the control periods reflected the null hypothesis and not the specific expectations that had been established by training. 3. The EEG phase patterns did not show a requisite convergence to synchrony (“binding”) with arrival of expected stimuli. 4. The EEG phase patterns did manifest the repeated nonlinear state transitions that enable the sensory cortices to construct the spatial patterns of amplitude appropriate for the conditioned stimuli and conditioned responses. A Theory of Trust It follows that each brain creates its own frames of reference, which are not directly accessible by any other brain.- IIow, then, can two or more brains be shaped by learning, so as to form cooperative pairs for reproduction and groups for survival? Evolution has provided a biological mechanism that first came under scientific scrutiny in the form of Pavlovian ‘brain washing’. Under now well known conditions of stress in the internal and external environments, a global transition takes place, following which brains sustain a remarkable period of malleability (Freeman, 1995). I believe that Pavlov manipulated a mammalian mechanism of pair bonding, for the nurture of altricial young through sexual orgasm and lactation, mediated by oxytocin, and that our remote ancestors evolved to adapt this mechanism for tribal bonding through dance, chanting, rituals, and evangelical conversions (Sargant 1957). These dimensions of human experience can be encompassed by a neurodynamical theory of intentionality, but not by theories of representation and symbol manipulation. References Dewey J (1914) Psychological doctine in philosophical teaching. Journal of Philosophy 11: 505-5 12. Freeman WJ (1975) Mass Action in the Nervous System. New York: Academic. Freeman WJ (1983) The physiology of mental images. Biological Psychiatry 18:1107-1125. Freeman WJ (1992) Tutorial in Neurobiology. International Journal of Bifurcation and Chaos 2: 451-482. Gibson JJ (1979) The Ecological Approach to Visual Perception. Boston: Houghton Mifflin. Freeman WJ (1995) Societies of Brains. Hillsdale NJ, Lawrence Erlbaum. Merleau-Ponty M (1942/1963) The Structure of Behavior (AL Fischer, Trans.). Boston: Beacon Press. Sargant W (1957) Battle for the Mind. Westport CT, Greenwood Press. Searle JR (1995) The Mystery of Consciousness. New York Rev 2-18 Nov, Skarda CA and Freeman WJ (1987) How brains make chaos in order to make sense of the world. Behav. & Brain Sci. 10: 161- 195. 1328 AAAI-96 | 1996 | 197 |
1,840 | Using Multi-Agent Systems to resent Uncertainty Joseph U. Halpern IBM Almaden Research Center San Jose, CA 95120 email: halpern@almaden.ibm.com Abstract I consider a logical framework for modeling uncer- tainty, based on the use of possible worlds, that incor- porates knowledge, probability, and time. This turns out to be a powerful approach for modeling many problems of interest. I show how it can be used to give insights into (among other things) several well- known puzzles. Introduction Uncertainty is a fundamental-and unavoidable- feature of daily life. In order to deal with uncertainty intelligently, we need to be able to represent it and rea- son about it. This invited talk describes one systematic approach for doing so. Reasoning about uncertainty can be subtle. Con- sider the following well-known puzzles. (These puzzles are presented under the assumption that the uncer- tainty is quantified in terms of probability, but the is- sues that they bring out arise whatever method we use to represent uncertainty.) The second-ace puzzle (Bar-Hillel & Falk 1982; Freund 1965; Shafer 1985): Suppose we have a deck with four cards: the ace and deuce of hearts, and the ace and deuce of spades. After a fair shuffle of the deck, two cards are dealt to Alice. It is easy to see that, at this point, there is a probability of l/6 that Alice has both aces, probability 5/6 that Alice has at least one ace, probability l/2 that Alice has the ace of spades, and probability l/2 that Alice has the ace of hearts: Out of the six possible deals of two cards out of four, Alice has both aces in one of them, at least one ace in five of them, the ace of hearts in three of them, and the ace of spades in three of them. Alice then says “I have an ace”. Conditioning on this information, Bob computes the probability that Alice holds both aces to be l/5. This seems reason- able: The probability of Alice having two aces goes up if we find out she has an ace. Next, Alice says “I have the ace of spades”. Conditioning on this new information, Bob now computes the probability that Alice holds both aces to be l/3. Of the three deals in which Alice holds the ace of spades, she holds both aces in one of them. As a result of learning not only that Alice holds at least one ace, but that the ace is actually the ace of spades, the conditional probabil- ity that Alice holds both aces goes up from l/5 to l/3. Similarly, if Alice had said “I have the ace of hearts”, the conditional probability that Alice holds both aces would be l/3. But is this reasonable ? When Bob learns that Al- ice has an ace, he knows that she must have either the ace of hearts or the ace of spades. Why should finding out which particular ace it is raise the con- ditional probability of Alice having two aces? The Monty Hall Puzzle (Savant 1990/91; Morgan et al. 1991): Suppose you’re on a game show and given a choice of three doors. Behind one is a car; behind the others are goats. You pick door 1. Before opening door 1, Monty Hall, the host (who knows what is behind each door), opens door 2, which has a goat. He then asks you if you still want to take what’s behind door 1, or to take what’s behind door 3 instead. Should you switch? There is certainly far more to representing uncer- tainty than dealing with puzzles such as these. Never- theless, the analysis of these puzzles will give us deeper insight into the process of reasoning under uncertainty and the problems involved with getting a good rkpre- sentation. So how do we represent and reason about uncer- tainty? I shall use the possible-worlds framework. This is the standard approach for giving semantics to modal logic. The intuition is that besides the true state of af- fairs, there are a number of other possible states of affairs or “worlds”, that an agent considers possible. We can view the set of worlds that an agent considers possible as a qualitative way to measure her uncer- tainty. The more worlds she considers possible, the more uncertain she has as to the true state of affairs, and the less she knows. This is not quite enough for dealing with the puzzles above. We need to add two more features to the picture: time and probability. To add time, we need to have possible worlds describing not only the current state of affairs, but the state of Invited Talks 1329 From: AAAI-96 Proceedings. Copyright © 1996, AAAI (www.aaai.org). All rights reserved. affairs at each time point of interest. As we shall see, it is also useful to assume that these states have some internal structure. This gives us the multi-agent sys- tems framework of (Fagin et al. 1995). To add proba- bility, we need to associate with each possible world a probability distribution over other possible worlds; this issue is discussed in detail in (Fagin & Halpern 1994; Halpern & Tuttle 1993). The resulting multi-agent systems provide a power- ful framework in which we can represent, in a natural way, time, knowledge, and probability. But where does the system come from ? Typically, it is generated by a protocol. An important theme in the talk is the im- portance of specifying clearly the protocol generating the system. In particular (as already pointed out by Shafer 1985), this is the key to understanding puzzles such as the second-ace puzzle. The material in this talk is largely covered in (Halpern 1995). References Bar-Hillel, M., and Falk, R. 1982. Some teasers con- cerning conditional probabilities. Cognition 11:109- 122. Fagin, R., and Halpern, J. Y. 1994. Reasoning about knowledge and probability. Journal of the A CM 41(2):340-367. Fagin, R.; Halpern, J. Y.; Moses, Y.; and Vardi, M. Y. 1995. Reasoning about Knowledge. Cambridge, Mass.: MIT Press. Freund, J. E. 1965. Puzzle or paradox? American Statistician 19(4):29-44. Halpern, J. Y., and Tuttle, M. R. 1993. Knowledge, probability, and adversaries. Journal of the ACM 40(4):917-962. Halpern, J. Y. 1995. A logical approach for reasoning about uncertainty. Research Report RJ 9972, IBM. Morgan, J. P.; Chaganty, N. R.; Dahiya, R. C.; and Doviak, M. J. 1991. Let’s make a deal: the player’s dilemma (with commentary). The American Statisti- cian 45(4):284-289. vos Savant, M. 1990/91. Ask Marilyn. Parade Mag- azine (Sept. 9, 1990; Dec. 2, 1990; Feb. 17, 1991). Shafer, G. 1985. Conditional probability. Interna- tional Statistical Review 53(3):261-277. 1330 AAAI-96 | 1996 | 198 |
1,841 | Refinement lanning: Status an Subbarao Kambhampati* Department of Computer Science and Engineering Arizona State University, Tempe, AZ 85287, rao@asu.edu Abstract Most current-day AI planning systems operate by iter- atively refining a partial plan until it meets the goal requirements. In the past five years, significant progress has been made in our understanding of the spectrum and capabilities of such refinement planners. In this talk, I will summarize this understanding in terms of a unified framework for refinement planning and discuss several current research directions. Introduction Developing automated methods for generating and reasoning about plans and schedules, whether in aid of autonomous or human agents, has been part and parcel of AI research from the beginning. The need for planning arises naturally when an agent is interested in controlling the evolution of its environment. Algorithmically, a planning problem has as input a set of possible courses of actions, a predictive model for the underlying dynamics, and a performance measure for evaluating the courses of action. The output or solution is one or more courses of action that satisfy the specified requirements for performance. A planning problem thus involves deciding “what” actions to do, and “when” to do them. The “when” part of the problem has traditionally been called the “scheduling’ ’ problem [20]. The simplest case of the planning problem, where the environment is static and deterministic, and the planner has complete information about the current state of the world, has come to be known as the classical planning problem. My talk is concerned with algorithms for synthesizing plans in classical planning. Generating plans for classical planners has received significant attention over the past twenty years. Most of the plan generation algorithms that have been developed are informally called “refinement planners”, in that they iteratively refine a partial plan until it meets the specified goals. In this talk, I will attempt to provide a coherent semantic *This research is supported in part by NSF research initiation award (RIA) IRI-9210997, NSF young investigator award (NYI) IRI-9457634 and ARPA/Rome Laboratory planning initiative grants F30602-93-C-0039 and F30602-95-C-0247. Special thanks to Bi- plav Srivastava, Gopi Bulusu, Suresh Katukam, and Laurie Ihrig for the many hours of discussions, David McAllester for his patient correspondence regarding SNLP and refinement search, and Dan Weld for his encouragement. Portions of this paper are borrowed from a recent overview of planning approaches, which I co-authored with Tom Dean. picture of refinement planning, and describe the various existing approaches in terms of this framework. I will also consider the tradeoffs inherent in refinement planning, and possible directions for developing more efficient refinement planners. Preliminaries of Modeling Change: Before proceeding further, let me briefly review how classical planning problems are modeled. In most classical planning approaches, a state is described in terms of a set of boolean state variables. Suppose that we have three boolean state variables: P, Q, and R. We represent the particular state s in which P and Q are true and R is false by the state-variable assignment, s = (P = true, Q = true, R = false}, or, somewhat more compactly, by s = (P, Q, lR}. An action is represented as a state-space operator Q de- fined in terms of preconditions (Pre(ar)) and postconditions (also called effects) (Post(a)). If an operator (action) is applied (executed) in a state in which the preconditions are satisfied, then the variables mentioned in the postconditions are assigned their respective values in the resulting state. If the preconditions are not satisfied, then there is no change in state. Several syntactic extensions can be added on top of this basic operator representation, facilitating conditional effects and effects quantified over finite universes. Pednault 1171 shows that this action representation is semantically equiva- lent to the largest subset of situation calculus for which we can get by without writing frame axioms explicitly. Goals are represented as state-variable assignments that assign values to subsets of the set of all state variables. By assigning values to one or more state variables, we designate a set of states as the goal. We say that a state s satisfies a goal (13, notated s k 4, just in case the assignment C# is a subset of the assignment s. Given an initial state SO, a goal C$J, and a library of operators, the objective of the planning problem is to find a sequence of state-space operators (at, . . . , cm) such that f(so, (~1,. . . , an>> j= 4. Semantic picture of Refinement planners 181 attempt to solve a planning problem by navigating the space of sets of potential solutions (action sequences). The potential solution sets are represented and manipulated in the form of “partial plans.” Syntactically, a partial plan T can be seen as a set of constraints (see below). Semantically, a partial plan is a shorthand notation for the set Invited Talks 1331 From: AAAI-96 Proceedings. Copyright © 1996, AAAI (www.aaai.org). All rights reserved. of action sequences that are consistent with its constraints. The set of such action sequences is called the set of candidates (or candidate set) of the partial plan. We define a generic refinement planning procedure, Refine(n), as follows 181. 1. If an action sequence (cY~, 02, . . . , a,) is a candidate of 7r and also solves the planning problem, terminate and return the action sequence. 2. If the constraints in zr are inconsistent, then eliminate 7r from future consideration. 3. Select a refinement strategy, and apply the strategy to x and add the resulting refinements to the set of plans under consideration. 4. Nondeterministically select a plan ?r’ from those under consideration and call Refine(n’). The first step of the search process is the “solution con- struction” process, where the planner attempts to extract a solution from the current partial plan’s candidate set. We shall see later that the solution constructor function checks only on the minimal candidates of the plan, since the candidate set of a partial plan can be infinitely large [8l. The second step is closely related to the first, and attempts to prune the plan from further refinement if it can be shown not to contain any solutions. The last two steps involve applying a refinement operator to the partial plan to generate new partial plans, and recursing on one of those refinements. Refinements can be understood as operations that split the candidate set of the partial plan to which they are applied. Specifically, a refinement strategy converts a partial plan x into a set of new plans {YQ, . . . , 7rn) such that the candidate set of each 7ri is a subset of the candidate set of X. A refinement operator is said to be complete if every solution belonging to the candidate set of the plan will be in the candidate sets of at least on of the plans generated by the refinement operator. A refinement operator is said to be systematic if the candidate sets of the refinements are disjoint. It is easy to see that the selection of refinement strategy does not have to backtracked over, as long as the refinement operators are complete. The specifics of a refinement planning algorithm will differ depending on the representation of the partial plans used (i.e., what specific constraints are employed) and the type of refinements employed on that representation. We already pointed out that syntactically, a partial plan is a set of constraints. The semantic status of a plan constraint is clarified by specifying when a given action sequence is said to satisfy the constraint. Within these broad guidelines, a large variety of syntactic representations can be developed. Once a representation for a partial plan is given, a refinement operator can be specified in terms of the types of constraints that it adds to a partial plan. If the constraint sets added by refinements are mutually exclusive and exhaustive, then the refinement operators will be systematic and complete. Representing Partial Plans To focus our discussion, we will start by looking at a specific partial plan representation that is useful for modeling most existing planners (later, we will consider alternative repre- sentations that are promising). In this representation, partial plan consists of a set of steps, a set of ordering constraints that restrict the order in which steps are to be executed, and a set of auxiliary constraints that restrict the value of state vari- ables over particular intervals of time. Each step is associated 1332 AAAI-96 Figure 1: This figure depicts the partial plan reg. The postconditions (effects) of the steps are shown above the steps, while the preconditions are shown below the steps in parentheses. The ordering constraints between steps are shown by arrows. The interval preservation constraints are shown by arcs, while the contiguity constraints are shown by dotted lines. \ GrO”“d Llncwizatlc.” ” I Ground fmmnmtons rhor ~oasfv oudmrv rmmm,n,, \ Safe Ground Llocmtullon 1 : sate @,mmd Lhemiution m : i Comspondr ID ,hc qmwtd oprmror sequence : . Syntactic View -------------_.-------------------------------------------.----------------. . i : i Semantic View I \ I \ \ / \ Union of these sets is the caddate set of the partial plan Figure 2: A schematic illustration of the relation between a partial plan and its candidate set. T with a state-space operator. To distinguish between multiple instances of the same operator appearing in a plan, we assign to each step a unique integer i and represent the ith step as the pair (i, CY;) where Q; is the operator associated with the ith step. Figure 1 shows a partial plan reg consisting of seven steps. The plan reg is represented as follows. ( ((0, ao), (1, al), (2, d, (3, ~3)r (4, ~41, (5, w)r (00, G}, ((0 7 I),(1 4 a,(1 4 4),(2 4 3),(3 4 3,(4 + 5),(5 2 W}, ((1 2 21, (3 fz 4) ) An ordering constraint of the form (i 4 j) indicates that Step i precedes Step j. An ordering constraint of the form (i 2 j) indicates that Step i is contiguous with Step j, that is Step i precedes Step j and no other steps intervene. The steps are partially ordered in that Step 2 can occur either before or P after Step 4. An auxiliary constraint of the form (i 1 j) is called an interval preservation constraint and indicates that P is to be preserved in the range between Steps i and j (and therefore no operator with postcondition -IP should occur between Steps i and j). In particular, according to the constraint (3 f oo), Step 4 should not occur between Steps 3 and 00. Figure 2 shows the schematic relations between a partial plan in such a representation and its candidate set, and we will illustrate it with respect to the example plan in Fig- ure 1. Each partial plan corresponds to a set of topological sorts (e.g. (1,2,3,4,5) and (I, 2,4,3,5)). The subset of these that satisfy the auxiliary constraints of the plan (e.g. (1,2,4,3,5)) are said to be the safe-ground linearizations of the plan. Each safe ground linearization of the plan corre- sponds to an action sequence which is a minimal candidate of the partial plan (e.g. ((~1, ~2, CY~, 03, cys)). An infinite number of additional candidates can be derived from each minimal z :stablishment candidate of the plan by augmenting (padding) it with addi- tional actions without violating the auxiliary constraints (e.g. (at, 02, a2, cy4, a3, 05)). Thus, the candidate set of a partial plan is infinite, but the set of its minimal candidates is finite. The solution constructor functions search the minimal candi- dates of the plan to see if any of them are solutions to the planning problem. Refinement process can be understood as ~~~~ I demooon promouon confrontahon incrementally increasing the size of these minimal candidates so that action sequences of increasing lengths are examined to see if they are solutions to the problem. The search starts with Figure 3: Example of plan-space refinement the null plan (((0, QO), (00, CL,&, ((0 + ~43, ()>, where a0 is a dummy operator with no preconditions and postconditions corresponding to the initial state, and Q, is a dummy operator made contiguous to the current head step and becomes the with no postconditions and preconditions corresponding the new head step. goal. As an example, one way of refining the plan Keg in Figure 1 using progression refinement would be to apply an instance Refining Partial Plans of the operator ~2 (either the instance that is currently in the plan (2, ~2) or a new instance) to the head state (recall that There are several possible ways of refining partial plans, corresponding intuitively to different ways of splitting the set of potential solutions represented by the plan. In the follow- ing sections, I outline several popular refinement strategies employed in the planning literature. State-Space Refinements it is (P, Q)). This is accomplished by putting a contiguity constraint between (2, ~2) and the current head step ( 1, Q 1) (thereby making the former the new head step). In realistic problems, many operators may be applicable in the head state and very few of them may be relevant to the top level goals. To improve efficiency, some planners use means-ends analysis to focus on relevant operators. The The most straightforward way of refining partial plans in- general idea is the following: Suppose we have an operator CY valves using progression to convert the initial state into a whose postconditions match a goal of the problem. Clearly, state satisfying the goal conditions, or using regression to Q is a relevant operator. If the preconditions of a, are satisfied convert a set of goal conditions into a set of conditions that in the head state of the current partial plan, we can apply it are satisfied in the initial state. From the point of view of par- directly. Suppose they are not all satisfied. In such a case, tial plans, this corresponds to growing prefix or the suffix of we can consider the preconditions of a, as subgoals, look the plan. The refinements are called state-space refinements for an operator Q’ whose postconditions match one of these since given either the prefix or the suffix of a plan, we can subgoals, and check if it is applicable to the head state. This uniquely determine the nature of the world state following the type of recursive analysis can be continued to find the set of prefix and preceding the suffix. relevant operators, and focus progression refinement [ 141. The set of steps (~t,c72, . . . , a;~) with contiguity con- We can also define a refinement strategy based on regres- straints ((a0 2 crt), (ai 2 02), . . . , (a,-~ 2 a,)) is called the header of the plan X. The last element of the header, a,, , is called the heud step. The state defined by f(so, (a,, , . . . , cyo,)), where Q,; is the operator associated with cri is called the head state. In a similar manner, we can define the tail, tail step, and tail state. As an example, the partial plan neg shown in Figure 1 has the Steps 0 and 1 in its header, with Step 1 being the head step. The head state (which is the state resulting from applying 01 to the initial state) is (P, Q). S imilarly, the tail consists of Steps 5 and 00, with Step 5 being the tail step. The tail state (which is the result of regressing the goal conditions through the operator ~5) is (R, U). Progression (or forward state-space) refinement involves advancing the head state by adding a step cr, such that the preconditions of Q, are satisfied in the current head state, to the header of the plan. The step Q may be newly added to the plan or currently present in the plan. In either case, it is sion, which involves regressing the tail state of a plan through an operator. For example, the operator (~3 is applicable (in the backward direction) through this tail state (which is (R, U)), while the operator ~4 is not (since its postconditions are in- consistent with the tail state). Thus, one way of refining reg using regression refinement would be to apply an instance of the operator ~3 (either the existing instance in Step 3 or a new one) to the tail state in the backward direction. This is accomplished by putting a contiguity constraint between (3,03) and the current tail step. In both progression and regression, solution constructor function can be simplified as follows: check to see if head state is a super set of the tail state, and if so, return the header concatenated with tail. Plan-Space Refinements State-space refinements have to guess correct answers to two questions up front: (a) whether a specific action is relevant to Invited Talks 1333 the goals of the planning problem and (b) where exactly in the final plan does the action take place. Often, it is easier to see whether or not a given action is relevant to a plan, but much harder to guess the precise position at which a step must occur in the final plan. The latter question more naturally falls in the purview of “scheduling” and cannot be answered well until all of the steps have been added. To avoid this premature forced commitment, we would like to introduce the new action into the plan, without committing to its position in the final solution. This is the intuition behind plan-space refinements. The refinement is named “plan-space” because when we allow an action to be part of a plan without constraining it to be either in the prefix or the suffix, the partial plan does not represent a unique world state. Thus, the search cannot be recast in terms of the space of world states. The main idea in plan-space refinement is to shift the attention from advancing or regressing the world state to establishing goals in the partial plan. A precondition P of a step (i, cyi) in a plan is said to be established if there is some step (j, CY~) in the plan that precedes i and causes P to be true, and no step that can possibly intervene between j and i has postconditions that are inconsistent with P. It is easy to see that if every precondition of every step in the plan is established, then that plan will be a solution plan. Plan-space refinement involves picking a precondition P of a step (i, CY~) in the partial plan, and adding enough additional step, or- dering, and auxiliary constraints to ensure the establishment of P. One problem with this precondition-by-precondition establishment approach is that the steps added in establish- ing a precondition might unwittingly violate a previously established precondition. Although this does not affect the completeness of the refinement search, it can lead to wasted planning effort, and necessitate repeated establishments of the same precondition within the same search branch. Many variants of plan-space refinements avoid this inefficiency by protecting their establishments using IPCs. When the plan- ner uses plan-space refinements exclusively, its refinement process can terminate as soon as any of the safe ground linearization of the plan correspond to solutions. Let me illustrate the main ideas in precondition establish- ment through an example. Consider the partial plan at the top in Figure 3. Step 2 in this plan requires a precondition Q. To establish this precondition, we need a step which has Q as its postcondition. None of the existing steps have such a postcondition. Suppose an operator ~3 in the library has a postcondition R + Q. We introduce an instance of a3 as Step 3 into the plan. Step 3 is ordered to come before Step 2 (and after Step 0). Since CX~ makes Q true only when R is true before it, to make sure that Q will be true following Step 3, we need to ensure that R is true before it. This can be done by posting R as a precondition of Step 3. Since R is not a normal precondition of (~3, and is being posted only to guar- antee one of its conditional effects, it is called a secondan, precondition [171. Finally, we can protect the establishment Q of precondition Q by adding the constraint 3 - 2. If we also want to ensure that 3 remains the sole establisher of Q in the final solution, we can add another auxiliary constraint 3 ? 2. In [131, McAllester shows that adding these two auxiliary constraints ensures systematicity of plan-space refinement. Tractability Refinements: Since the position of the steps 1334 AAAI-96 Figure 4: Step 2 in the partial plan shown on the left is reduced to obtain a new partial plan shown on the right. In the new plan, Step 2 is replaced with the (renamed) steps and constraints specified in the reduction shown in the center box. in the plan is not uniquely determined after a plan space refinement, there is uncertainty regarding (a) the state of the world preceding or following a step, (b) the relative order of steps in the plan and (c) the truth of IPC constraints in the plan. A variety of refinement strategies exist that attempt to make the reasoning with partial plans tractable by pushing the complexity into the search space. These refinements, called tractability refinements, fall into three broad classes: pre- positioning, pre-ordering and pre-satisfaction refinements. The first pick a pair of steps CYI and ~2 in the plan and generate two refinements one in which ~1 2 cr2, and the other in which CYI 7(: or2. The pre-ordering refinements do the same thing except they enforce ordering rather than contiguity constraints between the chosen steps. Finally, the pre-satisfaction refinements pick an IPC in the plan, and enforce constraints such that every ground linearization of the plan satisfies the IPC (see below). We can illustrate the pre-satisfaction refinements through the example in Figure 3, after we have introduced Step 3 and ensured that it produces Q as a postcondition, we need to make sure that Q is not violated by any steps possibly intervening between Steps 3 and 2. In our example, Step 1, which can possibly intervene between Steps 3 and 2, has a postcondition P * l&, that is potentially inconsistent with Q. To avert this inconsistency, we can either order Step 1 to come before Step 3 (demotion), or order Step 1 to come after Step 2 (promotion), or ensure that the offending conditional effect will not occur. This last option, called confrontation, can be carried out by posting 1 P as a (secondary) precondition of Step 1. Depending on whether protection strategies are used, and what tractability refinements are used, we can get a very large spectrum of plan-space refinements [81. The effectiveness of plan space refinement in controlling the search is determined by a variety of factors, including (a) the order in which the various preconditions are selected for establishment (b) the manner in which tractability refinements are applied during search. See [81 for a discussion of some of the trade-offs. Task-Reduction Refinements In both the state-space and plan-space refinements, the only knowledge that is assumed to be available about the planning task is in terms of primitive actions (that can be executed by the underlying hardware), and their preconditions and postconditions. Often, one has more structured planning knowledge available in a domain. For example, in a travel planning domain, we might have the knowledge that one can reach a destination by either “taking a flight” or by “taking a train”. We may also know that “taking a flight” in turn involves making a reservation, buying a ticket, taking a cab to the airport, getting on the plane etc. In such a situation, we can consider “taking a flight” as an abstract task (which cannot be directly executed by the hardware). This abstract task can then be reduced to a plan fragment consisting of other abstract or primitive tasks (in this case “making a reservation”, “buying a ticket”, “going to the airport’ ’ , “getting on the plane”). This way, if there are some high-level problems with the “taking flight” action and other goals, (e.g. there is not going to be enough money to take a flight as well paying the rent), we can resolve them before we work on low level details such as getting to the airport. The resolution is can be carried out by the generalized versions of tractability refinements used in plan-space refinement. This idea forms the basis for task reduction refinement. Specifically, we assume that in addition to the knowledge about prin&ive actions, we also have some abstract actions, and a set of schemas (plan fragments) that can replace any given abstract action. Task reduction refinement takes a partial plan x containing abstract and primitive tasks, picks an abstract task CT, and for each reduction schema (plan fragment) that can be used to reduce CT, a refinement of R is generated with o replaced by the reduction schema (plan fragment). As an example, consider the partial plan on the left in Figure 4. Suppose the operator CY~ is an abstract operator. The central box in Figure 4 shows a reduction schema for Step 2, and the partial plan shown on the right of the figure shows the result of refining the original plan with this reduction schema. At this point any interactions between the newly introduced plan fragment and the previously existing plan steps can be resolved using techniques such as promotion, demotion and confrontation discussed in the context of plan-space refinement. This type of reduction is carried out until all the tasks are primitive. Notice that the partial plans used in task reduction planning contain one additional type of constraint -- the non-primitive tasks. Informally, when a plan contains a non-primitive task t, then every candidate of the plan must have the actions comprising at least one concretization of t (where a concretization of a non-primitive task is the set of primitive partial plans that can be generated by reducing it using task reduction schemas). Tradeoffs in Refinement Planning Now that we looked at a variety of approaches to refine- ment planning, it is worth looking at the broad tradeoffs in refinement planning. There are two classes of tradeoffs -- the first arising from algorithmic modifications to the generic refinement search, and the second arising from the match between refinements and the characteristics of the planning domain. An example of the first class of tradeoffs is that between the cost of solution constructor vs. size of the search space. We can reduce the search space size by considering partial plans that can compactly represent a larger number of minimal candidates. From a planning view point, this leads to least commitment on the part of the planners. However, as the number of candidates represented by a partial plan grow, the cost of the picking a solution from the partial plan increases. This tradeoff is well represented in the refinements that we have looked at. Plans produced by state-space refinements will have single minimal candidates, while those produced by plan space refinements can have multiple minimal candidates (corresponding roughly to the many topological sorts of the plan). Finally, partial plans produced using task reduction refinements may have even larger number of minimal candi- dates since the presence of a non-primitive tasks essentially allows any action sequence that contains any concretization of the non-primitive task as a minimal candidate. There are also certain tradeoffs that arise from the match between the plan representations and refinements used, and the characteristics of the planning domain and problem. For example, it is known that the plan-space refinements can be more efficient compared to state-space refinements in domains where the ordering of steps cannot be guessed with reasonable accuracy a priori El; 161. The plan-space refinements also allow separation of action selection and establishment phases from the “scheduling” phase of the planning, thus facilitating easier adaptation of the plan to more situations [71, and to more closely integrate the planning and scheduling phases [41. On the other hand, state-space refinements provide a good sense of the state of the world corresponding to the partial plan, and can thus be useful to agents who need to do non-trivial reasoning about the world state to focus their planning and execution efforts [2; 141. Finally, task-reduction refinements facilitate user control of planner’s access to the primitive actions, and are thus the method of choice in any domain where the user has preferences among the solution plans [9l. Prospectus Although early refinement planning systems tended to sub- scribe exclusively to a single refinement strategy, our unifying treatment of refinement planning demonstrates that it is possi- ble to use multiple refinement strategies. As an example, the partial plan reg shown in Figure 1 can be refined with pro- gression refinement (e.g., by putting a contiguity constraint between Step 1 and Step 2), with regression refinement (e.g., by putting a contiguity constraint between Step 3 and Step 5), or plan-space refinement (e.g., by establishing the precondi- tion S of Step 3 with the help of the effect Step 2). Finally, if the operator ~4 is a non-primitive operator, we can also use task reduction refinement to replace ~24 with its reduction schema. There is some evidence that planners using multiple refinement strategies intelligently can outperform those using single refinement strategies [lo]. However, the question as to which refinement strategy should be preferred when is still largely open. We can be even more ambitious however. Most existing refinement planners have trouble scaling up to larger prob- lems, because of the very large search spaces they generate. While application of machine learning techniques to planning [15] hold a significant promise, we can also do better by improving the planning algorithms. One way of controlling the search space blow-up is to introduce appropriate forms of disjunction into the partial plan representation. By doing this, we can allow a single partial plan to stand for a larger number of minimal candidates. The conventional wisdom in refinement planning has been to keep the solution con- struction function tractable by pushing the complexity into the search space [8l. Some recent work by Blum and Furst 131 shows that partial plan representations that push all the complexity into the solution construction function may actu- Invited Talks 1335 Figure 5: To the left is the search space generated by a refinement planner using progression refinement. To the right is the partial plan representation, called plan graph, used in Graphplan 131. Each candidate plan of the plan graph must have some subset of the actions in jth level comin immediately before some subset of actions in the i + 1 t B level (for all i). The minimal candidates corresponding to all plans generated by the progression planner are compactly represented by a single partial plan (plan graph) in Graphplan. ally perform much better in practice. They describe a system called Graphplan in which the partial plan representation, called plan graph, corresponds to a disjunctive representation of the search space of a progression planner (see Figure 5) [l 11. The Graphplan refinement process (i.e., the process of growing the plan-graph) does not introduce any branching into the search space.-Thus, all the complexity is transferred to the solution construction process which has to search the plan graph structure for minimal candidates that are solutions. Empirical results demonstrate this apparently extreme solu- tion to the refinement and solution construction tradeoff in fact leads to significant improvements in performance. The success of Graphplan shows that there is a lot to be gained by considering other disjunctive partial plan represen- tations. An important issue in handling disjunctive partial plans is how to avoid losing all the search space savings in increased plan handling costs. One of the tricks in increasing least commitment without worsening the overall performance significantly seems to be to use constraint propagation tech- niques to enforce local consistency among the partial plan constraints. In CSP problems [181, refinement is used hand-in- hand with local consistency enforcement through constraint propagation to improve search performance. Although most refinement planning systems ignored the use of constraint propagation in planning, the situation is changing slowly. In addition to Graphplan [31, which uses the constraint propaga- tion process in both the partial plan construction, and solution construction phase, there are also systems such as Descartes 161, which attempt to incorporate constraint propagation tech- niques directly into existing refinement planners. Solution construction process can also be represented as an instance of propositional satisfiability problem, and there is some re- cent evidence E 121 that nonsystematic search techniques such as GSAT can give very good performance on such SAT instances. Summary ning. The framework explicates the tradeoffs offered by plan representation and refinement strategies. I have concluded by outlining several directions in which refinement planning algorithms can be made more efficient. These involve using disjunctive partial plan representations, and the using of CSP techniques for handling partial plans. [II 121 [31 [41 [51 I61 171 [81 [91 1101 Ill3 [121 I131 [141 iI51 [161 I171 [181 I191 [201 References A. Barrett and D. Weld. Partial Order Planning: Evaluating Possible Efficiency Gains. Artificial Intelligence, Vol. 67, No. 1, 1994. F. Bachus and F. Kabanza. Using Temporal Logic to Control Search in a forward chaining planner. In Proc European Planning Workshop, 1995. A. Blum and M. Furst. Fast planning throug planning graph analysis. In Proc. IJCAI-95, 1995. K. Cut-tie and A. Tate. O-Plan: The open planning architecture. Artificial intelligence, 5 1(1):49--86, 199 1. R. Fikes and N. Nilsson. Strips: A new approach to the application of theorem proving to problem solving. Artificial Intelligence, 2: 189--208, 197 1. D. Joslin and M. Pollack. Passive and active decision post- ponement in plan generation. In Proc. 3rd European Workshop on Planning, 1995. L. Ihrig and S. Kambhampati. Derivational replay for partial order planning. In Proc. AAAI-94. S. Kambhampati, C. Knoblock, and Q. Yang. Refinement search as a unifying framework for evaluating design tradeoffs in partial order planning. Artificial Intelligence, 76( l-2), 1995. S. Kambhampati. A comparative analysis of partial-order planning and task-reduction planning. ACM SIGART Bulletin. 6(l), 1995. S. Kambhampati and B. Srivastava. Universal Classical Plan- ner: An algorithm for unifying state space and plan space approaches. In Proc European Planning Workshop, 1995. S. Kambhampati. Planning Methods in AI (Notes from ASU Planning Seminar). ASU CSE TR 96-004. http://rakaposhi.eas.asu.edu:8001/vochan.html H. Kautz and B. Selman. Pushing the Envelope: Planning, Propositional Logic, and Stochastic Search In Proc. AAAI-96. D. McAllester and D. Rosenblitt. Systematic Nonlinear Plan- ning. In Proc. 9th AAAI, 1991. D. McDermott. A heuristic estimator for means-ends analysis in planning. In Proc. AIPS-96, 1996. Steve Minton, editor. Machine Learning Methodsfor Planning and Scheduling. Morgan Kaufmann, 1992. S. Minton, J. Bresina and M. Drummond. Total Order and Partial Order Planning: a comparative analysis. Journal of Artificial Intelligence Research 2 (1994) 227-262. E.P.D. Pednault. Synthesizing plans that contain actions with context-dependent effects. Computational Intelligence, 4(4):356--372, 1988. E. Tsang. Foundations of Constraint Satisfaction. Academic Press, San Diego, California, 1993. D.E. Wilkins. Practical Planning: Extending the Classical A I Planning Paradigm. Morgan Kaufmann, 1988. M. Zweben and M.S. Fox, editors. Intelligent Scheduling. Morgan Kaufmann, San Francisco, California, 1994. In this talk, algorithms I described the current state of refinement planning using a unified framework for refinement plan- 1336 AAAI-96 | 1996 | 199 |
1,842 | The ContactFinder agent: Answering bulletin board questions with referrals Bruce Krulwich and Chad Burkey Center for Strategic Technology Research Andersen Consulting 3773 Willow Drive, Northbrook, IL, 60062 { krulwich, burkey } @ cstar.ac.com Abstract ContactFinder is an intelligent agent whose approach to assisting users is valuable and innovative in the following four ways. First, ContactFinder operates proactively in reading and responding to messages on electronic bulletin boards rather than acting in response to user queries. Second, ContactFinder assists users by referring them to other people who can help them, rather than attempting to find information that directly answers the user’s specific question. Third, ContactFinder categorizes messages and extracts their topic areas using a set of heuristics that are very efficient and demonstrably highly effective. Fourth, ContactFinder posts its referrals back to the bulletin boards rather than simply communicating with specific users, to increase the information density and connectivity of the system. This paper discusses these aspects of the system and demonstrates their effectiveness in over six months of use on a large-scale internal bulletin board. 1. Electronic information systems The explosive growth of the Internet by individuals and corporations, and the growing use of corporate information repositories based on Internet technology or systems such as Lotus NotesTM, has led to an unprecedented number of people using network-based systems for finding solutions to problems. In addition to document browsing systems such as the Internet’s World Wide Web, a continuing interest remains in electronic bulletin board systems that allow large numbers of distributed users discuss issues, ask questions, and give answers. Unfortunately, a number of operational problems make it difficult to get high quality, fast answers to questions on bulletin boards, or to search bulletin boards for previous messages that can help solve a problem. First, the explosion in the number of bulletin boards makes it less likely that true experts will read any single bulletin board on a very frequent basis. Second, the increased volume of messages on these systems leads to more frequent routine deletion or migration of messages. For a user trying to find a quick solution to a problem, bulletin boards can be as frustrating as they are useful. This paper describes an intelligent agent called ContactFinder, that has been developed to address this problem. ContactFinder is similar to intelligent agents under development for question answering [Hammond et. al., 19951, e-mail filtering [Maes and Kozierok, 1993; Lashkari et. al., 19941, Usenet message filtering [Sheth, 19941, or other information search and retrieval domains [Holte and Drummond, 1994; Knoblock and Arens, 1994; Levy et. al., 1994; Pazzani et. al., 1996; Krulwich and Burkey, 19961. Like these other systems, ContactFinder extracts information from a large number of documents in order to present it to users in a more focused and productive fashion. Unlike these previous approaches, however, ContactFinder’s task is not to present the user with a subset of the information that can be used directly in problem solving. The agent instead keeps track of people who are key contacts in various topic areas, and helps question- askers by referring them to the appropriate key contact. More specifically, ContactFinder monitors the bulletin board for indications of message authors who are key contacts in some specific area, and stores these contacts for later use. The agent simultaneously watches for questions, and responds to the questions with a referral. This is a very valuable function for an intelligent agent to perform for several reasons. First, an agent that attempts to provide information that is directly relevant to the user’s goals will always be limited by the information that is available. While this is not a problem in solving problems that are very basic or frequently asked [Hammond et. al., 19951, it may make it difficult to be helpful in novel or very focused situations. In such a situation, however, a referral to a human contact will be available more often [Kantz and Selman, 19961. As we discuss in detail in section 5, ContactFinder has been able to make a referral for over 13% of the questions in a large-scale technology-related bulletin board, most of which were for very specific questions. Second, extracting contacts and facilitating human expertise transfer fits very well into current work styles and facilitates good learning from bulletin boards, which make the system easier to apply and test. 10 Agents From: AAAI-96 Proceedings. Copyright © 1996, AAAI (www.aaai.org). All rights reserved. In Response To Looking for a Netscape User Guide Discussion Topic Internet Technology Community: Network Solutions, <General> Key T bought: El’ Netscapa HTML User Reference Dots. Description: 0 r I Here are some manuals which 1 downloaded from Netscape’s Web Sites have to be opened by Netscape locally. If you want to make them into text such as “htmlcon 2.0”. These are *.htm files, so Lies. use convert softwarE Q (doclink to Tech. Attachments) ~ Contributed By: Ed Peter A. Glaser ~ GMU: 8’ Chicago 33W J Octet: El’ 51/79057j, Finure I: A bulletin board messaae as an indication of a contact ContactFinder’s processing happens in two phases. The first phase scans the new documents in the information repositories and searches for indications of key contacts in any technical area. It extracts the contacts and their technical areas, and stores them in its own database. In the second phase, ContactFinder scans on-line discussions for questions. It extracts the topics of the questions and checks if it has a contact to give as a referral on those topic areas. If it does, it responds to the question with a referral.’ This referral gives the name and contact information, along with a reference to the previous documents that served as the basis for the referral. 2. Extracting key contacts Figure 1 shows a bulletin board message that can provide indications of a key contact. A number of issues arise in looking at messages such as these as sources of contact information. First, while it is clear to people reading the message that it is a response to a question, and given the content is probably a good indication that the author is an expert or key contact in the relevant topic areas, it is not trivial to determine this in an automated fashion. Second, while this particular bulletin board system supports user- specified keywords, they do not contain enough detail to effectively classify the author’s areas of knowledge. ContactFinder approaches the problem of extracting key contacts from text messages by using heuristics that are specifically designed for extracting topic areas and contact information from text documents. Rather that attempt to process the document in a general fashion, it simply searches for indications of an appropriate contact, and looks locally at that point in the document for a name and contact information. This approach, very focused information extraction instead of general document understanding, has proven to be highly effective, as we discuss in section 5. ’ For our discussion in this paper we are omitting logistical details, such as human confirmation of contact accuracy and topic area prior to public referral. The most critical and difficult task that ContactFinder carries out in phase one is to extract topic areas from each document which serve as a description of the content areas for the extracted contact. The process used to achieve this in both phases of ContactFinder, for extracted contacts and for subsequent questions, is to look for phrases that are delimited by syntactic devices as central to the meaning of the message [Swaminathan, 1993; Krulwich, 19951. These methods are discussed in detail in section 4. Figure 2 shows ContactFinder extracting contact information from the discussion document like the one in Figure 1.2 The top of the screen shows the document information and its contents. The bottom shows the contact that was extracted and the topic areas, along with the heuristic basis for the extraction. In this case, the document in Figure 1 was a response to a message that ContactFinder examined and classified as a question, and the response itself was not a further question. Other bases for contact extractions include offers to be contacted and explicit referrals to third parties. 3. Answering questions wit ContactFinder’s second phase is to find questions in the bulletin board, extract their topic areas, and search for previously-extracted contacts to give as referrals. Figure 3 shows such a question in a discussion group which asks about the same product mentioned in Figure 1. ContactFinder finds this document, determines heuristically that it is a question, extracts the topic areas, and searches its database for an appropriate referral. In this case it finds the expert extracted in the previous section. For the message in Figure 3, ContactFinder realizes that this is a question based on the phrase “Does anyone” with the question mark at the end of the sentence. It first extracts the topic indicators using the same methods as are used in 2 Note that the display shown in Figure 2 will never be seen by a user, since the process is run on the information repositories in background. This display is used for explanation and demonstration only. Interaction I1 phase one (discussed in detail in section 4). It then proceeds to search its database of key contacts, as shown in Figure 4. It finds the contact which was extracted as a contact previously and sends the referral shown in Figure 5. A key point of this research initiative is that the documents that provide the original indication of expertise in phase one need not actually address the questions that are handled in the second phase. All that is required is that the topic areas that are extracted match close enough to make it likely that the contact person would be able to help the question asker or at least be able to refer the question asker to a third person. Because of this, our primary focus has been on the extraction of topic areas rather than on the details of the questions being asked or the expertise being provided. 4. Extracting topic indicators The most important step in both phases of the process described above is the extraction of semantically significant phrases. Previous research has attempted to perform document comparison using most or all of the words in a document (e.g., [Sheth, 1994; Hammond et. al., 1995; Pazzani, et. al, 1996]), but we are avoiding this approach for two reasons. First, very few of the words in a document reflect the underlying meaning and importance of the text, and moreover the distribution of words does not reflect the words or phrases that best characterize the document. Second, processing the entire text of a document is extremely costly in computational terms, and can be prohibitive for very large sample sets. Extracting semantically significant phrases and processing them is quite tractable. The critical step for extracting high quality phrases for documents is the set of heuristics for processing blocks of text. This is especially true for highly unstructured documents, which don ’ have many structured fields or keyword classifications. Even if a set of documents does have categorization keywords associated with each document, it is necessary to augment them with other significant phrases that the authors include in the document text. To accomplish this we are in the process of integrating and building upon the heuristics found in previous related research [Swaminathan, 1993; Krulwich, 19951 for extracting visually significant features from documents. This approach is built upon the observation that document authors typically use a variety of visual techniques to convey significant pieces of information to readers. Some examples are key points, lists of significant items, document structure, synopses, logical progression, and so on. Recognizing some of these visual patterns allows our agent to extract semantically meaningful phrases from bodies of text. For example, a simple heuristic is to extract any single word that is fully capitalized. Such a word is most likely an acronym or in some cases a proper technical name. In addition, there are a number of ways to find a definition of an acronym, such as looking for a parenthesized or quoted phrase immediately after the acronym, or at the words before the acronym if the acronym is itself in parentheses, or in the sentence or two preceding the acronym if neither is in parentheses. Another type of heuristic extracts sequences of multiple word that when taken as a unit can be used as a topic 12 Agents Didn’t Seem Vsrv Good To Me World Wide Web Access Thru AOL Internet Network Solutions I downloaded the 2.5 version last week for my father...l’m a loyal Netscape user. After I loaded it it seemed to be very slow and disconnected me several times. Through several attempts this situation did not get better. Please tell me if it needs a special configuration, otherwise I would not recommend it. Fiaure 3: A auestion in a discussion indicator. For example, a series of capitalized words will most often be a stronger indicator of topic than the constituent words treated independently. Phrase-level heuristics of this sort enable ContactFinder to operate on phrases that more closely resemble human usage, rather than on approximations such as word correlations. Another simple heuristic is to extract any short phrase, of l- 5 words, which appears in a different format from surrounding text and which is not a complete sentence. This heuristic takes advantage of the convention of italicizing or underlining significant phrases the first time that they’re used, or of capitalizing the first letters of proper names. A further condition for both of these heuristics to be applicable is that the phrase not appear on a fixed list of non-significant words and phrases. For example, the first heuristic should not extract the acronym TM that may follow a product name and the second should not extract words such as “not” or “certainly,” which are often italicized for emphasis. Other heuristics of this sort include recognition of lists of items (with bullet points or numbers), section headings, diagram labels, row and column headers in tables, and heavily repeated phrases. We are in the process of exploring heuristics such as extracting compound noun phrases (made up of three or more nouns in a row), which are frequently domain-specific phrases. Additionally, we are investigating the integration of a thesaurus, either commercial or domain-specific, to allow the agent to recognize that two words or phrases that have been extracted should be treated as equivalent. A key aspect of these heuristics is that they are completely free of domain and contextual knowledge, and rather focus entirely on the syntactic structure of the text. This allows them to be widely applicable, without relying on background knowledge, and to be computationally efficient. There will be situations, however, in which such knowledge is necessary to perform effectively. Types of knowledge that could be added include topic areas and their relationships. The next section discusses a particular situation in which this is necessary and undoubtedly more such cases will be uncovered as experimentation progresses. For the most part, however, ContactFinder will operate using knowledge-free heuristics of the sort described in this section. 5. To date, ContactFinder has operated for several months on an internal bulletin board for discussion of technical issues. Out of 2893 total documents processed from the bulletin board, ContactFinder extracted 762 key contacts on various topics. This reflects our desire that ContactFinder operate relatively conservatively and error on the side of false negative contact extractions (failing to extract contacts) rather than false positives (extracting people as contacts who in fact are not). Many messages that may be indications of expertise, such as those that are top-level (not responses) but are not questions, are skipped by ContactFinder for lack of certainty. Exploration of other heuristics for identifying contacts will improve this rate. Out of the same total set of messages the system extracted 611 questions for which it found 83 potential referrals. This rate of success (13.6%) reflects a number of aspects of the system’s operation. First, the system will never post a referral to someone who has already responded to the question, which will often be the case during early operation when most of the system’s set of key contacts have been extracted from the same set of messages. Second, the system requires a fairly strong match between topic areas of the question and the contact (90%) before considering a referral. Of these 83 referrals, 3 related to a particular technical topic (a system named SAP) that posed difficulties for ContactFinder’s approach. SAP is a very large system composed of many sub-systems and for the most part any individual will only work with a small number of these sub- systems. It’s necessary, therefore, for ContactFinder to correctly determine the relevant sub-systems for every contact and question. Unfortunately, this has proven difficult for a number of reasons. First, the sub-systems are often named as two-letter acronyms, without the use of punctuation as separators, such as SD, FL, HP, MM, PS, DB, and PP. Many of these two-letter names are also used in English messages for other purposes, such as PP being used for page number references or FL being the postal Interaction 13 Figure 4: ContactFinder processing a question code for Florida. For this reason it has been impossible for ContactFinder to extract these topic indicators in a knowledge-free and context-free fashion. Second, these sub-systems are sometimes referred to by their expanded names, requiring that ContactFinder know that the two- letter codes are synonyms for their expansions. In general, this problem is with the knowledge-free nature of ContactFinder topic extraction heuristics. Were ContactFinder to have specific knowledge of SAP and its sub-systems, it could look for the two-letter names only in the context of SAP and could know the relevant synonyms. While we have in fact included knowledge of some synonyms in ContactFinder, we have not yet explored broader domain knowledge such as system components and sub-systems. Future research will determine the degree to which knowledge of this sort is necessary. Out of the 80 remaining referrals, 39 of them have been approved by the contacts themselves and 11 of them have been refused, giving us a 78% success rate (after excluding SAP-related referrals). Continuing testing of the system will determine how this rate holds up over larger numbers of documents. Anecdotal feedback concerning ContactFinder has been very positive. Some bulletin board users have feared that the system will reduce the number of on-line responses and move the flow of knowledge off-line as people call contacts directly instead of waiting for them to respond on-line. In practice, however, it appears that just the opposite is true. In several cases the contacts referred by ContactFinder have posted information on-line having not seen the questions until ContactFinder contacted them. If this trend continues, it appears that ContactFinder will in fact increase the amount of information on-line as people who do not have a chance to read the bulletin board regularly are encouraged 14 Agents to respond to particular messages that relate to their areas of expertise. . Summary and discussion We have described an intelligent agent prototype that mines a heterogeneous information repository for key contacts in specific subject areas. This approach allows the agent to assist people seeking information without requiring deep understanding of the information source documents. It also allows the agent to fit well with typical work styles by facilitating transfer of expertise between people. Lastly, the advice and the reasoning behind it is very easily understood by the people involved because the referral can include a reference to the document that provided the contact. The agent has been designed to operate by responding to questions on discussion groups. This allows it to answer only those questions for which it has referrals and to operate in a background fashion appearing to users as simply another source of messages. The system currently leaves open a number of issues that will serve as the basis for our continuing research. How can a large variety of types of documents be successfully mined for indications of contacts? How can documents consisting of plain formatted text be processed effectively to extract contacts, questions, and indications of subject area? What types of background will be needed to operate effectively in a variety of domain areas? More generally, our approach raises the question of what other intelligent agent functionality can be achieved using document processing techniques such as significant phrase extraction, inductive learning, and document search. We are currently developing of several agents based on these techniques, such as an agent that learns the information interests of various users along with how to find new Trv contactino this aerson Discussion Topic Internet In Response To @World Wide Web Access Thru AOL Technology Community: Network Solutions A probable contact person for your question would be Peter A. Glaser ( Octel: 51/79057 ) of the Chicago 33W office. This referral is based on information posted to the Technology Discussion Database. This message has been sent by the ContactFinder intelligent agent prototype developed at CSTaR. For more information send mail to “CSTaR Intelligent Agent” with a subject field of “ContactFinder”. I hope this helps to get an answer to your question. Sincerely, ContactFinder Figure 5: ContactFinder’s referral message documents matching those interests [Krulwich and Burkey, 19961, an agent that interacts with on-line Internet services, and an agent that browses on-line documents to extract surnmary information. We are also investigating the application of other core document processing techniques, such as schema matching and message sequence modeling, to intelligent agent tasks. Future research will determine the range and effectiveness of intelligent agents that can be built on core document processing techniques such as these. eferenees Hammond, K., Burke, R., Martin, C., and Lytenin, S., 1995. FAQ Finder: A case-based approach to knowledge navigation. In Working Notes of the 1995 AAAI Spring Symposium on Information Gathering in Distributed Environments, Palo Alto, CA. Holte, R. and Drummond, C., 1994. A learning apprentice for browsing. In Working Notes of the 1994 AAAI Spring Symposium on SofnYare Agents, Stanford, CA, pp. 37-42. Kantz, H. and Selman, B., 1996. Agent Amplified Communication. In Proceedings of the 1996 National Conference on Artificial Intelligence, Portland, OR. Knoblock, C. and Arens, Y., 1994. An architecture for information retrieval agents. In Working Notes of the 1994 AAAI Spring Symposium on Software Agents, Stanford, CA, pp. 49-56. Krulwich, B., 1995. Learning user interests across heterogeneous document databases. In Working Notes of the 1995 AAAI Spring Symposium on Information Gathering in Distributed Environments, Palo Alto, CA. Krulwich, B. and Burkey, C., 1996. Learning user information interests through the extraction of semantically significant phrases. In Working Notes of the 1996 AAAI Spring Symposium on Machine Learning in Information Access. Lashkari, Y., Metral, M., and Maes, P., 1994. Collaborative interface agents. In Proceedings of the 1994 AAAZ Conference, Seattle, WA, pp. 444-449. Levy, A., Sagiv, Y., and Srivastava, D., 1994. Towards efficient information gathering agents. In Working Notes of the 1994 AAAI Spring Symposium on Software Agents, Stanford, CA, pp. 64-70. Maes, P. and Kozierok, R., 1993. Learning interface agents. In Proceedings of the 1993 AAAI Conference, Washington, DC, pp. 459-465. Pazzani, M., Muramatsu, I., and Billsus, D., 1996. Syskill and Webert: Identifying Interesting Web Sites. In Proceedings of the 1996 National Conference on Artificial Intelligence, Portland, OR. Sheth, B., 1994. A learning approach to personalized information filtering. M.S. Thesis, EECS Department, MIT. Swaminathan, K., 1993. Tau: A domain-independent approach to information extraction from natural language documents. DARPA workshop on document management, Palo Alto. Interaction 15 | 1996 | 2 |
1,843 | A Kernel-Oriented Model for Coalition-Formation in General Environments: Implementation and Results* Onn Shehory Sarit Kraus Department of Mathematics and Computer Science Bar Ran University Ramat Gan, 52900 Israel {shechory, sarit}@bimacs.cs.biu.ac.il Tel: +972-3-5318863 Fax: -l-972-3-5353325 Abstract In this paper we present a model for coalition forma- tion and payoff distribution in general environments. We focus on a reduced complexity kernel-oriented coalition formation model, and provide a detailed algo- rithm for the activity of the single rational agent. The model is partitioned into a social level and a strategic level, to distinguish between regulations that must be agreed upon and are forced by agent-designers, and strategies by which each agent acts at will. In addi- tion, we present an implementation of the model and simulation results. From these we conclude that im- plementing the model for coalition formation among agents increases the benefits of the agents with rea- sonable time consumption. It also shows that more coalition formations yield more benefits to the agents. Introduction An important method for cooperation in multi-agent environments is coalition formation. Membership in a coalition may increase the agent’s ability to satisfy its goals and maximize its own personal payoff. Game the- ory literature such as (Rapoport 1970) describes which coalitions will form in N-person games under different settings and how the players will distribute the bene- fits of the cooperation among themselves. These re- sults do not take into consideration the constraints of a multi-agent environment, such as communica- tion costs and limited computation time, and do not present algorithms for coalition formation. Our re- search presents a multi-agent approach to the coalition formation problem, and provides a coalition-formation procedure. The paper deals with autonomous agents, each of which has tasks it must fulfill and access to re- sources that it can use to fulfill these tasks. Agents can satisfy goals by themselves, but may also join together to satisfy their goals. In such a case we say that the agents form a coalition. *Kraus is also affiliated with the Institute for Advanced Computer Studies, University of Maryland. This material is based upon work supported in part by the NSF under grant No. IRI-9423967 and the Israeli Ministry of Science, grant No. 6288. We thank S. Aloni and M. Goren for their major contribution to the implementation of the model. 134 Agents In this paper we present a modification of the Kernel concept from game theory (Davis & Maschler 1965). The modified Kernel serves as a basis for a polynomial- complexity mechanism for coalition formation. The mechanism is partitioned into two levels - the so- cial level and the strategic level. The coordination- regulation protocols constitute the social leveli. Dif- ferent designers of agents must agree upon the regu- lation protocols of the social level in advance. The strategic level consists of strategies for the individual agent to act in the environment for maximization of its own expected payoff, given the social level, and can be decided upon by individual agents during the coalition formation process. Related work in DA1 Research in DA1 is divided into two basic classes: Dis- tributed Problem Solving (DPS) and Multi-Agent Sys- tems (MA) (Bond & G asser 1988; Durfee & Rosen- schein 1994). Our research is closer to MA since it deals with interactions among self-motivnted, ratio- nal and autonomous agents. However, any interaction among agents requires some regulations and structure. The minimal requirement for interactions in multi- agent systems is a common language or a common background (Gasser 1993). In coalition-formation, the need for regulations increases further(Shapley & Shu- bik 1973). Shoham, Tennenholtz and Moses (Moses & Tennen- holtz 1993; Shoham & Tennenholtz 1992) show that pre-compiled highly-structured “social laws” are able to coordinate agent activity. Agents are assumed to follow the social laws since they were designed to do so and not because they benefit individually from follow- ing these laws. In our research, we do not explicitly use social laws in order to coordinate agent activity. We provide the agents with a social level of coopera- tion. The designers of agents should agree in advance which regulations the agents in a given environment will use. These regulations are incorporated into all of the agents, but each agent chooses its strategy for the 1 We use the concep t of the social level and the notion s of regulations interchangeably. From: AAAI-96 Proceedings. Copyright © 1996, AAAI (www.aaai.org). All rights reserved. interaction individually and joins a coalition only if it increases its personal payoff. Our mechanisms are different from these that are applied in DPS environments. However, as in the Con- tract Net framework (Davis & Smith 1983; Malone et al. 1988) or the Functionally Accurate, Cooperative, (FA/C) paradigm (Durfee 1988; Decker & Lesser 1993) that considers DPS environments, we also enables co- operation of groups of agents, though not necessarily of all of them. Zlotkin and Rosenschein (Zlotkin & Rosenschein 1994) and Ketchpel (Ketchpel 1994) consider coalition formation in supper-additive environments. Our work discusses general multi-agent environments, and pro- vides the designers of agents with explicit coalition- formation mechanisms for these environments. Sandholm and Lesser (Sandholm & Lesser 1995) presented a coalition formation model for bounded- rational agents and present a general classification of coalition games. In their model, the value of a coalition depends on the computation time, however, all config- urations and possible coalitions are considered when computing a stable solution. We consider problems where the computational time of coalition values is polynomial. Therefore, we concentrate on the polyno- mial coalition-configuration formation and utility dis- tribution and provide polynomial concepts of stability. Environment Description We assume the following: (i) The d _ esigners of the agents agree upon the regulations in advance and in- corporate them into their agents, and these regulations are enforceable by deviation revealability and penal- ties. (ii) Various communication methods exist, so that the agents can negotiate and make agreements (Werner 1988). However, communications require time and ef- fort. (iii) Resources can be transferred among agents. (iv) There is a monetary system that can be used for side-payments. The agents use this monetary system in order to evaluate resources and productions that re- sult from the use of the resources. The money is trans- ferable among the agents and can be redistributed in a case of coalition formation. The monetary system is part of the regulation of the environment. We present it as a possible regulation since it increases the agents’ benefits from cooperation, although agents may reach agreements and form coalitions even if this assumption is not vLalid, e.g. (Kraus & Wilkenfeld 1991). I of n autonomous agents, We consider a‘group N N = (cz~,u~ ,..., a,). The or have access to, resources. resource vector pi = (q’, q”, quantities of resources that resources they have to exec come from task-execution is agents are provided with, Each agent ai has its own .“, 4:)) which denotes the it has. The agents use the :ute their tasks. ai’s out- expressed by a payoff func- tion from the resources domain Q to the reals. Such a function U” : Q - R exchanges resources into mone- tary units. Each agent tries to maximize its payoff. Individual self-motivated agents can cooperate bY forming coalitions. A coalition is defined as a group of agents-that have decided to cooperate and h&e also decided how the total benefit should be distributed among them. Formally: Definition 1 Coalition Given a group of agents N and a resource domain Q, a coalition is a quadrate C = (&,q&Uc), where NC C N; qc = (41,427.-d) is the coalition’s resource vector, where qj = CazENc q{ is the quantity of resource j that the coalition has. & is the set of resource vectors after the redistribution of qc among the members of NC (& sat- isfies qj = xa,ENc Q:). UC = (ul, u2, . . . , ulcl), is the coalitional payofl vector, where ui E R is the payofl of agent ai after the redistribution of the payofis. V is the value of C if the members of NC can jointly reach a payofl V. That is, V = CaSENc iY’(Qi) , where Ui is the payofl function of agent ui and Oi is its resource- vector after redistribution in C. The specific distribution of the resources among the members of the coalition strongly affects the results of the payoff functions of the agents and thus affects the coalitional value. It is in -equilibrium to reach a resource distribution that will maximize the coali- tional value. Therefore, the resources are redistributed within & in a way that maximizes the value of the coali- tion. Thus, the coalition value V of a specific group of agents NC is unique. The complexity of computing the redistribution of the resources and calculating the coalitional value depends on the type of payoff fun&ion of the coalition-members. For example, if the payoff functions are linear functions of resources, then poly- nomial calculation methods can be applied. Coalition-formation usually requires disbursement of payoffs among the agents. We define a payoff vector U = (u’ , u2, . . . , un) in which its elements are the pay- offs to the agents 2. In each stage of the coalition for- mation process, the agents are in a coalitional con- figuration. That is, the agents are arranged in a set of coalitions C= {Ci), that satisfies the conditions UiCi = N, VCi, Cj, Ci # Cj, Ci II Cj = 4. A pair of a payoff vector and a coalitional configuration are denoted by PC( U,C), or just PC (Payment Configura- tion). Since we assume individual rationality, we con- sider only individually rational payment configurations ( IRPC’s in game theory, e.g.,(Rapoport 1970)). A (rational) coalitional configuration space (CCS) is the set of all possible coalitional configurations such that the value-of each coalition within a configuration is greater or equal to the sum of the payoffs of the coalition-members3. The size of the CCS is O(nn). A 2Note that U is a payoff vector of all of the agents while UC is a payoff vector of the members of a specific coalition. 31t is very common to normalize the agents’ payoffs to zero. In such a case, the requirement on the sum becomes simpler - the coalitional values must be non-negative. Negotiation & Coalition 135 payment configuration space (PCS) is a set of possible individually-rational PCs. That is, a PCS consists of pairs (U, C) h w ere U is an individually-rational pay- off vector and C is a coalitional configuration in CCS. Since for each coalitional configuration t,here can usu- ally be an infinite number of payoff vectors, the number of PCs is infinite, and the PCS of all rational PC’s is an infinite space. We would like the resulting payoff vector of our coalition-formation model to be stable and Pareto- optimal. A payoff vector is Pareto-optimal if there is no other payoff vector that dominates it, i.e., there is no other payoff vector that is better for some of the agents and not worse for the others (Lute & Raiffa 1957). It seems in the best interest of individually rational agents to seek Pareto-optimal payoff vectors. However, a specific Pareto-optimal payoff vector is not necessarily the best for all of the agents. There can be a group of Pareto-optimal payoff vectors where dif- ferent agents prefer different payoff vectors. This may lead to difficulty when agents negotiate cooperation and coalition formation. Pareto-optimality is not suf- ficient for evaluating a possible coalition for a specific agent, hence we present the concept of stability. The issue of stability was studied in the game theory literature in the context of n-person games (Rapoport 1970; Lute & Raiffa 1957). These notions are useful for our purposes, when coalitions are formed during the coalition formation procedure. The members of such coalitions can apply these techniques to the dis- tribution of the coalitional value. Game theorists have given several solutions for n-person games, with several related stability notions. In this paper we concentrate solely on the Kernel solution concept. However, we shall discuss the other solution concepts in an extended version of this paper. The Kernel K The kernel (Davis & Maschler 1965) is a PCS in which the coalitional configurations are stable in the sense that there is equilibrium between pairs of individual agents which are in the same coalition. Two agents al, 132 in a coalition C, in a given PC, are in equilib- rium if they cannot outweigh one another from C, their common coalition. al can outweigh u2 if al is stronger than ~2, where strength refers to the potential of agent al to successfully claim a part of the payoff of u2 in PC. During the coalition formation, agents can use the kernel concept to object to the payoff distribution that is attached to their coalitional configuration. This ob- jection will be done by agents threatening to outweigh one another from their common coalition. Given a PC( U, C), agents can make objections based on the excess concept. We recall the relevant definitions. Definition 2 Excess The excess (Davis & Maschler 1965) of a coalition C with respect to the coali- tionat configuration PC is defined by e(C) = V(C) - c a,EC u 7 i where ui is the payofl of agent ui in PC. C is not necessarily a coalition in PC. Given a specific PC, the number of the excesses with respect to the specific coalitional configuration is 2n. Any change in the payoff vector U, either when the coalitional configuration changes or when it remains unchanged, may cause a change in the set of excesses. Such a change will require recalculation of all of the excesses. Agents use the excesses as a measure of their relative strengths. Since a higher excess correlates with more strength, rational agents must search for their highest excess, i.e., the surplus. Definition 3 Surplus and Outweigh The mazi- mum surplus Sub of agent a over agent b with respect to a PC is defined by Sab = MA&la~c,bgce(C), where e(C) are the excesses of all the coalitions that include a and exclude b, and the coalitions C are not in PC, the current coalitional configuration. Agent a outweighs agent b if Sub > Sba and ub > V(b), where V(b) is the value of b as a single agent. If two agents cannot outweigh one another, we say that they are in equilibrium. We say that a, b are in equilibrium if one of the following conditions is satis- fied: (i) Sab = Sba; (ii) Sab > Sba and ub = V(b); (iii) Sab < Sba and Ua = V(u). Using the concept of equilibrium, the kernel and its stability are: Definition 4 Kernel and K-Stability A PC is K-stable if ‘da, b agents in the same coalition C E PC, the agents a, b are in equilibrium. A PC is in the kernel ifl it is K-stable. The kernel stability concept provides a stable payoff distribution for any coalitional configuration in the ra- tional CCS (Davis & Maschler 1965; Aumann, Peleg, & Rabinowitz 1965)4. Using this distribution, the agents can compare different coalitional configurations. How- ever, checking the stability does not direct the agents to a specific coalitional configuration. The coalition formation model that we develop will perform this di- rection. The kernel leads symmetric agents to receive equal payoffs. Such symmetry is not always guar- anteed in other solution concepts (e.g., the bargaining set) (Rapoport 1970). Another property of the ker- nel - it is a comparatively small subset of the PCS of all rational PC’s. In addition, the mathematical formalism of the kernel allows one to divide its calcu- lation into small processes, thus simplifying it. Some exponentially-complex computing schemes for the ker- nel solution were provided, e.g., by (Aumann, Peleg, & Rabinowitz 1965). Stearns (Stearns 1968) presented a transfer scheme that, given a coalitional configuration and a payoff vector, converges to an element of the ker- nel. Due to its advantages, the kernel was chosen as 4The kernel does not provide coalitional configurations, it only determines how the payoffs will be distributed given a coalitional configuration. Therefore, it cannot be used as a coalition formation algorithm. 136 Agents the theoretical background for our coalition formation model. Coalition Negotiation Algorithm CNA The CNA is a coalition formation algorithm based on negotiation. It consists of steps in which coalitions transmit, accept and reject proposals for creating new coalitions. Initially, all agents are in single-membered coalitions. The CNA proceeds through a sequence of steps as described below. In each step, at least one coalition will make an attempt to improve the payoffs of its members by making a coalition formation pro- posal to another coalition. The CNA may continue either until all of the proposals of all of the agents are rejected or until a K-stable and Pareto-optimal PC has been reached, thus reaching a steady state. The CNA may also terminate when the allocated time- period ends. When the agents use the CNA, the coalitional con- figurations that are formed when the agents reach the steady state are stable according to a new stability con- cept that we define below, the polynomial-K-stability. However, since the CNA is an anytime algorithm (Dean & Boddy 1988) even if it is terminated after a limited number of steps before reaching a steady state, it will still provide the agents with a polynomial-K-stable PC. The Polynomial Approach We modify some concepts to adjust them to the polynomial-K-stability algorithm. Polynomial excesses are excesses that are calculated with respect to a poly- nomial subset of all 2n possible coalitions. The de- signers of agents must agree upon regulations that will direct their agents to a well defined polynomial set of coalitions. We suggest that the designers agree upon two integral constants iii , li2. These constants Ii1 5 1<z should not depend on n. Nevertheless, Iir, 1<z small w.r.t. n/2 should be preferred. In the regulation of the coalition formation model, the agents shall be allowed to consider excess calculations only for coalitions of sizes in the ranges [Ii’1 , K2]_ Choosing ii’s contradictory to the agreed upon Ii’r, 1<z by a specific agent shall be avoided, mainly because according to the regulations objections based on different li”s are not acceptable. Definition 5 A polynomial maximum surplus SP is a maximum surplus that is computed from a set of poly- nomial excesses. A coalition C in a coalitional con- figuration is polynomially-K-stable if for each pair of agents a, b E C, either one of the agents has a null normalized payoff in C, or /SP,b - SPbal 5 &, i.e., the agents are in equilibrium with respect to E, where SP are the polynomial surpluses, and E is a small prede- fined constant. Given a specific coalitional configuration with an arbitrary payoff distribution vector, it is possible to compute a polynomial-K-stable PC by using a trun- cated modification of the convergent transfer scheme in (Stearns 1968). We implement the Stearns-scheme by using the n-correction of Wu (Wu 1977)5 to initialize the process. The iterative part of the scheme is modi- fied so that it will terminate whenever a payoff vector that is close, according to the predefined small E, to an element of the polynomial kernel has been reached. A CNA Scheme The CNA scheme is aimed at advancing the agents from one coalitional configuration to the other, to achieve more cooperation and increase the agents’ pay- offs. Given a specific configuration, the agents attempt to find a correspondent stable payoff vector. The social level of the CNA will be constructed as follows: Regulation 1 Negotiation scheme 1. 2. 3. 4. 5. 6. 7. Initially, all entities are single agents. First stage: members of a coalition may receive proposals only as part of the coalition (thus, coali- tions can only expand in this stage). Each coalition will coordinate its actions either via a representative or by voting (or both) e.g., (Peleg 1984). Each coalition Cr, iteratively performs the following: e Decide which other coalitions it is interested in forming a joint coalition with. e Design proposals to be oflered to others: - A proposal of Cp to C,. is the details of the joint coalition Cneu, and coalitional configuration PCnetu: must be polynomially-K-stable, calcu- lated with respect to Ii’l,li’z. - Cp will design Cnew, Nneul = N,uN,., with payo# vectors Uneu, that increase Cneu, ‘s payofls. - In addition to C,,,, Cr, will design the coalitional configuration in which C,,, is included. Other coalitions stay unchanged in PC,,,. transmit one proposal to one target coalition at a time; wait for response. o When an ogler is accepted, the oflering coalition and the accepting coalition form a new coalition according to the details of the proposal6. The new PC determines the payofls of the agents. o Other active proposals will be canceled. If Cr, has no more proposals to make, it shall an- nounce it. If a steady state is reached, where all coalitions an- nounce that they have no proposals, proceed to second stage of the CNA. Otherwise, start a new iteration. If the agents run out of computation time before a steady state has been reached, the algorithm termi- nates and the last PC holds. 5While Wu uses this correction as a means for iterating in a transfer scheme for the core, we use it as a single correction in the beginning of our iterative algorithm. This n-correction is not necessary in the Stearns scheme. ‘The payoff vector U of the new PC is valid from now on and is used as the basis for future negotiation. Negotiation & Coalition 137 8. 9. 10. Second stage: the coalitions will follow the same sequence of steps as in the first stage of the CNA. However, proposals that involve destruction are al- lowed. When a new PC changes the payofls, dissatisfied agents may leave their coalitions. These coalitions will destruct. The second stage will end either when a steady state? is reached or when the computation time ends. The regulation above is enforceable since deviation from it is revealable. For example, proposals addressed to coalition-members will be detected immediately when accepted due to the coalition formation. The lim- itation that destruction of coalitions be avoided in the first stage will radically shorten the coalition formation process by avoiding most of the intra-coalitional com- putation and communication in this stage. The num- ber of iterations for reaching a steady state in the first stage of the CNA is O(n). If the agents have enough time and computational resources they will continue through stage 2. The number of steps until a steady state is reached in the second stage may be O(nn) due to the size of the PCS. An acceptance of a proposal implies an acceptance of the corresponding payoff vector by all agents. This may change the payoffs of agents who are not involved in the negotiation, because a proposal consists of a payoff vector to all of the agents. The aim of the change in the payoffs is to preserve the polynomial-K-stability. Since some of the agents may be dissatisfied with their new payoff, they can make new proposals according to which they receive a greater payoff. CNA Strategies According to part 4 of regulation 1, proposals for the generation of new coalitions should be designed by the current coalitions. We denote a coalition that designs and transmits a proposal by Cp and a coalition that receives a proposal by C,. Other coalitions will be denoted by C,. We suggest that Cp, that designs a proposal for C,, use the following strategy: Strategy 1 Proposal design Coalition Cp shall cal- culate the coalitional value of the joint coalition Vp+r. If VP + K L Vp+7. then Cp shall stop designing a pro- posal for Cr. Otherwise, Cp shall calculate the coali- tional values of all other coalitions of all sizes in the range [I<, , K2]. Cp shall calculate the payofl vector u new of the new PC wherein coalitions Cp and C, join to form Cnew, and all the other coalitions do not vary. These calculations will be done by using the truncated transfer scheme, starting from the initial payofl vector U. Coalition Cp will compare Uneu, to U. If the payofls to all of the members of C, and Cp in PC,,, are not smaller than their payoss in the current PC and are 7Note that the steady state in the second stage is differ- ent from the steady state in the first stage since there are different restrictions on the proposals in these two stages. 138 Agents also better than in all of the proposals that Cp has in the received-proposals queue then coalition Cp will send the resultant PCn.0, as a proposal to coalition C,. Other- wise, Cp shall stop the process of designing a proposal for C,. This strategy for proposal design shall be used by agents that are interested in reaching beneficial coali- tion formation and act under the regulations, which forces polynomial-K-stability of proposals. This is be- cause the calculation of the new coalitional value Vp+r and the comparison to the sum of original coalitional values VP and V, is done to avoid worthless propos- als in advance. In cases where Vp+r enables beneficial coalition joining, coalition Cp shall seek all coalitional values (of coalitions of sizes in the range [ICI, Iirz]) in order to use these values for PC calculations. Strategies are not enforced and agents can act with- out using our strategies. We propose them in order to increase the payoff to the individual agent and be- cause they satisfy the weak equilibrium requirement. A set of strategies is in weak equilibrium if none of the entities that act according to these strategies can guarantee, by deviating from its strategy, an increase in its benefits. An entity may be able to calculate all possible proposals and all of their consequences. How- ever, due to time and communication uncertainties, it cannot predict the exact results of the negotiation, and therefore it cannot guarantee an increase in its payoff. Complexity of the CNA The complexity of calculation of the polynomial set of coalitions, the coalitional values and the coalitional configurations is of the same order of the number of the coalitions which is given by ncoalitions = E ( Y ) = ?I i!cnnA i)! i= K1 which is a sum of polynoms of order O(ni). Computation of values and configurations The CNA requires the computation of n,.oalitions coali- tional values. It also requires the design of coalitional configurations, and the number of these depends on the time constraints. In a case where only the first stage of the CNA is performed, the number of coali- tional configurations which are treated is O(n). If the CNA proceeds through the second stage, the number of coalitional configurations increases. In each itera- tion of the CNA, when one coalitional configuration is treated, a polynomial-K-stable PC shall be calculated. Computation of polynomial-M-stable PC’s The CNA will employ the transfer scheme for calculat- ing polynomial-K-stable PC’s. The total complexity of one iteration of the transfer scheme is 0( n x n,,,litions). The number of iterations that should be performed to reach convergence depends on the predefined allowed error E. The resulting payoff vector of the transfer scheme will converge to an element of the polynomial- the number of coalition formations. This means that kernel (with a relative error not greater than E) within not only it is beneficial to form coalitions, formations n log,(e,,/E) iterations (Stearns 1968), where erO is the of more coalitions increase the average benefits of the relative error of the initial PC. agents. The transfer scheme will be performed for O(n) coalitional configurations in the first stage of the CNA and up to O(nn) in the second stage. Therefore, the complexity of the CNA is of at least 0(n2ncoalitiolas) and up to O(nnncoalitions) computations. If the com- putations are distributed among the agents, this order of complexity of computations is divided by n. There is an additional communicational complexity, which is of order 0( n2ncoalitions). 7o 1 Increasing profitability of cooperation S 60 z 50 E 40 B a 30 20 Implementation Coalitions’ mem bet-s 14 12 10 8 6 4 2 Membership in coalitions as function of value 100 200 400 800 i Value of coalitions figure 1 The performance of the CNA was tested with re- spect to different constants (K’s,E) and different envi- ronmental settings. Running the simulation has pro- vided several results as presented below. Initially, we have shown that the simulated CNA reaches a stable PC within a reasonable time (for the first stage of the CNA). In addition, it has been found (for the settings that we have examined) that the CNA continuously improves the agents’ payoffs, whether it is normally terminated or halted artificially. The main results of the simulation for 5 through 13 agents, without limita- tion on Ii’1 and Ii2 and with E = 1 (i.e., less than 1% of the average coalitional value) are as follows: 1. The number of agents that participate in coalitions is an increasing monotonic function of the average of potential coalitional values. The most appropri- ate analytical curve-fit to the results is a logarith- mic function, as in figure 1. We can also conclude that the increment in the coalitional values is not a sufficient condition for increasing the number of coalitions’ members. This may arise from unresolv- able conflicts that are present in non-super-additive environments. 2. In cases where cooperation is beneficial, we observe (figure 2) that the utility is growing as a function of 10 0 1 2 3 4 5 6 7 8 Number of coalition formations figure 2 3. As expected, the time necessary for coalition for- mation without bounding the K’s is an exponential function of the number of agents that comprise the agent-system (see figure 3). However, as can be ob- served from the graph, this exponent is not too steep. For example, in a system of 13 agents, where each agent is implemented on an Intel@486 processor, the computation time per agent is only a few minutes. 4. We also observed that the use of the algorithm by agents does not violate, in average, their individual rationality. According to these intermediate results, it shows that the CNA is a good coalition formation model for MA systems in general environments. We shall report more results in future work. Run-time Nagotiation time as function of 2500 l the number of agents 2000 1500 1000 500 0 5 6 7 8 9 10 11 12 13 Number of agents figure 3 Conclusion The CNA is useful for instances where the number of agents may be large (e.g., tens of agents), computa- tions are costly and time is limited. This is because Negotiation & Coalition 139 the model leads to coalition formation within a poly- nomial time and a polynomial amount of calculations. We introduce the polynomial K-stability. The original K-stability refers to a PC where agents cannot make justified objections, using (exponential) surplus calcu- lations. The new polynomial K-stability entails poly- nomial objection calculations. The CNA leads to distribution of both calculations and communications. In addition, it is an anytime al- gorithm: if halted after any negotiation step, it pro- vides the agents with a set of formed polynomial-K- stable coalitions. A deficiency of the CNA is that in polynomial time it cannot guarantee that a Pareto- optimal PC will be reached. However, for calculating a Pareto-optimal PC all of the coalitional configura- tions in the CCS shall be approached, and therefore there cannot be any polynomial method to find Pareto- optimality. An important advantage of our algorithm is that the average expected payoff of the agents is an increasing function of the time and effort spent by the agents performing the CNA steps. Therefore, if coop- eration is beneficial for the agents, using the CNA will always improve their payoffs. The last property, the anytime and distribution properties and other advan- tages of the CNA, have been proved via simulations. The model we present is not restricted to the super- additive environment, for which there are already several coalition formation algorithms in DAI (She- hory & Kraus 1993; Zlotkin & Rosenschein 1994; Klusch & Shehory 1996). However, the generality of the model does not make it inapplicable. As opposed to the majority of solution concepts presented in game theory, we present a detailed method for how the in- dividual agent should act in order to form coalitions that increase its personal payoff. References Aumann, R. J.; Peleg, B.; and Rabinowitz, P. 1965. A method for computing the kernel of n-person games. h!fathematics of Computation 19:531-551. Bond, A. H., and Gasser, L. 1988. An analysis of problems and research in DAI. In Bond, A. H ., and Gasser, L., eds., Readings in Distributed AI. Califor- nia: Morgan Kaufmann Publishers, Inc. 3-35. Davis, M., and Maschler, M. 1965. The kernel of a cooperative game. Naval research Logistics Quarterly 121223-259. Davis, R., and Smith, R. 1983. Negotiation as a metaphor for distributed problem solving. Artificial Intelligence 20:63-109. Dean, T., and Boddy, M. 1988. An analysis of time- dependent planning. In Proceedings, AAAI88, 49-54. Decker, K., and Lesser, V. 1993. A one-shot dy- namic coordination algorithm for distributed sensor networks. In Proc. of AAAI93. Durfee, E. H., and Rosenschein, J. S. 1994. Dis- tributed problem solving and multi-agent systems: Comparisons and examples. In Proc. of 13th Interna- tional Distributed AI Workshop, 94-104. Durfee, E. H. 1988. Coordination of Distributed Prob- lem Solvers. Boston: Kluwer Academic Publishers. Gasser, L. 1993. Social knowledge and social action. In 1JCAI93, 751-757. Ketchpel, S. P. 1994. Forming coalitions in the face of uncertain rewards. In Proc. of AAAI94, 414-419. Klusch, M., and Shehory, 0. 1996. Coalition forma- tion among rational information agents. In de Velde, W. V., and Perram, J. W ., eds., LNAI No. 1038, Agents Breaking Away. Springer-Verlag. 204-217. Kraus, S., and Wilkenfeld, J. 1991. Negotiations over time in a multi agent environment: Preliminary report. In Proc. of IJCAI-91, 56-61. Lute, R. D., and Raiffa, H. 1957. Games and Deci- sions. John Wiley and Sons, Inc. Malone, T. W.; Fikes, R. E.; Grant, K. R.; and Howard, M. T. 1988. Enterprise: A marketlike task schedule for distributed computing environments. In Huberman, B. A., ed., The Ecology of Computation. Amsterdam: North Holland. 177-205. Moses, Y., and Tennenholtz, M. 1993. Off-line rea- soning for on-line efficiency. In IJCA193, 490-495. Peleg, B. 1984. Game Theoretic Analysis of Voting in Commities. Cambridge University Press. Rapoport, A. 1970. N-Person Game Theory. Univer- sity of Michigan. Sandholm, T. W., and Lesser, V. R. 1995. Coalition formation among bounded rational agents. In Proc. of IJCAI-95, 662-669. Shapley, L. S., and Shubik, M. 1973. Game Theory in economics. California: Rand Corporation. Shehory, O., and Kraus, S. 1993. Coalition formation among autonomous agents: Strategies and complex- ity. In Castelfranchi, C., and M J. P., eds., LNAI No. 951; From Reaction to Cognition. Shoham, Y., and Tennenholtz, M. 1992. On the syn- thesis of useful social laws for artificial agent societies. In Proc. of AAAI-92, 276-281. Stearns, R. E. 1968. Convergent transfer schemes for n-person games. Transactions of the American Mathematical Society 134:449-459. Werner, E. 1988. Toward a theory of communication and cooperation for multiagent planning. In Proc. of the Second Conference on Theoretical Aspects of Reasoning about Know/edge, 129-143. Wu, L. S. 1977. A dynamic theory for the class of games with nonempty cores. Siam Journal of Applied Mathematics 32:328-338. Zlotkin, G., and Rosenschein, J. S. 1994. Coalition, cryptography, and stability: Mechanisms for coali- tion formation in task oriented domains. In Proc. of AAAI94, 432-437. 140 Agents | 1996 | 20 |
1,844 | Michael Kearns AT&T Research mkearns@research.att .com Difficulties in Comparing Machine Learning Heuristics One of the original goals of computational learning theory was that of formulating models that permit meaningful comparisons between the different machine learning heuristics that are used in practice [Kearns et aI., 19871. Despite the other successes of com- putational learning theory, this goal has proven elu- sive. Empirically successful machine learning algo- rithms such as 64.5 and the backpropagation algo- rithm for neural networks have not met the criteria of the well-known Probably Approximately Correct (PAC) model [Valiant, 19841 and its variants, and thus such models are of little use in drawing distinctions among the heuristics used in applications. Conversely, the algorithms suggested by computational learning theory are usually too limited in various ways to find wide application. The Theoretical Status of Decision Tree Learning As an illustration, let us review what has been dis- covered about decision tree learning algorithms in the computational learning theory literature. Consider the simple framework in which a learning algorithm re- ceives random examples, uniformly drawn from the hypercube (0, l)“, that are assigned binary labels ac- cording to some decision tree T that has at most s nodes. A natural goal would be to find an algorithm that can infer a good approximation to T in time and sample complexity that is bounded by a polynomial in n and S. ’ The existence of such an algorithm remains an ap- parently challenging open problem, so even with the various favorable and unrealistic assumptions (uniform input distribution, no noise or missing attributes in the data, the existence of a small “target” tree, and so on), computational learning theory has so far not provided ‘Here we are in the PAC model, where there is no noise in the sample data, with the additional restriction that the input distribution is uniform. vast advances in algorithm design for decision tree in- duction from random examples. On the other hand, in the framework under consideration, the heuristics for decision tree learning that are in wide experimental use do not fare much better. It is rather easy to show that CART and 64.5 will fail to meet the stated criteria, and for the usual reasons: if the target decision tree computes the parity of just two out of the n variables, top-down heuristics like CART and C4.5 may simply build a complete binary tree of depth n before achiev- ing non-trivial error. Of course, this particular con- struction does not rule out the possibility that slight modificutions of the standard heuristics might succeed - but a recent result [Blum et al., 19941 demonstrated that small decision trees can not be learned by any al- gorithm that works solely by “estimating conditional probabilities” [Kearns, 19931. The precise definition of this notion is slightly technical, but suffice to say that CART and 64.5 - which operate primarily by estimating the probabilities of reaching certain nodes in a decision tree, or the conditional distribution of the label given that a node is reached - are canoni- cal examples of the notion. Thus, although computa- tional learning theory has yet to suggest powerful al- gorithms for decision tree learning from random exam- ples, we can assert that if such algorithms exist, they will look nothing like the standard heuristics. Perhaps the more likely outcome is that the problem is sim- ply intractable. This would mean that the assumption that a small decision tree is labeling the data is not especially helpful when examining decision tree learn- ing algorithms, and we must seek alternative assump- tions if we wish to account for the empirical success of CART and 64.5. Provably efficient algorithms become available if we are willing to assume that the learning algorithm is provided with black-box access to the unknown tar- get decision tree (that is, membership queries, which let the learner actively choose the instances to be la- beled). A number of rather simple and elegant learn- ing algorithms have recently been proposed in this set- ting [Bshouty, 1993; Kushilevitz and Mansour, 19911 that will infer the unknown tree in polynomial time, Invited Talks 1337 From: AAAI-96 Proceedings. Copyright © 1996, AAAI (www.aaai.org). All rights reserved. in strong contrast to the case where only random ex- amples are available. However, because of the require- ment for a source of information rarely available in real appiications, these algorithms seem unlikely to replace the top-down heuristics, and their analysis sheds no light on why such heuristics succeed. Viewing Top-Down Decision Tree Heuristics as Boosting Algorithms The preceding summary indicates that some of the models of computational learning theory are unable to provide nontrivial insights into the behavior of CART and C4.5. One might be tempted to attribute this state of affairs to an inevitable chasm between theory and practice - that is, to claim that the standard heuristics succeed in practice due to some favorable structure possessed by real problems that simply can- not be captured by theory as we currently know it. Fortunately, some recent developments seem to demon- strate that such a defeatist position is not necessary. The weak learning or boosting model is a descen- dant of the PAC model in which, rather than directly assuming that the target function can be represented in a particular fashion, we instead assume that there is always a “simple” function that is at least weakly cor- related with the target function. We refer the reader to the literature for the precise technical definition, but for our informal purposes here, it suffices to assume that on any input distribution, there is an attribute whose value is correlated with the label. In this setting, nontrivial performance bounds have recently been proven for both CART and C4.5 [Kearns and Mansour, 1996]. More precisely, if we assume that there is always an attribute whose value correctly predicts the binary label with probabil- ity l/2 + 7 (thus, the attribute provides an advantage y over random guessing), then for CART it suffices to grow a tree of size 1 c/(7a~a log(l/e)) 0 - E (1) in order to achieve error less than e (where c > 0 is a constant), and for C4.5, a tree of size suffices (see [Kearns and Mansour, 19961 for detailed statements and proofs). These bounds imply, among other things, that if we assume that the advantage y is a fixed constant, then both algorithms will drive the error below any fixed E in a constant number of splits. Until the result of [Schapire, 19901, the existence of any algorithm - much less a standard heuristic - possessing this “boosting” behavior was not known. The results given by Equations (1) and (2) provide nontrivial peformance guarantees for CART and C4.5 in an independently motivated theoretical model. A Framework for Comparisons The theoretical results for CART and C4.5 in the weak learning model do more than simply reassure us that these empirically successful algorithms can in fact be proven successful in a reasonable model. As one might have hoped, these results also provide a techni- cal language in which one can attempt to make detailed comparisons between algorithms. Developing this Ian- guage further has been the focus of our recent exper- imental efforts [Dietterich et al., 19961, which we now summarize. First of all, notice that the bounds of Equations (1) and (2) predict that the performance of C4.5 should be superior to that of CART. In the analysis of [Kearns and Mansour, 19961, there are good technical reasons for this difference that are beyond our current scope, but that have to do with the differing concavity of the information gain splitting criterion used by 64.5 and the Gini splitting criterion used by CART. Further- more, again based on concavity arguments, they also suggest a new splitting criterion that enjoys an even better bound of / 1 \ c/y= I - 0 c (3) on the tree size required to achieve error E. In [Diet- terich et al., 19961 we demonstrate experimentally that this new splitting criterion results in small but statis- tically significant improvements in accuracy and tree size over C4.5, so the weak learning analysis seems to have pointed us to som standard algorithms. e modest improvements to the Another intriguing issue raised by the theoretical re- sults emerges if one compares any of Equations (l), (2) and (3) to the bounds enjoyed by the recently introduced Adaboost algorithm due to [Freund and Schapire, 19951, which requires only +ln! E (4) “rounds” (where each round is roughly analogous to a single split made by a top-down decision tree al- gorithm) to achieve error E. The naive interpreta- tion of this bound, which is only the logarithm of the best bound achieved by a top-down decision tree al- gorithm given by Equation (3), would lead us to pre- dict that Adaboost should vastly outperform, for in- stance, C4.5. In practice, the two algorithms are in fact rather comparable [Freund and Schapire, 1996; Dietterich et al., 19961. In the latter citation, we pro- vide extensive experimental evidence that this discrep- ancy between the disparate theoretical bounds and the parity of the algorithms on real problems can be ex- plained by our interpretation of the advantage param- eter y. Briefly, while theoretical boosting resuIts often assume for convenience that there is a simple function with a predictive advantage of y over random guess- ing on any input distribution, in reality this advan- tage varies from distribution to distribution (possibly 1338 AAAI-96 degrading to the trivial value of zero on “hard” distri- butions). Since Adaboost and C4.5 explore very dif- ferent spaces of input distributions as they grow their hypotheses, and since the theoretical bounds are valid only for the smallest advantage y that holds on the dis- tributions actually explored by the algorithm in ques- tion, y has different meaning for the two algorithms. In [Dietterich et al,, 19961, we plot the advantages for each algorithm and demonstrate that while the theo- retical bounds for a fixed advantage y may be worse for C4.5 than for Adaboost, the value of y achieved on real problems is better. This empirical fact largely reconciles the theoretical statements with the observed behavior. Thus, although the weak learning model provides what seems to be the right parameter to study (namely, the advantage y), experimental examination of this pa- rameter was required for real understanding of what the theory was saying and not saying. This kind of in- teraction - where the theory suggests improvements to the popular algorithms, and experimentation with these algorithms modifies our interpretation of the the- ory - seems like a good first step towards the goal mentioned at the outset. There is of course still much work to be done to further close the gap between theory and practice; but at least in the case of decision tree learning, the weak learning framework seems to have provided some footholds that were missing in previous models. In the bibliography, we provide some additional ref- erences on the topics discussed here. References Aslam, J. A. and Decatur, S. E. 1993. General bounds on statistical query learning and PAC learning with noise via hypothesis boosting. In Proceedings of the 35th. IEEE Symposium on the Foundations of Com- puter Science. IEEE Computer Society Press, Los Alamitos, CA. 282-291. Blum, A.; Furst, M.; Jackson, J.; Kearns, M.; Man- sour, Y.; and Rudich, S. 1994. Weakly learning DNF and characterizing statistical query learning us- ing Fourier analysis. In Proceedings of the 26th ACM Symposium on the Theory of Computing. ACM Press, New York, NY. Breiman, L.; Friedman, J. H.; Olshen, R. A.; and Stone, C. J. 1984. Classification and Regression Trees. Wadsworth International Group. Bshouty, N. and Mansour, Y. 1995. Simple learning algorithms for decision trees and multivariate poly- nomials. In Proceedings of the 36th IEEE Symposium on the Foundations of Computer Science. IEEE Com- puter Society Press, Los Alamitos, CA. 304-311. Bshouty, N. H. 1993. Exact learning via the monotone theory. In Proceedings of the 34th IEEE Symposium on the Foundations of Computer Science. IEEE Com- puter Society Press, Los Alamitos, CA. 302-311. Dietterich, Tom; Kearns, Michael; and Mansour, Yishay 1996. Applying the weak learning framework to understand and improve C4.5. In Machine Learn- ing: Proceedings of the Thirteenth International Con- ference. Morgan Kaufmann. Drucker, H.; Schapire, R.; and Simard, P. 1992. Improving performance in neural networks using a boosting algorithm. In Hanson, S.J.; Cowan, J.D.; and Giles, C.L., editors 1992, Advances in Neural Information Processing Systems. Morgan Kaufmann, San Mateo, CA. 42-49. Freund, Yoav and Schapire, Robert E. 1995. A decision-theoretic generalization of on-line learning and an application to boosting. In Second Euro- pean Conference on Computational Learning Theory. Springer-Verlag. 23-37. Freund, Y. and Schapire, R. 1996. Some experiments with a new boosting algorithm. In Machine Learning: Proceedings of the Thirteenth International Confer- ence. Morgan Kaufmann. Freund, Yoav 1995. Boosting a weak learning al- gorithm by majority. Information and Computation 121(2):256-285. Jackson, J. 1994. An efficient membership query al- gorithm for learning DNF with respect to the uniform distribution. In Proceedings of the 35th IEEE Sympo- sium on the Foundations of Computer Science. IEEE Computer Society Press, Los Alamitos, CA. Kearns, M. and Mansour, Y. 1996. On the boosting ability of top-down decision tree learning algorithms. In Proceedings of the 28th ACM Symposium on the Theory of Computing. ACM Press, New York, NY. Kearns, Michael J. and Vazirani, Umesh V. 1994. An Introduction to Computational Learning Theory. The MIT Press. Kearns, M.; Li, M.; Pitt, L.; and Valiant, L. 1987. Recent results on boolean concept learning. In Lan- gley, Pat, editor 1987, Proceedings of the Fourth In- ternational Workshop on Machine Learning. Morgan Kaufmann, San Mateo, CA. 337-352. Kearns, M. 1993. Efficient noise-tolerant learning from statistical queries. In Proceedings of the 25th ACM Symposium on the Theory of Computing. ACM Press, New York, NY. 392-401. Kushilevitz, E. and Mansour, Y. 1991. Learning de- cision trees using the Fourier spectrum. In Proc. of the 23rd Symposium on Theory of Computing. ACM Press, New York, NY. 455-464. Quinlan, J.R. 1993. C4.5: Programs for Machine Learning. Morgan Kaufmann. Schapire, R. E. 1990. The strength of weak learnabil- ity. Machine Learning 5( 2): 197-227. Valiant, L. G. 1984. A theory of the learnable. Com- munications of the A CM 27( 11): 1134-l 142. Invited Talks 1339 | 1996 | 200 |
1,845 | Challenge Problems for Artificial Intelligence Rodney A. Brooks MIT (panel statement) Bart Selman (moderator) AT&T Thomas Dean Brown University Eric Horvitz Microsoft Tom M. Mitchell Nils J. Nilsson CMU Stanford University Introduction: Bart Selman AI textbooks and papers often discuss the big ques- tions, such as “how to reason with uncertainty”, “how to reason efficiently”, or “how to improve performance through learning .” It is more difficult, however, to find descriptions of concrete problems or challenges that are still ambitious and interesting, yet not so open-ended. The goal of this panel is to formulate a set of such challenge problems for the field. Each panelist was asked to formulate one or more challenges. The em- phasis is on problems for which there is a good chance that they will be resolved within the next five to ten years. A good example of the potential benefit of a con- crete AI challenge problem is the recent success of Deep Blue. Deep Blue is the result of a research effort fo- cused on a single problem: develop a program to defeat the world chess champion. Although Deep Blue has not yet quite achieved this goal, it played a remark- ably strong game against Kasparov in the recent ACM Chess Challenge Match. A key lesson we learn from Deep Blue’s strength is that efficient brute-force search can be much more ef- fective than sophisticated, heuristically guided search. In fact, brute-force was so successful that it led Kas- parov to exclaim “I could feel - I could smell - a new kind of int.elligence across the table.” (Kasparov 1996) The experience with Deep Blue shows that a good challenge problem can focus research, lead to concrete 1340 AfMI-96 progress, and bring us important new insights. Many AI researchers may not like the particular lesson about the value of brute-force over more “intelligent” forms of search, but, nevertheless, it is a very tangible re- sult. In fact, the issue of general purpose ultra-fast search procedures versus heuristically guided domain- dependent methods is currently being revisited in the search and reasoning community. Finally, as a meta-issue, we will consider how to mea- sure progress in the field. Determining whether a par- ticular piece of work in AI actually brings us any closer to the ultimate goals of AI has proven to be quite dif- ficult. By introducing a set of well-defined challenge problems, we hope that this panel will help provide some benchmarks against which we can measure re- search progress. Eight Challenges for Artificial Intelligence: Rodney Brooks There are two very different sorts of challenges that I see for Artificial Intelligence - first, our systems are pathetic compared to biological systems, along many dimensions, and secondly, moderately good perfor- mance from some approaches has sociologically led to winner-take-all trends in research where other promis- ing lines of research have been snuffed out too soon. If we compare either software systems, or robotic systems to biological systems, we find that our cre- ations are incredibly fragile by comparison. Below I From: AAAI-96 Proceedings. Copyright © 1996, AAAI (www.aaai.org). All rights reserved. pose some challenges that are aimed at narrowing this distance. The challenges themselves are general in na- ture - they do not solve particular problems, but the acts of meeting these challenges will force the creation of new general purpose techniques and tools which will allow us to solve more particular problems. Challenge 1. Biological systems can adapt to new environments - not perfectly, they die in some envi- ronments, but often they can adapt. Currently our programs are very brittle, and certainly a program compiled for one architecture cannot run on another architecture. Can we build a program which can in- stall itself and run itself on an unknown architecture? This sounds very difficult. How about a program which can probe an unknown architecture from a known ma- chine and reconfigure a version of itself to run on the unknown machine ? Still rather difficult, so perhaps we have to work up to this by making some “blocks worlds” artificial architectures where we can do this. This might lead to some considerations of how future architectures might be designed so that software is self- configurable, and then even perhaps self-optimizing. Challenge 2. Minsky (1967) was foundational in es- tablishing the theory of computation, but after Hart- manis (1971) there has been a fixation with asymp- totic complexity. In reality lots of problems we face in building real AI systems do not get out of hand in terms of the size of problems for individual modules - in particular with behavior-based systems most of the submodules need only deal with bounded size prob- lems. There are other ways theory could have gone. For instance one might try to come up with a theory of computation based on how much divergence there might be in programs given a one bit error in either the program or data representation. If theory were based on this fundamental concern we might start to understand how to make programs more robust. Challenge 3. Recent work with evolutionary sys- tem has produced some tantalizing spectacular results, e.g., Sims (1994). But it is hard to know how to take things from successes and apply them to new problems. We do not have the equivalent of the Perceptron book (Minsky and Papert 1969) for evolutionary systems.’ We need such a new book of mathematics so that we understand the strengths and weaknesses of this excit- ing new approach. Challenge 4. We have been living with the basic for- malizations made by McCulloch and Pitts (1943) for over fifty years now. Their formalization included that ‘Note that these evolutionary systems are much more that straight genetic algorithms as there is both a variable length genotype and a morphogenesis phase that produces a distinctly different phenotype. the activity of the neuron is an “all-or-none” process, that a certain fixed number of synapses must be ex- cited within the period of latent addition in order to excite a neuron at any time, and this number is inde- pendent of the synapses’ previous activity and position on the neuron, that the only significant delay within the nervous system is synaptic delay, that the activity of any inhibitory synapse absolutely prevents excita- tion of the neuron at that time, and that the structure of the net does not change with time. With the addi- tion of changing synaptic weights by Hebb (1949) we pretty much have the modern computational model of neurons used by most researchers. With 50 years of ad- ditional neuroscience, we now know that there is much more to real neurons. Can newer models provide us with new computational tools, and will they lead to new insights to challenge the learning capabilities that we see in biological learning? Over time we become trapped in our shared vi- sions of appropriate ways to tackle problems, and even more trapped by our funding sources where we must constantly justify ourselves by making incremen- tal progress. Sometimes it is worthwhile stepping back and taking an entirely new (or perhaps very old) look at some problems and to think about solving them in new ways. This takes courage as we may be leading ourselves into different sorts of solutions that will for many years have poorer performance than existing so- lutions. With years of perseverance we may be able to overcome initial problems with the new approaches and eventually leapfrog to better performance. Or we may turn out to be totally wrong. That is where the courage comes in. Challenge 5. Despite some early misgivings (Sel- fridge 1956) back when chess playing programs had search trees only two deep (Newell et al. 1958), our modern chess programs completely rely on deep search trees and play chess not at all like humans. Can we build a program that plays chess in the way that a hu- man plays? If we could, then perhaps we could prove how good it was by getting it to play GO-tree search just cannot cut it with GO. Challenge 6. All of the competitive speech under- standing systems today use hidden Markov models. While trainable, these systems have some unfortunate properties. They have much higher error rates than we might desire, they require some restriction in do- main, and they are often inordinately sensitive to the choice of microphone. It seems doubtful that people use HMM’s internally (even if one doesn’t believe that generative grammars are the right approach either). Can we build a speech understanding system that is based on very different principles? Invited Talks 1341 Challenge 7. We live in an environment where we make extensive use of non-speech sound cues. There has been very little work on noise understanding. Can we build interesting noise understanding systems? Challenge 8. Can we build a system by evolution that is better at a non-trivial task than anything that has been built by hand? Integrating Theory and Practice in - Planning: Thomas Dean I issue a challenge to theorists, experimentalists, and practitioners alike to raise the level of expectation for collaborative scientific research in planning. Niels Bohr was first and foremost a theoretical physicist. Ernest Rutherford was first and foremost an experimentalist. It can be argued that neither Bohr nor Rutherford would have made as significant contributions to nu- clear physics without an appreciation and understand- ing of one another’s results. By analogy, I believe that a deeper understanding of planning problems is pos- sible only through the concerted efforts of theorists, experimentalists, and practitioners. The current in- terplay between these groups (perhaps factions is the appropriate word) is minimal. The pendulum of popular opinion swings back and forth between theory and practice. At different times, experimentalist have had to pepper their papers with equations and theorists have had to build systems and run experiments in order to get published. With the exception of the rare person gifted as mathematician and hacker, the requirement of both theory and prac- tice in every result is a difficult one to satisfy; difficult and, I believe, unnecessary. This requirement implies that every publishable result must put forth a theoret- ical argument and then verify it both analytically and experimentally. The requirement tends to downplay the need for a community effort to weave together a rich tapestry of ideas and results. A scientific field can nurture those who lean heav- ily toward theory or practice as long as the individ- uals direct their research to contribute to problems of common interest. Of course, the field must iden- tify the problems that it considers worthy of empha- sis and marshal its forces accordingly. I suggest plan- ning as such a problem and the study of propositional STRIPS planning a good starting point. By analogy to physics in the 1930’s, I recommend continued study of the STRIPS planning (the hydrogen atom of planning) and its stochastic counterparts (Markov decision prob- lems) in parallel with investigations into a wide range of more expressive languages for specifying planning problems (the whole of the periodic table). In terms of concrete proposals, I mention approaches from theoretical computer science that provide alterna- tives to the standard measures of performance, specif- ically asymptotic worst-case analysis. As an alterna- tive to worst-case analysis relying on all-powerful, all- knowing adversaries, I discuss the average-case per- formance measures and the properties of distribu- tions governing the generation of problem instances. Regarding asymptotic arguments, I consider sharp- threshold functions, the relevance of phase-transition phenomena, and the statistical properties of graphs of small order. The resulting perspective empha- sizes particular problem instances and specific algo- rithms rather than problem classes and complexity re- sults that pertain to all algorithms of a given order of growth. I also point out the embarrassing lack of ‘real’ planning problems, posit the reason for such a deficit, and suggest how recent progress in learning the- ory might provide a rich source of planning problems.2 Decisions, Uncertainty and Intelligence: Eric Horvitz To be successful in realistic environments, reason- ing systems must identify and implement effective ac- tions in the face of inescapable incompleteness in their knowledge about the world. AI investigators have long realized the crucial role that methods for handling in- completeness and uncertainty must play in intelligence. Although we have made significant gains in learning and decision making under uncertainty, difficult chal- lenges remain to be tackled. Challenge: Creating Situated Autonomous Decision Systems A key challenge for AI investigators is the develop- ment of comprehensive autonomous decision-making systems that are situated in dynamic environments over extended periods of time, and that are entrusted with handling varied, complex tasks. Such robust decision systems need the ability to process streams of events over time, and to continue, over their lifetimes, to pur- sue actions with the greatest expected utility. Mounting a response to this broad challenge imme- diately highlights several difficult subproblems, each of which may be viewed as a critical challenge in itself. Pursuing solutions to these subproblems will bring us closer to being able to field a spectrum of application- specific challenges such as developing robotic systems that are given the run of our homes, tractable medi- cal decision making associates that span broad areas of medicine, automated apprentices for helping people with scientific exploration, ideal resource management 2Additional details can be found ’ ftp://www.cs.brown.edu/u/tld/postscript/DeanetalAAA~ 96.ps. 1342 AAAI-96 in multimedia systems, and intelligent user interfaces that employ rich models of user intentions and can en- gage in effective dialogue with people. To address the broad challenge, we need to con- sider key phases of automated decision making, includ- ing the steps of perceiving states of the world, fram- ing decisions, performing inference to compute beliefs about the world, making observations, and, most im- portantly, identifying a best set of actions. Under lim- ited resources, we also need to carefully guide the allo- cation of resources to the different phases of analysis, and to extend decision making to the realm of monitor- ing and control of the entire decision-making process. I will dive into several problems associated with these components of decision making. Subproblem: Automated Framing of Decision Prob- lems Faced with a challenge, a decision-making system must rely on some rules or, more generally, a model that expresses relationships among observations, states of the world, and system actions. Several methods have been studied for dynamically building representations of the world that are custom-tailored to perceived chal- lenges. Framing a decision problem refers to identify- ing a set of relevant distinctions and relationships, at the appropriate level of detail, and weaving together a decision model. Framing a decision problem has re- sisted formalization. Nevertheless, strides have been made Lll”U~L b”LIU”L U”Vl”IL techniques, typically re- lying on the use of logical or decision-theoretic proce- dures to piece together or prune away distinctions, as a function of the state of the world, yielding manageable focused models. We have a long way to go in our under- standing of principles for tractably determining what distinctions and dependencies will be relevant given a situation. Subproblem: Handling Time, Synchronicity, and a- Streams of l3vents Autonomous systems must make decisions in an evolving environment that may change dramatically over time, partly in response to actions that a system has or will take. Most research on action under uncer- tainty has focused on models and inference procedures that are fundamentally atemporal, or that encode tem- poral distinctions as static variables. We must endow systems with the ability to represent and reason about the time-dependent dynamics of belief and action, in- cluding such critical notions as the persistence and dy- namics of world states. We also need to develop better means of synchronizing an agent’s perceptions, infer- ence, and actions with important events in the world. Subproblem: Modeling Preferences and Utility The axioms of utility give us the fundamental princi- ple of maximum expected utility: an agent should take actions that maximizes its expected (or average) mea- sure of reward. Although it is easy to state the prin- ciple, we are forced in practice to wrestle with sev- eral difficult problems. Where does information about the utility of states come from? Whose utility is be- ing maximized? How can we derive utilities associated with solving subproblems from assertions about high- level goals (e.g., “survive for as long as possible!“) or from utilities on goal states. ? What is the most reason- able utility model for evaluating a finite sequence of actions an agent might take over time (e.g., should we assume an infinite number of future actions and dis- count value of future rewards, or assume a finite set of steps and compute average reward?, etc.). Different assumptions about the specific structure of the utility model lead to different notions of the “best” behaviors and to different computational efficiencies with evalu- ating sequences of plans. Subproblem: Mastery of Attention and Architecture Perceiving, reasoning, and acting all require costly resources. Controlling the allocation of computational resources can be a critical issue in maximizing the value of a situated system’s behavior. What aspects of a problem and problem-solving strategy should a system attend to and when? We need to develop richer mod- els of attention. There is promise in continuing work that turns the analytic machinery of decision-theoretic inference onto problem solving itself, and using such measures as the expected value of computation (EVC) to make design-time and run-time decisions about the ideal quantities of computation and memory to allo- cate to alternative phases of reasoning - including to the control processes themselves. More generally, there is great opportunity in applying these methods in off- line and on-line settings to optimize the overall nature and configuration of a system’s architecture, including decisions about the compilation of results. Subproblem: Learning about Self and Environment Continual learning about the environment and about the efficacy of problem solving is critical for systems situated in complex, dynamic environments, especially when systems may wander into one of several special- ized environmental niches. We need to better under- stand how we can endow our systems with awareness of having adequate or inadequate knowledge about spe- cific types of problems so that they can allocate ap- propriate resources for exploration and active learning. There has been research on methods for computing the confidence in results given a model and problem instance. This work highlights opportunities for de- veloping methods that an agent could use to probe for critical gaps in its knowledge about the world. Invited Talks 1343 Continuing this research will be valuable for building decision-making systems that can perform active, di- rected learning. Subproblem: Living Life Fully-Harnessing Every Second To date, most of our reasoning systems have no choice but to idle away the precious time between their active problem-solving sessions. Systems immersed in complex environments should always have something to do with their time. We need to develop techniques that allow an agent to continuously partition its time not only among several phases of a pressing analysis but also among a variety of tasks that will help the agent to maximize the expected utility of its behavior over its entire lifetime. Tasks that can benefit from on- going attention include planning for future challenges, probing and refining a utility model, prefetching in- formation that is likely to be important, compilation of portions of expected forthcoming analyses, experi- menting (playing?) with its reasoning and motion con- trol systems, and learning about critical aspects of the external world. Think Big - AI Needs More Than Incremental Progress: Tom Mitchell 1. Let’s build programs that turn the WWWeb into the world’s largest knowledge-base. Doug Lenat had a good idea that we should have a large AI knowledge base covering much of human knowledge. One problem with building it is that it might take millions of people. And then we’d have to maintain it afterwards. The web is built and growing already, and it’s online, and people are already maintaining it. Unfortunately, it’s in text and images and sounds, not logic. So the challenge is to build programs that can “read” the web and turn it into, say, a frame-based symbolic representation that mirrors the content of the web. It’s hard, but there is no evidence that it’s impossible. And, even partial solutions will be of incredible value. 2. Apply Machine Learning- to learn to understand Natural Language. Natural language has always been considered to be too difficult to do for real. But things have changed over the past three years in an interesting way - for the first time in history we have hundreds of millions of supervised training examples indicating the meaning of sentences and phrases: those hyperlinks in all those web pages. Now agreed, they’re not quite the kind of supervised training data we’d ask for if we were to choose training data to learn natural language. But when it comes to training data you take what you can get, especially if it numbers in the millions (when there are less than 100,000 words to begin with). So each hy- perlink like my recent publications has a meaning that is revealed by the web page you get if you click on it. How can we learn something useful about natural language understanding from this kind of data? (We probably need to use more than just this kind of data to learn language, but once we have some basic ontol- ogy defined, this kind of data should be of great use in learning the details.) 3. Let’s build agents that exhibit life-long machine learning, rather than machine learning algorithms that learn one thing and then get rebooted. Consider people and consider current ML approaches such as decision tree or neural network learning. People mature, they learn things, then use these things they know to make it easier to learn new things. They use the things they know to choose what to learn next. ML has made good progress on approximating isolated functions from ex- amples of their input/output. But this is just a subrou- tine for learning, and it’s more of less a solved problem at this point. The next question is how can we build agents that exhibit long-term learning that is cumula- tive in a way more like people learn. Toward Flexible and Robust Robots: Nils J. Nilsson I start with the premise that it would be desirable to have mobile, AI-style robots that have continu- ous existence - ones that endure and perform useful work for long periods of time rather than ones that merely hold together long enough for a quick demo and video-taping session. Of course, some limited- capability robots - such as those currently used in automobile assembly and hospital-item delivery - do stay on the job for long periods of time. But all of these robots are far from being as robust and flexible as we want robots to be. My challenge problem is to produce a robot facto- turn and errand-runner for a typical office building - an office building that is not specially equipped to ac- commodate robots. The kinds of tasks that such a robot will be able to perform will depend, of course, on its effecters and sensors as well as on its software. There are plenty of important challenge problems con- cerned with sensors and effecters, but I leave it to oth- ers to pose those. Instead, my challenge is to AI people to develop the software for a robot with more-or-less state-of-the art range-finding and vision sensors and locomotion and manipulation effecters. Let’s assume that our robot can travel anywhere in the building - down hallways, into open offices, and up and down el- evators (but perhaps not escalators). Assume that it can pick up, carry, and put down small parcels, such as books and packages. For communication with humans, suppose it has a speech synthesizer and word or phrase 1344 AAAI-96 recognizer and a small keyboard/display console. Any such set of sensors and effecters would be sufficient for an extensive list of tasks if only we had the software to perform them. Here is the specific challenge: The robot must be able to perform (or learn to perform with instruc- tion and training - but without explicit post-factory computer programming) any tusk that a human might “reasonably” expect it to be able to perform given its effecter/sensor suite. Of course, some tasks will be im- possible - it cannot climb ladders, and it doesn’t do windows. And, I don’t mean the challenge to be one of developing a “Cyc-like” commonsense knowledge base and reasoning system. Our factotum will be allowed some lapses in commonsense so long as it can learn from its mistakes and benefit from instruction. It is not my purpose to set high standards for speech un- derstanding and generation. The human task-givers should be tolerant of and adapt to the current limited abilities of systems to process natural language. The second part of the challenge is that the robot must stay on-the-job and functioning for a year with- out being sent back to the factory for re-programming. What will be required to meet this challenge? First, of course, a major project involving the application and extension of several robotic and AI technologies and architectures for integrating them. I do not think that it will be feasible for the robot’s builders to send it to its office building with a suite of programs that an- ticipate allof the tasks that could be given. I think the robot will need to be able to plan and to learn how to perform some tasks that the building occupants (who know only about its sensors and effecters) might ex- pect it to be able to perform but that its programmers did not happen to anticipate. To plan and to learn efficiently, I think it will need to be able to construct for itself hierarchies of useful action routines and the appropriate associated perceptual processing routines for guiding these actions. Perhaps something like the “twin-tower” architecture of James Albus (1991) would be appropriate for overall control. But the towers will have to grow with instruction and experience. The computational models of developmental learning pro- posed by Gary Drescher (1991) seem to me to be a good place to start for the tower-building aspect of the problem. Work on this challenge problem would be good for AI. It would encourage progress on extending and in- tegrating the many disparate components of intelligent systems: reacting, planning, learning, perception, and reasoning. It might also connect the bottom-up and top-down AI approaches - to the benefit of both. (It could also produce a useful factotum.) Good luck! References J.S. Albus. Outline for a Theory of Intelligence. IEEE Systems, Man, and Cybernetics, Vol 21, No. 3, pp. 473-509, May/June 1991. G. Drescher. Made Up Minds: A Constructivist Ap- proach to Artificial Intelligence, Cambridge, MA: MIT Press, 1991. E.A. Feigenbaum, and J. Feldman. Computers and Thought, New York, NY: McGraw-Hill, 1963. M. L. Ginsberg. Do computers need common sense? Techn. report, CIRL, Univerity of Oregon, 1996. J. Hartmanis. Computational complexity of random access stored program machines. Mathematical Sys- tems Theory, 5:232-245, 1971. D.O. Hebb. The Organization of Behavior. John Wi- ley and Sons, New York, New York, 1949. G. Kasparov. The day that I sensed a new kind of intelligence. Time, March 25, 1996, p. 55. W.S. McCulloch and W. Pitts. A logical calculus of the ideas immanent in nervous activity. Bull. of Math. Biophysics, 5:115-137, 1943. Marvin Minsky. Computation: finite and infinite mu- chines. Prentice-Hall, 1967. Marvin Minsky and Seymour Papert. Perceptrons. MIT Press, Cambridge, Massachusetts, 1969. Allen Newell, J.C. Shaw, and Herbert Simon. Chess playing programs and the problem of complexity. Journal of Research and Development, 21320-335, 1958. Also appeared in (Feigenbaum and Feldman 1963). Oliver G. Selfridge. Pattern recognition and learning. In Colin Cherry, editor, Proceedings of the Third Lon- don Symposium on Information Theory, New York, New York, 1956. Academic Press. Karl Sims. Evolving 3d morphology and behavior by competition. In Rodney A. Brooks and Pattie Maes, editors, Artificial Life IV: Proceedings of the Fourth International Workshop on the Synthesis and Sim- ulation of Living Systems, pages 28-39. MIT Press, Cambridge, Massachusetts, 1994. Invited Talks 1345 | 1996 | 201 |
1,846 | The Database Approach to Knowledge Representation Jeffrey D. Ullman Department of Computer Science Stanford University Stanford CA 94305 ullman&s.stanford.edu http://db.stanford.edu/“ullman Abstract The database theory community, centered around the PODS (Principles of Database Systems) conference has had a long-term interest in logic as a way to rep- resent “data, ” “information,” and “knowledge” (take your pick on the term - it boils down to facts or atoms and rules, usually Horn clauses). The approach of this community has been “slow and steady,” preferring to build up carefully from simple special cases to more general ideas, always paying attention to how efficiently we can process queries and perform other operations on the facts and rules. A powerful theory has developed, and it is beginning to have some impact on applications, especially information-integration engines. Datalog The term Databug has been coined to refer to Prolog-like rules without function symbols, treated as a logic pro- gram. Unlike Prolog, however, the conventional least- fixed-point semantics of the rules is used whenever pos- sible. Example 1: The rules for ancestors can be written as the following Datalog program. anc(X,Y> :- par(X,Y) anc(X,Y> :- aJdx,z>, anc(Z,Y> That is, Y is an ancestor of X if Y is a parent of X or if there is some 2 that is an ancestor of X and a descen- dant of Y. Because of the least-fixed-point semantics, there is no question of this program entering a loop, as the corresponding Prolog program would. 0 Generally, we divide predicates into two classes. EDB (extensional database) predicates are stored as re- lations, while IDB (intensional database) predicates are defined by the heads (left sides) of rules only. Either EDB or IDB predicates can appear in subgoals of the bodies (right sides) of rules. Sometimes, Datalog is extended to allow negated subgoals. That extension causes the least-fixed-point semantics to become problematic when the rules are re- I346 A.&U-96 cursive, and several approaches such as stratified nega- tion and well-founded semantics have been developed to define suitable meanings for such Datalog programs. A survey of this subject, analogous to “nonmonotonic reasoning,” can be found in Ullman [1994], and we shall not address this set of issues further here. Example 2: A Datalog rule for ancestors that were not parents could be expressed as oldAnc(X,Y) :- anc(X,Y), NOT par(X,Y) 0 Conjunctive Queries A single Datalog rule in which an IDB predicate is de- fined in terms of one or more IDB and EDB predicates other than itself is called a conjunctive query (CQ). For instance, the rule of Example 2 is a CQ. Containment and Equivalence We say one CQ or Datalog program is contained in an- other if whatever the values of the EDB predicates (the “database”) is, the set of facts provable from the first is a subset of those provable from the second. CQ’s or Datalog programs are equivalent if the sets of provable facts are always the same for any database, i.e., the containment goes both ways. Example 3: Consider :I: ;$ :- arc(X,Y), arc(Y,X) . 20 -- arc(X,X) . We say that Q2 & Qr. Intuitively, &I defines the set of nodes of a directed graph that are on any cycle of two nodes or loop of one node, while Q2 defines only the set of nodes that have loops. It is also easy to check that Qr is not contained in Q2; that is, there are graphs with cycles of two nodes but no loops. Thus, Qi # Q2. Cl Chandra and Merlin [1977] first studied conjunc- tive queries and showed that there is a simple test for containment, and thus for equivalence. The question of whether one CQ is contained in another is NP-complete, From: AAAI-96 Proceedings. Copyright © 1996, AAAI (www.aaai.org). All rights reserved. but all the complexity is caused by “repeated predi- cates,” that is, predicates appearing three or more times fcor(X) :- par(a,A), par(A,B), par(C,B), par(D,C), Par&D) in the body. In the very common case that no predicate appears more than twice in any one query, containment can be tested in linear time (Saraiya [1991]). Moreover, CQ’s tend to be short, so in practice containment test- ing is not likely to be too inefficient. Also, in exponential time at most we can test whether a CQ is contained in a Datalog program (Ra- makrishnan et al. [1989]). The opposite containment, of a Datalog program in a CQ is harder, but still decidable (Chaudhuri and Vardi [1992]). When we extend rules by allowing negated sub- goals and/or by allowing arithmetic comparisons in the subgoals, things rapidly become undecidable. However, if we restrict ourselves to CQ’s, not Datalog programs, then the problem is exponential at worst. Levy and Sa- giv [1993] g ive a containment test for CQ’s with nega- tion, while the most efficient known algorithm for CQ’s with arithmetic comparisons (but no negation) is found in Zhang and Ozsoyoglu [1993]. Application to Information Integration A system for integrating heterogeneous information sources can be described logically by vievls that tell us what queries the various sources can answer. These views might be CQ’s or Datalog programs, for exam- ple. The “database” of EDB predicates over which these views are defined is not a concrete database but rather a collection of “global” predicates whose actual values are determined by the sources, via the views. Given a query Q, typically a CQ, one can ask whether it is possible to answer Q by using the vari- ous views in some combination. For example, Informu- tion Manifold at Bell Labs (Levy, Rajaraman, and Or- dille [1996]) searches for all combinations of views that answer a query, while Tsimmis at Stanford (Garcia et al. [1995]) t ries to find one such solution. Example 4: The following example is contrived but will illustrate the ideas. Suppose we have a global par- ent relation par(X, Y), meaning that Y is a parent of X. Suppose also that one source is capable only of pro- viding a grandparent view. That is, it gives us the view: gpoLn :- par(X,Z), par(Z,Y) A second source gives us a great-grandparent view: ggp(x,V :- par(X,A), par(A,B), par(BJ) Our query Q is “find the first cousins once removed of individual a. That is, we must go up two generations from a, then down three generations. Formally, in terms of the global “database”: We can answer Q in terms of the given views by: fcor(X) :- gp(a,B), ggp(X,B) cl Solving Queries in Terms of Views The fundamental work on how to find an expression for a given query in terms of views is Levy, Mendelzon, Sagiv, and Srivastava [1994]. This paper handles the case where both the views and the query are CQ’s. Rajaraman, Sagiv, and Ullman [1994] extends the latter to the case where the views have “binding pat- t ems” ; that is, the source can only answer a query in which one or more arguments are bound. An example of this situation is a bibliographic source that can find a book given an author or an author given a book, but cannot answer the query “tell be about all books and their authors.” In Example 4, there is a solution only if the source of view gp can handle a query with the first argument bound, and the source of ggp can handle a query with the second argument bound. There has also been some progress allowing the views to be described by Datalog programs rather than CQ’s. Specifically, each Datalog program can be ex- panded into a (possibly infinite) set of CQ’s, and we may suppose that a source will answer any one of these CQ’s. That model covers the case where the source is an SQL database, for instance. In Papakonstantinou et al. [1995], the test for con- tainment of a CQ in a Datalog program is exploited to find an expansion of the Datalog program that contains a given query. Levy, Rajaraman, and Ullman [1996] show how to decide equivalence of a query to some ex- pression built from (a finite subset of) the infinite set of views that are the expansions of a Datalog program. This result extends Levy, Mendelzon et al. [1995] to the situation where sources support infinite sets of views that are described by a Datalog program. Acknowledgements This work was supported by NSF grant IRI-92-23405, AR0 grant DAAH04-95-1-0192, and USAF contract F33615-93-1-1339. References Chandra, A. K. and P. M. Merlin [1977]. “Opti- mal implementation of conjunctive queries in relational databases,” Proc. Ninth Annual ACM Symposium on the Theory of Computing, pp. 77-90. Invited Talks 1347 Chaudhuri, S. and M. Y. Vardi [1992]. “On the equiva- lence of datalog programs,” Proc. Eleventh ACM Sym- posium on Principles of Database Systems, pp. 55-66. Garcia-MoIina, H., Y. Papakonstantinou, D. Quass, A. Rajaraman, Y. Sagiv, J. UUman, and J. Widom [1995]. “The TSIMMIS approach to mediation: data models and languages,” Second Workshop on Next-Generation Information Technologies and Systems, Naharia, Israel, June, 1995. Levy, A., A. Mendelzon, Y. Sagiv, and D. Srivastava [1995]. “Answering queries using views,” Proc. Four- teenth ACM Symposium on Principles of Database Sys- tems, pp. 113-124. Levy, A. Y., A. Rajaraman, and J. J. OrdiIIe [1996]. “Querying heterogeneous information sources us- ing source descriptions,” ATT Technical Memorandum, submitted for publication. Levy, A. Y., A. Rajaraman, and J. D. UIIman [1996]. “Answering queries using limited external processors,” to appear in PODS 1996. Levy, A. Y. and Y. Sagiv [1993]. “Queries indepen- dent of update,” Proc. International Conference on Very Large Data Bases, pp. 171-181. Papakonstantinou, Y., A. Gupta, H. Garcia-MoIina, and J. D. UIIman [1995]. “A query translation scheme for rapid implementation of wrappers,” Fourth DOOD, Singapore, Dec., 1995. Rajaraman, A., Y. Sagiv, and J. D. UIIman [1995]. “Query optimization using templates with binding pat- terns,” Proc. Fourteenth ACM Symposium on Princi- ples of Database Systems, pp. 105-112. Ramakrishnan, R., Y. Sagiv, J. D. UlIman, and M. Y. Vardi [1989]. “P roof tree transformation theorems and their applications,” Proc. Eighth ACM Symposium on Principles of Database Systems, pp. 172-181. Saraiya, Y. [1991]. “Subtree elimination algorithms in deductive databases,” Doctoral Thesis, Dept. of CS, Stanford Univ., Jan., 1991. UIIman, J. D. [1994]. “Assigning an appropriate mean- ing to database logic with negation,” in Computers as Our Better Partners (H. Yamada, Y. Kambayashi, and S. Ohta, eds.), pp. 216-225, World Scientific Press. Zhang, X. and M. Z. Ozsoyoglu [1993]. “On efficient reasoning with implication constraints,” Proc. Third DOOD Conference, pp. 236-252. 1348 A&U-96 | 1996 | 202 |
1,847 | Doing tasks with multiple mini-ro John Fischer and Paul Rybski and Dirk Edmonds and Maria Gini Department of Computer Science, University of Minnesota 200 Union St SE, Minneapolis, MN 55455 {jfischer, rybski, dedmonds, gini)@cs.umn.edu We are interested in building robots that are sim- ple and have limited computing power yet are capa- ble of surviving in an unstructured environment while achieving their assigned task. We have shown that even with limited computing small robots can learn how to achieve their task (Hougen et al. 1996)) pro- vided that the task is not extremely difficult and the learning algorithm is capable of fast learning. One of our mini-robots, named Walleye, was built to pick up cups and cans for the Mobile Robot Competi- tion that took place at IJCAI in August 1995 (Fischer & Gini 1996). Walleye, shown in Figure 1, is built out of a radio controlled car with the original electronics replaced by specially designed boards. All boards are built around the 68hcll microcontroller, and have 16k of ROM and 32k of RAM. The vision system uses a CCD chip with digital output, a wide-angle lens, and a frame grabber board on which the vision processing is done. Two 7.2 volt rechargeable batteries are used, one for the motors, one for the computer boards. All software is written in C, with some routines in assem- bly. Walleye is built with off-the-shelf components at a cost of approximately $500. The limited computing power has forced us to look for creative solutions that are simple and fast. This is particularly important considering that we do image processing on a 68hcll with limited memory. Our im- age processing algorithms are specialized to the task at hand and so extremely fast. In nature specialization is often the key to survival. One way of overcoming the limitations of a mini- robot is to construct a team of mini-robots. Unfortu- nately, there are a number of problems that come from having multiple robots. At the minimum, we have to ensure that the robots do not damage each other, do not interfere with each other, and can handle the pres- ence of other moving robots in the same environment. Partitioning the task is not always easy, and a poor partitioning of the task might make the task unsolv- able in the case a robot breaks down. Figure 1: Walleye, the trash collecting mini-robot The approach we are taking is to partition tasks in such a way that robots are independent of each other as much as possible and so have almost no need for communication. Take, for instance, a trash collect- ing task. Multiple independent robots are likely to work faster than a single robot, even though not as efficiently as robots that partition the space each has to cover, However, when each robot operates indepen- dently the overall system is more robust and less likely to fail catastrophically. We expect to demostrate mul- tiple mini-robots at the 1996 competition. Acknowledgements We wish to thank all the people who have helped building our mini-robots, Elena Beltran, Abraham Nemitz, Luis Or- tiz, Chris Smith, Erik Steinmetz, Maxim Tsvetovatyy, and Paul Zobitz, We would like to acknowledge the support of NSF under grant NSF/DUE-9351513, the UROP project at the University of Minnesota, and the AT&T Foundation. References Fischer, J., and Gini, M. 1996. Vision-based mini- robots. Robotics Practitioner. (to appear). Hougen, D.; Fischer, J.; Gini, M.; and Slagle, J. 1996. Fast connectionist learning for trailer backing using a real robot. In Proc. IEEE International Conference on Robotics and Automation. Robot Competition 1353 From: AAAI-96 Proceedings. Copyright © 1996, AAAI (www.aaai.org). All rights reserved. | 1996 | 203 |
1,848 | Lola, the mobile robot from NC State Ricardo Gutierrez-Osuna, Daniel S. SchudeI, Jason A. Janet and Ren C. Luo Center for Robotics and Intelligent Machines North Carolina State University Raleigh, NC 27695-791 I rgutier, dsschude, jajanet, luo [@eos.ncsu.edu] Introduction The North Carolina State Universit,y team intends to participate in the 1996 Mobile Robot Competition and Exhibition with Lola, a Nomad 200. This year marks our third entry in this competition. Robot description Lola is a Nomad 200 with a standard l6-transducer sonar ring and tactile bumper sensors. We have taken advantage of the Nomad’s modularity and moved 3 of the rear sonars to the front just above the bumper. This provides Lola with the ability t,o sense chairs and other potential obstacles that might otherwise be un- seen. Lola’s main processor is a 486DX2-66 running Linux (Unix). The combination of Linux and wireless Ether- net makes code development on Lola a real joy. That is, from any workstation on the network (including the Internet) we can telnet to Lola and export the display for developing, debugging and executing code while monitoring Lola’s status during operation, which beats having to wheel around a terminal and extension cord. The idea is to do all development without leaving your chair. The vision hardware consists of an on-board im- age processor and a single RGB camera mounted on a pan/tilt unit. Lola’s image processor was purchased from Traquair Data Systems and is blessed with two ‘C40 DSP’s running in parallel. Performing all com- putation on-board has several advantages: the video data is not corrupted by radio transmission noise, com- mands are not lost, and there is no communication lag that may result in Lola crashing into things. These findings are consistent with those of previous competi- tors. On the downside, the on-board image processor contributes significantly to the battery drain, which is partly due to its intended desktop use. Still, we are able to get about 2 hours of operation per charge. Software We intend to extend the software we used in the 1995 Mobile Robot Competition. As of tIllis writing not all 1354 AAAI-96 the modules have been implemented. The basic ap- proach for each event has been determined and is sum- marized in the following sections. Navigation architecture for Event I[ The navigation architecture consists of the following modules: High-level planning: We perform path planning on the topological map of the arena. The map is searched for an optimal path to the goal location. Navigation: We use a Partially Observable Markov Decision Process to estimate the location of the robot in the topological map from dead-reckoning and sonar information. Feature detection: We use a Certainty Grid to extract topological features from the sonar informa- tion and find the robot’s orientation with respect to the walls. Low-level control: Low-level control is based on “artificial forces”, where the robot is attracted by a desirable state (i.e., follow a direction) and repulsed by sensed obstacles. Perception/Manipulation for Event II For this event we will use the following approach: e Perception: The vision system on the robot and a color-histogramming technique are used to recognize and track tennis balls and the squiggle ball. e Manipulation: We intend to design and build a 5-degree-of-freedom arm to retrieve the balls. o Neural networks: It is expected that we will use a Region and Feature Based neural network for object detection/recognition and control of the arm. Team members Ricardo Gutierrez-Osuna, Daniel S. Schudel and Ja- son A. Janet are graduate students at NCSU’s Elec- trical and Computer Engineering Department. Dr. Ren C. Luo is Professor and Director of the Center for Robotics and Intelligent Machines at NCSU. From: AAAI-96 Proceedings. Copyright © 1996, AAAI (www.aaai.org). All rights reserved. | 1996 | 204 |
1,849 | Clementine: Colorado School of Mines Undergraduate Interdisciplinary Robotics Team Robin R. Murphy, Associate Director and Advisor Center for Robotics and Intelligent Systems Colorado School of Mines Golden, CO 80401-1887 phone: (303) 273-3874 fax: (303) 273-3975 rmurphy@mines.edu The 1996 Entry The Colorado School of Mines (CSM) is fielding a team comprised of undergraduates in Computer Science or Engineering who are enrolled in the Robotics and AI Minor. The intent is to provide a forum for the students to a) transfer what they have learned in the classroom to a more realistic setting, b) meet with top researchers in the field, c) have an undergraduate research experience, and d) have fun. The students work with the team advi- sor and graduate students at CSM to integrate and mod- ify code developed for NSF, ARPA, and NASA funded research projects. This will be the fourth year CSM has participated in the competition. The team’s platform is Oementine, a Denning-Branch MRV-4 research robot. She has a ring of 24 ultrasonics, a laser navigation system, and supports two cameras. All processing is done onboard by a 75MHz Pentium processor. A SoundBlaster board and speakers provides feedback on the robot’s activities. Clementine is used for research in indoor task domains such as the surveil- lance and maintenance of stockpiles of hazardous mate- rials, site assessment of dangerous environments such as a burning building or a collapsed mine, or security. A custom robot, C2, is used for outdoor environments. Objectives This year’s team is concentrating on Event 1. There are two primary pedagogical objectives. 1. Gain fu- miliarity with hybrid deliberative/reactive architectures by applying it to a well defined problem. This objec- tive is being met by having the students use a subset of the CSM hybrid architecture. The deliberative layer handles all activities which require knowledge about the robot’s task. The task manager receives the the topo- logical map, starting node, and goal node for the event via a human interface. It then activates the cartogrupher which produces a list of nodes, called the path plan, rep- resenting the best path between the start and goal. The task manager selects the behavior(s) for traveling be- tween the current node and the next node on the path plan. The reactive, or behavioral, layer is responsible for executing the constituent behaviors encapsulated by the abstract navigation behavior. 2. Gain practical experience in using multiple sensing modalities. As mobile robots are developed for more de- manding applications, it will become necessary to use multiple sensing modalities (e.g., vision, sonar, range finders, inclinometers, GPS, etc.). Accordingly, the stu- dents are required to use ultrasonics (sonar) for obstacle avoidance and basic navigation, and computer vision for identification of rooms. Research Innovations The students are incorporating two novel concepts from ongoing research at CSM: scripts for the coordination and control of concurrent and sequential activities in the reactive layer, and the partitioning of behaviors into strategic and tactical categories. Scripts, originally de- veloped as a representation for Natural Language Pro- cessing, serve as a template for coordinating and control- ling a collection of behaviors needed to perform a highly stereotyped task over time. The navigational activities needed for Event 1 have been collected into two abstract behaviors: NavigateHall and NavigateDoor. Another novel aspect of the CSM hybrid architecture is its organization of behaviors in the reactive layer into strategic and tactical activities. Strategic behaviors, such as NavigateHall, generate strategic directions or navigational goals for the robot based on large scale con- cerns. Tactical behaviors such as avoid-obstacle and fuzzy speed-control, interpret the robot’s strategic in- tent (e.g., go straight) in terms of the immediate situa- tion (e.g., there’s an obstacle directly ahead) and actu- ally produce the action for the robot to take. Acknowledgments The Undergraduate Interdisciplinary Robotics Team en- try is supported in part by Denning-Branch Interna- tional Robotics, Pittsburgh, PA. Robot Competition 1355 From: AAAI-96 Proceedings. Copyright © 1996, AAAI (www.aaai.org). All rights reserved. | 1996 | 205 |
1,850 | Mobile Robot Navigation and Control: A Case Study Nicholas Roy, Gregory Dudek, Michael Daum Research Centre for Intelligent Machines McGill University Montreal, Quebec, Canada 1 Introduction Robotic systems (and in particular mobile autonomous agents) embody a complex interaction of computational processes, mechanical systems, sensors, and communica- tions hardware. System integration can present significant difficulties to the construction of a real system, because the hardware is often built around convenience of design rather than convenience of system integration. Nonethe- less, in order for robots to perform real-world tasks such as navigation, localization and exploration, the different subsystems of motion, sensing and computation must be merged into a single, realisable unit. Our group is investigating particular problems in the do- main of computational perception, in the context of mo- bile robotics. In particular, we are concerned with envi- ronment exploration, position estimation, and map con- struction. We have several mobile platforms integrating different sensing modalities, which we are able to control simultaneously from a single source. 2 Methodology To support this work, we have developed a layered software architecture, that facilitates a modular approach to prob- lems, in addition to building an abstraction of a robotic system [l]. Our architecture involves three software lay- ers: on-board real-time subsystems, off-board hardware- specific systems that abstract away hardware dependen- cies, and top-level “client” processes. This abstraction al- lows external software to interact with either a simulated robot and environment or a real robot complete with sen- sors. The implementation is distributed across a network, and allows software to run on remote hardware, thus tak- ing advantage of specialized hardware available on the net- work. In appreciation of the necessity of simulation in addi- tion to real-robot control, we have developed a graphi- cal environment for the development of algorithms and software for mobile robotics. The environment was pro- duced as a result of the recognition that progress in mo- bile robotics entails a progression from basic implemen- tation of simple routines, through to the development of efficiently-implemented algorithms. With these considera- tions in mind, we have constructed a control and develop- ment interface for mobile robotics experiments that per- mits a single robot to be controlled and/or simulated us- ing any combination of manual experimentation, simple automation and high-level algorithms. *The support of NSERC and the Federal Centres of Excel- lence program is gratefully acknowledged. 3 Implementation In the context of the AAAI competition, we are tapping this infrastructure by rapidly constructing a set of client processes which embody task specific objectives for the meeting scheduling problem. The client processes group simple sonar measurements into clusters used to classify regions of the map according to a simple labelling hierar- chy. By recognizing and following corridors in the environ- ment, the system travels between open spaces, or corridors using a set of simple control heuristics. Our software tool allows us to transparently control and simulate several different types of mobile robots. In addi- tion, our work entails the use of a variety of sensing modal- ities, for example, sonar, laser-range, tactile sensing [5] and video images. Furthermore, we have developed a cus- tomized video and range-sensing platform called Quadris. The Quadris sensor can be used to further refine the la- belling hypotheses generated from sonar data. 4 Long-term Development Our long term objectives involve using these tools to exam- ine questions of spatial representation and exploration. In particular, we have performed image-based positioning [2], model-based localisation and exploration [3], and topologi- cal map representation and exploration. We are also work- ing on extending this work to collaborative multi-robot exploration, with several agents performing independent exploration and fusion of spatial information [4]. References PI PI PI 141 PI Gregory Dudek and Michael Jenkin. A multi-layer dis- tributed development environment for mobile robotics. In Proceedings of the Conference on Intelligent Autonomous Systems (IAS-3), pages 542-550, Pittsburgh, PA, February 1993. 10s Press. Gregory Dudek and Chi Zhang. Vision-based robot localiza- tion without explicit object models. In Proc. International Conference of Robotics and Automation, Minneapolis, MN, 1996. IEEE Press. Paul MacKenzie and Gregory Dudek. Precise positioning using model-based maps. In Proceedings of the International Conference on Robotics and Automation, San Diego, CA, 1994. IEEE Press. Nicholas Roy and Gregory Dudek. What to do when you’re lost at the zoo: Multi-robot rendezvous in unknown envi- ronments. CIM-96-900-1, McGill University, June 1996. Nicholas Roy, Gregory Dudek, and Paul Freedman. Surface sensing and classification for efficient mobile robot naviga- tion. In Proc. IEEE International Conference on Robotics and Automation, Minneapolis, MN, April 1996. IEEE Press. 1356 AAAI-96 From: AAAI-96 Proceedings. Copyright © 1996, AAAI (www.aaai.org). All rights reserved. | 1996 | 206 |
1,851 | YODA: The Young Observant Discovery Agent Wei-Min Shen, Jafar Adibi, Bonghan Cho, Gal Kaminka, Jihie Kim, Behnam Salemi, Sheila Tejada Information Sciences Institute University of Southern California Email: shen@isi.edu The YODA project at USC/IS1 consists of a group of young researchers who share a passion for autonomous systems that can bootstrap their knowledge of real envi- ronments by exploration, experimentation, learning, and discovery. Our goal is to create a mobile agent that can autonomously learn from its environment based on its own actions, percepts, and missions [l]. The current YODA system has a Denning MRV-3 mo- bile robot and an on-board portable personal computer. The robot is a three-wheel cylindrical system with sepa- rate motors for motion and steering. It is equipped with 24 long range sonar sensors, three cameras for stereo vi- sion, a speaker for sound emission, and a voice recognition system. The communication between the robot and the control computer is accomplished through an RS232 serial port using a remote programming interface [2]. The robot is controlled by a set of commands and the sensor readings include sonar ranges, motor status, and position vectors (vision is not used in this competition). As with any real sensing device, the sensor readings from the robot are not always reliable and this poses challenges for building a ro- bust system. YODA’s software is implemented in MCL3.0 on a Mac- intosh Powerbook computer. The control architecture is designed to integrate reactive behaviors with deliberate planning. It also has facilities to accommodate learning and discovery in the future. Currently, the architecture is divided into two layers: the lower layer contains modules for navigation and reactive behaviors; and the higher layer contains modules for mission (re)planning. The lower layer is a combination of production rules and behavior-based systems [3] with each behavior represented as a set of rules. These rules are different from traditional productions in that they have associated probabilities and make predictions. We believe the probabilities are impor- tant for robust behaviors in a real environment, and the predictions (i.e., the expected consequences of the robot’s actions) are the key for self-organized learning and discov- ery. Given a set of goals delivered from the higher layer, the behavior rules will compete with each other based on the current sensor readings. Actions associated with the winning rules are then executed (these actions are in some sense collaborating with each other). This cycle of percep- tion, decision, and action repeats itself until the goals are accomplished. In the case where the goals are impossible to reach, the lower layer will pass that failure to the higher layer for replanning. To deal with imprecision in sensor readings, the system fuses information from different sensors and treat them as a whole. Decisions are never made based on any sin- gle sensor but on “patterns” of sensed information. This gives the system the ability to incorporate the idea of “sig- gives the system the ability to incorporate the idea of “sig- natures” for recognizing locations and features in the en- natures” for recognizing locations and features in the en- vironment. vironment. For example, YODA has four basic built-in For example, YODA has four basic built-in signatures based the 24 sonar sensors for this competition: signatures based the 24 sonar sensors for this competition: corridor, corner, at-a-door, and at-a-foyer. These signa- corridor, corner, at-a-door, and at-a-foyer. These signa- tures, combined with the map and the (imprecise) position tures, combined with the map and the (imprecise) position vector, greatly increase the reliability of determining the vector, greatly increase the reliability of determining the current location of the robot. To detect whether a room current location of the robot. To detect whether a room is occupied or not, the system will enter the room and ask is occupied or not, the system will enter the room and ask verbally if any person in the room would like come closer verbally if any person in the room would like come closer to the robot. If this results some changes in the sonar read- to the robot. If this results some changes in the sonar read- ings, we conclude the room is occupied. Otherwise, after ings, we conclude the room is occupied. Otherwise, after several trials of asking, we conclude the room is empty. several trials of asking, we conclude the room is empty. The movement control system of YODA is designed to The movement control system of YODA is designed to bypass any obstacle during its movement towards a goal. bypass any obstacle during its movement towards a goal. However, if an obstacle is so large that the passage to a However, if an obstacle is so large that the passage to a goal position is completely blocked, the robot will stop goal position is completely blocked, the robot will stop and wait until the obstacle moves away. Currently, we and wait until the obstacle moves away. Currently, we are implementing and testing the system on a real floor are implementing and testing the system on a real- floor in the IS1 office building, with furniture and people traffic in the IS1 office building, with furniture and people traffic present. present. The higher layer in the architecture contains a planner The higher layer in the architecture contains a planner that controls the mission-oriented and long-term behav- that controls the mission-oriented and long-term behav- iors. This pl iors. This pl anner anner uses a given map and determines a uses a given map and determines a sequence of goals that must be accomplished by the robot. sequence of goals that must be accomplished by the robot. It can also replan based on information gathered by the It can also replan based on information gathered by the lower layer (such as that a corridor is blocked or a door lower layer (such as that a corridor is blocked or a door is closed). There are two alternate criteria for finding the is closed). There are two alternate criteria for finding the best solutions: the “risky” one for finding a shortest path best solutions: the “risky” one for finding a shortest path based on the current information, and the “conservative” based on the current information, and the “conservative” one for taking into account any dynamic information that one for taking into account any dynamic information that might be collected during a plan execution. We are cur- might be collected during a plan execution. We are cur- rently investigating the trade-off between the two. rently investigating the trade-off between the two. Finally, we would like to thank Dr. Ramakant Nevatia Finally, we would like to thank Dr. Ramakant Nevatia for providing us with the Denning Robot. Special thanks for providing us with the Denning Robot. Special thanks also to the various projects and people in ISI’s Intelligent also to the various projects and people in ISI’s Intelligent Systems Division for their moral support and their tol- Systems Division for their moral support and their tol- erance for sharing space (and occasionally “forces”) with erance for sharing space (and occasionally “forces”) with YODA. YODA. eferences [l] W.M. Shen. Autonomous Learning from ihe Environ- ment. W.H. Freeman, Computer Science Press, 1994. [2] Denning Robot Manual. Denning MRV-3 Product Maunal. Denning Mobile Robotics Inc. 1989. [3] Ronald C. A k r in. Motor Schema-Based Mobile Robot Navigation. International Journal of Robotics Re- search. 1987. Robot Competition 1357 From: AAAI-96 Proceedings. Copyright © 1996, AAAI (www.aaai.org). All rights reserved. | 1996 | 207 |
1,852 | Amelia Reid Simmons Sebastian Thrun Greg Armstrong Richard Goodwin Karen Haigh Sven Koenig Shyjan Mahamud Daniel Nikovski Joseph O’Sullivan School of Computer Science Carnegie Mellon University Pittsburgh, PA 15213-3891 mahamud@cs.cmu.edu Amelia was built by Real World Interface (RWI) using Xavier-a mobile robot platform developed at CMU on a B24 base from RWI-as a prototype. Amelia has substantial engineering improvements over Xavier. Amelia is built on a B21 base. It has a top speed of 32 inches per second, while improved integral dead-reckoning insures extremely accurate drive and position controls. The battery life is six hours, and are hot-swappable. Two Pentium-100s are the main CPU’s on board with special shock-mounted hard drives. A 75Mhz 486 laptop acts as onboard console. All these are inter- connected by an internal 10M Ethernet, and to the world via a 2MBs Wavelan wireless system. Sonar and infrared sensor arrays ring the robot, mounted on Smart Panels for quick and easy access to internal com- ponents. These Smart Panels also contain bump sen- sors. A Sony color camera is mounted on a Directed Perception pan/tilt head for visual sensing. Finally, an arm can extend for 4-degree of freedom manipulation of Amelia’s world. Like Xavier, Amelia has a distributed, concurrent software system, which runs under the Linux operat- ing system. All programming is done in C, and pro- cesses communicate and are sequenced and synchro- nized viathe TASK CONTROL ARCHITECTURE (TCA) (Simmons 1995). Communication with Amelia is graphical (via the laptop), remote (via zephyr), and speech-driven. An off-board Next computer runs the SPHINX real-time, speaker-independent speech recognition system and a text-to-speech board provides speech generation. Thus, we can give verbal commands to the robot and the robot can respond verbally to indicate its status. In addition, a graphical user interface is available for giving commands and monitoring the robot’s status. Amelia plans to participate in the “Call a meeting” event. She will use the ROGUE system (Haigh & Veloso 1996) for high-level task planning through PRODIGY, a planning and learning system (Veloso et al. 1995). ROGUE is able to create a branching plan based on the observations of the real world, select appropriate order- ings of goal locations, and monitor the robot’s progress 1358 AAAI-96 towards those goals. Planning actions are defined at the granularity of “goto-location” and “observe-room”, requiring more detailed schema for execution. Navigation is accomplished using a Partially Observ- able Markov Decision Process (POMDP) model of the environment (Simmons and Koenig 1995). POMDP models allow the robot to account for actuator and sensor uncertainty and to integrate topological map in- formation with approximate metric information. They also allow the robot to recover gracefully if they are uncertain about their current location. Amelia will use vision for detecting doorways as well as for detecting faces. Doorways are detected with- out the help of markers. Detection is accomplished with a combination of region-growing, appearance- based matching and sonar range readings. Faces are detected in two stages. The first stage uses a fast color histogram technique to identify candidate regions that have appropriate sizes and shapes. The second stage consists of a computationally costlier step that verifies whether the chosen regions are face-like or not. References Simmons, R., and Koenig, S. 1995. Probabilistic Robot Navigation in Partially Observable Environ- ments. In Proceedings of the IJCAI, 1080-1087. Veloso, M.; Carbonell, J. C.; Perez, A.; Borrajo, D.; Fink, E.; and Blythe, J. 1995. Integrating Planning and Learning: The PRODIGY Architecture. Journal of Theoretical and Experimental Artificial Intelligence 7(l). Haigh, K. Z., and Veloso, M. 1996. Interleaving Plan- ning and Robot Execution for Asynchronous User Re- quests. In the Proceedings of the AAAI-96 Spring sym- posium on Planning with Incomplete Information for Robot Problems. Simmons, R. 1995. Towards Reliable Autonomous Agents. In AAAI Spring symposium on Software Ar- chitectures. From: AAAI-96 Proceedings. Copyright © 1996, AAAI (www.aaai.org). All rights reserved. | 1996 | 208 |
1,853 | A reactive mobile robot based on a formal theory of action C. Baral: L. Floriano, A. Gabaldon, D. Morales, T. Son and R. Watson Department of Computer Science University of Texas at El Paso El Paso, Texas 79968, U.S.A. chitta@cs.utep.edu One of the agenda behind research in reasoning about actions is to develop autonomous agents (robots) that can act in a dynamic world. The early attempts to use theories of reasoning about actions and planning to for- mulate a robot control architecture were not successful for several reasons: The early theories based on STRIPS and its ex- tensions allowed only observations about the initial state. A robot control architecture using these the- ories was usually of the form: (i) make observations (ii) Use the action theory to construct a plan to achieve the goal, and (iii) execute the plan. For such an architecture to work the world must be static so that it does not change during the execu- tion of the plans. This assumption is not valid for a dynamic world where other agents may change the world and/or the robot may not have all the infor- mation about the environment when it makes the plan. Moreover, planning is a time consuming activity and it is not usually wise for the robot to spend a lot of time creating a plan, especially when it is supposed to interact with the environment in real time. This led to the development of several robot control architectures that were reactive in nature and usually were based on the paradigm of ‘situated activity’ which emphasized ongoing physical interaction with the envi- ronment as the main aspect in designing autonomous agents. These approaches were quite successful, es- pecially in the domain of mobile robots. But most of them distanced themselves from the traditional ap- proach based on theories of actions. Our intention in the AAAI 96 robot contest is to use re- active rules. But, we will show that the reactive rules we use are correct w.r.t. a formal theory of action called L’. Unlike STRIPS, the language L allows spec- ification of dynamic worlds. But it makes assumptions *Support was provided by the National Science Foun- dation under grant Nr. IRI-9211662 and IRI-9501577. ‘Please see the paper by Baral, Gabaldon and Provetti in this volume and the proceedings of the AAAI 96 work- such as: we know the effect of actions, the observations (sensor data) are correct, the robot has perfect control, etc. The last two assumptions are not consistent with the real world. Nevertheless, as we explain in the succeed- ing paragraphs, our approach based on this language is appropriate. Consider a reactive rule of the form if fi, . . . , fn then a, where, fi’s are fluents (that depend on the sensor read- ings) and a is an action. A simple reactive control mod- ule may consist of a set of such rules such that at any time the ‘if’ part of only one of the rules is satisfied. A robot equipped with this control after sensing finds a rule in the module whose ‘if’ part is satisfied and performs the corresponding action. We say a reactive rule is correct w.r.t. an action theory and a goal if for any situation that is consistent with the ‘if’ part of the rule, the action in the ‘then’ part is the first action in a minimal plan that will take the robot from the current situation to a situation where the goal is satisfied. The fact that we only have the first action of the min- imal plan in the reactive rule is important. Having a complete minimal plan will not work because of the dynamic nature of the world. By having only the first action of the minimal plan we can take into account the possibility of incorrect sensors, world unpredictability and imperfect control. After the robot executes an action based on its sensing and the reactive rules, it does not rely on a model of the world, rather it senses again. Hence the assumptions in L only mean that the minimal plan works if everything is perfect for a reasonable amount of time. Based on these ideas we are currently developing reactive control programs for the AAAI 96 robot contest on a B-14 mobile robot from RWI. A de- tailed report on our approach can be found through http://cs.utep.edu/chitta/chitta.html. shop on ‘Reasoning about actions, planning and control: Bridging the gap’. 1350 AAAI-96 From: AAAI-96 Proceedings. Copyright © 1996, AAAI (www.aaai.org). All rights reserved. | 1996 | 209 |
1,854 | Coping with Temporal Constraints in Multimedia Presentation Planning Elisabeth Andr6 and Thomas Rist German Research Center for Artificial Intelligence (DFKI) Stuhlsatzenhausweg 3, D-66123 Saarbriicken Email: <name>@dfki.uni-sb.de Abstract Computer-based presentation systems enable the re- alization of effective and dynamic presentation styles that incorporate multiple media. Obvious examples are animated user interface agents which verbally com- ment on multimedia objects displayed on the screen while performing cross-media and cross-window point- ing gestures. The design of such presentations must account for the temporal coordination of media output and the agent’s behavior. In this paper we describe a new presentation system which not only creates the multimedia objects to be presented, but also generates a script for presenting the material to the user. In our system, this script is forwarded to an animated pre- sentation agent running the presentation. The paper details the kernel of the system which is a component for planning temporally coordinated multimedia. Figure 1: Verbal Annotation of Graphical Objects Introduction The success of human-human communication undis- putably depends on the rhetorical and didactical skills of the speaker or presenter. Surprisingly enough, little attention has been paid to this aspect of computer- based presentation systems. Up to now, research has mainly focused on content selection and content encod- ing. Although multimedia documents synthesized by these systems might be coherent and even tailored to a user’s specific needs, the presentation as a whole may fail because the generated material has not been pre- sented in an appealing and intelligible way. This can often be observed in cases where multimedia output is distributed on several windows requiring the user to find out herself how to navigate through the presenta- tion. To enhance the effectivity of computer-based com- munication, we propose the use of a user interface agent which, in our case, appears as an animated char- acter, the so-called PPP Persona. This character acts as a presenter, showing, explaining, and verbally com- menting on textual and graphical output on a window- based interface. The use of such an animated agent to present multimedia material provides a good means of: o establishing cross-references between presentation parts which are conveyed by different media possibly being displayed in different windows guiding a user through a presentation and thus pre- venting her from orientation and navigation prob- lems and realizing new presentation styles that are dynamic and multimodal in nature. For illustration, let’s look at an example from one of our current application domains: the generation of in- structions for the maintenance, service and repair of technical devices such as a modem. Suppose the PPP system is requested to explain the internal parts of a modem. A strategy to accomplish this task is to gen- erate a picture showing the modem’s circuit board and to introduce the names of the depicted objects. Un- like conventional static graphics where the naming is usually done by drawing text labels onto the graphics (often together with arrows pointing from the label to the object), the PPP Persona enables the realization of dynamic annotation forms as well. The system first creates a window showing the circuit board. After the window has appeared on the screen, the PPP Persona takes up a suitable position for carrying out pointing 142 Art & Entertainment From: AAAI-96 Proceedings. Copyright © 1996, AAAI (www.aaai.org). All rights reserved. gestures. It points to the single objects one after the other and (cf. the screen shot shown in Fig. 1) utters the object names verbally (using a speech synthesizer). The advantage of this method over static annotations is that the system can influence the temporal order in which the user processes an illustration. Of course, it is also possible to combine this dynamic style with the standard annotation style; i.e. the PPP Persona attaches text labels to the depicted parts before the user’s eyes. The aim of this paper is to show (a) that such dy- namic presentation styles can be handled in a common framework for describing the structure of multimedia presentations, and (b) that a plan-based approach can be used to design such presentations automatically - provided it is able to handle timing constraints. The Structure of ultimedia Presentations In our previous work, we have developed principles for describing the structure of coherent text-picture com- binations (cf. (And@ & Rist 1993)). Essentially, these principles are based on a generalization of speech act theory (Searle 1980) to the broader context of com- munication with multiple media, and an extension of RST (Rhetorical Structure Theory, (Mann & Thomp- son 1987)) t o capture relations that occur not only between presentation parts realized within a particu- lar medium but also those between parts conveyed by different media. The rhetorical structure can be rep- resented by a directed acyclic graph (DAG) in which communicative acts appear as nodes and relations be- tween acts are reflected by the graph structure. While the top of such a DAG is a more or less complex com- municative act (e.g. to introduce an object), the lowest level is formed by specifications of elementary presen- tation tasks (e.g., pointing to an object). Coping with dynamic presentation styles as illustrated in the previ- ous section requires an extension of this framework. First of all, we have to address the temporal struc- ture of a dynamic presentation as well. Like other au- thors in the Multimedia community, e.g. (Buchanan & Zellweger 1993; Hardman, Bulterman, & van Rossum 1994; Hirzalla, Falchuk, & Karmouch 1995), we start from an ordered set of discrete timepoints. The tempo- ral structure is represented by timelines which position events along a single time axis where, in our case, an event corresponds to the start or the end of a commu- nicative act. Unlike systems where a human author has to spec- ify the multimedia material and the timing constraints, we are concerned with the automated generation of the presentation parts, too. Therefore, we refine the no- tion of communicative act by explicitly distinguishing between production and presentation 0cts.l Whereas Introduce Production Act Presentation Act “;;“Pgf$ S-Name 5-Name e Figure 2: Rhetorical Structure production acts refer to the creation of material, pre- sentation acts are display acts, such as S-Display-Text, or acts which are carried out by the PPP Persona, e.g. S-Point. Let’s consider the example shown in Fig. 1. The multimedia material was created by performing the following production acts: prod-act-l: Create a window containing a depiction of the circuit board prod-act-2: Create a specification for a pointing gesture referring to the transformer prod-act-3: Produce the sentence “This is the transformer” s . . To present the outcome of these production following presentation acts were carried out: acts, the pres-act-l: Expose the created window pres-act-2: Walk to the window pres-act-3: Point to the transformer pres-act-4: Say: “This is the transformer.” . . . pres-act-13: Wait a moment Fig. 2 exhibits the rhetorical structure of the sam- ple present ation. The presentation as a whole serves to introduce the circuit board of a modem. It con- sists of the presentation acts S-Show, S-Position and S-Wait and the complex communicative acts Create- Graphics and Elaborate-Parts. Create-Graphics is com- posed of two production acts (S-Create- Window and S-Depict). Elaborate-Parts is defined by several label- ing acts, which in turn are composed of two production acts (S-Name and S-Specify-Gesture) and two presen- tation acts (S-Speak and S-Point). Now let’s turn to the temporal structure of a multi- media presentation. Of course, to run the presentation in an intelligible way, the presentation acts need to be temporally coordinated. For example, pointing to an object depiction requires the exposure of the win- dow containing the depiction. In addition, the pointing gesture should be maintained while the name of the respective object is uttered. Concerning the tempo- ral relation between presentation acts and production acts, it is clear, that the presentation of material can- not start before its production. However, sometimes ‘Note that this distinction provides a further dimen- sion for characterizing communicative acts which has not been considered in previous classifications, 1993b). e.g. (Maybury Entertainment Introduce Create-Graph& S-Create-Window S-Depict S-Show S-Walt S-Position Elaborate-Parts Label .S;S ae;;v-Gesture If ggg” . . . Label .SS;me;y-Gesture “,%k Figure 3: Temporal Structure material is generated incrementally so that the presen- tation may start before the production is completed. For some purposes, it can also be reasonable to spec- ify the temporal order in which a set of production acts should be carried out. For example, when gener- ating a cross-media reference such as with prod-act-& it wouldn’t make much sense to start before the loca- tion (relative to the coordinate system of the display window) of the object depiction is known. To sum up, there are usually many temporal con- straints that must be satisfied by the communicative acts to produce and run a multimedia presentation. Fig. 3 shows a schedule for the communicative acts listed above. The durations of complex acts correspond to the length of the white bars, grey bars refer to du- rations of elementary acts. While it is easy to characterize the temporal rela- tions between communicative acts of a given presenta- tion, it is much harder to do the same during the plan- ning phase. The reason is that the temporal behavior of acts may be unpredictable. Among other things, it may depend on: Resource limitations of the computing environment For example, communicative acts such as Create- Graphics often involve the execution of programs with unpredictable runtimes. This may be aggra- vated by other factors such as network capacity and workload. The temporal behavior of other acts For instance, since we don’t know when the graphics generation process will be finished, the startpoint of S-Show, which immediately comes after Create- Graphics, is unknown as well. The current state of the presentation system In PPP, the user can alter the system state at any time, e.g. by having the PPP Persona move to an arbitrary position on the screen. Thus, we cannot anticipate where the Persona will stand at presenta- tion time, This makes it impossible to predict how long the Persona needs to walk to an appropriate position. 144 Art & Entertainment Figure 4: The PPP System In the next three sections, we will describe how presentation schedules can be automatically built up starting from possibly incomplete timing information. A View on PPP’S Architecture PPP’s major components are: a presentation planner, medium-specific generators, currently for graphics (cf. (Rist & Andre 1992)), text (cf. (Kilger 1994)) and gestures, the Persona Server and a constraint-based layout manager (cf. (Graf & Neurohr 1995)). The presentation planner (cf. (And& & Rist 1993)) is re- sponsible for determining the contents of multimedia material, selecting an appropriate medium combina- tion and designing a script for presenting the material. Elementary production acts are sent to the correspond- ing generators which, in turn, inform the presentation planner when they have accomplished their task and how they have encoded a certain piece of information. The results of the generators are taken as input for the design of the presentation script which is forwarded to the display components for execution. The task of the layout manager is the determination of effec- tive screen layouts and the maintenance of user inter- actions. The Persona Server (cf. (Andre, Miiller, & Rist 1996)) carries out Persona actions which, among other things, includes assembling appropriate anima- tion sequences. Both display components signal when they have accomplished their tasks and inform the pre- sentation planner about the occurrence of interaction events, such as mouse-clicks on windows. Representation of Design Knowledge In order to build up multimedia presentations, we have defined a set of presentation strategies that can be se- lected and combined according to a particular task. These strategies reflect general design knowledge or they embody more specific knowledge of how to present a certain subject. They are characterized by a header, a set of applicability conditions, a collection of inferior acts2, a list of qualitative and metric temporal con- straints and a start and an end interval. The header corresponds to a complex presentation act. The appli- cability conditions specify when a strategy may be used *For the sake of simplicity, we don’t distinguish between main and subsidiary acts as we did in our previous work. and constrain the variables to be instantiated. The in- ferior acts provide a decomposition of the header into more elementary presentation acts. Qualitative tem- poral constraints are represented in an “Allen-style” fashion which allows for the specification of thirteen temporal relationships between two named intervals: before, meets, overlaps, during, starts, finishes, equal and inverses of the first six relationships (cf. (Allen 1983)). Allen’s representation also permits the ex- pression of disjunctions, such as (A (before aper) I?), which means that A occurs before or after B. Metric constraints appear as difference (in)equalities on the endpoints of named intervals. They can constrain the duration of an interval (e.g., (10 5 Dzlr A2 5 do)), the elapsed time between intervals (e.g., (4 < End Al Start A2 < 6)) and the endpoints of an interval (e.g., (Start A2 < 6)). Examples of presentation strategies are listed below. The first strategy may be used to build up the presen- tation shown in Fig. 1. It only applies if the system believes that Zobject is a physical object. Besides acts for the creation of graphics and natural language ex- pressions, the strategy also comprises presentation acts to be executed by the PPP Persona, such as (S-Show, S-Position and S- Wait). Note that we are not forced to completely specify the temporal behavior of all pro- duction and presentation acts at definition time. This enables us to handle acts with unpredictable durations, start and endpoints, i.e. acts whose temporal behavior can only be determined by executing them. For exam- ple, in (Sl) we only specify a minimal duration for act A2 and a fixed duration for act A4. (Sl) Header: (Introduce S U ?object ?window) Applicability Conditions: (1313fii(zA ?object Physical-Object)) ((Al (Create-Graphics S U ?object ?window)) (A2 (S-Show S U ?window)) (A3 (S-Position S U)) (A4 (S-Wait S U)) (A5 (Elaborate-Parts S U ?object ?window))) Qualitative: ((Al (meets) A2) (A3 (starts) A2) (A3 (meets) A5) (A5 (meets) A4) (A4 (finishes) A2)) Metric: ((10 5 Dur A2) (2 5 Dur A4 5 2)) Start: Al Finish: A2 To enable iteration over finite domains, we rely on a Forall construct, which is used in Strategy (S2) to indicate for all parts Zpart of an object aobject one after the other to which class they belong. (S2) Header: (Elaborate-Parts S U ?object ?window) Applicability Conditions: (Be1 S (Encodes ?pic-part ?part ?window)) Inferiors: ((Al (Forall ?part With (*And* (Be1 S (Part-of ?part ?object)) (Be1 S (I-ISA ?part ?cIass))) Do Sequentially (Ai (Label S U ?part ?class ?window))))) (S-Show S U 7wlndow) (S-Depict S U circuit-boor&l ?wlndow) Figure 5: Propagating Multimedia Objects lanning of Communicative Acts To produce and present multimedia material, the strategies introduced above are considered operators of a planning system (cf. (Andre 8z Rist 1993)). Start- ing from a presentation goal, the presentation planner searches for plan operators and builds up a refinement- style plan in the form of a DAG. In the example shown Fig. 1, the system had to accomplish the task: (In- troduce S U circuit-board-l ?window), selected strat- egy (Sl) and set up the following subgoals: (Create- Graphics S U circuit-board-l Pwindow), (S-Show S U awindow), (S-Position S U), (Elaborate-Parts S U circuit-board-l ?window) and (S-Wait S U)* Whereas S-Show, S-Position, and S-Wait are ele- mentary acts, Create-Graphics and Elaborate-Parts are further expanded by the presentation planner. The refinement of Elaborate-Parts results in several point- ing gestures and speech acts. The speech acts are for- warded to the text generator which generates a name for each object to be introduced. The textual output is uttered using a speech synthesizer. Gestures are speci- fied by the gesture generator which determines the ges- ture type (in our case pointing with a stick) and the exact coordinates. The expansion of Create-Graphics leads to a call of the graphics generator, and the cre- ation of a window in which the resulting depiction of the modem’s circuit board will appear. During the planning process, multimedia objects are built up by performing production acts. These multi- media objects are bound to variables which are prop- agated in the DAG. In Fig. 5, the variable Zwindow is instantiated with a window structure by S-Create- Window. Performing the act S-Depict causes an up- date of the data strucure bound to ?window. The propagation mechanism ensures that the new value is accessible when executing the act S-Show which leads to the exposure of the window. To enable the process- ing of temporal constraints, we have combined PPP’s presentation planner PREPLAN with RAT (cf. Fig. 6). RAT is a system for representing and reasoning about actions and plans, which relies on an extended version of Kautz’s and Ladkin’s MATS system (Kautz & Ladkin 1991). F or each node of the presentation plan, the planner creates a local constraint network which includes the temporal constraints of the corre- sponding plan operators. During the planning process, Entertainment 145 Plan Nodes with Links to Local Temporal Constraint Networks Figure 6: Combining the Presentation Planner with a Temporal Description Logics RAT checks these local constraint networks for consis- tency and computes nu merit ranges on each endpoint and difference of endpoints and possible Allen relation- ships between each pair of intervals. In case of local inconsistencies, another presentation strategy is tried out. After the completion of the presentation planning process, a global temporal constraint network is built up by propagating the constraints associated with each planning node top-down and bottom-up in the DAG. If no global consistent temporal network can be built up, the presentation fails. Finally, a schedule is built up by resolving all disjoint temporal relationships be- tween intervals and computing a total temporal order. For example, the following schedules would be created for a network containing <he constraints (A (before af- ter) B), (A (equals) C), 1 5 Dur A < 1 and 1 5 Dur B 5 1: As mentioned earlier, the planning is aggravated by acts with an unpredictable temporal behavior. There- fore, RAT only builds up a partial schedule which has to be refined when running the presentation. That is for some communicative acts, RAT only indicates an interval within which they may start or end instead of prescribing an exact timepoint. The temporal behavior of a presentation is controlled by a presentation clock which is set to 0 when the sys- tem starts to show the planned material to the user and incremented by the length of one time unit3 un- til the presentation stops. For each timepoint, RAT indicates which events must or can take place. For instance, a communicative act whose starting point is between 0 and 2, may start at timepoint 0 or 1, but must start at timepoint 2 in case it has not yet started earlier. Whether the event actually takes place or not is decided by the PPP Persona. Currently, the Per- sona chooses the earliest possible timepoint. In order by 146 3The length of a time unit can be interactively the user. Art & Entertainment changed to satisfy the temporal constraints set up by RAT, the Persona may have to shorten a presentation, to skip parts of it or to make a pause. In some cases, this may lead to suboptimal presentations, e.g. if the Persona stops speaking in the midst of a sentence. As soon as the Persona has determined that a certain event should take place, RAT is notified of this decision because it may have influence on further events. RAT adds a new metric constraint to the global temporal constraint net- work and refines the schedule accordingly. Let’s assume that RAT informs the Persona that the creation of the window may be finished at timepoint 1 and that the Persona may show the window to the user. However, it turns out that 10 seconds4 are re- quired for the creation of the window. The Persona forwards this information to RAT, which adds 10 5 Dur Create-Window < 10 to the global temporal con- straint network. Since Create-Window meets S-Show, the display of the window can start only at timepoint 10. Efforts to develop time models for multimedia doc- uments have been made by (Buchanan 8z Zellweger 1993; Hardman, Bulterman, & van Rossum 1994; Hirzalla, Falchuk, & Karmouch 1995). But, in all ap- proaches the editing of a multimedia document is car- ried out by a human author who also has to specify the desired temporal relationships between the single document segments from which a consistent schedule is computed. Since there is no explicit representation of the contents of a document, it’s not possible to auto- matically determine a high-level temporal specification for a document. In contrast to this, our system is not only able to design multimedia material, but also plans presentation acts and their temporal coordination. A first attempt ot incorporate time into an auto- mated presentation system has been made by Feiner and colleagues (cf. (Feiner et al. 1993)). However, they only investigate how temporal information can be conveyed by dynamic media and don’t present a mechanism for synchronizing them. A second research area which is of interest for our work is the creation of lifelike characters (see e.g. (Takeuchi & Nagao 1993; Badler, Phillips, & Web- ber 1993; Kurlander & Ling 1995; Lashkari, Metral, & Maes 1994)). Th e work closest to our own is that being carried out by Microsoft Research in the Per- sona project (cf. (Kurlander & Ling 1995)). In the current prototype system, a parrot called Peedy acts as a conversational assistant who accepts user requests for audio CDs. In contrast to PPP, the presentation of material in their system is restricted to playing the selected CDs. 4Since we rely on a time model with discrete the actually needed time has to be rounded. timepoints, Conclusion In this paper, we have presented a plan-based approach for the automatic creation of dynamic multimedia pre- sentations. The novelty of PPP is that it not only de- signs multimedia material, but also plans presentation acts and their temporal coordination. This has been achieved by combining a presentation planning compo- nent with a module for temporal reasoning. To cope with unpredictable temporal behavior, we first build up a partial schedule while planning the contents and the form of a presentation which is refined when run- ning it. Our approach is particularly suitable for planning the behavior of animated user interface agents which have the potential of becoming integral parts of future intelligent presentation systems. However, it is not re- stricted to this class of applications. For instance, it can also be used for timing the display of static graph- ics and written text. In all existing presentation sys- tems, document parts are either shown to the user im- mediately after their production (incremental mode) or the systems wait until the production process is com- pleted and then present all the material at once (batch- mode). However, these systems are not able to flexibly change the presentation order depending on the cur- rent situation. In contrast, our approach makes it pos- sible to influence the order and the speed in which a user processes a document by explicitly specifying the time at which information should be shown. Further- more, multimedia material along with timing informa- tion specified by PPP can be used as input for existing presentation engines, such as the CMIFed environment (Hardman, Bulterman, & van Rossum 1994). Future work will concentrate on more complex user interactions. Currently, the system clock is stopped when the user interrupts a presentation and started again when he resumes it. However, we also want to explicitly represent temporal relationships between presentation acts and interaction acts. For example, clicking on menu items is only possible as long as the menu is visible. Acknowledgments This work has been supported by the BMBF under grant ITW 9400. We would like to thank H.-J. Prof- itlich and M. Metzler for the development of RAT and 9. Miiller for his work on the Persona server and the overall system integration. References Allen, 9. F. 1983. Maintaining Knowledge about Temporal Intervals. Communications of the ACM 26(11):832-843. Maybury, M., ed. 1993a. Intelligent Multimedia In- terfaces. AAAI Press. Maybury, M. T. 1993b. Planning Multimedia Ex- planations Using Communicative Acts. In Maybury (1993a). 59-74. Rist, T., and Andre, E. 1992. Incorporating Graph- ics Design and Realization into the Multimodal Pre- sentation System WIP. In Costabile, M. F.; Catarci, T.; and Levialdi, S., eds., Advanced Visual Interfaces. London: World Scientific Press. 193-207. Searle, J. 1980. Speech Acts: An Essay in the Philos- ophy of Language. Cambridge, England: Cambridge University Press. Andre, E., and Rist, T. 1993. The Design of Illus- Takeuchi, A., and Nagao, K. 1993. Communicative trated Documents as a Planning Task. In Maybury facial displays as a new conversational modality. In (1993a). 94-116. Proc. of ACM/IFIP IIVTERCHI’93, 187-193. Andre, E.; Miiller, J.; and Rist, T. 1996. The PPP Persona: A Multipurpose Animated Presenta- tion Agent. In Advanced Visual Interfaces. ACM Press. Badler, N.; Phillips, C.; and Webber, B. 1993. Simu- lating Humans: Computer Graphics, Animation and Control. New York, Oxford: Oxford University Press. Buchanan, M., and Zellweger, P. 1993. Automatically Generating Consistent Schedules for Multimedia Doc- uments. Multimedia Systems 1:55-67. Feiner, S. K.; Litman, D. 9.; McKeown, K. R.; and Passonneau, R. J. 1993. Towards Coordinated Tem- poral Multimedia Presentations. In Maybury (1993a). 139-147. Graf, W. H., and Neurohr, S. 1995. Constraint-based layout in visual program design. In Proc. of the 1 lth International IEEE Symposium on Visual Languages. Hardman, L.; Bulterman, D.; and van Rossum, G. 1994. The Amsterdam Hypermedia Model: Adding Time and Context to the Dexter Model. Communi- cations of the ACM 37(2):50-62. Hirzalla, N.; Falchuk, B.; and Karmouch, A. 1995. A Temporal Model for Interactive Multimedia Scenar- ios. IEEE Multimedia 2(3):24-31. Kautz, H. A., and Ladkin, P. B. 1991. Integrating metric and qualitative temporal reasoning. In Proc. of AAAI-91, 241-246. Kilger, A. 1994. Using UTAGs for Incremental and Parallel Generation. Computational Intelligence 10(4):591-603. Kurlander, D., and Ling, D. 1995. Planning-Based Control of Interface Animation. In Proc. of CHI’95, 472-479. Lashkari, Y.; Metral, M.; and Maes, P. 1994. Collabo- rative interface agents. In Proc. of AAAI-94,444-449. Mann, W. C., and Thompson, S. A. 1987. Rhetorical Structure Theory: A Theory of Text Organization. Report ISI/RS-87-190, Univ. of Southern California, Marina de1 Rey, CA. Entertainment 147 | 1996 | 21 |
1,855 | CoMRoS: Coo erative MO obots Stuttgar Thomas Brkml, Martin Kalbacher, Paul Levi, tinter Mamier Universitat Stuttgart, IPVR Applied Computer Science - Computer Vision, Prof. Levi Breitwiesenstr. 20-22, D-70565 Stuttgart, Germany http://www.informatik.uni-stuttgart.de/ipvr/bv/comros Project CoMRoS has the goal to develop intelligent coop- erating mobile robots. Several different vehicles are to solve a single task autonomously by exchanging plans without a central control (Levi, Braunl, Muscholl, Rausch. 1994). We use “ II” vehicles from Robosoft France, adapted to our needs. The standard vehicle has very little local intelligence (VME bus system) and is controlled re- motely by wireless Ethernet for sending steering com- mands and receiving sonar sensor data. A wireless video link is used to transmit camera images. Data exchange be- tween vehicles is then performed among the corresponding workstations. The remote control is basically used to sim- plify testing and debugging of robot programs. However, each vehicle can also be driven completely autonomous by using a laptop PC. (Bayer, Braunl, Rausch, Sommerau, Levi 1995). Figure: The three CoMRoS “musketeers” On the PC, we are using the Linux version of Unix as operating system, while the robot itself has a real-time op- erating system (Albatros or D Oberon). Quite a number of libraries for Ethernet connection, polling sonar sensor data, and digitizing video frames have been implemented by members of our group for various tasks on various ma- chines, including our massively parallel MasPar MP- 1216 system. We also implemented a physically-based simula- tion and animation system for our vehicles (Stolz, Braunl, Levi 1995). For the robot competition tasks, we configured one of our robots completely autonomous. For event 1, “ call a meeting”, we will almost exclusively rely on a belt of 24 sonar sensors around the robot, to detect walls, doors, hallways, and obstacles. However, it is planned to use a simple vision difference algorithm to determine whether a conference room is empty by asking any people inside the room to wave their hands. An alternative being investigated is implementing voice input. The Floyd algorithm is used to determine shortest driving paths. Using standard sound tools, the robot will tell the audience about its actions and plans. For event 2, “ up the tennis court”, we first intended to use a robot with an on-board manipulator. However, this approach required a lot of time consuming image process- ing and manipulator control. Then, we took on an approach similar to a harvester machine. We developed a rotating cylinder to be mounted in front of the vehicle, in order to pick up any balls in the drive path of the robot. With this device, the ball collecting task is reduced to an area filling algorithm. Collected balls are being unloaded by reversing the rotation direction of the harvester. Only in a subsequent step will image processing be used to look for balls missed (e.g. the “ ball”). The vehicle will then drive di- rectly towards remaining balls and pick them up. Acknowledgments The authors would like to thank all assistants and students participating in the CoMRoS project, especially Alexander Rausch, Norbert Oswald, Michael Vogt, Marco Sommerau, Niels Mache, Hans-Georg Filipp, Ralf Taugerbeck, Frank Doberenz, Wolfgang Hersmann, and Normann Ness. References Levi, Braunl, Muscholl, Rausch. Architektur der Koopera- tiven Mobilen Robotersysteme Stuttgart. In Levi, Braunl (Eds.) 10. AMS, Stuttgart, Oct. 1994, pp. 262-273 (12) Bayer, Braunl, Rausch, Sommerau, Levi. Autonomous Ve- hicle Control by Remote Computer Systems. In Proceed- ings of the 4th Intl. Conf. on Intelligent Autonomous Systems, IAS-4, Karlsruhe, March 1995, pp. 158-165 (8) Stolz, Braunl, Levi. A Mobile Robot Simulation System. In Proc. ISATA Intl. Symposium on Automotive Technology & Automation, Boblingen, Sep. 1995, pp. 377-382 (6) Robot Competition 1351 From: AAAI-96 Proceedings. Copyright © 1996, AAAI (www.aaai.org). All rights reserved. | 1996 | 210 |
1,856 | McMaster University’s Artificial Computing System Andrew Dawes, Mark Bentley McMaster University Department of Engineering Physics 1280 Main St. W Hamilton, Ontario, Canada LSS 4L7 ~93 125 19@muss.cis.mcmaster.ca Introduction This will be McMaster University’s first entry into the AAAI Mobile Robotics competition. As such, this year will serve as a testing ground for future developments. It is the goal of the designers to experiment with new techniques and approaches based on their engineering background. Project Objectives The developers of this robot intend to simplify the complex task of making a robot intelligent by classifying the types of problems faced by the robot. In that way, the robot can determine the best course of action depending upon the task required. Obviously, a lot of time will be spent determining the key factors necessary to identify each type of problem. To simplify the task of classifying the problems faced by the robot, we intend to use our engineering physics background to develop an approach. By understanding the key underlying physical factors specific to a situation, and using these to develop a solution, the amount of processing required of the system will be greatly reduced. Only essential data will be collected, with the actual system determining what data and how much data is needed to solve the problem. In keeping a general approach to problem solving, the robot will need to identify situations that have been previously encountered. The key to identifying similar problems is collecting the proper data to determine the robot’s situation. For example, finding four solid objects enclosing the robot could be interpreted as a room. What to do in this room would then be the new problem faced by the robot. The initial task of finding the four walls would be simplified by first getting the correct data to identify the objects as walls and not obstacles to be avoided. We hope to avoid relying on properties specific only to one goal to keep the algorithms general. 1352 AAAI-96 Complex tasks would then be accomplished by separating them into components that can be dealt with. Only then will specific properties of the task be used to allow the system to make assumptions in determining solutions to certain situations. System Components In keeping with the theme of dividing the programming task into manageable pieces, the hardware will follow the same design philosophy. Specific computations will be carried out in certain areas of the system. The main processor, where the decision making is carried out, will communicate with the sub-processors to request and receive data. The central intelligence of the system will run on a standard desktop computer, on board the robot platform. Other tasks such as data collection and motion control will be accomplished through use of embedded controllers. Existing technology will be used for the sensors of the system. Specific components used for data collection will include ultrasonic ranging for quick obstacle avoidance. As well, laser ranging and scanning will be used for acquiring positional data and information about surroundings. The intelligence of the robot will. be distributed throughout the system to minimize the load on each processor. The sub-processors will then notify the main system that important information is available. In this way only when new information is ready will the main processor look for the data. This is done to mimic the brain, or the main processor collecting information from the senses, or sub-processors. Acknowledgements This project is being supported by the Engineering Physics Department at McMaster University. It is through the support and dedication of its professors and staff that this project will be a success. From: AAAI-96 Proceedings. Copyright © 1996, AAAI (www.aaai.org). All rights reserved. | 1996 | 211 |
1,857 | Botond Virginas University of Portsmouth, Department of Information Science Locksway Road, Southsea PO4 8.TF Hampshire, United Kingdom botond@sis.port.ac.uk Introduction Several accidents during the last decade have emphasized the need to properly analyse and manage the low probability high consequence risks associated with plant operations in the nuclear, chemical and other industries. Formal risk assessment methods are vital tools for this task, and include a number of qualitative and quantitative techniques. The increasing need to perform risk analyses induces a growing interest in standardised analysis procedures and corresponding computerised supporting tools. The amount and type of information handled in risk assessment calls for the application of knowledge-based systems. The research investigates how knowledge-based systems techniques can be used in conjunction with conventional risk assessment methods in order to develop a Knowledge-Based Risk Assessor (KoBRA) supporting the various phases of risk assessment. ‘The KoBRA environment takes the form of a toolkit containing a variety of tools performing the different tasks involved in risk assessment. The following research area problems have been iden tificd: A safety oriented process model needs to be developed for a given type of process within a certain application domain. Different knowledge representation formalisms have to be investigated. Different reasoning strategies have to be explored for the construction of different logic models. A mixed programming language approach will be examined. A platform concept will be analysed. The research concentrates mainly on: Review of the risk assessment field Evaluation of the existing computer support for risk assessment Design of a risk assessment framework within an overall risk management strategy Design of an integrated set of computer aided risk assessment tools Implementation and testing of KoBRA Review of the work completed A general review of the risk assessment field has been completed. In parallel, computerised tools supporting various phases of the risk assessment process have been explored It has been decided that an application domain from the-chemical process industry will be chosen and the toolkit will be tested on several particular processes within that domain. The design of the risk assessment framework has been completed. A substantial part of KoBRA has been implemented on a Sun SPARC workstation. The different risk assessment tools are being developed and tested incrementally. A mixed language programming environment (POPLOG) is used for the implementation. As the development of the different tools from KoBRA continues, so a safety oriented process model is being built incrementally. This model takes the form of an object oriented database with functional and structural links between the objects. Generic objects describe what is known in general about a particular type of system or component. The user builds his process description with process specific instantiations of these objects. The objects are represented with a hybrid formalism using rules and frames. The different tools in the toolkit perform the v<arious risk assessment tasks on this model. A platform concept will be used for testing KoBRA . A pilot system will be built for one chosen process maintaining a distinction between process-specific and process-independent (but still domain-specific) knowledge. The process-specific knowledge will then be removed from the pilot system leaving the knowledge that will form the basis of the platform. The platform will then be used to build a subsequent pilot system for another process. References 1. Center for Chemical Process Safety 1989. Guidelines .for Chemicd Process Qucrntitcrtive Risk Andy.&. New York, N.Y.: American Institute of Chemical Engineers. 2. Apostolakis, G.E. ed. 1990. The Role and Use of Personal Computers in Probabilistic Safety Assessment and Decision Making. Relicrhility Enginc~ering X System Sqfety 30. Elsevier Applied Science. 3. Poucet, A. 1992. Knowledge Based Systems for Risk Assessment and Monitoring. In Computer Applications in Ergonomics, Occupational Safety and Health, 29-36. Elsevier Science Publishers B .V. 4. Barett, R.; Ramsay, A.; and Sloman, A. 1986. Popll: A Pructicul Lungutrge ji,r Art@kid Intelligence. Chichester, England: Ellis Hot-wood Limited. 1374 SIGART/AAAI From: AAAI-96 Proceedings. Copyright © 1996, AAAI (www.aaai.org). All rights reserved. | 1996 | 212 |
1,858 | Selection of Passages for Information Reduction * Jody J. Daniels Department of Computer Science University of Massachusetts Amherst, MA 0 1003 USA Email: daniels@cs.umass.edu There currently exists a bottleneck in extracting informa- tion from pre-existing texts to generate a symbolic represen- tation of the text that can be used by a case-based reasoning (CBR) system. Symbolic case representations are used in legal and medical domains among others. Finding similar cases in the legal domain is crucial because of the impor- tance precedents play when arguing a case. Further, by examining the features and decisions of previous cases, an advocate or judge can decide how to handle a current prob- lem. In the medical domain, remembering or finding cases similar to the current patient’s may be key to making a cor- rect diagnosis: they may provide insight as to how an illness should be treated or which treatments may prove to be the most effective. This thesis demonstrates methods of locating, automati- cally and quickly, those textual passages that relate to pre- defined important features contained in previously unseen texts. The important features are those defined for use by a CBR system as slots and fillers and constitute the fratne- based representation of a text or case. Broadly, we use a set of textual “annotations” associated with each slot to generate an information retrieval (IR) query. Each query is aimed at locating the set of passages most likely to contain information about the slot under consideration. Currently, a user must read through many pages of text in order to find fillers for all the slots in a case-frame. This is a huge manual undertaking, particularly when there are fifty or more texts. Unfortunately, full-text understanding is not yet feasible as an alternative and information extract techniques themselves rely on large numbers of training texts with manually encoded answer keys. By locating and presenting relevant passages to the user, we will have significantly reduced the time and effort expenditure. Alter- natively, we could save an automated information extraction system from processing an entire text by focusing the sys- tem on those portions of the text most IikeIy to contain the desired information. This work integrates a case-based reasoner with an IR engine to reduce the information bottleneck. SPIRE [Se- *This research was supported by NSFGrant no. EEC-9209623, State/Industry/University Cooperative Research on Intelligent In- formation Retrieval, Digital Equipment Corporation and the Na- tional Center for Automated Information Research. 1360 SIGARTMAAI lection of Passages for Information REductionI works as follows: the CBR system evaluates its case-base relative to a current problem situation. It passes along to the IR engine the identifiers of the documents that describe fact situations the most similar to the current problem. The IR engine treats these documents as though they were hand-marked as relevant and uses them to generate a query against a larger corpus of texts (Daniels and Rissland 1995). After retrieving additional relevant texts, we might wish to add them to the CBR system’s knowledge base. However, the documents must be converted from their original text into a frame-based representation, a time-consuming and error-prone activity. To assist the knowledge engineer, we save a set of “annotations”, which we derive when creating the original case-base. An annotation contains the words and phrases that describe the value of a particular slot filler and the annotation is associated with its respective slot. An annotation may be a segment of a sentence, an entire sentence, or several sentences. For example, for the slot that contains the value of someone’s monthly-income, sample annotations from SPIRE’s case-base are: “net disposable monthly income for 1979 averaged $1,624.82” and “His cut-rent gross income is $24,000 per year.” SPIRE passes the case-base of annotations for each slot to the IR system. Using these annotations, the IR component generates a new query aimed at retrieving small relevant passages from the documents just retrieved. By combining into a query those descriptive terms and phrases used to identify the slot fillers within the current case-base, we can locate relevant passages within novel texts. By retrieving passages for display to the user, we have winnowed a text down to several sets of sentences. This process is repeated for each slot in the case-based reasoner’s representation of the problem. By locating and displaying these important passages to a user, we have reduced reading an entire document to examining several sets of sentences, resulting in a tremendous savings in time and effort. References Jody J. Daniels and EdwinaL. Rissland. A Case-Based Ap- proach to Intelligent Information Retrieval. Inf roceedings of the 18th Annual International ACM/SIGIR Conference, pages 238-245, Seattle, WA, July 1995. ACM. From: AAAI-96 Proceedings. Copyright © 1996, AAAI (www.aaai.org). All rights reserved. | 1996 | 213 |
1,859 | Towards a Unified A roach to Come Pedro Domingos* Department of Information and Computer Science University of California, Irvine Irvine, California 92717, U.S.A. pedrod@ics.uci.edu http://www.ics.uci.edu/“pedrod Rule induction (either directly or by means of deci- sion trees) and case-based learning (forms of which are also known as instance-based, memory-based and nearest-neighbor learning) arguably constitute the two leading symbolic approaches to concept and classifica- tion learning. Rule-based methods discard the indi- vidual training examples, and remember only abstrac- tions formed from them. At performance time, rules are applied by logical match (i.e., only rules whose pre- conditions are satisfied by an example are applied to it). Case-based methods explicitly memorize some or all of the examples; they avoid forming abstractions, and instead invest more effort at performance time in finding the most similar cases to the target one. There has been much debate over which of these two approaches is preferable. While each one can be extended to fit the results originally presented as evi- dence for the other, it typically does so at the cost of a more complex, less parsimonious model. In classifi- cation applications, each approach has been observed to outperform the other in some, but not all, domains. In recent years, multistrategy learning has become a major focus of research within machine learning. Its main insight is that a combination of learning paradigms is often preferable to any single one. How- ever, a multistrategy learning system typically oper- ates by calling the individual approaches as subproce- dures from a control module of variable sophistication, and again this is not completely satisfactory from the point of view of parsimony. In my thesis work I argue that rule induction and case-based learning have much more in common than a superficial examination reveals, and can be unified into a single, simple and coherent model of symbolic learning. The proposed unification rests on two key observations. One is that a case can be regarded as a maximally specific rule (i.e., a rule whose precondi- tions are satisfied by exactly one case). Therefore, no syntactic distinction need be made between the two. The second observation is that rules can be matched approximately, as cases are in a case-based classifier ( i.e., a rule can match an example if it is the closest one to it according to some similarity-computing pro- cedure, even if the example does not logically satisfy *Partly supported by a PRAXIS XXI scholarship. all of the rule’s preconditions). A rule’s extension, like a case’s, then becomes the set of examples that it is the most similar rule to, and thus there is also no necessary semantic distinction between a rule and a case. The RISE algorithm is a practical, computationally efficient realization of this idea. RISE starts with a rule base that is simply the case base itself, and gradually generalizes each rule to cover neighboring cases, as long as this does not increase the rule base’s error rate on the known cases. If no generalizations are performed, RISE acts as a pure case-based learner. If all cases are generalized and the resulting set of rules covers all regions of the instance space that have nonzero proba- bility, it acts as a pure rule inducer. More generally, it will produce rules along a wide spectrum of generality; sometimes a rule that is logically satisfied by the target case will be applied, and in other cases an approximate match will be used. This unified model is more elegant and parsimonious than a subprocedure-style combina- tion. Experiments with a large number of benchmark classification problems have also shown it to consis- tently outperform either of the component approaches alone, and lesion studies and experiments on artificial domains have confirmed that its power derives from its ability to simultaneously harness the strengths of both components (Domingos 1996). The remaining proposed thesis work will focus on further elucidating what the bias of RISE is compared to that of its parent paradigms (and thus determining when it will be the more appropriate algorithm to use), on applying the ideas contained in RISE to problems like context dependency, feature selection, fragmenta- tion avoidance, and data mining, and on bringing the use of domain knowledge into the RISE framework. eferences Domingos, P. 1996. Unifying Instance-Based and Rule- Based Induction. Machine Learning. Forthcoming. Doctoral Consortium Abstracts From: AAAI-96 Proceedings. Copyright © 1996, AAAI (www.aaai.org). All rights reserved. | 1996 | 214 |
1,860 | A Computational Theory of T’urn-taking Toby Donaldson University of Waterloo Department of Computer Science Waterloo, Ontario, Canada tjdonald@neumann.uwaterloo.ca A Turn-taking Framework My research is concerned with the problem of turn- taking in discourse, especially as applied to intelligent interfaces, such as advice-giving systems or software help systems. A limitation of many discourse systems is their need for explicit turn-ending signals (e.g. press- ing a return key). In such systems, mid-turn inter- ruptions are impossible, although there are practical examples of where mid-turn interruptions are highly desirable. For example, an interface agent should promptly inform the user of important pieces of in- formation, such as a lack of disk space or the loss of a network connection, especially if the user is enaged in some activity that relies on that information Interruptions are a particularly useful instance of turn-taking, and we have outlined a general three-part goal-oriented model of turn-taking: Motivation An agent must first have some motivat- ing reason to take a turn. Motivations include, for example, recognition of an inconsistency in the be- liefs of the speaker, or a desire for plan clarification; Goal Adoption It is often inappropriate to take a turn the moment you have something to say - you should wait until the other person has finished speaking. Thus, motivations trigger the adoption of turn-taking goals; Turn Execution Conversants typically coordinate turn-taking by giving and receiving various vocal and semantic signals. Par example, decreased speak- ing volume can indicate that .a speaker is willing to give up the floor (Orestrom 1983). Time-bounded Persistent Goals We have designed a goal-based framework for con- trolling an agent’s actions based on the idea of time- bounded persistent goals, a time-sensitive variation of Cohen and Levesque’s persistent goals (Allen 1983; 1362 SIGART/AAAI Cohen & Levesque 1990). In their most general form, time-bounded persistent goals looks like this: Bounded-persistent-goal( $,‘I’) While: simple-goal (4) Adopt-when: B(hoZds(B+,some-head-of (2’))) Drop-when: B(hoZds(B4,some-tail-of(T))) B(hoZds(B+, some-tail-of(T))) B(after(T, now)) A bounded persistent goal to make 4 hold over T is adopted when the agent has a simple goal to achieve 4, and the agent believes 4 does not already hold at the start of 2’. The goal is dropped when the agent believes 4 holds over some interval that ends T, or that 14 holds over some interval that ends T, or T is in the past. By defining different kinds of simple goals, the bounded persistent goals can be used to help a rational agent decide how to manage its turn-taking activities. We have considered applying bounded-persistent goals to the problem of the initiation of clarification dialogs in advice-giving settings (for cases of miscon- ceptions and plan ambiguity). While it is typically assumed that, for example, a possible misconception should be dealt with immediately, time-bounded per- sistent goals allow certain turns to be put aside, and not actually executed until absolutely necessary. Such “lazy” turn-taking thus allows for the possibility that perceived problems may actually be corrected by the speaker, and thus no clarification dialog need be en- tered into at all. eferences Allen, J. 1983. ‘Maintaining knowledge about temporal intervals. Communications of the ACM 26(11):832-843. Cohen, P., and Levesque, H. 1990. Intention is choice with commitment. Artificial Intelligence 42:213-261 e Orestriim, B. 1983. Turn-taking In English Conver- sation. CWK Gleerup. From: AAAI-96 Proceedings. Copyright © 1996, AAAI (www.aaai.org). All rights reserved. | 1996 | 215 |
1,861 | Learning in multi-agent systems Claudia V. Goldman * Institute of Computer Science The Hebrew University Givat Ram, Jerusalem, Israel clag@cs.huji.ac.il Learning agents acting in a multi agent environment can improve their performance. These agents might decide upon their course of action by learning about other agents with whom they interact. The learning agents can learn about the others’ information and rules of behavior. The agents will not need to plan their actions beforehand, each time they are asked to solve the same problem they have already solved or when dealing with similar problems. Our research focusses in finding learning algorithms for multi agent environments. In particular, we want to develop agents that learn how to cooperate with other agents and learn how to become experts. I am studying these themes in several domains to achieve a more broader understanding of how learning is influ- enced by different multi-agent domains and thus how learning algorithms can be developed. The research aims at answering the following questions: 1. How do agents learn how to act? I am looking at a teacher-learner model in which the teacher might also play the role of the learner, and the learner might also behave as a teacher. I am investigating two main directions. One direction regards a 2-phase model and we investigate how agents can be trained by other agents and then how they generalize the knowledge they acquired (Goldman & Rosenschein 1995). The second direction focusses on a dynamic learning model in which the agents learn from the others incrementally and also act based on the cur- rent knowledge they have. I am also investigating the behavior of learning agents that are implemented as automata (deter- minisitic and probabilistic ). We investigated (Mor, Goldman, & Rosenschein 1995) how one learning agent can decide which action to play while he learns his opponent’s strategy. Currently, we are focussing on mutual learning In both cases, I am looking at reinforcement learn- ing, where the feedback is given to the agents by the *This research is supervised by Dr. Jeffrey S. Rosen- schein and supported by the Eshkol Fellowship, Israeli Min- istry of Science 2. other agents, in contrast to most methods used in reinforcement learning in which the agents receive their feedbacks from the world in which they per- form their actions. How do agents learn about information (i.e. under- stand, share and compose new information)? Cur- rently, I am working on three projects: Musag (Gold- man, Langer, & Rosenschein 1996): a system based on four software agents, that learns about concepts by “reading” html documents in the Web. I am cur- rently working on expanding the system to include a group of such systems that will interact in order to learn more about the concepts they are learning, by sharing information they have, with the others. Courtz (Goldman, Mor, & Rosenschein 1996): a software agent dedicated to looking for information about a topic given by a user in a fuzzy way, i.e. the information is not well defined and might have ambiguous meanings. NetNeg (Goldman, Gang, & Rosenschein 1995): a hybrid system built on a neural network module and an agents based module. This agents deal with multi-media information. References Goldman, C. V., and Rosenschein, J. S. 1995. Mu- tually supervised learning in multiagent systems. In Workshop on Adaptation and Learning in MAS at IJ- CAI. Goldman, C. V.; Gang, D.; and Rosenschein, J. S. 1995. Netneg: A hybrid system architecture for com- posing polyphonic music. In Workshop on AI and Music at IJCAI. Goldman, C. V.; Langer, A.; and Rosenschein, J. S. 1996. Musag: an agent that learns what you mean. In PAAM96. Goldman, C. V.; Mor, Y.; and Rosenschein, J. S. 1996. Courtz: an agent that pleases you. In PAAM96. Mor, Y.; Goldman, C. V.; and Rosenschein, J. S. 1995. Learn your opponent’s strategy (in polynomial time)! In Workshop on Adaptation and Learning in MAS at IJCAI. Doctoral Consortium Abstracts 1363 From: AAAI-96 Proceedings. Copyright © 1996, AAAI (www.aaai.org). All rights reserved. | 1996 | 216 |
1,862 | Bounding the Cost of Learned Rules: A Transformational Approach Jihie Kim Information Sciences Institute and Computer Science Department University of Southern California 4676 Admiralty Way, Marina de1 Rey, CA 90292, U.S.A. jihie@isi.edu My dissertation research centers on application of ma- chine learning techniques to speed up problem solving. In fact, many speed-up learning systems suffer from the util- iv problem; time after learning is greater than time before learning. Discovering how to assure that learned knowl- edge will in fact speed up system performance has been a focus of research in explanation-based learning (EBL). One way of finding a solution which can guarantee that cost after learning is bounded by cost of problem solving is to ana- lyze all the sources of cost increase in the learning process and then eliminate these sources. I began on this task by decomposing the learning process into a sequence of trans- formations that go from a problem solving episode, through a sequence of intermediate problem solving/rule hybrids, to a learned rule. This transformational analysis itself is important to understand the characteristics of the learning system, including cost changes through learning. Such an analysis has been performed for Soar/EBL(Kim & Rosenbloom 1995). The learning process has been de- composed into a sequence of transformations from the prob- lem solving to learned rule. By analyzing these transfor- mations, I have identified three sources which can make the output rule expensive. First, ignoring search-control rules which constrained the problem solving can increase the cost. For example, PRODIGY/EBL (Minton 1993) and Soar ignore a large part of the search-control rules in learning to increase the generality of the learned rules. The consequence of this omission is that the learned rules are not constrained by the path actually taken in the problem space, and thus can perform an exponential amount of search even when the original problem-space search was highly directed (by the control rules). By incorporating search control in the ex- planation structure, this problem can be avoided (Kim & Rosenbloom 1993). Second, when the structure of the problem solving differs from the structure of the match process for the learned rules, time after learning can be greater than time before learning. During problem solving, the rules that fire tend to form a hierarchical structure in which the ewly rules provide infor- mation upon which the firing of later rules depends. This hierarchical structure is reflected in EBL most obviously in the structure of the explanation (an1 the more general 1364 SIGART/AAAI explanation structure). However, if this hierarchical struc- ture is then flattened into a linear sequence of conditions for use in matching the rule that is learned, the time after learning can be greater than the time before learning. If instead, the learning mechanism is made sensitive to the problem-solving structure, this source of expensiveness can be avoided (Kim & Rosenbloom 1996). Third and finally, ignoring the optimization employed in the problem solving can increase the cost. In Soar, work- ing memory is a set. Whenever two different instantiations create the equivalent working memory elements, they are merged into one. Eliminating this process in learning, and keeping equivalent set of partial instantiations separately can increase the cost. By preprocessing the partial instan- tiations, and merging the equivalent instantiations into one, this cost increase can be avoided. The results on a set of known expensive-rule-learning tasks have shown that such modifications can effectively eliminate the identified set of sources of expensiveness. My future work would be extending the experimental results to a wider range of tasks, both traditional expensive-rule tasks and non-expensive-rule tasks. Also, experiments on a practical domain rather than a toy domain would allow a more realistic analysis of the approach. In addition to the sources of expensiveness which have found so far, I am working toward identifying other potential sources of expensiveness, should they exist. By finding the complete set of sources of expensiveness and avoiding those sources, the cost of using the learned rules should always be bounded by the cost of the problem solving episode from which they were learned. eferences Kim, J., and Rosenbloom, P. S. 1993. Constraining learn- ing with search control. In Proc. 10th Int’l ConJ on Ma- chine Learning, 174-l 8 1. Kim, J., and Rosenbloom, P. S. 1995. Transformation analyses of learning in Soar. Technical Report ISVRR-95- 422 1, USC-ISI. Kim, J., and Rosenbloom, P. S. 1996. Learning efficient rules by maintaining the explanation structure. In Proc. 13th Nat’1 Con. on Artificial Intelligence (to appear). Minton, S. 1993. Personal communication. From: AAAI-96 Proceedings. Copyright © 1996, AAAI (www.aaai.org). All rights reserved. | 1996 | 217 |
1,863 | Agent-Centered Search: Situated Search wit Small Look-Ahea Sven Koenig School of Computer Science Carnegie Mellon University Pittsburgh, PA 15213-3891 skoenig@cs.cmu.edu Situated search is the process of achieving a goal in the world. Traditional single-agent search algorithms (such as the A* algorithm) usually assume completely known, stationary domains with deterministic actions. These assumptions favor search approaches that first determine open-loop plans (sequences of actions) that can then be executed blindly in the world. Conse- quently, single-agent search in AI is often performed in a mental model of the world (state space): states are represented as memory images and search is a pro- cess inside the computer (search-in-memory). Situated search can interleave or overlap search- in-memory with action execution. This approach has advantages over traditional approaches in non- deterministic, non-stationary, or only partially known domains. It allows one, for example, to gather infor- mation by executing actions. This information can be used to resolve uncertainty caused by missing knowl- edge of the domain, inaccurate world models, or ac- tions with non-deterministic effects. For instance, one does not need to plan for every contingency in non- deterministic domains; planning is necessary only for those situations that actually result from the execution of actions. My research focuses on the design and analysis of agent-centered search methods. Agent-centered search methods are situated search methods that restrict the search-in-memory to a small part of the state space that is centered around the current state of the agent before committing to the execution of the next action. Since agent-centered search methods perform a small amount of search-in-memory around the current state of the agent, they can be characterized as “situated search with small look-ahead.” Examples of agent- centered search methods include real-time heuristic search methods (they search forward from the current state of the agent), many on-line reinforcement learn- ing algorithms, and exploration approaches. A major difference .between search-in-memory and agent-centered search is that search-in-memory meth- ods can “jump around in the state space.” If, after they have expanded a state, a state in a different part of the state space looks promising, they can expand this state next. In contrast, in situated search, if an agent wanted to explore a state, it would have to execute ac- tions that get it there. The further the state is from the current state of the agent, the more expensive it is to get to that state. Furthermore, backtracking to a pre- vious state might not be easy: it can be very expensive in state spaces with asymmetric costs and the agent might not even have learned how to “undo” an ac- tion execution. Thus, one can expect situated (agent- centered) search methods to have different properties than search-in-memory methods. To date, agent-centered search methods have mostly been studied in the context of specific applications. The few existing results are mostly of empirical na- ture and no systematic or comparative studies have yet been conducted. It is therefore unclear when (and why) agent-centered search algorithms perform well. My research explores, both theoretically and experi- mentally, how agent-centered search methods behave. This includes an analysis of the factors that influ- ence their performance. Such an analysis is helpful, for example, for predicting their performance in non- deterministic domains, for distinguishing easy from hard search problems, for representing situated search problems in a way that allows them to be solved effi- ciently, and for developing more adequate testbeds for agent-centered search methods than sliding tile puz- zles, blocks worlds, and grid worlds. The second step then is to use these results to extend the functional- ity of agent-centered search algorithms to better fit typically encountered situated search situations. This includes probabilistic domains with non-linear reward structures, which can be caused, for instance, by the presence of dead-lines or risk attitudes. For more de- tails and references to the literature, see (Koenig SC Simmons 1996). eferences Koenig, S., and Simmons, R. 1996. Easy and hard testbeds for real-time search algorithms. In Proceed- ings of the National Conference on Artificial Intelli- gence (AAAI), this volume. Doctoral Consortium Abstracts 1365 From: AAAI-96 Proceedings. Copyright © 1996, AAAI (www.aaai.org). All rights reserved. | 1996 | 218 |
1,864 | Recurrent Expert Networks Cathie LeBlanc Dept of Computer Science Florida State Univ Tallahassee, FL 32306-4019 leblancQcs.fsu.edu Research has shown that computational techniques such as neural networks often provide classification abilities that are more accurate than methods which rely on explicit knowledge acquisition alone (Ben- David & Mandel 1995). On the other hand, because no “reason” for a particular classification can be given when a computational technique has been used, hu- man experts tend to be skeptical of such systems. As a result, many researchers have developed tools, called hybrid systems, which combine the pattern recognition capabilities and parallel processing of neural systems while retaining the domain knowledge encoded in ex- pert systems (Medsker 1994). Because the widely known “knowledge acquisition bottleneck” makes explicit knowledge acquisition tools (such as expert systems) expensive to create, Kunci- cky, Hruska and Lather (Kuncicky, Hruska, & Lather 1992) have developed expert networks, which eliminate the need for the expert to associate a certainty factor with each rule. Instead, the expert system rules, ac- quired from the human expert, are translated into the topology of a computational network, called an expert network. The individual nodes in an expert network are not all identical but instead have functionalities that match the part of the knowledge base which they encode. For example, a node which encodes an AND between two pieces of knowledge in the knowledge base might take the minimum of its inputs and if that min- imum is above a certain threshold, outputs that min- imum (Lather & Nguyen 1994). The certainty fac- tors correspond to the trainable weights between the nodes. Example data are presented to the network and the weights are learned via a backpropagation-like algorithm (Lather, Hruska, & Kuncicky 1992). The topologies and learning algorithms developed for ex- pert networks thus far have been strictly feed-forward. Some classification tasks, however, will be difficult to complete using a strictly feed-forward architecture. In particular, the solution to many problems requires that “state” information be maintained. State information is the context in which the problem is currently being solved. The context of a problem solution will change as the solution proceeds. Certain sets of rules need only be considered in certain contexts. For example, if the problem is to read email, the state must include 1366 SIGARTIAAAI information about whether the computer is on or off. The rules to turn on the computer will only be considered if the state tells us that the computer is off. Such state information will be difficult to manage using feed-forward architectures. In fact, in standard artificial neural networks, such state information is handled by the addition of recurrent connections in the topology of the network (Hertz, Krogh, & Palmer 1991). The recurrent connections allow the context information to be input to the network at succeeding steps. Therefore, I will extend the notion of expert networks so that they will be able to maintain state in- formation via recurrent connections while at the same time encoding previously discovered expert knowledge. The problem domain to which I will apply this tech- nology is the protein folding problem. In this prob- lem, state information about secondary structure pre- dictions for amino acids earlier in the protein’s primary sequence will play an important role in the secondary structure prediction for the current amino acid. References Ben-David, A., and Mandel, J. 1995. Classification accuracy: Machine learning vs. explicit knowledge ac- quisition. Machine Learning 18: 109-l 14. Hertz, J.; Krogh, A.; and Palmer, R. G. 1991. Intro- duction to the theory of neural computation. Redwood City, California: Addison-Wesley Publishing Com- pany- Kuncicky, D.; Hruska, S.; and Lather, R. 1992. Hy- brid systems: the equivalence of expert system and neural network inference. International Journal of Expert Systems 41281-297. Lather, R., and Nguyen, K. 1994. Hierarchical ar- chitectures for reasoning. In Sun, R., and Bookman, L., eds., Computational Architectures for Integrating Neural and Symbolic Processes, 117-150. Boston: Kluwer Academic Publishers. Lather, R.; Hruska, S.; and Kuncicky, D. 1992. Backpropagation learning in expert networks. IEEE Transactions on Neural Networks 3~62-72. Medsker, L. R. 1994. Hybrid Neural Network and Ex- pert Systems. Boston: Kluwer Academic Publishers. From: AAAI-96 Proceedings. Copyright © 1996, AAAI (www.aaai.org). All rights reserved. | 1996 | 219 |
1,865 | Declarative Camera Contra tomatic Cinematograp David B. Christianson Sean E. Anderson Li-wei He David H. Salesin Daniel S. Weld Michael F. Cohen* Department of Computer Science and Engineering *Microsoft Research University of Washington One Microsoft Way Seattle, Washington 98195 Redmond, WA 98052 ( dbcl, lhe,salesin, weld) @es. Washington. edu, seander@cs.stanford. edu, mcohen@microsoft. corn Abstract Animations generated by interactive 3D computer graphics applications are typically portrayed either from a particular character’s point of view or from a small set of strategically-placed viewpoints. By ignor- ing camera placement, such applications fail to realize important storytelling capabilities that have been ex- plored by cinematographers for many years. In this paper, we describe several of the principles of cinematography and show how they can be formal- ized into a declarative language, called the Declaru- tive Camera Control Language (DCCL). We describe the application of DCCL within the context of a simple interactive video game and argue that DCCL represents cinematic knowledge at the same level of abstraction as expert directors by encoding 16 idioms from a film textbook. These idioms produce compelling anima- tions, as demonstrated on the accompanying video- tape. Introduction The language of film is a complex one, which has evolved gradually through the efforts of talented film- makers since the beginning of the century. As a re- sult the rules of film are now so common that they are nearly always taken for granted by audiences; nonetheless, they are every bit as essential as they are invisible. Most interactive 3D computer graph- ics applications (e.g., virtual chat managers, interac- tive fiction environments, and videogames) do not ex- ploit established cinematographic techniques. In par- ticular, most computer animations are portrayed ei- ther from a particular character’s point of view or from a small set of strategically-placed viewpoints. By restricting camera placement, such applications fail to realize the expository capabilities developed by cinematographers over many decades. Unfortu- nately, while there are several textbooks that con- tain informal descriptions of numerous rules for film- ing various types of scenes (Arijon 1976; Lukas 1985; Mascelli 1965), it is difficult to encode this textbook knowledge in a manner that is precise enough for a computer program to manipulate. In this paper, we describe several of the principles of filmmaking, show how they can be formalized into a declarative language, and then apply this language to 148 Art & Entertainment the problem of camera control in an interactive video game. Specifically, we describe the Declarative Cam- era Control Language (DCCL) and demonstrate that it is sufficient for encoding many of the heuristics found in a film textbook. We also present a Camera Planning System (CPS), which accepts animation traces as input and returns complete camera specifications. The CPS contains a domain-independent compiler that solves DCCL constraints and calculates the dynamical con- trol of the camera, as well as a domain-independent heuristic evaluator that ranks the quality of the can- didate shot specifications that the compiler produces. We demonstrate a simple interactive video game that creates simulated animations in response to user input and then feeds these animations to CPS in order to pro- duce complete camera specifications as shown on the accompanying videotape. Our prototype video game serves as a testbed for appli- cations of DCCL and CPS. However, there are number of alternative applications to which both DCCL and CPS might be applied. Within the realm of video games, Multi-user Dungeons (MUDS), and interactive fiction, automated cinematography would allow an applica- tion to convey the subjective impression of a particu- lar character without resorting to point-of-view shots.’ Because many MUDS operate over long periods of time, an automated cinematography system could provide users with customized summaries of events they had missed while they were away. Alternatively, automated cinematography could be used to create natural inter- actions with the “intelligent agents” that are likely to take part in the next generation of user interfaces. Au- tomated cinematography could also be used to assist naive users in the creation of desktop videos, or for building animated presentations. In the latter case, Karp and Feiner have shown (Karp & Feiner 1990; 1993) that animated presentations can be effectively designed on a computer, reducing costly human in- volvement and allowing presentations to be customized for a particular viewer or situation. ‘Most current games, of which Doom is the classic ex- ample, still provide each participant with a single point-of- view shot; however, a number of games such as Alone in the Dark, Fade 2 Black and Virtua Fighter have begun to employ a wider variety of perspectives. From: AAAI-96 Proceedings. Copyright © 1996, AAAI (www.aaai.org). All rights reserved. Principles of Cinematography Although a film can be considered to be nothing but a linear sequence of frames, it is often helpful to think of a film as having structure. At the highest level, a film is a sequence of scenes, each of which captures some specific situation or action. Each scene in the film is composed of one or more shots. A single shot covers the small portion of a movie between when a camera is turned on and when it is turned off. Typically, a film is comprised of a large number of individual shots, with each shot’s duration lasting from a second or two in length to perhaps tens of seconds.2 Camera Placement Directors specify camera placements relative to the line of interest, an imaginary vector connecting two inter- acting actors, directed along the line of an actor’s mo- tion, or oriented in the direction the actor is facing. Figure 1 shows the line formed by two actors facing each other. d v Apex Figure 1: Camera placement is specified relative to the “line of interest.” (Adapted from figure 4.11 of (Arijon 1976)) Shooting actor X from camera position b is called a paruEEeZ camera placement. Filming X from position c yields an internal reverse placement. Shooting from position d results in an apex shot that shows both ac- tors. Finally, filming from g is called an external re- verse placement. Cinematographers have identified that certain “cutting heights” make for pleasing compositions while others yield ugly results (e.g., an image of a man cut off at the ankles). There is a set of (roughly) five useful camera distances (Arijon 1976, p. 18). An extreme closeup cuts at the neck; a closeup cuts under the chest or at the waist; a medium view cuts at the crotch or under the knees; a fuZZ view shows the entire person; and a long view provides a distant perspective. Heuristics and Constraints Filmmakers have articulated numerous heuristics for selecting good shots and have informally specified con- 2A notable exception is Alfred Hitchcock’s Rope, which was filmed in a single shot, albeit with disguised breaks. straints to be placed on successive shots to lead to good scenes. Several of the more important rules in- clude (Arijon 1976): Parallel editing: Story lines (visualized as scenes) should alternate between different characters, loca- tions, or times. Only show peak moments of the story: Repetitive moments from a narrative should be deleted. Don’t cross the line: Once an initial shot is taken from the left or right side of the line, subsequent shots should maintain that side, unless a neutral, establishing shot is used to show the transition from one side to the other. This rule ensures that suc- cessive shots of a moving actor will maintain the direction of apparent motion. Let the actor lead: The actor should initiate all movement, with the camera following; conversely, the camera should come to rest a little before the actor. Break movement: A scene illustrating motion should be broken into at least two shots. Typically, each shot is cut so that the actor appears to move across half the screen area. A change of the camera-to- subject distance should also be made in the switch. Idioms Perhaps the most significant invention of cinematogra- hers is the notion of an idiom - a stereotypical way to capture some specific action as a series of shots. For example, in a dialogue between two people, a film- maker might begin with an apex view of both actors, and then alternate views of each, at times using in- ternal reverse placements and at times using external reverse. While there is an infinite variety of idioms, film directors have learned to rely on a small subset of these. Indeed, film books (e.g., (Arijon 1976)) are pri- marily a compilation of idioms along with a discussion of the situations when a filmmaker should prefer one idiom over another. Figure 2 presents a three-shot id- iom that serves as an extended example throughout the remainder of this paper. The idiom, adapted from Fig- ure 13.2 of Arijon’s text (1976), provides a method for depicting short-range motion of one actor approaching another. The first shot is a closeup; actor X begins in the center of the screen and exits left. The second shot begins with a long view of actor Y; actor X enters from off-screen right, and the shot ends when X reaches the center. The final shot begins with a medium view of Y, with actor X entering from off-screen right and stop- ping at center. DCCL This section provides an informal description of the Declarative Camera Control Language DCCL. The specification of DCCL is important because it allows CPS to formalize, encode, and implement common film idioms, such as the one presented in Figure 2. Entertainment 149 (AcFilmIdiom name Arijon-13-2 :parameter (AcParamApproach :actori :actor2 :start :stop) :line (AcLineIdiom :primary ?actori :other ?actor2 :side left) (AcFilmShot name shot1 (AcFragGoBy name fragl :time ?start :primary-moment beginning :entry-pos center :exit-pos out-left :placement (AcPlaceInternal :primary ?actorl :other ?actor2 :range closeup :primary-side center)>) (AcFilmShot name shot2 (AcFragGoBy name frag2 :time ?frag3.first-tick :primary-moment end :entry-pos on-right :exit-pos center :placement (AcPlaceExternal :near ?actorl :far ?actor2 :primary-subject near :range longshot :primary-side center>>> (AcFilmShot name shot3 (AcFragGoBy name frag3 :time ?stop :primary-moment end :entry-pos out-right :exit-pos right12 :placement (AcPlaceApex :primary ?actorl :other ?actor2 :range mediumshot :primary-side rightl2)))) Figure 3: DCCL code corresponding to the idiom depicted in Figure 2. Figure 2: (Adapted from figure 13.2 of (Arijon 1976)). A common idiom for depicting short range movement of one actor approaching another. Camera positions are shown on the left side of the figure; the resulting image is shown on the right. Arrows indicate motion of actors into and out of the screen. There are four basic primitive concepts in DCCL: frag- ments, views, placements, and movement endpoints; these primitives are combined to specify higher-level constructs such as shots and idioms. Fragments In the previous section, we discussed how cinematog- raphers treat a shot as the primitive building block in a film. For our automated system, we have found it useful to further decompose each shot into a collec- tion of one or more fragments. A fragment specifies an interval of time during which the camera is in a static position and orientation or is performing a sin- gle simple motion. DCCL defines five fragment types (illustrated schematically in Figure 4). A fully-specified fragment requires additional informa- tion. Some of these arguments are obvious - for exam- ple, to track the motion of an actor, one must specify which actor to track and over what time interval to roll the film. In addition, one must also specify the desired camera range - extreme, closeup, medium, full, or long, as well as the placement of the camera relative to the 150 Art & Entertainment actor (or actors) - internal, external, parallel, and apex. Three of the fragments, go-by, panning, and tracking, require an additional argument, called the primary mo- ment, which specifies the moment at which the place- ment command is to take effect during a shot in which there is motion. Finally, two of these fragments, go-by and panning, re- quire another argument called a movement endpoint, which is used to indicate the range of motion to be cov- ered by the actor relative to the screen.3 As Figure 5 illustrates, DCCL recognizes seven movement-endpoint keywords. Note that although the movement-endpoint keywords refer to locations on the screen, they are used to calculate the temporal duration of go-by and panning fragments.4 For example, the first shot of the idiom of Figure 2 can be defined as a go-by moving from center to out-left. Shots and Idioms In many cases, a shot is composed of a single frag- ment, but sometimes it is effective to merge several fragments together to form a more complex shot. For example, one can start with a panning fragment in 3To understand why this argument is necessary, recall that a go-by fragment results in a static camera position directed at an actor who moves across the field of view. An example of a go-by fragment is the first shot of Figure 2. Note that Arijon (1976) expresses the shot not by specify- ing the temporal duration, but rather by indicating (with arrows) the range of motion that the actor should cover while the film is rolling. DCCL uses movement endpoints to allow the same type of declarative specification and relies upon the compiler to calculate the temporal bounds that will yield the proper range of motion. *This explains the definition of out-right and out-left. Arijon (1976) specifies that shots in which an actor moves off-screen (or onto the screen) should be cut three frames after (of before) the actor disappears (or appears). As are- sult we define out-right and out-left in terms of the distance traveled while three frames transpire. *\ \ #’ v Static Fragment . , Actor moves across screen , ..-.J..-. +! \ w -...-...-.-..*.. ;<. line (,f achon \ I \ \ t’ \ \ ,’ \ I’ ‘\ \ I’ ‘0’ Go-hy Fragment Actor stays near center of sceen as camera turns . . . “‘ --“-‘ 7 ~~~~~~~**-~~~ line of action Panning Fragment Actor stays near center of screen as camera moves parallel to line \ ....---~.-;“‘.--.“.‘ .“‘ . line “faction \ \ I’ \ \ I’ .\ , \ \ I’ \ \ 4‘ ‘0’ Tracking Fragment -----. line of action ‘. camera move together ‘\ ‘\ \ ‘. \ Point of View (POV) Fragment Figure 4: DCCL fragments specify the type of camera motion. which a running actor moves from out-left into the center of the screen, then shift to a tracking shot by terminating camera rotation and increasing its lateral motion to match that of the actor (Figure 6). Multi- fragment shots typically combine panning, tracking and go-by fragments in different orders. In multifragment shots, it is often important to be able to synchronize (in “simulation time”) the end of one fragment with the beginning of the next. For this reason, DCCL supports the ability to export computed variables, such as the starting or ending time of a frag- ment , for use by other fragments in the same idiom. The duration of a scene can decrease if fragments do not cover the entire scene, producing “time contrac- tion.” To define an idiom, one must specify the activities for which the idiom is deemed appropriate, a name, ar- guments (e.g., actors and times), and a list of shot descriptions. For example, Figure 3 shows the actual DCCL encoding of the idiom illustrated in Figure 2; this idiom is a good choice for showing one actor approach- ing anot her. principles of camera placement for the case of simple movement sequences on the part of one or two actors. As input, CPS requires an animation trace: informa- tion about the positions and activities of each charac- ter, as well as directives stating which actors are to be filmed over what intervals of time. In our interactive game, this information is produced by a simple com- puter simulation generated in response to a user com- mand. Given this trace information, the CPS chooses which subintervals of time to capture with the camera and from which vantage points and with which lenses (i.e., what field of view). The animation can then be played back to the user using the intervals and camera placements selected by the CPS. The primary data structure used by CPS is the film tree (Figure 7), which represents the film being generated. Of primary consequence are the scene and candidate idiom levels of the tree: each scene in the film is associ- ated with one or more possible idioms, with each idiom representing a particular method of filming the parent scene. The CPS operates by expanding the film tree until it has compiled detailed solutions for each idiom, and then selecting appropriate candidate idioms and frames for each scene. Internally, the CPS is implemented as a three-stage pipeline involving a sequence planner, DCCL compiler, and a heuristic evaluator, as shown in Figure 8. An id- iom database (not shown) provides idiom specifications relevant to each scene in the animation being filmed. The Camera Planning System The Sequence Planner The Camera Planning System (CPS) codifies and im- The current implementation of the CPS sequence plan- plements the previously described cinematographic ner is quite simple. Unlike the other portions of the CPS Figure 5: DCCL allows the user to delimit the temporal duration of go-by and panning fragments by specifying the desired initial and terminal locations of the actor on the screen. Panning Tracking Figure 6: Schematic illustration of a shot composed of two fragments: a panning fragment that melds impercep- tibly into a tracking fragment. Entertainment 151 Figure 8: The CPS is implemented as a three-stage pipeline. Film Sequences Scenes Candidate Idioms Candidate Frames Figure 7: Successive modules in the CPS pipeline incre- mentally expand the film tree data structure. the code implementing the sequence planner is specific to the domain (plot) of the application (e.g.chase and capture interactions). As input the planner receives an animation trace describing the time-varying behavior of a group of actors. As output, the sequence planner produces a film tree (Figure 7) that is specified down to the scene level. The animation trace specifies position, velocity, and joint positions for each actor for each frame of the an- imation. The trace also labels the activity being per- formed by each actor in each frame, as well as higher- level information encoded as a set of film sequences, with each film sequence including an interval of time, an actor to use as the protagonist, and (optionally) a second actor. In the current application, multiple film sequences are used to create parallel editing effects by having the CPS intermix scenes featuring one set of ac- tors with scenes featuring a different set of actors (see accompanying videotape). Given the information in the animation trace, the se- quence planner generates scenes by first partitioning each film sequence according to the activities per- formed by the protagonist during the given sequence. For the current application we have identified ten ac- tivity types (Table 1). After partitioning a sequence, the sequence planner generates scenes parameterized by the activity, actors, and time interval of each parti- tion. Once the sequence planner has created the scene nodes, the CPS must instantiate the idiom schemata relevant for each scene. Relevance is determined by matching the scene activity against a list of applicable activities defined for each idiom. The current implementation of the database contains 16 idioms (Table 1). The plan- ner instantiates idioms by substituting actual param- eters (actor names, and scene start and ending times) Activity Solitary Idioms Stopping/Starting Y 1 X-Approaches-Y N 2 X-Retreats-From-Y N 2 X-Follows-Y N 2 Moving Y 2 Turning Y 1 HeadTurning Y 1 Stationary Y 1 Looking Y 1 Noticing N 1 Picking Up N 1 Holding N 1 Table 1: Activity classifications for prototype game. for the placeholders specified in the idiom definitions. References to actor placements on the right or left sides will be automatically mirrored later, during the idiom solving process. The DCCL Compiler The DCCL compiler uses information about the move- ment of the actors to expand the fragments in each candidate idiom into an array of frame specifications, which can be played directly. Since a frame is fully constrained by the combination of its camera’s 3D po- sition, orientation, and field of view, the compiler need only generate an array of these values for each fragment in each shot in each candidate idiom. In its simplest form, an idiom consists of a single shot that is composed of a single fragment, so we cover that case first. If the fragment has type pov, then compi- lation is trivial, so we assume that the fragment has type static, tracking, go-by, or panning. We decompose the compiler’s job into four tasks: 1. Determine the appropriate primary moment(s). 2. Determine the set of possible frame specifications for each primary moment. 3. Calculate the temporal duration (length of the frame array) of the fragment given an initial frame specification. 4. Generate the interpolated specification of frame n from that of frame n - 1. Once these tasks have been completed, the compiler simply has to generate a frame array for each primary moment and frame specification. In the current ver- sion of CPS there are typically only two frame arrays corresponding to placing the camera on one side of the line of interest or the other. The task of choosing the 152 Art 8c Entertainment appropriate side is left up to the heuristic evaluator.5 The primary moment of a fragment defines the point in time at which the camera placement should conform to the placement specified for the fragment, and varies with the type of fragment. The go-by, tracking, and panning fragments specify the primary moment as ei- ther the first or last tick of the fragment. The static fragments, on the other hand, do not specify a primary moment, so in the current version of CPS we solve the placement for the first, last, and midpoint ticks of the fragment’s time interval; the heuristic evaluator will later determine which solution looks best and prune the rest. The fragment’s placement (e.g., internal, external, parallel, or apex), as specified in the idiom, combined with the location of the actors (from the animation trace) constrains the camera’s initial position to lie on one of two vectors, according to the side of the line being used (Figure 1). The actual location on this vector is determined by the desired distance between the camera and the primary subject. This distance, in turn, is specified by the fragment’s range (extreme, closeup, medium, full, or long) and the lens focal length. The compiler attempts to generate a set of appropriate placements using a normal lens (e.g., a 60-degree field of view). The vector algebra behind these calculations is explained in (Blinn 1988). The temporal duration of static, tracking, and pov frag- ments is specified explicitly as part of the DCCL spec- ification. However, the duration of go-by and panning fragments must be computed from the movement- endpoint specification in conjunction with the actor’s velocity. The function used to update the camera position and orientation from one frame to the next depends on the type of fragment involved and the change in the actors’ positions. Static and go-by fragments do not change camera position or orientation. The camera in tracking fragments maintains its orientation, but changes its position based on the actor’s velocity vec- tor. The camera in panning fragments maintains its position, but changes its orientation with angular ve- locity constrained by actor velocity and the distance at the closest approach to the camera (as determined by the primary moment). Note that unlike the sequence planner, the DCCL com- piler is completely domain-independent in that the compiler depends only on geometry and not on the plot or subject of the animation. Furthermore, the DCCL specifications in the idiom database are applica- ble across various animations; for example, the idioms in our database should apply to any animation with two-character interactions. 5Typically, the camera is restricted to one side of the line of interest. However, opportunities to “switch sides” sometimes present themselves, such as when an actor turns to walk in a neutral direction. euristic Evaluator Since the film tree is an AND/OR graph, it represents many possible films. 6 The heuristic evaluator chooses the candidate idiom that renders a scene best, assign- ing each idiom a real-valued score by first scoring and pruning the frame arrays of its fragments. Note that it is not possible to estimate the quality of a fragment or idiom before it is compiled, because visual quality is determined by the complex, geometric interactions be- tween the fragment type and the motions of the actors. A given idiom might work extremely well when the ac- tors stand in one position relative to one another, yet work poorly given a different configuration. The scoring mechanism is primarily focused towards evaluating inter-fragment criteria, namely: @ maintaining placements; smooth transitions between camera e eliminating fragments cross the line. which cause the camera to In addition, the scoring mechanism deals with certain intra-fragment behaviors that sometimes arise from the compilation phase of the CPS such as: penalizing very short or very long fragments; Q eliminating backwards. fragments in which the camera pans After the evaluator has selected the best idiom for each scene to be included in the film, the camera planning process is complete. CPS concatenates the frame ar- rays for all idiom nodes remaining in the film tree and outputs the corresponding sequence of frames to the player for rendering. Note that while the evaluator’s rules are heuristic, they are also domain-independent within the domain of film and animation: each rule encodes broadly-applicable knowledge about which types of shots look good on film, and which do not. Sample Application We are particularly interested in interactive uses of au- tomatic cinematograpy. Therefore, we decided to build a simple interactive game that would use CPS to film actions commanded by a human player. The basic plot of the game is very simple. The main character, Bob, must walk around a virtual world (in our case, SGI’s Performer Town) looking for the Holy Grail. The game is made more interesting by the introduction of Fido, an evil dog that seeks to steal the Grail. From time to time, the game generates animations of Fido’s ac- tivities and instructs CPS to edit these animations into the action. Eventually, Bob and Fido interact: if Fido gets the grail, Bob has to chase Fido in order to get it back. The user commands Bob through a pop-up menu. These commands fall into four basic categories: ‘Indeed, if there are n scenes and each scene has k can- didate idioms as children, then the film tree represents nk possible idiom combinations. Entertainment 153 telling Bob to look in a particular direction, to move to a particular point on the screen, to pick up an object, or to chase another actor. User Commands Movies I 0 7 Figure 9: Overall context of CPS The implementation of the game and its various anima- tion/simulation modules was done in C++ using the IRIS Performer toolkit running on SGI workstations, and based partially on the perfly application provided by SGI (a Performer-based walkthrough application). The game operates as a finite-state machine that pro- duces animation traces as the user issues commands to the game engine, with the CPS acting as a separate library whose sole inputs are the animation trace and the database of idioms (Figure 9). The game itself (not counting CPS or the code present in the existing per- fly application) required approximately 10,000 lines of C++ code. The CPS system is also written in C++ (despite the Lisp-like appearance of DCCL) and imple- mented with about 19,000 lines of code. The sample game interaction presented at the end of our video is intended to demonstrate a number of the activities possible in the game, as well as the various DCCL idioms. For presentation purposes, the planning time required by CPS was edited out of the video; Ta- ble 2 gives performance data taken from a similar run- through of the game on an SGI Onyx. Related Work The subject of using principles from cinematography to control camera positions and scene structure has re- ceived relatively little attention in the computer graph- ics or AI communities. We survey most of the related work here. He, Cohen, and Salesin ( 1996) have developed a sys- tem for controlling camera placement in real-time us- ing some of the ideas behind DCCL. Their work focuses on filming dialogues between multiple animated char- acters, and uses a finite state machine to select and generate camera positions. A number of systems have been described for auto- matically placing the camera in an advantageous posi- tion when performing a given interactive task (Gleicher & Witkin 1992; Mackinlay, Card, & Robertson 1990; Phillips, Badler, & Granieri 1992). However, these sys- tems neither attempt to create sequences of scenes, nor do they apply rules of cinematography in developing their specifications. In work that is closer to our own, Karp and Feiner (Karp & Feiner 1990; 1993) describe an animation planning system for generating automatic presenta- tions. Their emphasis is on the planning engine itself, whereas the work described in this paper is more con- cerned with the problem of defining a high-level declar- ative language for encoding cinematic expertise. Thus, the two approaches complement each other. Strassman (Strassman 1994) reports on Divaldo, an ambitious experiment to create a prototype system for “desktop theatre.” Unlike our focus on camera place- ment, Strassman attempts to create semi-autonomous actors who respond to natural language commands. CPS is also complementary to Divaldo. Drucker et al. (Drucker, Galyean, & Zeltzer 1992; Drucker & Zeltzer 1994; 1995) are concerned with the problem of setting up the optimal camera position for individual shots, subject to constraints. Specific cam- era parameters are automatically tuned for a given shot based on a general-purpose continuous optimization paradigm. In our work, a set of possible cameras is fully specified by the shot descriptions in DCCL and the geometry of the scene. The final selection from among this set of different shots is made according to how well each shot covers the scene. Our approach avoids the need for generic optimization searches, and it is guar- anteed to result in a common shot form. The cost is a greatly reduced set of possible camera specifications. Several useful texts derive low-level camera parame- ters, given the geometry of the scene (Foley et al. 1990; Hearn & Baker 1994; Blinn 1988). Conclusion We close by summarizing the contributions of this pa- per and describing the directions we intend to pursue in future work. The main contributions of this paper include: o Surveying established principles from filmmaking that can be used in a variety of computer graphics applications. e Describing a high-level language, DCCL, for specify- ing camera shots in terms of the desired positions and movements of actors across the screen. We have argued that DCCL represents cinematic knowledge at the same abstraction level as expert directors and producers by encoding sixteen idioms from a film textbook (Arijon 1976) (e.g., Figure 2). 154 Art & Entertainment Command Pick Up Grail 733 9 47.63 Pick Up Net 451 4 44.74 Catch Dog 603 4 32.74 Walk (long range) 701 4 11.52 Walk (med range) 208 4 6.74 Look Right 87 3 5.6 Walk (short range) 158 4 5.12 Num. Frames Scenes Generated CPU Time (s) Table 2: Typical CPS Performance Presenting a domain-independent compiler that solves DCCL constraints and dynamically controls the camera. Describing a domain-independent heuristic evalua- tor that ranks the quality of a shot specification us- ing detailed geometric information and knowledge of desirable focal lengths, shot durations, etc. Describing a fully-implemented film camera plan- ning system (CPS) that uses the DCCL compiler and heuristic evaluator to synthesize short animated scenes from 3D data produced by an independent, interactive application. Incorporating CPS into a prototype game, and demonstrating sample interactions in the game (see videotape). Acknowledgements This research was funded in part by Office of Naval Research grants N00014-94-1-0060 and N00014-95-1- 0728, National Science Foundation grants IRI-9303461 and CCR-9553199, ARPA/Rome Labs grant F30602- 95-I-0024, an Alfred P. Sloan Research Fellowship (BR-3495), and industrial gifts from Interval, Mi- crosoft, Rockwell, and Xerox. References Arijon, D. 1976. Grammar of the Film Language. New York: Communication Arts Books, Hastings House, Publishers. Blinn, J. 1988. Where am I? What am I looking at? IEEE Computer Graphics and Applications 76-81. Christianson, D. B., Anderson, S. E., He, L., Weld, D. S., Salesin, D. H., and Cohen, M. F. 1996. Declar- ative camera control for automatic cinematography. TR UW-CSE96-02-01, University of Washington De- partment of Computer Science and Engineering. Drucker, S. M., and Zeltzer, D. 1994. Intelligent cam- era control in a virtual environment. In Proceedings of Graphics Interface ‘94, 190-199. Banff, Alberta, Canada: Canadian Information Processing Society. Drucker, S. M., and Zeltzer, D. 1995. Camdroid: A system for intelligent camera control. In Proceed- ings of the SIGGRAPH Symposium on Interactive 30 Graphics ‘95. Drucker, S. M., Galyean, T. A., and Zeltzer, D. 1992. CINEMA: A system for procedural camera move- ments. In Zeltzer, D., ed., Computer Graphics (1992 Symposium on Interactive 3D Graphics), volume 25, 67-70. Foley, J. D., van Dam, A., Feiner, S. K., and Hughes, J. F. 1990. Computer Graphics, Principles and Prac- tice. Reading, Massachusetts: Addison-Wesley Pub- lishing Company, second edition. Gleicher, M., and Witkin, A. 1992. Through-the-lens camera control. In Catmull, E. E., ed., Computer Graphics (SIGGRAPH ‘92 Proceedings), volume 26, 331-340. He, L., Cohen, M. F., and Salesin, D. H. 1996. Vir- tual cinematography: A paradigm for automatic real- time camera control and directing. To appear at SIG- GRAPH ‘96. Hearn, D., and Baker, M. P. 1994. Computer Gruph- its. Englewood Cliffs, New Jersey: Prentice Hall, sec- ond edition. Karp, P., and Feiner, S. 1990. Issues in the automated generation of animated presentations. In Proceedings of Graphics Interface ‘90, 39-48. Karp, P., and Feiner, S. 1993. Automated presenta- tion planning of animation using task decomposition with heuristic reasoning. In Proceedings of Graphics Interface '93, 118-127. Toronto, Ontario, Canada: Canadian Information Processing Society. Lukas, C. 1985. Directing for Film and Television. Garden City, N.Y.: Anchor Press/Doubleday. Mackinlay, J. D., Card, S. K., and Robertson, 6. G. 1990. Rapid controlled movement through a virtual 3D workspace. In Baskett, F., ed., Computer Graph- ics (SIGGRAPH ‘90 Proceedings), volume 24, 171- 176. Mascelli, J. V. 1965. The Five C’s of Cinematography. Hollywood: Cine/Grafic Publications. Phillips, C. B., Badler, N. I., and Granieri, J. 1992. Automatic viewing control for 3D direct manipula- tion. In Zeltzer, D., ed., Computer Graphics (1992 Symposium on Interactive 30 Graphics), volume 25, 71-74. Strassman, S. 1994. Semi-autonomous animated ac- tors. In Proceedings of the AAAI-94, 128-134. Entertainment 155 | 1996 | 22 |
1,866 | Semi-Deterministic Reasoning Chengjiang Mao Department of Computer and Information Sciences University of Delaware Newark, DE 19716 mao@cis.udel.edu In typical AI systems, we employ so-called non- deterministic reasoning (NDR), which resorts to some systematic search with backtracking in the search spaces defined by knowledge bases (KBs) . An eminent property of NDR is that it facilitates programming, especially programming for those difficult AI problems such as natural language processing for which it is dif- ficult to find algorithms to tell computers what to do at every step. However, poor efficiency of NDR is still an open problem. Our work aims at overcoming this efficiency problem. There exists a lot of work done for this problem. For example, separation of domain knowledge from con- trol knowledge to facilitate pruning search spaces has been employed in many knowledge based systems such as blackboard systems. However, the improvement is restricted by the fact that it is difficult for program- mers to acquire precise control knowledge by analyzing search spaces with only the help of their intuition. Meanwhile, researches in machine learning, such as chunking and explanation-based learning, provide some means of automatic construction of control knowledge to prune search spaces. Unfortunately, im- provement of efficiency is limited by a so-called utility problem. The common strategy underlying these various ex- isting techniques is that they prune search spaces at every step of search by matching control knowledge a- gainst the current working memory (i.e., the current state in a search space). This “on-line” pruning of search spaces does not change the static organization of search spaces, and it keeps the NDR behavior through- out. Different from the above techniques, our work intro- duces a way of “off-line” pruning of search spaces. It computationally analyzes the search space defined by a KB by learning NDR implementation results based on the KB, and then reorganizes the search space into a new organization which supports semi-deterministic reasoning (SDR) as defined in the following. For any problem instance to be solved, SDR first determinis- tically chooses a sub-search space and then does non- deterministic search in the chosen sub-search space un- til the end of the whole process of search. Reorganization of a search space defined by a KB can be realized by reorganizing the KB into independent sub-knowledge bases (SKBs). Two SKBs are indepen- dent i# throughout the problem solving process for any problem instance, at most one of the two SKBs is ac- tivated. The KB organization with independent SKBs is called a solution-oriented KB organization (SOK) (Mao, 1992). B ase on the SOK, SDR first determin- d istically chooses an SKB, and then does NDR based on the chosen SKB. The determinism of SDR indicates that for any problem instance, at most one SKB can be activated. If an activated SKB fails to deduce a so- lution, then, without checking any other SKBs, SDR will claim that there is no solution for the problem instance. In our work, we choose Prolog as an NDR inference engine. To transform a typically unorganized first- order logic based Prolog KB (i.e., a Prolog program) into an SOK organized KB, an integrated learning system THOUGHT-PROLOG (Mao & Chester, 1996) first calls a Prolog interpreter to do NDR and records the inference paths, and then classifies these inference paths into disjoint groups which form SKBs, and finally generates control knowledge to describe what problem instances can be solved in each SKB. A small scale experiment on a Prolog program with 22 rules and 33 facts shows that the average activated rules are reduced by 41.3% by THOUGHT-PROLOG. Our work can also be viewed as learning for knowl- edge base organization. It recognizes knowledge base organization as the meta-meta knowledge which de- fines the ways for control knowledge (met a-knowledge) to guide the use of domain knowledge. References Mao, C. (1992). THOUGHT: An Integrated Learning System for Acquiring Knowledge Structure. In Pro- ceedings of the Ninth International Machine Learning Conference. 300-309. Aberdeen, Scotland. Mao, C. and Chester, D. (1996). Transformation of Non-Deterministic Problem Solvers into Semi- Deterministic Problem Solvers. Technical Report, NLP-HCI-AI-96-01. Dept. of Computer and Informa- tion Sciences, Univ. of Delaware. Doctord Consortium Abstracts 1367 From: AAAI-96 Proceedings. Copyright © 1996, AAAI (www.aaai.org). All rights reserved. | 1996 | 220 |
1,867 | A Connectionist Model f Instructed Learning David C. Noelle Computer Science & Engineering University of California, San Diego La Jolla, California 92093-0114 dnoelle@cs.ucsd.edu The focus of this research is on how people blend knowledge gained through explicit instruction with knowledge gained through experience. The product of this work will be a cognitively plausible computational learning model which integrates instructed learning with inductive generalization from examples. The suc- cess of this model will require the attainment of both a technical and a scientific goal. The technical goal is the design of a computational mechanism in which induction and instruction are smoothly integrated. The design of such a multistrat- egy learner might be implemented within a symbolic rule-based framework (Huffman, Miller, & Laird 1993), within a framework strong in inductive generalization, such as connectionism (Noelle & Cottrell 1995), or within a hybrid architecture (Maclin & Shavlik 1994). This work pursues the second of these three general approaches. Following an intuition concerning the pri- macy of induction (which precedes linguistic rule fol- lowing both phylogenetically and ontogenetically) and with an eye on future reduction to neurological ex- planations, the model proposed here is built within a wholly connection& framework. The scientific goal of this research is to account for certain interaction effects between instruction and in- duction that have been observed in humans, including: e the performance improvements brought by experi- ence following direct instruction. o the apparent superiority of providing direct instruc- tion prior to the presentation of examples. o the relative insensitivity to contingencies exhibited by subjects when following a simple rule. m the intrusion of similarity into instructed category learning, resulting in a failure to follow instructions. If successful, our computational model will exhibit all of these phenomena, and it will produce testable hy- potheses concerning the accuracy and time course of human learning in various contexts. Typical connectionist models learn via the modifica- tion of connection weights in response to an external error signal. Such methods, however, are essentially too slow to account for the rapid effects of “learning 1368 SIGART/AAAI by being told”. Instead, our approach captures the effects of direct instruction in the dynamic activation state of a recurrent network. The combinatoric space of valid instruction sequences is represented by the space of articulated attractors in the network’s dynam- ics. Each fixed-point attractor in this structured acti- vation space encodes a unique collection of instructed knowledge. Linguistic advice is viewed as input ac- tivity which pushes the network into an appropriate basin of attraction, allowing the network to settle to a representation of the received knowledge. By model- ing explicit instruction in this way, connection weight modification is left in the capable hands of standard inductive learning techniques. This allows induction and instruction to operate in tandem, and it opens the door to complex interactions between them. We have already demonstrated the ability of our net- works to learn a simple instructional language in the service of a task and to thereafter exhibit rapid in- struction following behavior. We have also begun ap- plying our model to experimental results from the cate- gory learning and implicit/explicit learning literature. A detailed comparison of our model with some sym- bolic models of instructed learning, such as Instructo- SOAR (Huffman, Miller, & Laird 1993), will also be conducted. References Huffman, S. B.; Miller, C. S.; and Laird, J. E. 1993. Learning from instruction: A knowledge-level capa- bility within a unified theory of cognition. In Pro- ceedings of the 15th Annual Conference of the Cog- nitive Science Society, 114-119. Boulder: Lawrence Erlbaum. Maclin, R., and Shavlik, J. W. 1994. Incorporating advice into agents that learn from reinforcements. In Proceedings of the 12th National Conference on Arti- ficial I n e i ence, 694-699. Seattle: AAAI Press. t 11 g Noelle, D. C., and Cottrell, G. W. 1995. A con- nectionist model of instruction following. In Moore, J. D., and Lehman, J. F., eds., Proceedings of the 17th Annual Conference of the Cognitive Science Society, 369-374. Pittsburgh: Lawrence Erlbaum. From: AAAI-96 Proceedings. Copyright © 1996, AAAI (www.aaai.org). All rights reserved. | 1996 | 221 |
1,868 | Symptom Manage ent for Schizo hrenic Agents Phoebe Sengers Department of Computer Science and Program in Literary and Cultural Theory Carnegie Mellon University 5000 Forbes Ave. Pittsburgh, PA 15213-3891 phoebe@cs.cmu.edu Behavior-based paradigms are a promising avenue towards creating full-blown integrated autonomous agents. However, until now they have had a major stumbling block: programmers can create robust, sub- tle, and expressive behaviors, but the agent’s overall behavior gradually falls apart as these behaviors are combined. For small numbers of behaviors, this disin- tegration can be managed by the programmer, but as more behaviors are combined their interactions become so complex that they become at least time-consuming and at worst impossible to manage. One of the characteristic modes of breakdown that occurs in such an agent is that it engages in stereotyped behavior with abrupt switching between relatively ho- mogeneous modes of behavior. I term this symptom, inspired by cultural theory, schizophrenia, and identify its cause in a methodology of atomization. Atomiza- tion is the reduction of a complex and not necessarily well-defined phenomenon to a set of relatively simple parts with limited interaction; it is what makes be- havior switching so abrupt. Unfortunately, the imple- mentation of agents depends on this reduction, since complicated, wholistic agents are nearly impossible to design, build, and debug. While this may mean that we must build atomistic agents, it does not necessarily consign us to the schizophrenic scrapheap-provided we take a fresh perspective. Cultural theory suggests that when agents appear schizophrenic, it may be because their social and cul- tural environment is ignored. While alternative AI (in- cluding Artificial Life, situated action, and behavior- based AI) insists that agents must be thought of in terms of their environment, this environment is usu- ally thought of only in terms of the physical objects the agent encounters, leaving out the designer and au- dience of the agent. The problem with ignoring part of the agent’s environment is that it obscures the origin of various technical problems, thereby making them harder to solve. I propose an AI methodology that does not ignore the social situation of the agent, but instead uses it to help design the agent. In particular, “socially situated AI” may help solve schizophrenia. In schizophrenia, behaviors are too at- omized, causing agents to switch abruptly between behaviors. From a social perspective, schizophrenia means the user can see the way the designer has broken the agent up. Therefore, schizophrenia may go away if we make the breaks somewhere the user is unlikely to look. To be more specific, behavior-based agents are typi- cally broken up into visible behaviors. But if users are meant to recognize these behaviors, abrupt switching between them will be obvious. Instead, internal switch- ing should occur without changing the visible behavior of the agent. These internal switches may be less obvi- ous, since the user sees the agent doing the same thing before and after the switch. I am building an architecture in which “atoms” of agents are not just behaviors but also behavior trunsi- tions, special behaviors that change from an old high- level activity to a new one. Switches among such ac- tivities are now implemented smoothly as behaviors, instead of occuring abruptly between them. Behavior transitions need to allow programmers to manage the complexity of combining many activities while making the agent’s behavior look more smooth and natural. Potential technical problems with behavior tran- sitions include quadratic explosion of code; large amounts of state to communicate between behaviors and the transitions taking over for them; and compli- cated, inter-related, undebugable code. I am devel- oping solutions to these problems based on a socially situated approach, which integrates, not the agent’s be- havior, but the e#ect of the behavior on the user. The designer is given the capability to manage the agent’s eflect, instead of just the agent’s behavior. I hope to show that a behavior-transition architecture based on these principles will allow designers to use the bene- fits of atomization (modularization; clean code; under- standability) without the drawback of schizophrenia, thereby creating larger, more coordinated agents. Acknowledgments This work was supported by the Office of Naval Re- search under grant N00014-92-J-1298. Many thanks to my advisors, Joseph Bates and Camilla Griggers. Doctoral Consortium Abstracts 1369 From: AAAI-96 Proceedings. Copyright © 1996, AAAI (www.aaai.org). All rights reserved. | 1996 | 222 |
1,869 | Adaptive Shared Control for an Intelligent Power Wheelc Richard C. Simpson, M.S. and Simon P. Levine, Ph.D. University of Michigan Rehabilitation Engineering Program 1 C335 University of Michigan Hospital Ann Arbor, Michigan 48 109-0032 {rsimpson, silevine}@umich.edu The NavChair Assistive Navigation System (Levine, Koren, & Borenstein 1990) is being developed to increase the mobility of severely handicapped individuals by providing navigation assistance for a power wheelchair. While designing the NavChair it became clear that obtaining the full range of desired functionality required several different “operating modes,” each of which was appropriate in different contexts. This also necessarily created a need for a method of choosing between these modes. One solution is for the user to manage the task of mode determination, which may place unacceptable performance burdens on NavChair users with severe disabilities. Instead, a means for the NavChair to automatically choose the proper operating mode is being sought. Research to develop such an adaptation method impacts a general class of human-machine systems that change their behavior based on fluctuations in the needs, goals, or capabilities of their human operator. Automatic adaptation is important for the NavChair because it allows both the operator and system to achieve levels of performance neither could reach alone. Researchers have produced a variety of methods to perform automatic adaptation. Often, even though several sources of information relevant to the adaptation process may exist for a given man-machine system, very little effort is made to combine multiple information sources together within one system. An adaptation method that can make use of more information should make better adaptation decisions than one that is limited in the information it can consider. This research will examine one promising approach called Bayesian networks, which provide a method of probabilistically modeling a situation in which causality is important, but our knowledge of what is actually going on is not complete (Charniak 1991). Experiments employing the NavChair are proposed to evaluate the performance of Bayesian networks in adaptation tasks through the following specific aims: 1. Test the hypothesis that combining multiple information sources improves adaptation in a man- machine system. 1370 SIGART/AAAI 2. Determine whether reasoning about adaptation degrades the NavChair’s performance in situations where no mode change (adaptation) is required. This research will implement an adaptation system within the NavChair that makes use of Bayesian networks. The information available to the network will include the identities of objects in the NavChair’s environment and an internal map of the larger environment in which the NavChair is moving. This work represents a preliminary effort in the development of adaptation methods that are applicable to a wide variety of man-machine systems. The immediate impact on the NavChair will be an improved ability to make adaptation decisions and an increased ability to meet the needs of people with disabilities. Beyond the scope of this research, Bayesian networks could be used to drive adaptation in applications in fields such as intelligent vehicle control, aviation, factory automation and human- computer interaction. For example, an adaptive control system similar to the NavChair’s could provide assistance to automobile drivers to improve driving safety and traffic flow on the highways. Several of the pieces needed for the proposed research have been completed. Mechanisms for automatically steering the NavChair through doorways and along walls have been developed, and integrated into a method for recognizing environmental cues and automatically adapting the NavChair’s behavior in response to them. Finally, a mapping mechanism for use in location-based adaptation has been completed. References [I] Levine, S.; Koren, Y.; and Borenstein, J. 1990. NavChair Control System for Automatic Assistive Wheelchair Navigation. In Proceedings of the 13th Annual RESNA International Conference, 193- 194. Washington, D.C.: RESNA. [2] Charniak, E. 1991. Bayesian Networks Without Tears. A/ Magazine 12(4): 50-63. From: AAAI-96 Proceedings. Copyright © 1996, AAAI (www.aaai.org). All rights reserved. | 1996 | 223 |
1,870 | Induction of Selective Bayesian Networks from Data Moninder Singh* Department of Computer and Information Science University of Pennsylvania, Philadelphia, PA 19104-6389 msingh@gradient.cis.upenn.edu Bayesian networks (Pearl 1988), which provide a compact graphical way to express complex probabilis- tic relationships among several random variables, are rapidly becoming the tool of choice for dealing with uncertainty in knowledge based systems. Amongst the many advantages offered by Bayesian networks over other representations such as decision trees and neural networks are the ease of comprehensibility to humans, effectiveness as complex decision making models and elicitability of informative prior distributions. However, approaches based on Bayesian networks have often been dismissed as unfit for many real-world applications because they are difficult to construct and probabilistic inference is intractable for most problems of realistic size. Given the increasing availability of large amounts of data in most domains, learning of Bayesian networks from data can circumvent the first problem. This research deals primarily with the second problem. We address this issue by learning selective Bayesian networks - a variant of the Bayesian net- work that uses only a subset of the given attributes to model a domain. Our aim is to learn networks that are smaller, and hence computationally simpler to evalu- ate, but display accuracy comparable to that of net- works induced using all attributes. We have developed two methods for inducing selec- tive Bayesian networks from data. The first method, K2-AS (Singh & Provan 1995), selects a subset of at- tributes that maximizes predictive accuracy prior to the network learning phase.The idea behind this ap- proach is that attributes which have little or no in- fluence on the accuracy of learned networks can be discarded without significantly affecting their perfor- mance. The second method we have developed, Info- AS (Singh & Provan 1996), uses information-theoretic metrics to efficiently select a subset of attributes from which to learn the classifier. The aim is to discard those attributes which can give us little or no infor- mation about the class variable, given the other at- tributes in the network. We have showed that rel- ative to networks learned using all attributes, net- works learned by both K2-AS and Info-AS are signif- icantly smaller and computationally simpler to evalu- ate but display comparable predictive accuracy. More- *This work is funded by an IBM Cooperative fellowship. over, they display faster learning rates, hence requiring smaller datasets to achieve their asymptotic accuracy. We have also shown that both methods significantly outperform the naive Bayesian classifier, one of the most widely-studied Bayesian methods within the ma- chine learning community. These results have several important ramifications. First, they give us a way of applying Bayesian net- works to problems where it was not possible to do so previously, due to computational intractability. Sec- ond, they show that decreasing the size of the net- works does not significantly reduce the classification accuracy which may be very important in some appli- cations (e.g. medicine). Third, in real world appli- cations, features may have an associated cost (e.g. a feature representing an expensive test). The learning algorithms proposed can be modified to prefer removal of such high-cost tests. Since databases from most real life domains, espe- cially medicine, are replete with missing data, we are also working on extending our learning algorithms to deal with such data.Previous work in this area basi- cally deals with learning the conditional probability tables assuming that the Bayesian network structure is known. We are trying to extend this work to learn both network structure as well as the probability tables from data that has missing values/variables and in- corporate it with the feature selection approaches pre- sented in this paper. Moreover, we would like to test our methods of learning selective Bayesian networks on two real-world databases in the domain of acute abdominal pain. This domain is relatively hard, yield- ing about 65% accuracy with other learning methods. Thus, it offers a good test bed for our ideas. References Pearl, J. 1988. Probabilistic Reasoning in Intelligent Systems. San Mateo, CA: Morgan Kaufmann. Singh, M., and Provan, G. M. 1995. A comparison of induction algorithms for selective and non-selective Bayesian classifiers. In Proc. 12th Id. Conference on Machine Learning, 497-505. Singh, M., and Provan, G. M. 1996. Efficient learning of selective Bayesian network classifiers. To appear in Proc. 13th Intl. Conference on Machine Learning. Doctoral Consortium Abstracts 1371 From: AAAI-96 Proceedings. Copyright © 1996, AAAI (www.aaai.org). All rights reserved. | 1996 | 224 |
1,871 | Why Dissect a Frog When You Can Simulate a Lion? Brian K. Smith School of Education and Social Policy & The Institute for the Learning Sciences Northwestern University Evanston, IL 60208 bsmith@ils.nwu.edu We are concerned with creating computer-based learning environments which provide students with opportunities to develop causal explanations of complex phenomena through experimentation and observation. We combine video and simulation to facilitate such exploration in high school biology classrooms. Specifically, we focus on issues in behavioral ecology and the predation behaviors of the Serengeti lion. The question explored by students concerns the effec- tiveness of the lion’s hunting strategies. Although popular- ized as a skilled predator, only 1530% of the hunts at- tempted by lions result in the successful capture of prey (Schaller 1972). Explaining the causal factors underlying this statistic requires an understanding of cooperative be- havior, optimality, resource competition, and variation. The conjecture is that active explanation will result in a greater understanding of these concepts than passive lec- ture/textbook teaching approaches. Students first watch a collection of video clips showing lions employing various hunting strategies. Their task is to create narratives by explaining and comparing features of the videos. Video is useful for detecting dynamic features such as speed or use of cover, but factors that are “implicit” (e.g., time of day), invisible (e.g., wind direction), or ob scured (e.g., spatial positioning of predator/prey) in the film can be difficult to isolate and understand. To highlight such salient events, we are building a simu- lation of the lion hunt called the Animal Landlord. Animals are behavioral agents possessing perceptual, motor, and ac- tion selection routines. Students observe an aerial view of the creatures and can manipulate parameters influencing their behaviors (e.g., number of predators, vegetation den- sity). Thus, the simulation provides a setting in which fur- ther data can be collected to investigate questions raised during the video exercise. Students require assistance in open-ended experimenta- tion tasks, so unlike similar systems (e.g., Resnick 1994), we provide domain-specific guidance. We monitor the progress of agents to structure a debriefing episode (Johnson 1994) upon completion of a hunt. The rule system and logic-based truth maintenance system described in (Everett & Forbus 1996) records logical dependencies be- tween aspects of the environment, creatures and their ac- tions, and justifications for these actions. Agents can be queried about all actions taken during the hunt through a multimedia interface. The dependencies are 1372 SIGARTIAAAI traced to provide justifications of these actions, justifica- tions that might otherwise go unnoticed by students (e.g., “Warthog- became alert when it heard a noise due north of it.“). In addition, the dependencies are used as indices into a library of investigation prompts, suggestions about possible exploration paths based on ecological assumptions and methodological strategies. For a stalking lion, the sys- tem would suggest considering the significance of the aver- age stalk distance and comparing conditions between stalk- ing hunts and ambushes. Investigation prompts are re- trieved using probes combining local dependency informa- tion and global properties of the world. By using this con- text information to present appropriate guidance, we hope to assist students in generating and exploring hypotheses. We have piloted the video portion of the learning envi- ronment with a small group of students, and further testing begins in classrooms in the spring of 1996. The focus of these evaluations is to determine the types of guidance re- quired in the simulation to assist student investigations. We are also refining and evaluating our agent models in accor- dance with ecological literature on the lions and their prey. This work is being advised by Brian J. Reiser and has benefited from discussions with Aggelici Agganis, John Everett, Ken Forbus, Hans Landel, and David Scheel. References Everett, J. O., and Forbus, K. D. 1996. Scaling Up Logic- Based Truth Maintenance Systems via Fact Garbage Collection. In Proceedings of the Thirteenth National Conference on Artificial Intelligence. Menlo Park, CA: AAAI Press. Johnson, W. L. 1994. Agents That Learn How to Explain Themselves. In Proceedings of the Twelfth National Conference on Artificial Intelligence, 1257-l 263. Menlo Park, CA: AAAI Press. Resnick, M. 1994. Turtles, Termites, and TrafJic Jams: Explorations in Massively Parallel Microworlds. Cambridge, MA: The MIT Press. Schaller, G. B. 1972. The Serengeti Lion: A Study of Predator-Prey Relations. Chicago, IL: University of Chicago Press. From: AAAI-96 Proceedings. Copyright © 1996, AAAI (www.aaai.org). All rights reserved. | 1996 | 225 |
1,872 | Algorithm volution for Sig al Understan Astro Teller Computer Science Department Carnegie Mellon University Pittsburgh, PA 15213 astro@cs.cmu.edu http://www.cs.cmu.edu/ astro Automated program evolution has existed in some form for over thirty years. Signal understanding (e.g., signal classification) has been a scientific concern for even longer than that. Interest in generating, through machine learning techniques, a general signal under- standing system is a newer topic, but has recently at- tracted considerable attention. First, I have proposed to define and create a machine learning mechanism for generating signal understanding systems independent of the signal’s type and size. Second, I have proposed to do this through an evolutionary strategy that is an extension of genetic programming. Third, I have pro- posed to introduce a suite of sub-mechanisms that not only contribute to the power of the thesis mechanism, but are also contributions to the understanding of the learning technique developed. Existing machine learning techniques have some ad- vantages and disadvantages with respect to finding so- lutions to the general signal-to-symbol problem. Con- cretely, the goal of this thesis work is to overcome some of these disadvantages without losing any of the impor- tant advantages of existing systems. Two particularly prominent disadvantages of exist- ing machine learning techniques for signal understand- ing are that the input must almost always be prepro- cessed and that domain knowledge must be input in the form of preprocessing or technical details that are not obvious to a signal expert. These two disadvantages can be avoided by the evolution of programs that use parameterized signal primitives. Three prominent advantages of existing machine learning techniques for signal understanding are: that “real-world” signals can be handled; that, even when learning must be done off line, the learned function can be run in real time; and that the technique mech- anisms are well understood, thereby generating faith in the method. One of the thesis goals is to transfer these advantages to the evolution of algorithms. My thesis work involves iteratively improving the representation, evolutionary environment, and coordi- nation of programs. These evolved programs are each expected to learn to discriminate one signal type from all others in a set of signal training examples. Then multiple, highly fit programs from each “discrimina- tion pool” are orchestrated in a signal understanding system. I have called this paradigm PADO: Parallel Algorithm Discovery and Orchestration. My prelimi- nary work developing the PAD0 approach and system can be seen in papers such as (Teller & Veloso 1995b; 1995c; 1995a; Teller 1996). My work concentrates on these learning mechanism innovations and in real world signal domains where the signals are typically large and/or poorly understood. This thesis work is unique in three aspects: No sys- tem currently exists that can learn to classify signals with no space or size penalties for the signal’s size or type. No genetic programming system currently exists that purposefully generates and orchestrates a variety of experts along problem specific lines. There is cur- rently no analytically sound mechanism for explaining and reinforcing specific parts of an evolved program. The main question that this thesis will answer is: Can the algorithm evolution paradigm be extended (and how far) to apply success- fully as a machine learning technique to the general signal-to-symbol problem? Acknowledgements This research is supported through the generosity of the Fannie and John Hertz foundation. eferences Teller, A., and Veloso, M. 1995a. Algorithm evolu- tion for face recognition: What makes a picture dif- ficult. In Proceedings of the International Conference on Evolutionary Computation. IEEE Press. Teller, A., and Veloso, M. 1995b. PADO: A new learn- ing architecture for object recognition. In Ikeuchi, K., and Veloso, M., eds., Symbolic Visual Learning. Ox- ford University Press. 81-116. Teller, A., and Veloso, M. 1995c. Program evolution for data mining. In Louis, S., ed., The International Journal of Expert Systems. Third Quarter. Special Is- sue on Genetic Algorithms and Knowledge Bases. JAI Press. Teller, A. 1996. Evolving programmers: The co- evolution of intelligent recombination operators. In Kinnear, K., and Angeline, P., eds., Advances in Ge- netic Programming II. MIT Press. Doctoral Consortium Abstracts 1373 From: AAAI-96 Proceedings. Copyright © 1996, AAAI (www.aaai.org). All rights reserved. | 1996 | 226 |
1,873 | Neural Network Gui 1 Order Plan Terry Zi erman Advisor: Subbarao Kambhampati Department of Computer Science and Engineering Arizona State University, Tempe AZ 85287-5406 (zim@asu.edu) The development of efficient search control methods is an active research topic in the field of planning (Kambham- pati, Katukam, & Qu 1996). Investigation of a planning program integrated with a neural network (NN) that assists in search control is underway, and has produced promising preliminary results. Project Overview: The UCPOP partial order planner (Penberthy & Weld 1992) was used in this project and the initial experiments were limited to “blocks world” problems with up to 3 blocks. Experimentation was done with several candidate sets of “partial plan” parameters or metrics in the search for one that is useful to the NN in discerning whether a partial plan is likely to evolve into a solution plan. This parameter set functions as the input vector to the NN. The planning program was modified to automatically produce an input vector for every partial plan visited in its search process. When search is complete the program classifies each vector associated with plans lying on the solution path as positive examples and all other vectors generated as negative examples. The modified planner is run on a representative set of problems, generating two sets of input vectors -one for network training, one for testing. The next project phase involved designing and training a NN using the training set of input vectors. The network should correctly classify a maximum number of its training vectors and still be able to generalize over vectors not yet presented. The trained network is subsequently tested on the set of input vectors not used in training. The input vector and/or network design is revisited if the trained network does not perform well in classifying these partial plan vectors. The final phase is to develop a version of the UCPOP that incorporates the threshold function representing the suc- cessfully trained NN in such a manner that it can efficiently guide the planner’s solution search. The performance of the modified planner on a variety of problems can then be compared with the original planner. Current Design: The modified UCPOP program gener- ated roughly 500 input vectors for training from 6 problems and 1000 testing vectors from a different 13 problems. UCPOP’s default Best First Search (BFS) algorithm was used for search control during the input vector generation phase. Designing a “good” candidate set of input parameters for the NN is a major (and ongoing) issue requiring careful analysis and iteration. But the process in itself provides interesting insight into the open question of the “goodness” of a partial plan. The candidate set on which the most 1418 AA41-96 extensive experimentation was performed contained 27 partial plan parameters: Number of nodes expanded at the current search stage Number of “refinements” in the current plan Number of unsatisfied subgoals on the goals queue 12 parameters that can take on 1 of 3 values representing whether each of the possible states for blocks (such as on(A B) ) are present in the initial and/or goal state 12 related parameters indicating whether a series of (possibly temporary) causal links to the initial state has been established for each goal state literal In this case the best performance was obtained with a multilayer feedforward network with 20 hidden nodes and one output node. The network was trained using a variation of the Gauss-Newton algorithm (Haykin 1994). Results and Current Directions: When presented with the 1000 vectors from problems not trained on the NN correctly classifies 82% of them. That is, the trained NN exhibits a success rate of 82% in discriminating between those partial plans which will ultimately lie on the solution path and those which won’t for plans created and visited by the planner during its solution search for test problems not previously “seen” by the neural net. This suggests the NN as configured has a “forecasting” capability that may be useful in improving the planner’s search control. Work is currently still underway on the last project phase discussed above, but initial attempts to use the “NN forecaster” in combination with UCPOP’s BFS algorithm to provide search control have produced mixed results. It is apparent that UCPOP performance is very sensitive to how the NN forecasting is used to guide its search control. Various techniques for combining the BFS and NN guidance as well as methods for using the NN guidance alone are being investigated. There are also a number of interesting possibilities and techniques for improving the input vector with respect to partial plan parameters likely to be most useful to the neural network forecasting process. References Haykin, S. 1994. Neural Networks -A Comprehensive Founda- tion. Macmillan College Publishing, Inc. 2 15-2 17. Kambhampati, S.; Katukam, S.; and Qu, Y. 1996. Failure driven dynamic search control for partial order planners: An explanation based approach. Artificial Intelligence (forthcoming). Penberthy, J., and Weld, D. 1992. Ucpop: A sound, complete, partial order planner for adl. Proceedings of KR-92 103-- 114. From: AAAI-96 Proceedings. Copyright © 1996, AAAI (www.aaai.org). All rights reserved. | 1996 | 227 |
1,874 | ynamic Ma : Representation of ions between 04;s Christian Zanardi GRPR, Ecole Polytechnique de Montreal P.O. Box 6079, succ. “Centre-ville”, H3C 3A7, Montreal, CANADA email:zanardi@ai.polymtl.ca Introduction As robotics applications become more complex, the need for tools to analyze and explain interactions be- tween robots has become more acute. We introduce the concept of Dynamic Map (DM), which can serve as a generic tool to analyze interactions between robots or with their environment. We show that this concept can be applied to different kinds of applications, like a predator-prey situation, or collision avoidance. The Dynamic Map (DM) For a dynamic system, represented by the state equa- tion li: = f(z, u, t), where x is the state of the system, u the control vector, and t time, T-reachable regions are commonly defined as the set of all states that can be reached within time 2’ from a same initial position x(O), with an acceptable control vector u. Although these regions indicate which points can be reached, they do not inform about (the quality of) the trajec- tories leading to these points. This information is pro- vided by a goodness functional g-depending upon the task at hand-defined at each point of the T-reachable regions, and that must be maximized. We call this ex- tension of the reachable regions Dynamic Map (DM) of the system. Constructing and Using the DM With a simple model of a car-like robot, where the control is the steer- ing angle, the Dynamic Map can be constructed along two methods. First, all points that can be reached are exhaustively generated using bang-bang controls, along with the value on the associated functional. Sec- ond, it is possible to contruct the map, using only “lim- its curves.” The points on those curves are defined to be the external boundaries of reachable regions. It is then simple to generate the shape of the Dynamic Map based on those curves. Ideally the goodness functional g should take into account important criteria such as presence of obsta- cles, energy consumption, and relative density of tra- jectories. Indeed, the values corresponding to points that belong to an obstacle should be negative in order to prevent trajectories to pass by them. Energy con- sumption is an important criteria, if a robot need to join periodically with an other to regain power. Since planned trajectories may have to change according to new environmental conditions while the initial goals remain, it would be important to know the relative density of trajectories leading to the neighboorhood of a point4.e. the number of different trajectories lead- ing to a similar position-as a mesure of confidence in the planned trajectory. Figure 1 presents an example of a DM, with such a functional-represented as shades of gray-for a car-like robot. For example, if a mobile robot R is trying to evade its pursuer F, it would be safe to know which places R can reach before its le adversary, and how Figure 1: Example of DM long before F does. For such a problem, it is simply a matter of composing Dynamic Maps, e.g. through an addition. Thus, the robot will just have to combine an estimation of its adversary’s map with its own, according to their relative pose. Then, using a simple maximization algorithm, the robot will be able to plan an evasion trajectory. Conclusions We presented here the Dynamic Map which is a tool to analyze interactions between mobile robots. One of the great advantages of the dynamic map-in a learning scheme context-is that it is built at the scale of the robot itself. Furthermore, it remains that the DM is a general concept that can be extended to a large number of dynamic systems, such as a robot manipulator. Student Abstracts 1417 From: AAAI-96 Proceedings. Copyright © 1996, AAAI (www.aaai.org). All rights reserved. | 1996 | 228 |
1,875 | Optimal Factory Scheduling using Stm Peter R. Wurman Artificial Intelligence Laboratory University of Michigan 1101 Beal Avenue Ann Arbor, MI, 48109-2110 pwurman@umich.edu Generating optimal production schedules for manu- facturing facilities is an area of great theoretical and practical importance. During the last decade, an effort has been made to reconcile the techniques developed by the AI and OR communities. The work described here aims to continue in this vein by showing how a class of well-defined stochastic scheduling problems can be mapped into a general search procedure. This ap- proach improves upon other methods by handling the general case of multidimensional stochastic costs We consider scheduling in a multi-product, sin- gle machine factory with sequence independent setup times. The machine processing and setup times are random variables. The factory is faced with a set of jobs that incur a late penalty if not completed by their deadlines, The optimization problem is to generate a static schedule that minimizes the expected penalty. Given that factory performance is stochastic, it is easy to show that an optimal solution to a determin- istic model using the expected run times will lead to sub-optimal schedules. Instead, we tackle the stochas- tic problem directly using an algorithm called Stochas- tic Dominance A* (Wellman, Ford, & Larson 1995). SDA* is designed for problems with path-dependent, stochastic operator costs. In SDA”, paths can be pruned only if their path cost is stochastically domi- nated by an already discovered path to the same state. For heuristics to be admissibile, the actual remaining cost must be stochastically dominated by the heuristic estimate. In addition, we impose a benign consistency condition to retain validity. Our work extends SDA* to the scheduling prob- lem. We formulate the problem as a state space search where the state is defined by the current inventory, the current machine setup, and the orders that have been filled. Three classes of operators are allowed. Make operators increment the inventory of the state. Ship operators decrement the inventory and change orders from unfilled to filled. Setup operators change the cur- rent product that can be built. 1416 MI-96 The path costs are defined by the pair <time, pen&y>. We must keep the full probability distri- bution for the time attribute, but, assuming we are risk-neutral in penalty, it is sufficient to store only the expected value of the penalty. The make and setup operators increment only the time element of the path cost. The ship operator incurs a penalty as a function of the current time. The definition of costs as a two-attribute structure requires some small elaborations to the SDA* algo- rithm. First, the priority queue is sorted by the esti- mated expected penalty of the paths. The estimated expected penalty is the sum of the penalty accumu- lated so far plus the heuristic estimate of the rest of the penalty. Second, paths can be pruned only if a path has already been found to the same state that dominates it in both time and penalty. This notion of dominance over multi-element costs is the same used in multi-objective A* (Stewart & White 1991). To compare the algorithm’s performance versus the deterministic model, we ran it on randomly generated problems where the number of orders and the total requested capacity were varied. We solved 400 such problems with both SDA* and A* using a deterministic model based on mean run and setup times. We found that in more than 70% of the problems, the solution found using SDA* had a lower expected cost than the deterministic solution applied to the stochastic data. We also found that the number of nodes expanded and the percentage of pruning between the stochastic and deterministic algorithms were very similar. References Stewart, B., and White, III, C. 1991. Multiobjec- tive A”. Journal of the Assocaation for Computang Machinery 38(4):775-814. Wellman, M. P.; Ford, M.; and Larson, K. 1995. Path planning under time-dependent uncertainty. In Proc. 11th Conf. on Uncertainty in AI, 532-539. From: AAAI-96 Proceedings. Copyright © 1996, AAAI (www.aaai.org). All rights reserved. | 1996 | 229 |
1,876 | el of Poetic Comprehension Kenneth EIaase MIT Media Laboratory 20 Ames Street Cambridge, Massachusetts 02 139 haase@media.mit.edu Abstract This article introduces an account of aesthetic comprehension and experience together with an implemented miniature which generates analogical interpretations from a semi-automatic parse of Wordsworth’s “Lines Written in Early Spring”. In our account, a poem serves as an analogy teaching machine by using formal structure to cue the formation of novel analogies. This account builds on an analogical model of comprehension previously applied to large corpora of newspaper summaries. In the miniature, an automatic grammatical and semantic analysis of the text is augmented with information about rhyme and rhythm. These formal cues allow the system to determine analogies which it would not otherwise consider. The article describes the comprehension framework, the annotated piece, and the matcher’s performance on the piece. It closes with a discussion of possible objections to aspects of the thesis or experiment and suggested directions for future work. Introduction This article introduces an account of aesthetic comprehension and describes an implemented miniature inspired by the account. The account begins with a general model of comprehension where routine and systematic analogies among descriptions takes the place of translation into canonical form. In this framework, semantic correspondence is based on dynamic analogizing rather than structural or nominal alignment of canonical descriptions. In other work, this model has been applied to indexing and analyzing a large (- 10 million word) corpus of short news summaries (Haase 1995). In this article, we discuss the application of the same mechanisms to a transcription of “Lines Written in Early Spring” by William Wordsworth ( 1770- 1850). Our thesis is that aesthetic experience involves the identification of new analogies and similarities and that the formal structure of a piece provides the cues which enable such analogies to be considered. Given annotations describing rhythmic and rhyming structure, our analogical matcher constructs mappings consistent (in many cases) with the metaphor 156 Art & Entertainment Wordsworth is invoking. In our (partially implemented) comprehension framework, these associations enable the future identification of similar analogies in the absence of formal cues. To follow on Minsky’s analysis of musical understanding (Minsky 1981), we propose that a poem functions as an analogy teaching machine. Our miniature illustrates a characterization of aesthetic experience as involving certain radical reorganizations of memory based on the consideration of new analogies. We propose that aesthetic experiences change the way our minds match and index subsequent experiences. This account assumes that our daily experience is organized around a dynamic memory (Schank 1982) and that aesthetic experiences are those which change the indexing and matching strategies of this memory. Though this may seem an oversimplification if we consider the small dynamic memories our programs have had to date, it seems less objectionable if we consider the kinds of rich dynamic memories which humans really have, accumulating years of lived experience. In the rest of this article, we introduce the analogical comprehension model, describe the representations of prose and poetry it operates over and discuss its performance on Wordsworth’s Lines and its sensitivity to different formal cues (rhyme, rhythm, etc.). We then discuss how the system may operate differently in the future based on its partial comprehension of Lines and lead into a discussion of the general model of aesthetic experience which we are proposing. Analogical Comprehension This section sketches the model of analogical comprehension applied to the interpretation of Lines. Analogical comprehension replaces the reduction to and comparison of canonical forms with the determination of dynamic analogies between individual non-canonical descriptions. Descriptions in this framework consist of nodes in a semantic network connected to one another by two kinds of relations: micro-relations capture the structure of individual descriptions; associations capture the significance of individual nodes by connection either to From: AAAI-96 Proceedings. Copyright © 1996, AAAI (www.aaai.org). All rights reserved. individuals in other descriptions or to nodes in a shared ontology. In the description of prose text, for instance, sentences are descriptions, phrases are nodes, grammatical structure is represented by micro-relations, and possible word meaning is encoded as association relations. Descriptions are matched at two levels: 8 8 cognate matches are based solely on associations structural matches extend cognate matches based on micro-relational structure Cognate matches are pairings having some unique common association with respect to their contexts. For example, in the descriptions of the sentences “Chris embraced Terry” and “Pat hugged Robin” where each word is a node, the nodes representing “embraced” and “hugged” would be cognates because they have a common association (e.g. “embrace, hug, bosom, or squeeze”) in WordNet) shared by none of the other nodes. The other possible pairings, however, would not be cognates because any associations they have in common (e.g. “person”) are common to all of them. Structurall matches extend cognate matching based on structural systematicity (Gentner 1983) of micro-relations. In the example above, subject and object relations of identified cognates (“embraced” + “hugged”) generate the mappings (“Chris” - “Pat”) and (“Terry” + “Robin”). Such extensions remain constrained associational information, requiring some common association between the linked nodes. When micro-relational structure is ambiguous (e.g. the subject of a verb phrase is uncertain), unique common associations are required with respect to the candidates. The overall comprehension framework includes a (still experimental) indexing facility which associates each new description with previously encountered descriptions of similar associational and micro-relational structure. In this framework, cognate matches through these structural prototypes will reflect structural roles as well as term similarity. Association of structural roles with each other can then reflect semantic similarity of structural roles, as in: where particular semantic associations (the heavy lines) combined with automatic structural associations (the light lines) yield semantically significant cognate relations (the dashed lines and arc). From this example, we can see that the formation of prior semantic associations (the heavy lines) is the basis for subsequent comprehension. Our model of aesthetic comprehension is an account of the role aesthetic experience Plays in the formation of some of these prior analogies. arisen to other analogicall ers Unlike the base level of matching in SME (Falkenhainer, Forbus, and Gentner 1989), the cognate relation is contextually sensitive and can be changed without modifying or extending the matcher itself. Unlike ACME (Holyoak and Thagard 1989) or Copycat (Mitchell 1993), where matching is also contextually sensitive, the cognate relation is also defeasible: a match depends on the existence of unique common associations which can be added or removed rather than on a combination of weights and activation levels. The representation of poetic text extends the representation of prose text with associations based on rhyme and meter. We retain the grammatical micro-relations and add associations for each node (phrase) representing: 8 possible meaning, based on WordNet (Miller 1990) 8 final phoneme (representing rhymes) e metrical position in the line (1” beat, 2nd beat, etc.) The meaning representation starts with a node based on surface form and part of speech for each word in the phrase. This is in turn associated with all of the WordNet senses for that combination and the WordNet senses are then associated with their hypernyms (generalizations). For instance, the phrase “in a grove” is associated with nodes in.preg, the.det, and grovs.noun; grove.noun is then associated with WordNet synsets for “grove” and “grove, woodlet, or orchard”. These are then associated with their respective hypernyms: “forest, wood, or woods” for “grove” and “garden” for “grove woodlet or orchard” and so on for their hypernyms (generalizations).. The representation explicit preserves ambiguity in both grammatical structure and word meaning. In prose understanding, this allows interpretation to be delayed until disambiguating contextual cues are available, much as in (Hirst 1987). In poetic interpretation, this provides the “play” which allows metaphorical interpretations to emerge. epresenting Lines To our pleasant surprise, our parser did a passable job of analyzing the Wordsworth poem, which consists of six stanzas of two couplets each. The couplets were treated as sentences and passed to our parser. The parser produced a node for each phrase and these were then annotated with associations based on rhyme, rhythmical position, broad syntactic category (thing, action, etc.) and possible word meaning (representing ambiguously via WordNet). A node representing a phrase was counted as being on a beat if its head (e.g. the noun or verb) fell on the beat, leaving some beats unaccounted for. Some analogies were precluded by the use of a phrase-level representation; e.g. a promising Art 157 match between “blended” and “pleasant” in the first stanza was not recognized because the phrases, rather than the words, were distinguished as individuals. The Engine Matches Given this representation, our program takes each represented stanza and attempts to match its two couplets.’ This particularly suits Lines (though we didn’t realize this until examining the program’s analysis) because each stanza is divided into a ‘naturalistic’ and ‘contemplative’ couplet. The matcher determines analogies which sometimes (but not always) fits the mapping of mental and natural realms which Wordsworth is trying to invoke. In the listing to the right, we show each stanza of the poem annotated with the information available to the matcher. Alternating italic and roman text indicate phrase structure, subscripts indicate metrical position of a phrase’s head, and underline words indicate rhymes. Some rhymes are duplicated within a stanza, particularly “Man” in the second and last stanza. To the right of each stanza are the matches the system found with cognate matches (in italics) listed first and structural matches listed subsequently. The described matches were based on all the sorts of associations described above, but in order to determine the role which different annotations played in the final match, we processed the text several times while selectively disabling different kinds of association (e.g. ignoring rhyme). We will refer to some of these variations in our discussion of the stanzas2. Note that adding new associational information can either add or remove existing matches by introducing new connections or making existing connections ambiguous. In the first stanza, cognate matching pairs “reclined” and “mind” (silly, but based on meter and rhyme) and “grove” and “mood” (more interesting and based on their common role as “settings”). Structural extension of this second match yields a pairing of “heard” and “bring” which in turn yields a match between “notes” and “thoughts”. This trio of matches seems consonant with Wordsworth’s intended juxtaposition of nature and thought in the poem. In the second stanza, as in the first, meter and rhyme cue an initial analogy between “link” and “think”; the rhyme between “ran” and “Man” doesn’t result in an initially analogy because the second occurrence of “Man” keeps the rhyme from being unique. Both rhythm and WordNet suggest the link between “Nature” and “heart”. The connection between “did” and “grieved” doesn’t make a lot ‘The actual mappings and transcripts of the program can be found on the World Wide Web at http://mu.www.media.mit.edu/projects/poetry 158 Art & Entertainment 1 heard1 a thousand blended notes4 + mood While in a grove6 I sat7 reclined8 reclined a mind In rhat sweet mood2 when pleasant thowhtsq =s bring =$ thoughts Bring sad thoughts to the mind2 To herfair works2 did Nature3 Ii& The human SO&j that through me rang And much1 it grieved2 my heart3 to Ud Through primrose tufts2 in that sweet bower4 The periwinkle6 trailed7 its wreaths3 And ‘tis my faith2 that everyflower Enjoys1 the air-1 it breathes1 Nature 3 heart link j think did a grieved soul *Man ran ti made The birds1 around me hopped3 and p- - Their thoughts5 I cannot measure1 But the least motion2 which3 they &4 It seemed5 a thrill6 of pleasure1 The budding twigs2 spread out theirm4 To catch5 the breezy a&z And 11 must thinkz, do all I can_4 That there5 was plemure6 -1 Ifthis belief2 from heaven3 be =Q If such5 be Nature’s holy p& Have 11 not reuson2 to lament4 what h@5 huS made6 Of &&q The Poem and Its Matches of sense but is justified by a relatively obscure path through WordNet. The connection between “soul” and “Man” is striking and is based on the proximate link of the verbs “link” and “think”. It in turn generates the link between “ran” and “made” which has a vague sort of thematic consistency though not as striking as some of the other metaphors determined elsewhere. The third stanza is singularly unproductive; the problem is that there are no cognate relations to start with because there are too many unique common associations. The link between “bower” and “flower” based on rhyme and rhythm competes with a link between “tufts” and “flower” based on WordNet while metrical and rhyme matches are also in conflict. As we take information away, certain mappings emerge, but none of them are very striking; most are the consequence of straightforward alignment of meter or rhyme. Part of the problem is also that this stanza mixes the descriptions of nature and thought; this may be intentional on Wordsworth’s part, but it’s not something which simple couplet to couplet matching can handle. In the fourth stanza, despite a similar mingling of action and thought, the system does establish an interesting correspondence between the birds in the first couplet and the motion he perceives in the second. All the connections here are cognates and the overall analogy does not have any satisfying systematicity. In the fifth stanza, the links are all cognate relations based on rhythm or, in the case of “air” and “pleasure” on the WordNet synset for “activity or behavior,” which is quite a stretch (as in ‘air one’s views’ and ‘pleasure oneself’). However, after a dry spell, the matcher does a better job on the final stanza, where WordNet, meter, and part of speech conspire to create the map between ‘belief’ and ‘reason’ and meter and rhyme establish the other initial mappings. The initial mappings seem to make a certain narrative sense; the active actions (sent and lament) coincide and the match between “belief’ and “reason” is consistent with the metaphor Wordsworth has been using. The generated match between “plan” and “Man” might be reflect Wordsworth’s expression of sadness at some lost potential and the link between “be” and “made” suggests that this loss is somehow intentional. In brief summary, the matcher produced relatively reasonable matches between the couplets of stanzas 1, 2, and 6; it did little of interest with 3, 4, and 5, possibly because they lacked significant inter-couplet structure. Aesthetic Function How do the matches determined in our miniature relate to our model of aesthetic comprehension? The answer lies in how new situations are processed in the analogical comprehension model. Briefly, when a new description is encountered, the system searches for previous descriptions with similar associations and micro-relations; it then attempts to analogize between the new description and these previous descriptions and these analogies become associations fixed in memory. Consider the analogies determined in the first stanza between ‘Nature link’ (1) and ‘Heart think’ (2); given these analogies fixed as associations, consider the case where two new descriptions arrive: ‘thunderstorms cause’ (3) and ‘leaders decide’ (4). These might be associated (through search and analogy) to (1) and (2) based on the similarity of causing and linking and thinking and deciding. However, through this connection and the precedent connection of (1) and (2), an association between the (3) and (4) might be determined based on the analogy and association established by exposure to the poem. The point here is not that such an association is always valid or sensible, but that the exposure to the match of (1) and (2) enables the consideration of a match between (3) and (4) even in the absence of strong rhythmic or rhyming cues. The analogies set up in stanza 1, in this case, enable future cases to be seen differently. This is the core of our account of aesthetic experience: it makes us see things differently by changing the structure of our memory. This work has shown how artistic form can introduce analogies; the tantalizing hypothesis that this changes future comprehension is yet to be shown. To implement the example above, while possible, would be a contrivance: the real test must come in indexing and matching against a larger corpus. Anticipate bjections This section discusses the previous sections by considering objections which might be made to either the question itself, the approach taken, or the results and their significance. bjections to the question Two obvious objections to the very enterprise begun in this article come from two different camps. One is a humanist objection that to “reduce” aesthetic experience to some computational model will rob it of its power and make the world a much poorer place in which to live. This is certainly true in the sense that any critical analysis of an experience ‘takes us out of the experience and thus diminishes it in certain ways. On the other hand, such analysis can also enrich experience in other ways. “To explain” and “to explain away” are not necessarily the same thing. While it is vitally important to maintain respect for aesthetic experience, analysis need not be disrespectful. Indeed our thesis is that aesthetic experience is basically about changing the way the world is seen, according it a central role in human understanding. A different objection may come from colleagues concerned that trying to understand aesthetic experience is “setting our sights too high”. Such concerns might be phrased thus: “get a handle on conventional reasoning and cognition and then start worrying about aesthetic experience; understanding literal meaning first and then start worrying about metaphor!” This concern is a valid one if aesthetic experience in fact builds upon a foundation of literal understanding and everyday problem solving. However, if the dependency goes the other way or if the two phenomena are co-dependent, understanding aesthetic experience is both comparable to and necessary for understanding everyday cognition. Indeed, given the model of aesthetic experience as memory reorganization, “deep learning” (of representations, for instance) may always be an aesthetic experience. bjectisns to the approach There can be little argument that the choice of poetic understanding and of Lines in particular was somewhat contrived. Our focus on poetry came from the existence of parsers and matchers built for handling prose and also from the accessibility of the medium to a wide audience of readers. Lines was not selected entirely at random: dozens of poems were looked at with an eye towards pieces with rich rhythmic and rhyming structure to provide cues for matching. There were roughly half a dozen candidates Art 159 originally selected and Lines was chosen more or less arbitrarily from this set. Because annotation of rhyme and meter had to be done by hand, a single poem was selected at this point. The need for structure reflects another possible objection: human understanding of a poem plays on a rich articulated background of experience which any program necessarily (at this point) lacks. WordNet, for all its scope and detail, is a poor substitute for the lived life which humans bring to bear when reading poetry. Our system would be confounded by less structured verse, where the connections and associations rely on understanding text with respect to experience. Interpreting ‘shall I compare thee to a summer’s day, thou art more lovely and more temperate’ requires knowing properties of summer and properties of lovers instead of mere associations of words. Such knowledge is as often experiential and poetry often evokes such common experience as well as formal structures to establish the analogies it teaches. Objections to the results Our results are of two sorts: the mappings determined between stanzas in Lines and the way in which these mappings influenced future understanding. I’ll address some possible objections to each of these in order. The problem with evaluating the matches determined by the system is in the character of poetic language: there is no ‘objective’ standard to hold the program’s performance against. For instance, the sensibility of the mapping between ‘Man’ and ‘plan’ in the last stanza depends on an interpretation of the second occurrence of ‘Man’ as denoting ‘What Man could have been’. Nonetheless, the goal is not to get the “correct” mapping because there isn’t one; instead, the goal is to get a mapping which is plausible. The second substantial problem is the argument about how exposure to Lines changed the analogies which the system would draw in the future. As a demonstration, the single case is unconvincing: in particular, it is not clear that the ‘new perspective’ generalizes (applies to many different descriptions) or doesn’t over-generalize (apply to everything in sight). However, the case is mostly intended to be illustrative rather than demonstrative: the real test comes in a rich background with a larger corpora of understood ‘real-life’ texts. We cannot do this currently because our text database does not have the intra-textual mappings which would enable aesthetic experiences to transform them. We look forward to examining this question in the future. Objections to the thesis Our definition of aesthetic experience is that it changes the way in which our memories are indexed and matched. It is important that this definition be at the right level: it should not exclude some aesthetic experiences nor should it include experiences which are not aesthetic. Because this is an inductive question, new cases may always break it, but in this section I address some obvious ones. Non-aesthetic memory reorganization. Are our memories ever reorganized without being an aesthetic experience? First, let’s distinguish reorganizing changes to memory from mere ‘additive’ changes to memory. For instance, the experience of being served by a new person at the cafe does not generally (by my definition) reorganize memory. The following definition may make this clear: a change to memory counts as a reorganization when two previously similar or dissimilar experiences change their relation (i.e. become dissimilar or similar) by virtue of the change. Experiencing the new server at the coffee shop does not normally count as aesthetic, but if the new server is a former congressman or department head, it might! One related objection takes the form ‘a mugger certainly changes the organization of my memory, but I wouldn’t call it an artistic experience!’ This criticism is somewhat allayed by introducing the distinction between aesthetic and artistic experience: artistic experience is an intentionally aesthetic experience. When you’re frightened by a mugger, it may be an aesthetic experience but it is certainly not artistic. However, when you’re frightened by a Spielberg film, the experience is both aesthetic and artistic. Both cases make the ‘dark of night’ look different by changing the memories which are triggered and the interpretations placed on shadows. But Spielberg is seeking that effect (and affect) while the mugger just wants your money (actually Spielberg might also want some of your money, but that’s not the point). Aesthetic experience without reorganization. Can aesthetic experiences exist which do not reorganize our memories? When I hear ‘Blackbird’ (Lennon & McCartney) playing in the background for the 59th time (in my life), it is probably not reorganizing my memory, but it might still be an aesthetic experience. However, I would argue otherwise. It is important to distinguish properties of the piece and properties of the experience. Experiences are aesthetic while pieces are artistic. In particular, pieces are artistic by virtue of their intent and the aesthetic experiences they invoke. The first time I heard ‘Blackbird,’ it was an aesthetic (and artistic) experience. However, as it plays in the background as I type against a deadline, it is certainly pleasant but probably not ‘aesthetic’ --- I don’t have the time or attention to experience it aesthetically. Aesthetic Feeling. Am I misappropriating the word ‘aesthetic’ from its feeling-based roots by my definition? We say that experiences are ‘aesthetically pleasing’ --- can we talk about an experience being aesthetic without discussing a corresponding feeling? Well, “aesthetic” is not necessarily tied to “pleasing”: a given ‘good’ and terrifying scene (for instance, the shower scene from Hitchcock’s 160 Art & Entertainment ‘Psycho’), has a clear aesthetic dimension, but ‘pleasurable’ would certainly not be an applicable predicate for anyone I’d care to know. Some of us can also, I argue, draw a certain aesthetic sense from a clever proof or abstract painting without any associated emotion. Our use of the word aesthetic, given these cases, must in some way be distinguished from the emotions associated with it. Of course, aesthetic experiences are often connected with emotion because emotion is a particularly potent device for invoking and changing the structure of our memories. (Gelernter 1994) proposes that emotion (through “affect linking”) drives much of the ‘low focus’ indexing and analogizing that underlies art and creativity. Likewise, (Ortony 1988) suggests that emotions provide cues for learning. For this reason, the experiences which access and then change our memory structures often invoke pleasure, fear, or other emotions. In addition, the experience of memory reorganization has a certain “felt” character of its own; it is not entirely (or even primarily) an intellectual experience but it does change the world (as we see it) and such changes often bring their own emotional charge quite independently of the events which bring about the change. Future Work In the final response to possible objections, we wandered far from a relatively small program matching skeletal descriptions of parsed poems. Future work will strike out across this gap, seeking to expand the performance of the program, the background it works against, the range of forms to which it is exposed, and a serious assessment of how the structures created by aesthetic comprehension change comprehension and performance in everyday life and problem solving. Two near-term steps are applying these mechanisms to more pieces and the automation of the rhyme and rhythm annotation of parses. If the second were accomplished, it would be interesting to do matching in larger corpora of poems, among the works of a particular poet or across poets within a school or style. It would be interesting if exposure to some works in a genre enabled analogies in other works. Finally, in order to assess the role which these mappings play in affecting “everyday comprehension,” we hope to have a version of our ‘news database’ indexed together with our ‘poetry database’ in order to look at the deep or fanciful mappings generated by their combination. Applying these techniques to other artistic forms is another area of future work. The native representation of ambiguity we use may suit the overlapping and ambiguous aspects of musical structure. (Ruttenberg 1994) proposes to use methods like these to analyze musical pieces. Visual and plastic arts are more complicated, because of the increased difficult of parsing. Another interesting area of research would be to consider critical theory and practice in light of these technologies and results. (Holyoak 1984) suggests that critical theory might be informed by efforts in analogical reasoning; this work can be considered a step towards such a connection. Another open and interesting question is whether we can use this characterization of aesthetic experience to help us design aesthetic experiences which work in new interactive media. Most new media start by copying existing media until they discover the new effects which they alone can produce. If we better understand how new modes of experience connect to the organization of memory we may be better able to use those modes in the construction of rich aesthetic experiences. nowledgments This work builds on research supported by the News in the Future consortium at the MIT Media Laboratory. I would also like to thank Jay Keyser, Roz Picard, Whitman Richards, and Janet Cahn for lively discussion and insightful comments on both this work and its presentation. eferences Falkenhainer, B., Forbus, K., and Gentner, D., “The Structure-Mapping Engine: Algorithms and Examples”, Artificial Intelligence (41), 1989 Gelernter, D., The Muse in the Machine, The Free Press, Macmillan, 1994 Gentner, D., The Structure-Mapping Engine, Cognitive Science, 7 (2), 1983 Haase, K., Analogy in the Large, Proceedings of IJCAI-96, MIT Press 1995. Hirst, G., Semantic Interpretation and the Resolution of Ambiguity, Cambridge University Press, 1987. Holyoak, K. and Thagard, P. “A computational model of analogical problem solving,” In Similarity and Analogical Reasoning (edited by S. Vosniadou and A. Ortony), Cambridge University Press 1989 Holyoak, K., An analogical framework for literary interpretation, Poetics 11: 105- 126, 1984. Miller, G. A. 1990, “WordNet: An On-line Lexical Database,” International Journal of Lexicography, 3(4). Minsky, M., “Music, Mind, and Meaning,” in Computer Music Journal, Volume 5, No. 3, Fall 198 1 Mitchell, M., Analogy Making as Perception, MIT Press 1993 Ortony, A.,Clore, G., and Collins, A., The Cognitive Structure of Emotions, Cambridge University Press 1988 Ruttenberg, A. Memory Representation and Deployment in Musical Thinking, Unpublished PhD Thesis proposal. Schank, R., Dynamic Memory, Cambridge University Press, 1982. Al?t 161 | 1996 | 23 |
1,877 | Agents Modeling Agents in Information Economies Jo& M. Vidal* and Edmund H. Durfee Artificial Intelligence Laboratory, University of Michigan. 1101 Beal Avenue, Ann Arbor, Michigan 48109-2110 jmvidal@umich.edu Our goal is to design and build agents that act in- telligently when placed in an agent-based information economy, where agents buy and sell services (e.g. the- saurus, search, task planning services, etc.). The econ- omy we are working in is the University of Michi- gan Digital Library (UMDL), a large scale multidis- ciplinary effort to build an infrastructure for the deliv- ery of library services [2]. In contrast with a typical economy, an information economy deals in goods and services that are often derived from unique sources (au- thors, analysts, etc.), so that many goods and services are not interchangeable. Also, the cost of replicat- ing and transporting goods is usually negligible, and the quality of goods and services is difficult to mea- sure objectively: even two sources with essentially the same information might appeal to different audiences. Thus, each agent has its own assessment of the quality of goods and services delivered. Our emphasis, therefore, is not on developing mar- ket mechanisms for traditional economies with inter- changeable goods/services, but rather for those where each participant might be unique and inscrutable from the perspective of others. In order to make good deci- sions, an agent in such an economy must: (A) Deter- mine exactly what services other agents provide and at what price. (B) Avoid agents whose services are not reliable or needed, and form teams with those that pro- vide needed services. (C) Decide how much to charge and whom to target. We believe these can be accom- plished if an agent builds models of how others appear to be assessing quality and/or establishing prices, and even how others are modeling others in these ways. Our previous work [l] considered such recursive mod- els, and gave algorithms that trade-off the time costs of using the deeper recursive models versus the costs of taking a possibly inferior action. However, it assumed the agents already had correct deeper models of others. The next problem to solve is how agents can ac- Supported by NSF/DARPA/NASA DL Initiative. quire models of other agents. We have built agents that provide some useful services in the UMDL, but do not charge for them. As we incorporate the abil- ity to buy and sell, the purchasing agents will have to make more complex decisions when deciding who to buy from, while the sellers will need to make decisions about how much to charge. Our research plan is to ex- plore this economy, by expanding our agents into three types, based on their modeling capabilities. That is, we built agents that: (1) Do not build models of other agents. (2) Build one-level or “policy” models of other agents, based on observations of their behavior. (3) Build intentional/two-level models of others, which are composed of an intentional model of the agent being modeled, and the one-level models it has of others. It should be clear that agents with no models of other agents cannot predict what others will do. Therefore, they either try to maximize their expected payoffs, given their ignorance of others’ behaviors, or try to minimize their possible losses. Agents that are capa- ble of building simple models are able to determine which agents have, in the past, delivered the best ser- vice. They have an advantage over agents that do not model and, in fact, are able to cheat them (i.e. deliver a lower quality service) and get away with it since, with no models, no records of their actions are kept. Agents with two-level models should be able to better predict what the competition will do (i.e. to “get inside their heads”) giving them a small advantage over the com- petition. We are testing these assertions to determine exactly when it is advantageous to use deeper models. Preliminary results show correlations between the het- erogeneity of the population, price volatility, and the benefits of using deeper models. [l] J .M. Vidal and E.H. Durfee. Recursive Agent Modeling Using Limited Rationality. ICMAS 95. [2] M. Wellman, et.& Toward Inquiry-based Ed- ucation through Interacting Software Agents. IEEE Computer. May 96. Student Abstracts 1415 From: AAAI-96 Proceedings. Copyright © 1996, AAAI (www.aaai.org). All rights reserved. | 1996 | 230 |
1,878 | Constructive Neural Network Learning Algorithms Rajesh Parekh, Jihoon Yang, and Vasant Honavar Artificial Intelligence Research Group Department of Computer Science Iowa State University, Ames, IA 50011 {parekhlyanglhonavar}@cs.iastate.edu Introduction Constructive Algorithms offer an approach for in- cremental construction of potentially minimal neural network architectures for pattern classification tasks. These algorithms obviate the need for an ad-hoc a- priori choice of the network topology. The construc- tive algorithm design involves alternately augment- ing the existing network topology by adding one or more threshold logic units and training the newly added threshold neuron(s) using a stable variant of the per- ceptron learning algorithm (e.g., pocket algorithm, ther- mal perceptron, and barycentric correction procedure). Several constructive algorithms including tower, pyra- mid, tiling, upstart, and perceptron cascade have been proposed for a-category pattern classification. These algorithms differ in terms of their topological and con- nectivity constraints as well as the training strategies used for individual neurons. Multi-Category Pattern Classification Several applications involve assigning patterns to one of M (M > 2) classes. The above constructive al- gorithms are known to converge to zero classification errors on a finite, non-contradictory, 2-category clas- sification task. We have developed provably conver- gent multi-category extensions of the above construc- tive algorithms. Simulations on artificial and real world datasets have resulted in fairly compact net- works. More recently, we have developed fast construc- tive algorithms that exhibit superior generalization. Real world data sets often have continuous valued attributes. Q uan iza ion t t can be used to transform continuous valued patterns to an equivalent set of bi- nary/bipolar valued patterns. Our experiments with quantization show that relatively difficult classification tasks can be transformed into simpler ones by effec- tive quantization of the input space (albeit with the added expense of increased dimensionality in the pat- tern space). 1398 AAAI-96 Pruning Strategies Network pruning involves elimination of redundant neurons and connections. With appropriate pruning strategies for constructive algorithms, it is possible to obtain smaller networks in terms of number of neu- rons and the number of connection weights. Smaller networks are known to possess better generalization ability. Pruning can be interleaved with the network construction phase or can take place once the entire nctv, VL v--k has been constructed. Some promising ideas on pruning include, removal of auxiliary neurons not contributing to the faithful representation of a layer in the tiling network; elimination of redundant daugh- ter neurons in the upstart and cascade networks; re- training newly added output neurons in the tower and pyramid algorithms to improve classification accuracy at each successive layer. A systematic analysis of prun- ing strategies is a topic of ongoing research. Current Research Each constructive learning algorithm has a set of in- ductive and representational biases implicit in the de- sign choices that determine where a new neuron is added and how it is trained. Our goal is to systemati- cally characterize these biases and design constructive learning algorithms that can dynamically add, train, and prune neurons in a manner that is best suited to the needs of each individual classification task. Acknowledgements This research is partially supported by the NSF grant IRI-0409580 to Vasant Honavar. References Parekh, R., Yang, J., and Honavar V. (1995). Multi- category Constructive Neural Network Learning Al- gorithms for Pattern Classification, Computer Science Department, Technical Report TR 95-15a, ISU. (Refer http://www.cs.iastate.edu/Nhonavar/publist.html) From: AAAI-96 Proceedings. Copyright © 1996, AAAI (www.aaai.org). All rights reserved. | 1996 | 231 |
1,879 | An Incremental Interactive Algorithm for Regular Grammar Inference Rajesh Parekh and Vasant Honavar Artificial Intelligence Research Group Department of Computer Science Iowa State University, Ames, IA 50011 {parekhlhonavar)@cs.iastate.edu Introduction Grammar inference, a problem with many applications in pattern recognition and language learning, is defined as follows: For an unknown grammar G, given a finite set of positive examples S’ that belong to L(G), and possibly a finite set of negative examples S-, infer a grammar G* equivalent to G. Different restrictions on S+ and S- and the interaction of the learner with the teacher or the environment give rise to different vari- ants of this task. We present, an interactive incremen- tal algorithm for inference of a finite state automaton (FSA) corresponding to an unknown regular grammar. Search Space A set of positive examples (strings of the unknown lan- guage) is structurally complete if each production rule of the unknown grammar is used at least once in the generation of some string in the set. If S+ is struc- turally complete, it implicitly defines a lattice (w) of candidate grammars that is guaranteed to contain the target grammar. At the base of the lattice is the max- imal canonical automaton (MCA) that accepts exactly the set S+. Other elements of the lattice (ordered by the grammar covers relation) represent progressively more general languages i.e., supersets of S+ and are generated by successively merging states of the MCA. This (exponential sized) search space can be concisely represented by two sets S and G which correspond to the most specific and most general FSA respectively. A version space based technique is used to search the hypothesis space. FSA corresponding to two lattice el- ements (one from S and the other from G) are com- pared for equivalence. If the two FSA are not equiva- lent, the shortest string y belonging to the symmetric difference of their languages is posed as a membership query to the teacher. Based on the teacher’s response to the query the learner is able to prune the search space without eliminating the desired solution. For example, if the teacher’s response to a query is nega- tive, the FSA accepting the negative example and all FSA that cover it (and thus also accept the same neg- ative example) are eliminated. This elimination is car- ried out implicitly by modifying the two sets S and S as needed. This interaction between the teacher and learner continues till the hypothesis space is reduced to one (or a set of equivalent) FSA. The resulting FSA is provably equivalent to the target FSA. Incremental Algorithm The incremental version of the algorithm relaxes the structural completeness assumption. The teacher may provide a few positive examples to start with. The learner performs candidate elimination by posing safe membership queries to the teacher. After seeing more positive examples, the learner incrementally updates the lattice to incorporate the new examples and con- tinues with candidate elimination. Eventually, when the set of positive examples provided by the teacher in- cludes a structurally complete set for the target FSA, no more lattice updates take place and all queries are treated as safe. The algorithm then converges to the target FSA. The necessary and sufficient conditions for guaranteed convergence of the algorithm to the correct solution are identified. Future Directions Promising directions for further research include: heuristics for informative query generation to speed up learning; inference of regular tree grammars and at- tributed grammars; empirical estimates of the expected case time and space complexity of the proposed gram- mar inference algorithm and its extensions. References Parekh R.G., and Honavar V.G. An Incremental In- teractive Algorithm for Regular Grammar Inference. Computer Science Department, Technical Report TR 96-03, Iowa State University. Student Abstracts 1397 From: AAAI-96 Proceedings. Copyright © 1996, AAAI (www.aaai.org). All rights reserved. | 1996 | 232 |
1,880 | A Computational Model of ersistent Beliefs Sunju Park Artificial Intelligence Laboratory The University of Michigan Ann Arbor, MI 48 109-2 110 boxenju@eecs.umich.edu The persistence of beliefs has been assumed in many research effotrs, either explicitly or implicitly, but a computational model is hard to find. For instance, in his well-known AOP article (Shoham 1993), Shoham suggests a formal language for beliefs and states that beliefs persist by default. The author writes (Be/A3 Be&‘O Like(A,B)7) means that at time 3 agent A believes that at time 10 agent B will believe that at time 7 A liked B. Shoham, however, does not elaborate on how to formally interpret the persistence of beliefs. Moreover, in his implemented agent language, AGENT-O, both the temporal aspect of beliefs (e.g., At time t, I believe . ..) and the nested beliefs (e.g., I believe you believe I believe . ..) have been omitted. In this abstract, we summarize our work on developing a computational model of persistent beliefs, which supports both the temporal information and the nested belief model. First, we propose a time-interval representation for nonambiguous interpretation of persistent beliefs. The main idea is rather simple: to have explicit lower and upper time-bounds when representing facts and beliefs. The time-interval representation clarifies the meaning of persistence without tedious elaboration of each implied belief. For example, the time-interval representation, (Be/f0 -I on(paper, table)l” -‘,, elaborates the implied persistence of the belief, (Be/,” on(paper, table)“), without ambiguity. It is read as “Agent A believes at t 110 that on(paper,tab/e) will be true at t 2 IO”. In addition, it can represent the history of beliefs, which was not possible in AOP. Secondly, we have developed an algorithm for checking consistency between two beliefs. The basic idea is that two beliefs are always compatible with each other unless all the following four conditions are satisfied. 0 Negated, same facts: Two beliefs (without any conjunction, disjunction, and deduction) can be potentially inconsistent only if they are about the same facts, one of which is negated. 0 Same depth of nested beliefs: If the depth of two nested beliefs are different, they are always consistent. 0 Beliefs of same agents: Two beliefs are always compatible if the agents holding two beliefs are different. 0 Overlapping time-intervals: Only the overlap and subsume relations can have a potential for conflicts. We have developed a consistency-checking algorithm between two beliefs, whose time complexity is O(d), where d is the smaller nested depth between two beliefs. If we consider d as a large constant, the complexity is O(I). Thirdly, to incorporate a new set of beliefs into its old beliefs, the agent needs a belief-revision algorithm. At present, we consider two revision methods: one where new beliefs override old beliefs in the case of inconsistency, and the other where an agent chooses to believe the maximal number of consistent beliefs. The former case is an extension of AGENT-O, since beliefs now can have temporal information and can be nested. The consistency-checking between two sets of beliefs will take O((n+mfxd), where n and an represent the number of new beliefs and old beliefs, respectively. On the other hand, the problem of finding the maximally consistent beliefs is transformed to the independent-set problem, which is NP-complete (Garey & Johnson 1979). If we assume internal consistency of the new belief set and of the belief DB, respectively, however, a polynomial-time algorithm can be possible (Park 1996). Our research shows some promising early results. First, the interval-based representation is able to represent history and allows nonambiguous interpretation of persistent beliefs. Second, a computationally simple consistency- checking algorithm has been developed. Finally, although finding a maximally-consistent belief set is NP-complete, a polynomial-time revision algorithm is possible under the assumption of internal consistency. In the future, we will work on relaxing our assumptions of not allowing disjunction and conjunction, and will develop a belief DB that supports basic operations, such as add, delete, update, and query (e.g., what the agent believes, believed, or will believe at time t). This research has been funded in part by the Digital Libraries Initiative under CERA IRI-94 11287. References Garey, M. R., and Johnson, D. S. 1979. Computers and Intractability: A Guide to the Theory of NP-Completeness: W. II. Freeman. Park, S. 1996. Belief DB: A Computational Model for Persistent Beliefs. Forthcoming. Shoham, Y. 1993. Agent-oriented Programming. Artificial Intelligence 605 l-9 1. Student Abstracts 1399 From: AAAI-96 Proceedings. Copyright © 1996, AAAI (www.aaai.org). All rights reserved. | 1996 | 233 |
1,881 | Contracting Strategy based on Markov mcess Modeling Sunju Park and Edmund W. Durfee Artificial Intelligence Laboratory The University of Michigan Ann Arbor, MI 48109-2110 { boxenju, durfee} @eecs.umich.edu One of the fundamental activities in multiagent systems is the exchange of tasks among agents (Davis & Smith 1983). In particular, we are interested in contracts among self- interested agents (Sandholm & Lesser 1995), where a contractor desires to find a contractee that will perform the task for the lowest payment, and a contractee wants to perform tasks that maximize its profit (payment received less the cost of doing the task). Multiple, concurrent contracts take place such that a contract may be retracted because of other contracts. In our work, we are asking the question: What payment should a contractor offer to maximize its expected utility? If the contractor knows the costs of the agents and knows that the agent(s) with the minimum cost are available, then it can offer to pay some small amount above that cost. But the contractor usually will face uncertainty: it might have only probabilistic information about the costs of other agents for a task, and also about their current and future availability. A risk-averse contractor therefore needs to offer a payment that is not only likely to be acceptable to some contractee, but which also is sufficiently high that the contractee will be unlikely to retract on the deal as other tasks are announced by other contractors. A risk-taking contractor, on the other hand, may want to pay a little less and risk non-acceptance or eventual retraction. This abstract defines the contractor’s decision problem, and presents a contracting strategy by which the contractor can determine an optimal payment to offer. The contractor’s decision problem in the contracting process is to find a payment that maximizes its expected utility. The contractor’s utility for the payment, p, is defined as Ps x U(Payofss(p)) + PF x U(PayofliAp)), where UC.) is the utility function, Ps/F denote the probability of success (S) and failure (F) of accomplishing a contract, and Payofw are the payoff of S and F, respectively, given p. We have developed a four-step contracting strategy for the contractor to compute Psn; and Payo& and thus to find the best payment to offer. First, the contractor models the future contracting process stochastically as a Markov Process (MP). An example MP model is shown in Figure l-(a). StateZ is the initial state, and state A is the announced state. State C is the contracted state, where the contractor has awarded the task to one of those who accepted its offer. State Sand F are success and failure states, respectively. From A, the 1400 AAAI-96 process goes to C if at least one agent accepts the offer. If no agent accepts the offer, the process goes to F. The process may go back to Z, if there are some agents who can perform the task but are busy at the moment. If the contractee retracts the contractor’s task (to do other more profitable task(s)), the process goes from C to I. Second, the contractor computes the transition probabilities between the MP states. The transition probability from state i to state j is a function of many factors, such as the payment, the potential contractees’ costs, the payments of other contracts, and so on. Third, having the model and its transition probabilities, the contractor computes P, and Payoff,,,. We have developed a theoretically-sound method of computing those values based on MP theory @hat 1972). Finally, when Ps,F and Payofl& are known, finding the optimal payment is an optimization problem. At present, the contractor uses a simple generate-and-test. An example of a contractor’s expected utility is plotted in Figure l-(b). In this case, the contractor will receive the highest expected utility when it proposes a payment of 9. :‘. . .;. . -. : l-l- : --o- ~ : : .I . . .-I I . (a) Markov process model (b) Expected utility vs. payment Figure 1: A Markov Process Model We have applied our approach to cases with two tasks, and are currently building a m-task model. This research has been funded in part by Digital Libraries Initiative under CEPA RI-941 1287. eferenees Bhat, U. 1972. Elements of Applied Stochastic Processes: John Wiley (a Sons Inc. Davis, R., and Smith, R. 1983. Negotiation as a Metaphor for Distributed Problem Solving. AZ 20:63-109. Sandholm, T., and Lesser, V. 1995. Issues in Automated Negotiation and Electronic Commerce: Extending the Contract Net Framework. In Proc. of ICMAS-95, 328-335. From: AAAI-96 Proceedings. Copyright © 1996, AAAI (www.aaai.org). All rights reserved. | 1996 | 234 |
1,882 | earning roeedura Douglas J. Pearson Artificial Intelligence Laboratory The University of Michigan, 1101 Beal Ave. Ann Arbor, MI 48109, USA dpearson@umich.edu, http://ai.eecs.umich.edu Autonomous agents functioning in complex and rapidly changing environments can improve their task performance if they update and correct their world model over the life of the agent. Existing research on this problem can be divided into two classes. First, reinforcement learners that use weak inductive meth- ods to directly modify an agent’s procedural execution knowledge. These systems are robust in dynamic and complex environments but generally do not support planning or the pursuit of multiple goals and learn slowly as a result of their weak methods. In con- trast, the second category, theory revision systems, learn declarative planning knowledge through stronger methods that use explicit reasoning to identify and cor- rect errors in the agent’s domain knowledge. However, these methods are generally only applicable to agents with instantaneous actions in fully sensed domains. This research explores learning procedural planning knowledge through deliberate reasoning about the cor- rectness of an agent’s knowledge. As the system, IM- PROV, uses a procedural knowledge representation it can efficiently be extended to complex actions that have duration and multiple conditional effects, taking it beyond the scope of traditional theory revision sys- tems. Additionally, the deliberate reasoning about cor- rectness leads to stronger, more directed learning, than is possible in reinforcement learners. An IMPROV agent’s planning knowledge is repre- sented by production rules that encode preconditions and actions of operators. Plans are also procedurally represented as rule sets that efficiently guide the agent in making local decisions during execution. Learn- ing occurs during plan execution whenever the agent’s knowledge is insufficient to determine the next action to take. This is a weaker method than traditional plan monitoring, where incorrect predictions trigger the cor- rection method, as prediction-based methods perform poorly in stochastic environments. IMPROV’s method for correcting domain knowledge is primarily based around correcting operator precon- ditions. This is done by generating and executing alter- native plans in decreasing order of expected likelihood of reaching the current goal. Once a successful plan has been discovered, IMPROV uses an inductive learn- ing module to correct the preconditions of the opera- tors used in the set of k plans (successes and failures). Each operator and whether it lead to success or fail- ure is used as a training instance. This k-incremental learning is based on the last k instances and results in incremental performance which is required in do- mains that are time-critical. K-incremental learning is stronger than traditional reinforcement learning as the differences between successful plans and failed plans lead to better credit assignment in determining which operator(s) were incorrect in the failed plans and how the operator’s planning knowledge was wrong. Actions are corrected by recursively re-using the precondition correction method. The agent’s domain knowledge is encoded as a hierarchy of operators of pro- gressively smaller grain size. The most primitive op- erators manipulate only a single symbol, guaranteeing they have correct actions. Incorrect actions at higher levels are corrected by changing the preconditions of the sub-operators which implement them. For exam- ple, the effects of a brake operator are encoded as more primitive operators which modify the car’s speed, tire condition etc. IMPROV’s correction method is recur- sively employed to change the preconditions of these sub-operators and thereby correct the planning knowl- edge associated with the brake operator’s actions. This method allows IMPROV to learn complex actions with durations and conditional effects. The system has been tested on a robotic simula- tion and in driving a simulated car. We have demon- strated that k-incremental learning outperforms sin- gle instance incremental learning and that a procedu- ral representation supports correcting complex non- instantaneous actions. We have also shown noise- tolerance, tolerance to a large evolving target domain theory and learning in time-constrained environments. Student Abstracts 1401 From: AAAI-96 Proceedings. Copyright © 1996, AAAI (www.aaai.org). All rights reserved. | 1996 | 235 |
1,883 | MarketBayes: A Distributed, Market-Based Bayesian Network David M. Pennock Artificial Intelligence Laboratory, University of Michigan 1101 Beal Avenue Ann Arbor, MI 48109-2110 dpennock@eecs.umich.edu This paper presents initial work on a system called MarketBayes, a computational market economy where distributed agents trade in uncertain propositions. For any Bayesian network, we have defined a corresponding economy of goods, consumers and producers that es- sentially “computes” the same information. Although our research thus far has only verified the existence of a market structure capable of Bayesian calculations, our hope is that such a system may address a variety of in- teresting problems of distributed uncertain reasoning. For example, the economic framework should be well suited for belief aggregation, since the bids of numer- ous agents with varying beliefs, confidence levels and wealth are concisely “summarized” in the going prices of goods. A Bayesian network structure consists of a set of related propositions with information about how the probabilities of the propositions depend on one an- other. In a MarketBayes economy, the goods to be bought and sold correspond to these propositions. If a proposition is true, the corresponding good is worth one “dollar”; if the proposition is false, it is worth noth- ing. Then if the proposition is uncertain, its worth should be exactly the probability that it is true (Han- son 1995), assuming risk neutrality. A MarketBayes economy is a set of goods along with a mix of con- sumers and producers that trade in these goods. After equilibrium is reached, the prices of the propositions should equal the probabilities that the propositions are true. In a Bayesian network, links between propositions encode conditional probabilities. For example a single link from proposition A to proposition B is accompa- nied by the information P(BIA) = k where k is some probability. The same equation can be rewritten as: Pr(AB) = kPr(A) (1) In a MarketBayes economy, the consumers effectively implement equations of the form (1). AB and A are propositions or goods, and the consumer’s preference 1402 AAAI-96 for AB is k times that of A. If the ratio of the prices Pr(AB)/Pr(A) d’ g f lver es ram k, the consumer will buy or sell according to its preference, driving the ratio toward k. In a Bayesian network, the laws of probability are in- herent in the inference mechanism. In a MarketBayes economy, producers ensure that the laws of probabil- ity are not violated. For example, the following is an identity in probability theory: Pr(A) = Pr(AB) + Pr(AB) (2) Equations of the form (2) are enforced by producers that have the technology to “transform” one .4 into one AB and one AD, and vice versa. If the price Pr(A) diverges from the price Pr(,4B) + Pr(AB), a producer will transform one good into the other in order to cap- italize on the potential profits-thus driving the two prices together. This type of producer is an arbitrageur since it capitalizes on inconsistencies between related prices. We have found that consumers of the form (1) and producers of the form (2) are sufficient to encode any Bayesian network with binary propositions. We have built the initial MarketBayes system on top of a distributed auction mechanism called WAL- RAS (Wellman 1993). 0 ur next research goal is to bet- ter characterize any advantages that a market-based probabilistic reasoning mechanism may have over tra- ditional Bayesian networks. We conjecture that the market system will offer a concise and principled way to aggregate beliefs of multiple distributed agents. References Hanson, R. D. 1995. Could gambling save science? Encouraging an honest consensus. Social EpistemoG ogy 9( 1):3-33. Wellman, M. P. 1993. A market-oriented program- ming environment and its application to distributed multicommodity flow problems. Journal of Artificial Intelligence Research l:l-22. From: AAAI-96 Proceedings. Copyright © 1996, AAAI (www.aaai.org). All rights reserved. | 1996 | 236 |
1,884 | The Kritzel System for Handwriting Interpretation * Gaofeng Qian The University of Texas at Dallas, Artificial Intelligence Lab P. 0. Box 830688, Richardson, Texas 75083-0688 gqian@utdallas.edu Introduction We present a new system for recognizing on-line cursive handwriting. The system, which is called the Kritzel System, has four features. First, the system characterizes handwriting as a se- quence of feature vectors. Second, the system adapts to a particular writing style itself through a learning process. Third, the reasoning of the system is formu- lated in propositional logic with likelihoods. Fourth, the system can be readily linked with other English processing systems for lexical and contextual checking. System Structure The Kritzel System is organized into three modules: Interpretation module, Partition module and Learning module. The Interpretation module extracts local topologi- cal features along the pen trajectory, yielding a se- quence of time-ordered feature vectors. By scanning the sequence against character templates, it identifies all possible characters and provides a confidence value for them. Using lexical and contextual checking, it se- lects reasonable words from these ‘candidates. The Partition module is invoked when the Interpre- tation module fails to provide the correct word. It asks the user for the exact word written, then partitions the handwriting into discrete segments which correspond to each character of the word. Using the results of the Partition module, the Learn- ing module analyzes the mistakes of the Interpretation module and adjusts the templates. The Learning mod- ule has three alternative ways to adjust the templates. Each of the alternatives is evaluated by a logic program and the best one is selected. System Implementation The Kritzel System is implemented in the C program- ming language. The logic programs are compiled by the Leibniz System. *Dissertation research supervised by Professor Klaus Truemper and supported in part by the Office of Naval Research under Grant N00014-93-1-0096. The Interpretation module implements a five-layer decision pyramid. At layer 1, it reads X-Y coordinate values of the pen trajectory. At layer 2, it extracts lo- cal topological features such as maxima, minima, slope and curvature. This yields a sequence of time-ordered feature vectors. At layer 3, it does template matching to find all possible characters along the sequence. At layer 4, it figures out the relationships between each pair of those characters. At layer 5, it searches for the written word, using the Laempel System which per- forms lexical and contextual checking. The Partition module is invoked when the result of the Interpretation module is incorrect. First, it lo- cates the tall characters in the word. Then, using all feature vectors of the templates and a reasoning pro- gram, it searches for the remaining characters. The search is performed on each range separated by the al- ready identified tall characters. If the word has not been fully segmented, it searches the possible ranges again using selected feature vectors of the characters. The Learning module analyzes the mistakes made by the Interpretation module. For each mistake, it either does a monotone adjustment, or replaces an in- frequently used template in the database, or adds a new template to the database. Monotone adjustment involves changing the thresholds or weights of some pa- rameters of the feature vectors. By checking for con- flicts with history and reasoning involving likelihoods, it selects the option which will improve the system the most. Results To-Date The Interpretation module, the Partition module and the Learning module have been developed. The system is under integration testing. References 1. Leibniz System for Logic Programming, Version 4.0, Leibniz, Plano, Texas 75023, 1994 2. Laempel System, Version 1.1, University of Texas at Dallas, Richardson, Texas, 75083-0688, 1995 Student Abstracts 1403 From: AAAI-96 Proceedings. Copyright © 1996, AAAI (www.aaai.org). All rights reserved. | 1996 | 237 |
1,885 | SplitNet: A Dynamic Hierarchical Network Model Jiirgen Rahmel University of Kaiserslautern, Centre for Learning Systems & Applications PO Box 3049, 67653 Kaiserslautern, Germany e-mail:rahmel@informatik.uni-kl.de Graph Properties and Retrieval We investigate the information that is contained in the structure of a topology preserving neural network. We consider a topological map as a graph G, propose cer- tain properties of the structure and formulate the re- spective expectable results of network interpretation. The scenario we deal with is the nearest-neighbor approach to classification. The problems are to find the number and positions of neurons that is useful and efficient for the given data and to retrieve a list L of m nearest neighbors (where r-r1 is not necessarily known in advance) for a presented query q that is to be classified. First, we assume a complete storage of data records in the graph G, i.e. each data record is represented by a neuron, and a perfect topology preservation, which means that an edge between neurons Ni and Nj is in G iff Ri n Rj # 8 where Ri denotes the Voronoi region of node 2ri. Thus, the graph corresponds to the Delaunay triangulation of the nodes in G. For this situation, we can formulate an algorithm that is complete at any stage of its incremental retrieval. Considering complete storage but imperfect topol- ogy preservation, we deal with a subgraph of the above mentioned Delaunay graph. We use a topographic function (Villmann et al. 1994) to measure the topol- ogy preservation and describe it by the characteristic number t+ which is the size of the largest topological defect. We can reformulate the previous retrieval algo- rithm for this case and again its completeness can be shown. The efficiency of the algorithm depends expo- nentially on the value oft+ , so a good topology preser- vation of the network is needed. However, if incomplete storage is investigated, we can show that we have to restrict the neuron distribu- tion in the data space. If we use a quantizing method, we can minimize the probability of incompleteness of the retrieved list of nearest neighbors to a given query. Guided by these insights, we developed the SplitNet model that provides interpretability by neuron distri- bution, network topology and hierarchy. The SplitNet Model SplitNet is a dynamically growing network that cre- ates a hierarchy of topologically linked one-dimensional Kohonen chains (Kohonen 1990). Topological defects in the chains are detected and resolved by splitting a chain into linked parts, thus keeping the value of t+ fairly low. These subchains and the error minimizing insertion criterion for new neurons (similar to the one presented in (Fritzke 1993)) provide the quantization properties of the network. The figure depicts the behaviour of SplitNet for a two-dimensional sample data set (left). SplitNet de- velops a structure of interconnected chains (middle) and quickly approximates the decision regions of the data records as they appear in the Voronoi diagram (left and right, bold lines). The hierarchy cannot be seen from the figure, but on one hand speeds up the training remarkably and on the other hand provides an additional way of interpreting the grown structure. Different levels of generalization and abstraction are naturally observed. References B. Fritzke. Growing cell structures. Technical Report TR-93-026, ICSI, 1993. T. Kohonen. The self-organizing map. Proceedings of the IEEE, 78(9):1464-1480, 1990. Th. Villmann, R. Der, and Th. Martinetz. A new quantitative measure of topology preservation in Ko- honen’s feature maps. In Proc. of the ICNi!!, 1994. 1404 AAAI-96 From: AAAI-96 Proceedings. Copyright © 1996, AAAI (www.aaai.org). All rights reserved. | 1996 | 238 |
1,886 | Symbolic erformance & Learning in Continuous Environ ents Seth 0. Rogers University of Michigan AI Lab 1101 Beal Avenue Ann Arbor, MI 48109-2110 USA srogers@eecs.umich.edu Introduction of We present an approach which enables an agent to learn to achieve goals in continuous environments us- ing a symbolic architecture. Symbolic processing has an advantage over numerical regression techniques be- cause it can interface more easily with other symbolic systems, such as systems for natural language and planning. Our approach is to endow an agent with qualitative “seed” knowledge and allow it to experi- ment in its environment. learned. The new action is the linear interpolation the two closest results straddling the current goal. Results Continuous environments consist of a set of quanti- tative state variables which may vary over time. The agent represents goals as a user-specified desired value for a variable and a deadline for its achievement. To de- termine the correct action given the current situation and goals, the agent maps the numbers to symbolic regions, then maps these regions to an action. The learning task of the agent is to develop these mappings. Since SPLICE does not explicitly represent domain law equations, it performs about as well in very complex nonlinear domains as simple domains. SPLICE was tested on three successively more detailed and com- plex automotive simulations. Figure 1 illustrates its performance where the agent is trying to find the right throttle setting for a desired speed. At first, the agent requires several attempts, but over time performance improves. The added detail and complexity of domains 2 and 3 does not slow the learning rate. Domain 1 - 4.5 ------ i \, 4 \ : Domain 2 Domain 3 Performance and Learning Our system, SPLICE (Symbolic Performance & Learn- ing in Continuous Environments), incorporates a per- formance module and a learning module. The perfor- mance module applies the agent’s knowledge to its situ- ation. First the agent maps the numeric variables to hi- erarchical symbolic regions of varying generality, both for the perceived state and the agent’s goals. Then the agent searches the action mappings from specific to general for a match with the symbolic situation and goals. If there is no matching action mapping, the agent uses a qualitative domain model to suggest an action. Finally the agent takes the suggested action. 0 10 20 30 40 50 60 70 80 90 100 Problems Figure 1: SPLICE performance in three domains. Conclusion and Future Instead of numerical techniques for adaptive control, SPLICE symbolically represents state variables and searches for mappings, allowing SPLICE to incorpo- For learning, the agent waits until the deadline, then rate other symbolic systems, such as a qualitative rea- evaluates the action’s effect to create a new action soning module. Our current research has focussed on mapping (a new condition and a new action). Failures learning action mappings and left region mapping to may be caused either by overgeneral goal or state con- a static binary division scheme. We are extending ditions. In the first case, the new condition is the most SPLICE in a number of dimensions, including multiple general goal not achieved by the action. In the second goals and hierarchical control structures. Our final ob- case, the new condition is the most general state region jective is to demonstrate SPLICE in complex realistic different from the situation when the action was first environments, such as airplane flight. Student Abstracts 1405 From: AAAI-96 Proceedings. Copyright © 1996, AAAI (www.aaai.org). All rights reserved. | 1996 | 239 |
1,887 | ework for Plot Control i nteractive N. M. Sgouros, G. Papakonstantinou, sanakas Department of Electrical & Computer Engineering National Technical University of Athens Zographou Campus, Greece, 157 73 sgouros@dsclab. ece. ntua. gr Abstract This paper presents a framework for plot control in interactive story systems. In this framework, the user takes the place of the main character of the story, the protagonist. The rest of the cast consists of discrete characters, each playing a specific role in the story. A separate module in this system, the plot manager, controls the behavior of the cast and specifies what the protagonist can do. The story plot is dynamically shaped by the interference between cast members and their social interactions. The system accepts as input a story map which provides the main metaphor for organizing the plot and localizes the interaction of the protagonist with the rest of the cast. We are implementing this framework in PEGASUS, an interactive travel story environment for G-reek mythology. Introduction Research in interactive story systems aims to create a new computer-based art form, providing experiences that are both meaningfUlly interactive and good stories (Waters 1995). Current interactive story systems can be classified into two categories: (i) story graphs and (ii) simulated worlds. In a story graph the user follows links between predefined episodes. In a simulated world, the user interacts with computer-simulated characters in a virtual environment. Unfortunately, story graphs are only minimally interactive, while, in the majority of cases, the interaction with a simulated world is not coherent enough and it does not have a temporal structure that could classify it as a story. Therefore, research in plot control is crucial for further development of interactive story systems. Travel narratives are stories in which the main character progresses through a series of episodes or encounters, until it reaches a final destination. Some of the oldest and most popular story forms can be classified as travel stories (e.g. the Odyssey, Alice in Wonderland, etc.). The fact that a lot of travel stories have weakly constrained plots ailows their presentation in highly interactive forms (e.g. rhapsodies in ancient Greece). Consequently, we believe that the richness of material that travelogues can encompass, the nature of their plot, and 162 Art & Entertainment the wealth of characters they can support, makes them ideal for the creation of computer-based interactive stories. This paper presents a framework for plot control in interactive story systems. Figure 1 shows the framework structure. The main module in this system is the plot manager (PM). PM shapes the protagonist’s interaction with the story system by controlling what the cast members do and specifying what the user can do. PM consists of a set of rules for social action, the specifications for each character role in the story, and a user interface manager. The story plot is dynamically shaped by the interference between cast members and their social interactions. Cast members interfere with each other, either by managing resources that influence the service of their goals, or by reacting to social norms or values :vant to their actions. Local Plot ,*- Plot Manager r Transfer Plan I Figure 1: Architecture of the interactive story environment. PM accepts as input a story map consisting of a set ol points (Pl, P2 and P3 in the figure). This map provides the main metaphor for understanding the narrative. Furthermore, it serves as the organization for storing plot control and background knowledge structures. More specifically, each location in the map has a plot structure associated with it. This structure contains the local cast members, their roles and relations, and it is activated when the user reaches this location. Furthermore, pairs of points in the map have transjir plans associated with them. These describe the conditions under which the user can move between these points. In From: AAAI-96 Proceedings. Copyright © 1996, AAAI (www.aaai.org). All rights reserved. addition, they specify the resources required for the execution of the transfer plan and ways for acquiring them. At each point in the story, PM communicates with the user and coordinates the interaction between its different subsystems. This framework strikes a balance between opposing features such as interactivity and plot control. In particular, it allows the user to have a lot of control over what happens, by taking the place of the main character of the story and deciding on its current set of goals and actions. On the other hand, it controls the behavior of the cast using constraints derived from their role specifications and a set of social action rules that govern their social interactions. The rest of this paper is organized as follows. Section 2 describes the methods by which character interference takes place in the story environment, along with the specifications for the roles played by the cast. Section 3 describes the rules for social action in the PM. Section 4 describes the user interface. Section 5 presents the algorithm for the plot manager, while section 6 presents an example of a story created in PEGASUS and gives some details on the implementation of this environment. Finally. section 7 presents related work, while section 8 is a conclusions and future work section. ales There are two types of plot interference between cast members; goal and normative. In goal interference a character X tries to influence the service of the goals of some other cast member Y in any of the following ways: e Favorable Interference. X helps Y through the acquisition or use of exceptional resources or plans that may serve Y’s goals. Examples in PEGASUS include X using weapons endowed with special powers (e.g. the shield of Athena) on the side of Y during a battle, or X helping Y win the support of Poseidon, the sea god, if Y is to embark on a sea journey. e Unfavorable Interference. In order to make life difficult for Y, X introduces two kinds of problems affecting Y’s resources and social relations. 1. Loss. Possible choices include theft, breakdown or physical destruction of resources (e.g. fire). 2. Shortages. X creates shortages over necessary resources for Y. These may range from forbidding their use, if X plays the king role, to forcing Y to share these resources with other cast members. In normative interference, a character X seeks to influence positively or negatively the compliance of Y to a set of social norms or values. In the case of positive influence, X checks whether Y’s behavior violates any moral, or legitimate rules. If any transgression is observed, Y is notified and proper punishment is planned by X. In the case of negative influence, X tries through ingratiation or reciprocity to lure Y away from compliance. The story map assigns a set of thematic roles to each cast member (except for the user). Each character pursues only types of interference consistent with its role. Currently we have developed seven thematic roles. The Judge. This role seeks to enforce legitimate power over the story characters. It checks for transgressions of character behavior from the law and enforces penalties in cases of deviation. The judge pursues positive normative interference. The King is concerned with political aspects of the behavior of story characters. It checks for political repercussions of each character’s actions. These can include conflicts over public resources, disagreement over authority rulings or public policies. In these cases the king is the voice of the political establishment and tries to maintain the status quo. Furthermore, the king can issue new orders with which the rest of the cast should comply. Both goal and normative interference are consistent with this role. The Rebel. The purpose of this role is to oppose the actions of the political establishment. It disagrees with what the king proposes and uses unfavorable interference strategies to impede the king’s goals. This may bring him in opposition with the judge and the priest as well. The rebel pursues negative normative interference. The Priest. This role has the goal of making the characters comply with religious norms. The priest suggests to the protagonist plans consistent with religious beliefs and practices. Furthermore, it informs the user of divine attitudes regarding its actions. Finally, the priest evaluates character behavior in terms of religious beliefs. This role pursues positive normative interference. The Messenger. The goal of this role is to inform the user on off-stage developments. PM uses these character types to present to the user a coherent and entertaining account of the current story context. The Sage. This role offers expert advice in aid of the protagonist. PM supports different instantiations of this role based on the current story context. Thus in a battle scene the sage can be an oracle which advises the user on the battle outcome, or in the case of a sea journey it can be an experienced sailor. Only favorable goal interference for the protagonist are consistent with this role. The Villain. This role represents all things evil in the story. The villain checks for profit opportunities independent of the legitimacy or the morality of its actions. The villain practices negative normative and unfavorable goal interference. Al-t 163 In addition to these role features, PM also contains a set of conflict management rules. These are domain-specific rules that specify a set of conditions under which a character playing a given role prevails over its opponents. For example, in the case of Greek mythology, one such rule says that cast members supported by at least one god, prevail over opponents with no divine support. Social Action The behavior of each particular character is a combination of the types of interference associated with its role and its interaction with the rest of the cast. This interaction is shaped by the application of a series of rules governing social action in the story environment and described below. Cooperation The cooperation rule states that whenever two characters X and Y mutually believe they have a common goal G and complementary resources for it, then they pursue favorable goal interherence between them with respect to ti. Reciprocation The positive reciprocation rule states that character X seeks to favorably interfere with some of character Y’s goals relativized to the belief that Y has or will adopt some of X’s goals. The negative reciprocation rule describes states of mutual aggression in which X seeks to unfavorably interfere with some of Y’s goals, believing that Y has done the same thing. Cooperation and positive reciprocation leads to the formation of groups of allies in the story, i.e. characters tied with common goals or positive reciprocity commitments. PM assigns a leader to every group of allies. This is the cast member with the largest number of common goals or reciprocation commitments with the rest of the group (i.e. the one who cares most about the group). In addition, the group has an agenda consisting of the common goals of the group members. The leader assigns tasks to group members based on this agenda. Group members have to follow their task assignments and favorably interfere with the service of each goal in it. In the case of unfavorable interference from X against the protagonist, PM presents two options to the protagonist: 0 Replenish resources using the exchange rule. 0 Retaliate against X based on the negative reciprocation rule. Group Performance Each character X monitors the progress of the goal agenda The user has the option of replenishing its resources in of the group it belongs relative to its own goals. case of loss. Choosing this option leads the user to a set of Furthermore, X monitors the compliance of the rest of the possible trade agreements with characters that have the group with this agenda. If X discovers that its goals cannot necessary resources. If the user decides to retaliate against be served by the current group, then X defects from its X, it can do it either directly, by confronting X, or by group and seeks new allies. X also defects, if it discovers joining groups that oppose X’s allies. that any of the other group members (e.g. Y) unfavorably interferes with any of the goals in the agenda. In the later case, X and Y become enemies through the application of the negative reciprocation rule. Exchange The exchange rule states that character X can use the resources supplied by some other cast member Y, if he offers some other resource or service specified by Y. The exchange rule governs the way characters trade the resources at their disposal. X can override this rule through theft or deception. In both cases, X and Y become enemies according to the negative reciprocation rule. The goal of the user interface is to specify what the user can do, as the story unfolds. The user interface manager consists of two subsystems: the action menu and the storyboard. Action Menu At each point in the story, PM composes dynamically a menu of possible user actions based on the protagonist’s actions and the behavior of the rest of the cast. These actions include possible reactions to interference from other cast members, character-specific actions for interacting directly with a cast member and resource- specific actions for following transfer plans. User reactions. User reactions are consistent with the character behavior rules described in the previous section. In the case of favorable interference from another character X, PM applies the positive reciprocation rule and proposes to reward X. If the protagonist chooses this option, possible actions include joining any of the groups led by X, or providing X with resources or services necessary for accomplishing its goals. Choosing the later leads the user to a set of character-specific actions determined by the thematic roles played by X and its current goals. 164 Art & Entertainment In the case of normative interference from X, the user can choose either to obey the constraints introduced by X or to transgress the norm. a~~~te~-s~e~~~c actions. PM composes a set of actions specializing the decisions made by the protagonist using the reactions of the previous section. These actions are geared towards a specific character X. Composition takes into account the thematic role played by X, the groups X belongs to, its current goals and a set of plans stored in PM’s memory for achieving these goals. For example in PEGASUS, if the protagonist asks the help of a priest to appease some god, then this priest will suggest a sacrifice or a sponde to this god. Both actions are role-specific to the priest. Resource-specific actions. Each time the protagonist follows a particular transfer plan in the story, PM composes a set of buttons prompting the user to acquire all the resources specified in the plan. Storyboard The storyboard has three main goals: e Provide the user with a metaphor for understanding the development of the story. 8 Give to the user concise directions for navigating in the story environment. e Dramatize character behavior. The interface satisfies all these requirements by diving the screen space into distinct functional areas (Figure 2). The first area (upper left of Fig. 2) depicts the story map, which in PEGASUS coincides with a geographical map of the Aegean.. This graphical description indicates the current location of the protagonist, its final destination, the places he has already visited and, depending on the current context, possible intermediate destinations. Whenever the user decides to move between places, the user interface instantiates the transfer plans connecting these points. In PEGASUS, for example, the map contains a Take-Sea-Journey-Plan with the resources shown in Figure 3c for traveling between mount Pelion and Troy, two places lying on opposite coasts in the Aegean. The second area (upper right of Fig.2) displays the action menu composed by PM. This is a button-based interface that provides the user with alternatives for interacting with the rest of the cast. Finally, PM uses the rest of the screen for dramatizing character behavior using a set of multimedia techniques (e.g. animation, still images, text clips etc.). Figure 2: Screen design for the PEGASUS interactive lot Manager Algorithm PM is controlled by an event-driven algorithm, with the story map as the focus of activity. In particular, when the user chooses to move to a point in the map, PM executes the following commands: 1. Instantiate the transfer plan connecting the two points. 2. Instantiate the local plot structure at the destination point. 3. Run role and social action rules. 4. Update the user interface. When the user reaches its current destination, PM executes the following sequence in a loop: 1. Update the user interface. 2. Run role and social action rules. 3. Run conflict management rules. The loop terminates when the user decides to move to another point in the map. An Example In the beginning, PEGASUS displays a text clip that sets the context for the current story. According to this introduction, the gods warn Cheron, the wisest of the Centaurs, that Achilles, his favorite grandson, will die in the battle of Troy. Worried about the fate of the famous hero, he asks the user to help him make the journey from its home, in mount Pelion, to Troy, to protect Achilles. When the user accepts this invitation the following things happen. PM instantiates the plot structure for Pelion (see Figure 3a). This plot structure suggests that Cheron is the only cast member active in Pelion. Furthermore, it suggests that Cheron is an ally of Achilles and that the protagonist and Cheron have a common goal Art 165 to protect Achilles. These goal statements satisfy the cooperation rule, therefore PM creates the first group in the story with the members and the agenda shown in Figure 3b. Finally, a button showing Troy on the map appears and the transfer plan from Pelion to Troy is instantiated. This plan suggest that our heroes should travel by boat. PM composes and presents a text clip on screen in which Cheron, in its sage role, suggests to the user a sea journey. In addition, PM creates a button for this option in the interface. (defPlotStructure Pelion Cast ((Sage Cheron)) 6) G&Relations ((Allies Cheron Achilles) (Goal User (Protect Achilles)) (Goal Cheron (Protect Achilles)))) (Group #:groupl (Cheron User)) (Goalff:groupi (Protect Achilles)) (defPlanResources (Take-Sea-Journey ?start ?dest) (b) (c) ((Resource Ship (Take-Sea-Journey ?start ?dest)) (Resource Crew (Take-Sea-Journey ?start ?dest)) (Resource (Supply Food) (Take-Sea-Journey ?start ?dest)) (Resource (Supply Water) (Take-Sea-Journey ?start ?dest)) (Resource (Support Poseidon) (Take-Sea-Journey ?start ?dest)))) (defPlotStructure Corinth (4 Cast ((Priest Anacleon Poseidon Corinth) (King Eumeneas Corinth) (King Agamemnon Mycinae) (Sage Cheron)) CastRelations ((Allies Eumeneas Agamemnon) (Allies Cheron Achilles) (Enemies Achilles Agamemnon))) Figure 3: Plot elements in PEGASUS. When the user clicks on this option PM instantiates the Take-Sea-Journey plan and activates the structure representing its prerequisites (Fig. 3~). Based on this structure, PM composes an action menu that prompts the protagonist to acquire all these resources. One of the prerequisites for this plan is that Poseidon, the sea god, should bless this journey, When the user clicks the button corresponding to this option, the goal of worshipping Poseidon is pushed onto the agenda for #:groupl. Furthermore, PM composes a text clip in which Cheron, in his sage role, suggests to the user possible Poseidon temples. The location of these temples automatically appear as buttons on the map. Let us assume that user clicks on Corinth, a big port near Athens with one of the biggest Poseidon temples. Then PM loads the plot structure associated with it (Fig. 3d) and an icon on the map starts moving from Pelion to Corinth. This plot structure suggests that there are four cast members active in Corinth, Anacleon the local priest for Poseidon, Eumeneas the king of Corinth, Agamemnon the king of Mycenae and Cheron as the sage. Furthermore, it suggests that Eumeneas and Agamemnon as well as Cheron and Achilles are allies, while Achilles and Agamemnon are enemies. These relations and #group1 ‘s goal which is to protect Achilles, an enemy of Agamemnon, trigger the negative reciprocation rule. Based on this rule, Eumeneas opposes #:groupl and seeks to unfavorably interfere with its goals. Eumeneas notices that a shortage type of interference is possible and, as a king, issues an order forbidding Cheron and the user to worship Poseidon in Corinth. The priest on the other hand, interprets the royal order as a transgression from the religious belief that everyone has the right to worship the gods. As a result, Anacleon pursues positive normative interference and seeks to oppose this order. PM uses its conflict management rules and decides that Anacleon will prevail in this conflict because he is supported by Poseidon, while Eumeneas receives no divine support. Then it instantiates the Wrath- of-Poseidon dramatic script which involves sound effects of a sea storm. Finally, it informs the user that Poseidon had its way and now Eumeneas allows our heroes to worship the sea god. The story continues and our heroes try to obtain all the resources specified in the Take-Sea- Journey plan. PEGASUS runs on a Windows PC. The plot manager is built on top of an ATMS-based rule engine written in C-++, similar to the one described in (Forbus & de Kleer 1993). The multimedia interface was built using Toolbook. a multimedia authoring system for the PC. We have implemented a Dynamic Link Library (DLL) in Windows that allows PM to communicate with Toolbook. Currently, we have implemented the user interface manager, the cooperation and reciprocation rules and the specifications for the king and the priest roles in the PM. We work on implementing the rest of the framework. Related Work Recently, there has been considerable research in the creation of believable interactive characters. This work has concentrated mainly on portraying the emotional state of these characters (Bates 1994) on supporting mll-body interactive video environments (Maes 1995). or on developing directed improvisation paradigms in which computer characters improvise a joint course of behavior following users’ directions (Hayes-Roth & Brownston & Sincoff 1995). Our research complements this work, focusing on effective plot control and higher level social interactions between story characters. Our model for the social behavior of story characters views social action as a complex phenomenon emerging from the interaction between cognitive agents (i.e. agents endowed with cognitive representations of goals and the capacity to pursue them). This is a view shared by many social and cognitive psychology researchers (Castelfranchi & Conte 1995, Hogg & Vaughan 1995, Ortony et al. 166 Art & Entertainment 1988) and forms the basis for constructing the social action rules in this framework . Map or labyrinths have been used as the primary metaphor for organizing stories in hypertext, MUD or MOO systems (Barrett & Redmond 1995). We have adopted this metaphor in our work and showed how it can be used for interactive story generation. Storytelling systems have been constructed for use in educational applications (Schank & Fano. 1992). These programs support sophisticated indexing mechanisms that allow the user to find and access stories relevant to its needs, from a large database. Our work provides a framework that can enrich the storytelling capabilities of these systems by allowing for interactive renditions of the stories they deliver. Conclusions & Further Wor We have described a framework for plot control in interactive story systems. In this framework, the story plot is dynamically shaped by the interference between cast members and their social interactions. A separate module in this system, the plot manager, controls the behavior of the cast using a set of specifications for the roles played by the cast members and a set of rules for social action. Furthermore, the plot manager specifies what the user can do via the user interface. As a result, the system is meaningfully interactive while achieving adequate plot control. We are extending this framework by developing higher level plot control techniques that will preserve thematic coherence and prevent the role and social action rules from overgeneration. These techniques consist of local character behavior filters that restrict the set of possible actions of the cast to those that are ‘relevant’ to the current plot. Moreover, we are developing more complex social action rules that describe concepts such as altruism or love and we are extending the repertoire of character roles to enrich the character types supported in this environment. Finally, we are exploring ways of incorporating plot elements from traditional story environments in our interactive system, based on the work cited in (Lehnert 1981, Lakoff 1972). This work is part of a bigger project for developing the next generation of electronic books. We are cooperating with pedagogical institutions in the University of Athens to integrate our framework into a series of activities that foster creativity and enhance the enjoyment of standard reading material. Ackuowledgements This work has been partially funded by the Greek Secretariat for Research & Technology (GSRT). eferences Waters, R. C. 1995. The 1995 AAAI Spring Symposia Reports; Interactive Story Systems: Plot and Character, AI Magazine, 16(3): 8-9. Barrett, E., Redmond, M. eds. 1995. Contextual Media. Cambridge, Mass.: MIT Press. Bates, J. 1994. The Role of Emotion in Believable Agents, Communications of the ACA4, 37(7): 122-125. Conte, R., and Castelfranchi, C. 1995. Cognitive and Social Action. London: UCL Press. Forbus, K. D., and de Kleer, J. 1993. Building Problem Solvers. Cambridge Mass. : MIT Press. Hayes-Roth, B., Brownston, L., Sincoff, E. 1995. Directed Improvisation by Computer Characters, Technical Report, KSL-95-04, Dept. of Computer Science, Stanford Univ. Lakoff, G. 1972. Structural Complexity in Fairy Tales, The Study ofMan 1: 128-150. Lehnert, W. 1981. Plot Units and Narrative Summarization, Cognitive Science, 4:293-33 1.. Maes, P. 1995. Artificial Life meets Entertainment: Lifelike Autonomous Agents, Communications of the ACM, vol. 38( 11). Hogg, M. A., and Vaughan, G. M. 1995. Social Psychology: An Introduction. London: Prentice Hall. Schank R., and Fano A. 1992. A Thematic Hierarchy for Indexing Stories in Social Domains, Technical Report, # 29, The Institute for the Learning Sciences, Northwestern Univ. Ortony A., Glore G. L., Collins A. 1988. The cognitive structure of emotions. New York: Cambridge University Press. Art 167 | 1996 | 24 |
1,888 | Effects of local information on group behavior Shounak Roychowdhury, Neeraj Arora, & Sandip Sen Department of Mathematical & Computer Sciences, University of Tulsa 600 South College Avenue, Tulsa, Ok 74104-3189. e-mail: {roy,arora,sandip}@euler.mcs.utulsa.edu Researchers in the field of Distributed Artificial In- telligence have studied the effects of local decision- making on overall system performance in both cooper- ative and self-interested agent groups (Bond & Gasser, 1988) . The performance of individual agents depends critically on the quality of information available to it about local and global goals and resources. Whereas in general it is assumed that the more accurate and up-to-date the available information, the better is the expected performance of the individual and the group, this conclusion can be challenged in a number of sce- narios. The populace in human societies tend to look for op- portunities and search for better opportunities in their environment (Bartos, 1967) . The theory of migration in social behavior and occupational mobility suggests that the stability of the population depends on how an individual chooses its action based on the prevail- ing circumst antes. As agent designers, we are faced with the problem of developing decision mechanisms that allow agent societies to stabilize in states where system resources are effectively utilized. In this research, we focus on a particular aspect of distributed decision-making: the effect of limited global knowledge on group behavior. The research question that we are asking is the following: Can lim- ited local knowledge be a boon rather than a bane in a multiagent system ? To investigate this issue we use a resource utilization problem where a number of agents are distributed between several identical resources. We assume that the cost of using any resource is directly proportional to its usage. This cost can be due to a delay in processing of the task in hand, or a reduc- tion in the quality of the resource due to congestion. Hence, there is a justified urge in agents to seek out and move to resource with lesser usage. Other researchers have shown that such systems can exhibit oscillatory or chaotic behavior where agents move back and forth between resources (Hogg & Huberman 1991; Kephart, Hogg & Huberman 1989) resulting in ineffective uti- 1406 AAAI-96 lization of system resources. Whereas asynchrony, het- erogeneous agent decision mechanisms, etc. has been suggested as possible means for solving the instabil- ity problem, our proposed solution of using locally dif- fering global views is a novel mechanism to introduce asymmetry in group decisions that expedites group sta- bility. We have developed a decision mechanism to be used by individual agents to decide whether to continue us- ing the same resource or to relinquish it in the above- mentioned resource utilization problem. We show that a spatially local view of an agent can be effectively used in a decision procedure that enable the system to quickly converge to a stable optimal global state (in terms of effective resource utilization). In addition, in- creasing the information available to an agent increases the time taken to reach the desired equilibrium state. We explain this phenomenon with a probabilistic anal- ysis. This analysis also suggests a promising line of fu- ture work where adaptive agents use varying amounts of global information to further accelerate convergence. References Otomar J. Bartos. Simple Models of Group Behavior. Columbia University Press, New York, NY, 1967. Alan H. Bond and Les Gasser. Readings in Distributed Artificial Intelligence. Morgan Kaufmann Publishers, San Mateo, CA, 1988. Tad Hogg and Bernard0 A. Huberman. Control- ling chaos in distributed systems. IEEE Transactions on Systems, Man, and Cybernetics, 21(6), December 1991. (Special Issue on Distributed AI). J. 0. Kephart, T. Hogg, and B. A. Huberman. Dy- namics of computational ecosystems: Implications for DAI. In Michael N. Huhns and Les Gasser, editors, Distributed Artificial Intelligence, volume 2 of Re- search Notes in Artificial Intelligence. Pitman, 1989. From: AAAI-96 Proceedings. Copyright © 1996, AAAI (www.aaai.org). All rights reserved. | 1996 | 240 |
1,889 | Automated Formulation of Constraint Satisfaction Mihaela Sabin and Eugene C. Freuder Department of Computer Science University of New Hampshire Durham, New Hampshire 03824, USA mcs,ecf@cs.unh.edu A wide variety of problems can be represented as constraint satisfaction problems (CSPs), and once so represented can be solved by a variety of effective al- gorithms. However, as with other powerful, general AI problem solving methods, we must still address the task of moving from a natural statement of the prob- lem to a formulation of the problem as a CSP. This research addresses the task of automating this problem formulation process, using logic puzzles as a testbed. Beyond problem formulation per se, we address the issues of effective problem formulation, i.e. finding for- mulations that support more efficient solution, as well as incremental problem formulation that supports rea- soning from partial information and are congenial to human thought processes. A CSP is defined by a set of variables with their associated domains of values and a set of constraints which restrict the combinations of values allowed. The example in Fig. 1 shows a logic puzzle text and a cor- responding constraint network representation. In CSP terms, associated with the introductory portion of the logic puzzle there are 8 variables, each variable with the same domain of values, the tower positions. All the variables are nodes in the constraint network with the values labeling them. Due to the structure of the problem (no two acrobats, as well as no two items, cor- respond to the same position in the tower), the CSP variables are partitioned into two cliques, Acrobats and Items, with disequality constraints (#) between every pair of variables in each clique (drawn as edges in the constraint network and marked as initial con- straints in the figure). ~ E ; ; ; ‘ i ~ , . ~ ~ ; . ~ e x ; ‘ a n b ” . i i Zeke are 4 acrobats; : each wearing one of ; i the4 items: chaps, : i holster,.hat, or ve+, ; : and berng placed in : i the lst, j2nd, 3rd,or i ; 4thposltlon of the : : tower they form I such that: i .“..“....““.........~-.-......~.~. Loe1c I .-- -- 11 Jed isnot the lst, but ; is abavethe one with ; thehat. 2Zekedoesrwt wear / the holster. F3The one in the vest is : not the 1st. !4Theone in the chaps is f below Zeke, but above cc%. ..-...-f _.-_ ..-...-...-........ 1 ‘uzzle Text The current implementation of the translation tool handles the translation of the clues into the CSP con- straints. The translation scheme recognizes patterns such as not, above and below, that match logic puzzle clues in the input, and applies the translation rules to generate corresponding clue constraints. The figure shows the binary constraints as bold continuous lines, drawn as edges and labeled #, < and >, and the unary constraints as deleted (hashed) values. More efficient CSP formulations are possible by ex- ploiting the inherent structure of the problem (e.g. the two cliques in our example) or the semantics of the constraints. The enhanced translation scheme defines specialized consistency functions attached to each con- straint. As we build the representation, with each clue parsed, corresponding local consistency can be per- formed that may add inferred constraints, examples of which are shown in the figure as bold, dotted lines. Postponing search as much as possible while locally propagating the available, partial information seems to reflect human problem solving behavior. The acquired reasoning power is encoded in the form of restricted do- mains and additional constraints which support more efficient problem solving. Acknowledgments This material is based on work supported by the National Science Foundation under Grant No. IRI- 9207633 and Grant No. IRI-9504316 and on work sup- ported by Digital Equipment Corporation. This work profited from discussion with Mohammed H. Sqalli. We thank Nancy Schuster and Dell Magazines for per- mission to reproduce material from Dell Logic Puzzles. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..~.............~................................. Constramt Network Figure 1. From logic puzzle statement to CSP formulation i initial constraint Student Abstracts 1407 From: AAAI-96 Proceedings. Copyright © 1996, AAAI (www.aaai.org). All rights reserved. | 1996 | 241 |
1,890 | ynamic Constraint- lanning in Moninder Singh” Dept. of Computer and Information Science University of Pennsylvania, Philadelphia, PA 19104-6389 msingh@gradient.cis.upenn.edu This research deals with planning in domains with dynamically changing, multiple, interacting goals. What distinguishes this work from reactive planners (e.g. (Firby 1987)) is the fact that the goals for which planning is done are not known in advance; rather, goals are formed and change rapidly during the plan- ning process itself. Although planners that produce appropriate plans exist for such domains (Rymon et al. 1993), we want a planner that also provides a basis for explaining why some action is chosen over another or why some goal is no longer relevant etc., which is necessary for effective decision support (Gertner 1994). I am developing an efficient, 3-level, dynamic con- straint based planner for one such domain, trauma management. The three levels show how plans are naturally formed in this domain. The top level cor- responds to goals, the second level corresponds to the various, alternative procedures that can be used to ad- dress these goals, while the third level corresponds to the actions that constitute these procedures. Different kinds of constraints are added at each level. For exam- ple, urgency constraints (e.g. “shock” must be treated first) hold between goals, while precedence constraints (e.g. perform IVP before arteriogram) hold at the ac- tion level. Constraints at higher levels are inherited by the lower levels. The network is dynamically up- dated by adding/deleting nodes and/or modifying con- straints as goals are created, discarded or achieved. If an action/procedure cannot be done, or a goal cannot be addressed, the corresponding nodes are deactivated, but not removed. Then, if the situation changes later, the nodes are reactivated and again considered during the planning process. This structure, similar to dy- namic constraint networks (Dechter et ad. 1988), offers several distinct advantages. First, by maintaining consistency and recording jus- tifications for elimination/selection of actions, easy cri- tiquing is facilitated. Consider a scenario involving, among others, the goals of treating a tension pneu- mothorax (TP) as well as ruling out a pericardial tam- ponade (PT) and a renal injury (RI) in a patient in shock. Urgency constraints are added requiring that TP, RI and PT be addressed before realizing other goals or performing time consuming actions. For treat- *This work is funded by an IBM Cooperative fellowship. 1408 AAAI-96 ing the TP, a chest tube is the only option. However, making the network consistent and recording justifica- tions not only causes a needle aspiration to be chosen for diagnosing the PT (over the normally preferred but lengthy ultrasound due to urgency constraints), but also enables the system to explain its choice. Second, the three level structure with different, dy- namic constraints at each level makes the planning pro- cess very efficient. In the above example, for ruling out a RI, an IVP is chosen over the normally preferred CT-scan (never done if patient in shock). Now, sup- pose that the planner considers scheduling an arteri- ogram to address a non-urgent goal. Although, prece- dence constraints require that an arteriogram precede an IVP, urgency constraints requiring the IVP to be performed first cause the arteriogram to be deacti- vated. This information is propagated upwards, thus deactivating all procedures involving an arteriogram. This causes alternative procedures to be chosen for all goals corresponding to the deactivated procedures; if there is no other procedure to satisfy some goal, the goal is left unaddressed. If planning is done using back- tracking or a ‘flat’ constraint network, this would not be discovered until the later goals are planned for. Third, the current plan can be easily modified to in- corporate new information and/or goal changes, with- out re-planning from scratch. In the above example, assume that a chest tube relieves the shock. This causes the urgency constraints to be eliminated. Since nodes corresponding to the ultrasound, arteriogram and IVP (along with the relevant constraints) were only deactivated and not removed from the network, they are simply reactivated. This enables the plan- ner to easily modify the current plan and schedule the ultrasound (preferred over needle aspiration) followed by an arteriogram (since CT-scan and IVP can now be done later) and a CT-scan (preferred over IVP). References Dechter, R. and Dechter, A. 1988. Belief maintenance in dynamic constraint networks. In Proc. AAAI, 178-183. Gertner, A. 1994. Ongoing critiquing during trauma man- agement. AAAI Spring Symp. on AI in Medicine. Firby, R. J. 1987. An investigation into reactive planning in complex domains. In &XX. AAAI, 202-206. Rymon, R., Webber, B., and Clarke, J. 1993. Progressive horizon planning. IEEE Trans. on SMC 23(6):1551-1560. From: AAAI-96 Proceedings. Copyright © 1996, AAAI (www.aaai.org). All rights reserved. | 1996 | 242 |
1,891 | locking as a iplav Srivastava Advisor: Subbarao Kambhampati Department of Computer Science and Engineering Arizona State University, Tempe AZ 85287-5406 (biplav,rao) @ asu.edu Partial order planners commit only to the relative posi- tions of the steps in the plan, and leave both their absolute positions as well as the relative distance between the differ- ent steps unspecified until the end of planning. Although this is seen as an advantageous feature of partial order planning, it can sometimes be a mixed-blessing. Because the relative distances between the steps are unspecified, any unordered step may be able to come between any existing steps and cause interactions and the planner may spend inordinate ef- fort considering all possible interleavings of the subplans of the individual goals. This happens in cases where top-level goals are serializable but have long sub-plans which have internal interactions, plan-space planners would consider all simple-establishments and threats between steps of a the subplan of a top-level goal gi (represented by P,;) and Ps3 which could affect its performance drastically. State-space planners, on the other hand, fix both the distance and po- sition, and this is often more commitment than is needed, causing extensive backtracking (Barrett & Weld 1994). We are investigating a middle ground between these two approaches, that we call “blocking” refinement. The essen- tial idea is to work on individual subgoals one after another in LIFO fashion using the plan-space refinements. Once the complete subplan for one individual toplevel goal is constructed, the steps comprising that goal are “blocked” together by posting contiguity constraints between steps (Kambhampati & Srivastava 1995). To ensure complete- ness, we also leave the un-blocked version of the plan in the search space. Since no other steps can come between blocked steps, the planner will not waste time considering all possible interleavings of the subplans. Once all the top level goals are handled this way, any inter-block interac- tions between the blocked subplans are resolved (thereby implicitly sequencing the subplans of the individual goals). Notice that in contrast to partial order planning this ap- proach fixes the distance between two steps in Psi but not the position of Psi with respect to Pg, . Variant of D’S2 domain OP Prec Add Del Ai (i odd) Ii Mi,he hf Ai (i even) Ii l&,hf he Bi (i odd) Mi,he Gi he Bi (i even) M;, hf Gi hf We implemented blocking on Universal Classical Plan- ner (UCP). The implementation required distinguishing DlS2 Variation DiS.2 Variation NWBBex.B”dBd Timtaken lWc.m 0 I Fi ure 1: Plots illustrating the performance of blocking in D F S2 variation. Blocking performed the best among PS, Blocking, FSS and MEA between inter-step, inter-block, and block-step conflicts in the plan and handling them at appropriate times. We tested the effectiveness of this approach in a variation of the D’ S2 domain used in (Barrett & Weld 1994). Our domain con- tains a set of goals of the form g; which can be achieved by action Bi. Bi in turn needs condition A& given by action Ai. Ai also provides he to Bi, and Bi deletes he. Because of this latter condition, the subplans for individual toplevel goals will have many interactions, even though the overall plans are serializable. Figure 1 shows that blocking of steps of a top-level goal in a serializable domain improves performance over plan-space (PS) refinement. As UCP allows forward state-space (FSS) and means-ends analysis (MEA) refinements, we also tested them on this domain. Blocking not only visits lesser nodes than FSS, MEA and PS refinements, but also takes lesser time. These results show the promise of the blocking refine- ment as a middle-ground between state-space and plan- space approaches in terms of step order commitment. In our future work, we will attempt to provide a better char- acterization of the classes of domains which could benefit from blocking, using theoretical and empirical analyses. eferences Barrett, A., and Weld, D. 1994. Partial order planning: Evaluat- ing possible efficiency gains. Artificial Intelligence 67:71--l 12. Kambhampati, S., and Srivastava, B. 1995. Universal classical planning: An algorithm for unifying state space and plan space planning approaches. Current trends in AI Planning: EWSP 9.5, IOS Press. Student Abstracts 1409 From: AAAI-96 Proceedings. Copyright © 1996, AAAI (www.aaai.org). All rights reserved. | 1996 | 243 |
1,892 | Experimentation-Driven Operator Learning Kang Soo Tae Department of Computer Science and Engineering University of Texas at Arlington Arlington, TX 76019, Box 19015 tae@cse.uta.edu Expert-provided operator descriptions are expen- sive, incomplete, and incorrect. Given the assumptions of noise-free information and an completely-observable state, OBSERVER can autonomously learn and re- fines new operators through observation and practice (Wang 1995). WISER, our learning system, relaxes these assumptions and learns operator preconditions through experimentation utilizing imperfect expert- provided knowledge. Our decision-theoretic formula calculates a probably best state S’ for experimentation based on the imperfect knowledge. When a robotic ac- tion is executed successfully for the first time in a state S, the corresponding operator’s initial preconditions are learned as parameterized S. We empirically show the number of training examples required to learn the initial preconditions as a function of the amount of in- jected error. The learned preconditions contain all the necessary positive literals, but no negative literals. Unless given a rule like arm-empty --+ +&ding(X), a robot may believe a state, {(arm-empty), (holding bo~j}, to be possible due to unavoidable perceptual alias. The plan execution in this state is unreliable. To make a planner robust in a noisy state, WISER learns constraining negative preconditions by inter- connecting two types of logic systems used in planning systems. The state representation uses two-value logic plus the Closed-World Assumption and the operator representation uses three-value logic. Let L represent all the predicates known to WISER and P the predi- cates true in S. By CWA, N, the predicates not true in S, is defined as {L - P}. WISER induces precondi- tions from S* = {P U TN}, which prevent inconsistent actions in a noisy state. Let the instantiated operators, iopi and iopj, be ob- tained from an operator op by instantiating each pa- rameter of op to the same object respectively except that one parameter is instantiated to different objects of the same type, say a for iopi and b for iopj. iqj is obtained by substituting b for a. In any state S, if the preconditions for iopi and iopj are both satisfied (or not satisfied) and their respective actions have the same (parameterized) effects on S, the two objects a 1410 AAAI-96 and b are called homogeneous in terms of op. For ex- ample, if both bowl and boz2 can be picked up by an agent, they are homogeneous to the PICKUP operator. Homogeneous objects of a type form an equivalence re- lation, partitioning the type into subtypes. If an object of a subtype is tested for the operator, no additional object of the subtype needs to be tested further. WISER adopts the abstract domain assumption that every object in a type belongs to one and only one sub- type. Therefore, we can experiment with an operator using only one object of a type for a parameter. This assumption drastically reduces the size of the search space. The abstraction of objects is inevitable in a real domain. The resulting representation is, however, in- herently incomplete. The robot must be able to adjust its initial definitions, learned under the abstract do- main assumption, in complex environments composed of heterogeneous objects by decomposing a type into many subtypes. The traditional approach, where a human-provided fixed type hierarchy is given, cannot handle this problem. We handle this type classification problem using C4.5. Some unobservable predicates, such as carri- able(X), are used in classifying a type into subtypes. An object is described as a vector of observable fea- tures. The robot initially learns overly-general precon- ditions by assuming that every object of a box type is carriable. Experimentation acts as an environmental feedback and produces positive and negative training examples for [carriable] subtype of the box type. C4.5 selects the features to distinguish between positive and negative examples. The unobservable functional con- cept is represented by structural descriptions consist- ing of observable features, satisfying the operational criterion. We empirically demonstrate the learner’s ability to generate more accurate definitions with each learning iteration. eferences Wang, X. 1995. Learning by observation and practice: An incremental approach for planning operator acquisition. In Proceedings of the 12th International Conference on Machine Learning. From: AAAI-96 Proceedings. Copyright © 1996, AAAI (www.aaai.org). All rights reserved. | 1996 | 244 |
1,893 | Hybrid Knowledge- and Databases Merwyn Taylor Department of Computer Science Univeristy of Maryland College Park, MD 20742 USA mtaylor@cs.umd.edu In the modern era, databases have been created spanning many domains. However, these databases do not contain general knowledge about their respective domains. For example, whereas a medical database could contain an entry for a patient with some medi- cal disorder, it would not normally contain taxonomic information about medical disorders, known causal agents, symptoms, etc. Collections of this sort of gen- eral information are usually called knowledge bases and powerful tools have been developed for querying these collections in complex and flexible ways. The research described in this abstract aims to develop method- ologies for merging existing databases with knowledge bases, so that the power and flexibility of knowledge base technology can be applied to existing collections of data.l In traditional DB’s, queries are limited to those that do not reference general information about the do- mains in which the DB’s are created. To support queries that reference such information, one can merge a DB created in some domain with a knowledge base of the domain to create a hybrid knowledge base / database. Merging a knowledge base with a database extends the range of queries that can be issued. In some cases, ad hoc queries can be simplified when ap- plied to the hybrid KB/DB. Besides expanding the range of queries that can be issued, a hybrid KB/DB also lends itself to research in attribute oriented data mining techniques. In this research, a frame based semantic network, Parka, developed by the PLUS group at the University of Maryland was used to facilitate the merger of a KB with a DB. As an example, we are working with a medical database of about 20,000 OB/GYN patients. The DB is a single table with 70+ columns. Using the origi- nal DB, a user could query the DB to list all patients with a particular type of infection. But if the user wanted to query the DB to list all patients with in- fections known to be caused by any form of bacteria, the query would have to be expressed as a disjunction ‘This research was supported by AF contract F49609410422 over all the infections that the user knows to be caused by bacteria. Since the DB does not contain a general concept of bacteria, the query would have to be con- structed without any references to that term. Rather it would look something like: select patient from ptable where infection = ‘Peritonitis ’ OR infection = ‘Pyelonephritis’ OR . . . This requirement limits the number of users that can issue interesting queries to those that have an extensive background in OB/GYN. With a hybrid KB/DB, a broader spectrum of users can issue interesting queries in a form much simpler than the one mentioned above. We have developed a hybrid OB/GYN KB/DB from the DB. (It contains 60,000 frames and 1.3 million assertions!) Figure 1 illustrates how the previous query can be expressed in a simpler form using Parka’s Graphical Query By Example System. Figure 1: Sample Parka Query In my current research I am developing general methodologies for merging DB’s and KB’s that allow this technique to be more widely applied. I am investi- gating techniques that automatically convert relational DB’s to semantic networks by analyzing the schemas of relational DB’s. I am also investigating techniques to automatically extract and integrate parts of existing KB’s, such as NIH’s medical term ontology, UMLS. Student Abstracts 1411 From: AAAI-96 Proceedings. Copyright © 1996, AAAI (www.aaai.org). All rights reserved. | 1996 | 245 |
1,894 | Learning Models for Multi-Source Integration Sheila Tejada Craig A. Knoblock Steven Minton University of Southern California/IS1 4676 Admiralty Way Marina de1 Rey, California 90292 { tejada,knoblock,minton}@isi.edu Because of the growing number of information sources available through the internet there are many cases in which information needed to solve a problem or answer a question is spread across several information sources. For example, when given two sources, one about comic books and the other about super heroes, you might want to ask the question “Is Spiderman a Marvel Super Hero?” This query accesses both sources; therefore, it is necessary to have information about the relationships of the data within each source and between sources to properly access and integrate the data retrieved. The SIMS information broker captures this type of information in the form of a model. All the information sources map into the model providing the user a single interface to multiple sources. Presently, models are manually constructed by human experts who are familiar with the data stored in the sources. Automation of this task would improve efficiency and accuracy, especially for large information sources. We have conducted preliminary work in automating model construction. There has been related work conducted in this area (Perkowitz & Etzioni, 1993, which has focused on learning the attributes of sources. Their approach assumes that it has an initial model of the information to be The diagram depicts a partial model of two sources. The ovals in the diagram represent classes of information and the lines describe the relationships between the classes. The two types of classes are basic classes and composite classes. Members of a basic class are single-valued elements, such as strings; while the members of a composite class are objects, which can have several attributes. Comic Book Illustrator and Super Hero Creator are basic classes; and Comic Book, Marvel Comic Book, and Super Hero are all represented as composite classes. Arrows represent the attributes of objects, and the thick lines represent the superclass/ subclass relationships between classes. 1412 AAAI-96 Our approach to learning models examines all of the data to extract the relationships needed. Learning the model is an iterative process which involves user interaction. The user can make corrections or specify an information source to be added or deleted. The model proposed to the user is generated by heuristics we have developed to derive the necessary relationships from the data. These heuristics involve creating abstract descriptions of the data. We have assumed the data is stored as tables. Properties of the data, such as type and range, are contained in an abstract description. For each column of data, an abstract description is computed. Each description corresponds to a basic class. For the basic class Super Hero Creator an example of an abstract description would be alphabetic strings with minimum length 8 and maximum 12. These descriptions are then used to help determine the superclass/subclass relationships that exist between the basic classes, like between Comic Book Illustrator and Super Potentially, all basic classes would need to be checked with every other basic class for a superclass/subclass relationship, but now only classes which have similar descriptions are tested. So, for example, the basic class Super Nero Creator would only be checked with other basic classes that contain alphabetic strings whose lengths are within the range specified in the abstract description. Our experimental results show that in every case using these restrictions reduced the running time and number of comparisions performed. We are planning to apply statistical methods to assist in constructing the model, as well as integrating a natural language knowledge base into this process to help determine whether classes are semantically related. The relationships between the basic classes will be useful in the later steps of the modeling process which are to determine the relationships between composite class. References Ambite, J., Y. Arens, N. Ashish, C. Chee, C. Hsu, C. Knoblock, and S. Tejada. The SIMS Manual, Technical Report ISVTM-95428,1995. Perkowitz, M. & Etzioni, 0. Category Translation: Learning to Understand Information on the Internet. The International Joint Conference on Artificial Intelligence, Montreal, 1995. From: AAAI-96 Proceedings. Copyright © 1996, AAAI (www.aaai.org). All rights reserved. | 1996 | 246 |
1,895 | Rabbi: exploring t e inner world t Marina Umaschi MIT Media Laboratory E15-320R, 20 Ames St. Cambridge, MA 02 139 marinau @ media.mit.edu In the oral tradition, stories were told by the elder sages in order to give indirect advice. Today most stories are told in order to entertain. While some research on storytelling systems has focused on drama/theater metaphors and adventure/mystery simulation games (Bates et al., 1995), my research emphasizes the counseling and self-awareness possibilities of storytelling. I am exploring how storytelling systems can enable people to tell and listen to personal stories in order to learn about themselves. I am developing an authoring tool that allows children to create their own storytellers in order to start thinking about the complexity of meaning in communication. In the light of this interest, I designed “Rabbi”, a conversational storytelling system that simulates a rabbi with a repertoire of Hasidic stories to offer as indirect counsel. It approaches narrative through a psycho-social theory of discourse which looks not only at the linguistic structures of the text but also at the socio-cultural context in which the contract between teller and listener occurs. “Rabbi” uses rich interaction instead of complex matching and indexing techniques (Schank et al. 198 1; Domeshek, 1992) to provide a meaningful experience for the user. Research showed that the construction of emotional believable characters is requisite to maintain the suspension of disbelief (Weizenbaum, 1976; Bates, 1995). “Rabbi” constructs his “persona” through the conversational turns in order to set a socio-cultural context without limiting the interaction and breaking expectations. System Design The system parses user input and assigns prominence to keywords. To make the match with the Hasidic story, nouns and verbs in the user’s story are augmented with synonyms, hyponyms and hypernyms found through WordNet (Miller et al 1993). Each Hasidic story in the data-base is indexed with a set of three descriptors: nouns, verbs and Commandments, that are weighted more heavily because they set up the story domain according to one of the universal values stated in the Ten Commandments. The conversational interaction is built around seven phases of an encounter with a rabbi (Schachter-Shalomi, 1991) : Conclusions People using “Rabbi” expressed their need to tell personal stories. The fact that subjects found the Hasidic stories to be coherent with their own stories, in spite of the primitive matching technique, shows that given the appropriate socio-cultural context, construction of meaningful interpretation is possible. The system need not to be intelligent, but rather appears believable by projecting a personality to the user. Current work focuses on what kind of discourse structures the system should parse in order to abstract the point made by a personal story. eferences Bates, J. et al. 1995 Interactive Story Systems: Plot & Character,. In AAAI Working Notes Spring Symposium. Domeshek, Eric. “Do the right thing: a component theory for indexing stories as social advice”. PHD Thesis Northwestern University, 1992. Miller, 6. et al. 1993.WordNet: An On-line Lexical Database. (paper found through ftp claritz.princeton.edu) Schachter-Shalomi, 2. 1991 Spiritual intimacy, a study of counseling in Hasidism, NJ: Jason Aronson. Schank, R. and Riesbeck, C. 1981 Inside Computer Understanding. Lawrence Elbaum. Weizenbaum, J. 1976.Computer power and human reason. SF: Freeman and Co. Student Abstracts 1413 From: AAAI-96 Proceedings. Copyright © 1996, AAAI (www.aaai.org). All rights reserved. | 1996 | 247 |
1,896 | Constructive Induction of Features for Planning Michael van Lent Artificial Intelligence Laboratory University of Michigan 1101 Beal Avenue Ann Arbor, MI, 48109-2110 vanlent@umich.edu Constructive induction techniques use constructors to combine existing features into new features. Usually the goal is to improve the accuracy and/or efficiency of classification. An alternate use of new features is to create representations which allow planning in more efficient state spaces. An inefficient state space may be too fine grained, requiring deep search for plans with many steps, may be too fragmented, requiring separate plans for similar cases, or may be unfocused, resulting in poorly directed search. Modifying the representa- tion with constructive induction can improve the state space and overcome these inefficiencies. Additionally, since most learning systems depend on good domain features, constructive induction will compliment the action of other algorithms. This abstract describes a system that uses construc- tive induction to generate new state features in the Tic-Tat-Toe (TTT) domain. The system generates features like win, block, and fork that are useful for planning in TTT. TTT has been chosen as an initial domain because it is simple, the features are well de- fined, and it is clear what domain knowledge has been added. Additionally, previous work on constructive in- duction in the TTT domain provides a starting point. The CITRE system (Matheus & Rendell 1989) cre- ates a decision tree with primitive TTT features (the contents of the board positions) and uses a binary and constructor to incrementally combine features selected to improve the decision tree. Domain knowledge fil- ters out less promising features. CITRE minimally im- proves classification of board positions but cannot gen- erate some types of planning features. Our system has fewer constraints and includes extensions that expand the space of constructible features. For example, n-ary conjunction (not binary) is used to combine existing features into more complex features and n-ary disjunc- tion groups symmetrical versions of the same feature. Also, constructors are applied to all pairs of features, including a new “player to move” feature, without us- ing domain knowledge as a filter. 1414 MI-96 The perfect “X win” feature is a disjunction of the 24 ways two X’s in a row can appear in TTT. Each such row is represented by a conjunction of primitive fea- tures (e.g. and (posll=X) (posl2=X) (to-move=X)). To construct new features, the system calculates the information gain of each existing feature. Due to the symmetry of TTT states, symmetrical features have equal information gains and the same parent primi- tive features and can be correctly grouped into dis- junctions (e.g. or (posll=X) (poslS=X) (pos31=X) (pos33=X)). Taking advantage of symmetrical fea- tures is one example of correcting a fragmented state space with constructive induction. The system then constructs more sets of features from the conjunc- tion of elements of the disjunctive features (e.g. and (posll=X) (posl2=X)). These sets of conjunctive fea- tures are again combined into symmetrical disjunctions by comparing information gain and parent features used. Conjunction and disjunction alternate, incre- mentally building complex features. This algorithm generates and selects features, such as win and block, which CITRE cannot generate. Win and block are each represented by 3 features covering the symmetries of the diagonal lines, the middle lines, and the outer lines. The fork and block-fork features are generated but criteria for selecting them need to be developed. This constructive induction system creates useful features for planning in TTT. An immediate next step is to apply the system to more complex domains Also, the present system inefficiently applies the conjunction constructor to all pairs of features. In the same way CITRE selects features to improve an existing deci- sion tree, our system could use planning knowledge to direct its selection of features. References Matheus, C., and Rendell, L. 1989. Constructive induction on decision trees. In Proceedings of the Eleventh International Joint Conference on Artificial Intelligence, 645-650. . From: AAAI-96 Proceedings. Copyright © 1996, AAAI (www.aaai.org). All rights reserved. | 1996 | 248 |
1,897 | Ad Hoc Attribute-Value Prediction Gabor Melli Simon Fraser University British Columbia, Canada, V5A lS6 melli@cs.sfu.ca The evolving ease and efficiency in accessing large amounts of data presents an opportunity to execute prediction tasks based on this data (Hunt, Marin, & Stone 1964). Research in learning-from-example has addressed this opportunity with algorithms that in- duce either decision structures (ID3) or classification rules (AQ15). Lazy learning research on the other hand, delay the model construction to strictly satisfy a prediction task (Aha, Kibler, & Albert 1991). To support a prediction query against a data set, current techniques require a large amount of preprocessing to either construct a complete domain model, or to de- termine attribute relevance. Our work in this area is to develop an algorithm that will automatically return a probabilistic classification rule for a prediction query with equal accuracy to current techniques but with no preprocessing requirements. The proposed algorithm, DBPredictor, combines the delayed model construc- tion approach of lazy learning along with the informa- tion theoretic measure and top-down heuristic search of learning-from-example algorithms. The algorithm induces only the information required to satisfy the prediction query and avoids the attribute relevance tests required by the nearest-neighbour measures of lazy learning. Given a data set in some domain, an attribute- value prediction query requests the prediction of an attribute’s value for some partially described event drawn from this domain. Applicable classification rules for an attribute-value prediction query are shown to be based on the exponential number of combinations of the attribute-values specified in the query. DBPre- dictor performs an informed top-down search that in- crementally specializes one attribute-value at a time to locate a maximally valued classification rule. In a sense the algorithm iteratively selects the next most relevant attribute-value for this query. If interrupted, the algo- rithm reports the best encountered classification rule to date. Given a query with n instantiated values the search is contained to n + (n - 1) + . . . + 1 = n(n + 1)/2 evaluations. We use the information-theoretic J- Measure (Smyth & Goodman 1992) to evaluate the quality of a probabilistic classification rule. A corresponding program based on DBPredictor has been developed to satisfy ad hoc attribute-value pre- 1396 AAAI-96 diction queries against SQL-based relational database management systems. The performance and accuracy of DBPredict,or is empirically tested against, both real- world and synthetic data. Furthermore the results are contrasted to the ID3, AQ15, and ITRULE algorit,hms. DBPredictor commonly requires t$wo orders of magni- tude less processing than the other algorithms for a single query. Finally, as expected, DBPredictor’s ac- curacy is equivalent to that of the other algorithms. To exemplify the increased access to large amounts of information and the opportunities provided by this technique, an interactive version of the program has be placed on the World Wide Web. Users first choose the real-world database they want, to query against, next they choose the att,ribute whose value will be predict& and finally, with the use of pull down menus, describe the events’s instantiated att,ributes. Because of the in- conclusive nature of most prediction queries based on real-world dat,abases, the generated report provides a ranked distribution of the values to expect, rather t,han only the most likely value. Several interesting future research directions are pos- sible for DBPredictor. First,, the search t,echnique could be extended to support requests for more accu- rate classification rules. Finally the caching of discov- ered rules could be used to expedite future queries and eventually to better understand the underlying model of a large data set. The DBPredictor algorithm’s flexibility and effi- ciency can support ad hoc attribute-value prediction queries against large and accessible data sets of the near future. References Aha, D. W.; Kibler, D.; and Albert, M. K. 1991. Instance-based learning algorithms. Machine Learn- ing 6:37-66. Hunt, E. B.; Marin, J.; and Stone, P. J. 1964. Exper- iments in Induction. New York: Academic Press. Smyth, P., and Goodman, R. M. 1992. An infor- mation theoretic approach to rule induction. IEEE Transactions on Knowledge and Data Engineering 4(4):301-316. From: AAAI-96 Proceedings. Copyright © 1996, AAAI (www.aaai.org). All rights reserved. | 1996 | 249 |
1,898 | &me Fargier, JMhne Lang IRIT - Universite Paul Sabatier 3 1062 Toulouse Cedex (France) Abstract Constraint satisfaction is a powerful tool for representing and solving decision problems with complete knowledge about the world. We extend the CSP framework so as to represent decision problems under incomplete knowledge. The basis of the extension consists in a distinction between control- lable and uncontrollable variables - hence the terminology “mixed CSP” - and a “solution” gives actually a conditional decision. We study the complexity of deciding the consis- tency of a mixed CSP. As the problem is generally intractable, we propose an algorithm for finding an approximate solution. Introduction Most of the time, solving a constraint satisfaction problem (CSP) means finding an assignment of the variables satis- fying all the constraints, if such an assignment exists. Such a request often relies on the assumption (generally implicit) that the variables are controllable, i.e., that the agent has the power to fix their values. However, one may think to give variables a status of uncontroZZabZe: the value of an uncon- trollable variable is fixed by the external world and thus it is outside the control of the agent. In the rest of the paper we will refer to controllable variables as decision variables and to uncontrollable as parameters. If all variables of a CSP are controllable, the purpose of the resolution of the CSP is to find a good decision which the agent may apply: we will refer to this kind of CSP (which is the most common in the CSP literature) as decision-oriented CSP. If all vari- ables are uncontrollable, the constraints represent pieces of knowledge about the possible actual values of the variables. In this case, what is expected is not necessarily to find a consistent assignment, but rather to find the actual value of each variable. This means that the resolution process has to be interpreted not in terms of decision but in terms of reasoning (as this is the case in most logical approaches for knowledge representation). We will refer to this kind of CSP as reasoning-oriented CSP. Examples of reasoning- oriented CSPs in the literature are far less numerous. For instance, CSP used for scene interpretation clearly belong in this class. The fundamental difference between these two kinds of CSPs thus lies in the interpretation of the data (variables and constraints) and in the output of the resolution pro- homas Sehiex INRA 3 1320 Castanet Cedex (France) cess; interpreting the number of solutions is particularly worth of interest: if a decision-oriented CSP has several soZutions, this is rather good news since the agent will be (in principle) happy with any of them; on the contrary, if a reasoning-orientedCSP has several solutions, which means that the knowledge is incomplete, giving only one solution means arbitrary selecting one possible situation, which is not necessarily the real one - giving all solutions could be more adequate. Now, if a decision-oriented CSP has no solution, the problem is over-specified; inconsistency for reasoning-oriented CSP has to be interpreted in a different way, since the variables do have a value in the actual world and therefore at least one of the constraints is not sound. Now, beyond pure decision or reasoning-oriented prob- lems, there are practical problems where a decision has to be taken, while the actual world is not completely known (which is the basic assumption in applications of decision theory). For instance, in scheduling, the decision (sequenc- ing the tasks) may depend on ill-known parameters such as the reliability of some machines. Our motivation is to show that the CSP framework, which has been successfully developed and applied in the past years, can be extended in such a way that it can represent and solve problems of deci- sion making under incomplete knowledge. These problems will be represented by so-called mixed CSPs involving both controllable and uncontrollable variables. The definition of a solution of a mixed CSP depends cru- cially on the assumptions concerning the agent’s awareness of the parameter values (the state of the world) at the time the decision must be made (deadline for acting) - where a decision is an assignment of decision variables. We will consider these two epistemological assumptions: (full observability): the actual world will be completely revealed to the agent before the deadline is reached (possi- bly, just before), so that it is useful to compute “off-line” a ready-to-use conditional decision, mapping possible cases to decisions, that the agent will be able to instantiate in real time, as soon as the actual world is known. NO (no observability): the agent will never learn anything new about the actual state of the world before the deadline; all it will ever know is already encoded in the initial specifi- cation of the problem. In this case, it is useless to provide a conditional decision, and a suitable solution of the problem Constraint Satisfaction 175 From: AAAI-96 Proceedings. Copyright © 1996, AAAI (www.aaai.org). All rights reserved. consists in pure, unconditional decision to be taken in any outside the scope of the paper, we assume that K is consis- possible case. tent. The intermediate case where some partial knowledge about the state of the world may be learned (for instance through information-gathering actions) is not considered in this pa- per. The two extreme assumptions we consider lead to two definitions of a solution of a mixed CSP and two different notions of consistency, that we study in Section 2. In Sec- tion 3 we propose an anytime algorithm for solving mixed CSP under assumption (FO). Possible extensions and ap- plications are mentioned in conclusion. A mixed CSP defines a collection Candidates(P), of classical, decision-oriented CSPs, one for each possible world w - it will be called the set of candidate problems induced by P. Briefly (we omit technical details for space considerations), the candidate problem induced by a world w is a CSP over X whose constraints are the original pure decision constraints plus those obtained by reducing each mixed constraint C to a pure decision constraint by letting its parameters take their value as specified in wl. Mixed constraint satisfaction Definitions A classical constraint satisfaction problem is a triple P = (X, D, C), where X is a set of variables, each of which takes its possible values in a domain Di (supposed here finite, we note D = x iDi), and C is a a set of constraints, each of which restricting the possible values of some variables. Sol(P) denotes the set of all assignments satisfying all constraints of P. Roughly speaking, a mixed CSP is a CSP equipped with a partition between (controllable) decision variables and (uncontrollable) parameters. Definition 1 (mixed CSP) A mixed CSP is defined by a 6-uple P = (A, L, X, D, ic,C) where: - A== {A,,... , X,) is a set of parameters; - L=L1x . + e x L,, where Li is the domain of Xi; - x = (Xl,... , xn) is a set of decision variables; - D=D1 x . . . x Dn, where Di is the domain of xi; - K is a set of constraints involving only parameters; - C is a set of constraints, each of them involving at least one decision variable. A complete assignment of the parameters will be called a world and will be denoted by w. A complete assignment of the decision variables will be called a decision and will be generally denoted by d. The complete assignment of both parameters and decision variables resulting from the concatenation of w and d will be denoted by (w, d). The important distinction between the constraints of K and those of C is due to their very different interpretations: the constraints of K involve only parameters and thus rep- resent the knowledge we have about the real world. Now, C contains decision constraints, which restrict the allowed values of decision variables, possibly conditioned by some parameters - if so they will be called mixed (or equivalently parameterized) constraints. If A = 0 (resp. X = m), P is a classical decision-oriented (resp. reasoning-oriented) CSP. Definition 2 A solution of the reasoning-oriented CSP (A, L, K> is a possible world. The set of all possible worlds for P is denoted by Pass(P). Now, a decision d is said to cover a world w iff it satisfies the candidate problem induced by w; this means that if the real world appears to be w, then d will be a suitable decision. Among the possible worlds, those which are covered by at least one decision, i.e., whose induced candidate problem is consistent, are called good worlds - and the others are called bad worlds. Definition 3 A decision d is said to cover a world w iff (w, d) is a solution of (AUX, L x D, C). The set of all worlds (possible or not) covered by d is denoted by Covers?(d). Definition 4 Good(P) = Pass(P) n (&=oCoversp(d)) Bad(P) = Pass(P) \ Good(p) Equivalently, Good(p) = &D{W E Poss(P)((w,d) satisfies C}. Example: let P be the following mixed CSP: - X = {x1,x2,x3); Dl = Dz = D3 = {1,2,3} - A = {A,, A,}; Ll = La = {a, b, c, d} - K: contains only one constraint: Co: XI # X2 - C contains three constraints Cl, C2, C3: Cl c2 c3 f’o4’) = {(a, b), (a, 4 (~4, @+4 (W, @A Cc, 4 (c, b), (c, d), @,a), (d, b), (d, c)}. tit WI = (d, b) and w = (d, c); the decisions covering w1 are (3,1,2) and (3,1,3). It can be checked that no decision covers ~2, and that all other possible worlds are good: Bad(P) = (~2). Lastly, let dl = (1,3,3), then Coversp(dl) = {a, b} x {a, c}. Conditional and unconditional decisions The case of full observability In the case of full observ- ability, solving a mixed CSP consists in giving a conditional strategy (from a planning point of view,-a conditional plan) associating, to each possible world, a decision that satis- fies the corresponding candidate problem: once the actual If 1c: were inconsistent, we would get Pass(P) = 0. But this means that at least one of the constraints of K: is not sound. A non-trivial handling of this inconsistency being ‘Note that among the CSPs of Candidates(P), some may be inconsistent, which means that the actual problem may be inconsistent. 176 Constraint Satisfaction values of the parameters are known, we have an appropri- ate decision. More reasonably, we may consider condi- tional decisions defined on a subset of Pass(P) (ideally on Good(P)). Definition 5 (conditional decisions) A conditional deci- sion s for P is a map from a subset W, of Pass(P) to D such that VW E W,, s(w) covers w. Zf W, = Good(p), s is optimal. If W, = Good(P) = Pass(P) (implying Bad(P) = 0), s is complete. Obviously, a complete conditional decision is also opti- mal, but the converse is generally not true: while a mixed CSP always has optimal conditional decisions, it does not always have a complete conditional decision. Definition 6 (consistency) A mixed CSP is consistent ifit has a complete conditional decision. Clearly, P is consistent H Bad(P) = o ti Good(P) = Pass(P). The consistency of P means that all candidate problems are consistent. When it is not the case, a weaker request consists in giving an optimal conditional decision, which covers all situations which are possible to cover. Example (continued): the following s is an optimal conditional decision for P2: I XI L I X2 -+ II a I b I c I d I -. , - a b : The case of no observability The previous approach as- sumes that the actual world will be known before the deci- sion has to be taken (FO). Clearly, under the other extreme assumption (NO), it is useless to look for a conditional decision: consequently, in this case one has to look for a pure (unconditional) decision. Intuitively, this pure deci- sion should cover “as many worlds as possible”; if there is no other decision covering more worlds covered by d (w.r.t. set inclusion), d is undominated; if furthermore d covers all good (resp. possible) worlds, it is universal (resp. strong universal). Definition 7 A decision d E D is a undominated for P iff there is no d’ E D such that Coversp(d’) strictly con- tains Coversp(d); it is an universal decision for P if Coversp (d) = Good(P) and a strong universal decision for P if Coversp(d) = Pass(P). While a mixed CSP always has an undominated decision (and generally has several incomparable ones), it is not always the case that it has a universal decision (and a fortiori a strong universal decision). Note that a pure decision is an extreme case of a conditional decision, where all considered worlds are mapped to the same decision. Here is a summary of the relations between different notions of decisions: 2The table has to be read this way: a world w is defined by the values assigned to both parameters X1 and X2 and corresponds thus to a row and a column, at the intersection of which is the decision s(w) when defined i.e., when w E W(s). Here, W(s) contains all good worlds, so s is optimal. strong universal decisions 5 universal decisions E undominated decisions C conditional decisions and strong universal decisions s complete conditional decisions C optimal conditional decisions 5 conditional decisions Definition 8 (uniform & strong consistency) A mixed CSP is uniformly consistent ifs it has a universal decision, and strongly consistent iffit has a strong univer- sal decision. Clearly, P is strongly consistent if and only if it is both consistent and uniformly consistent. The converses are gen- erally not true. Note that universal (resp. strong universal) consistency implies that any undominated decision is a uni- versal decision (resp. a strong universal decision). Example (continued): P has no universal decision. Consider the mixed CSP P’ obtained from P by replacing Ca : z1 + 22 5 4 by Ci : 2x1 + 22 5 4. Then, Go&(?‘) = {a, b} x {a, b, d} and d = (1,2,3) is a universal decision (but not a strong universal decision). Complexity Interestingly, the problems of deciding con- sistency and strong consistency fall in complexity classes which are above /VP in the polynomial hierarchy. We recall (Stockmeyer, 1977) that a decision problem is in NPNP = Cc iff it can be solved in polynomial time on a non- deterministic Turin 5 machine using N P-oracles; a decision problem is in co-C, = II: iff its complementary problem is in Cc. The role of the “canonical” complete problem (which is played by SAT for NP) is played by 2-QBF for Cr; 2-QBF is the problem of deciding whether the follow- ing quantified boolean formula is true: XVb’F(Z, c). The complementary problem 2-QBF (vz %F(a’, 6)) is complete3 for II<. Strong consistency (resp. consistency) looks sim- ilar to 2-QBF (resp. to 2-QBF) since it involves the same alternation of quantifiers. roposition 1 The problem MIXED-SAT of deciding consis- tency of a binary mixed CSP is II:-complete. Sketch of proof: the problem is in l-I: since its complementary problem can be solved by the following algorithm: guess a world w, check that it is possible and that the corresponding candidate problem is inconsistent (co-NP-complete oracle). The complete- ness can be proven by reducing 2-QBF restricted to 3-CNF to the MIXED-SAT problem (see (Fargier et al., 1996)). Note that MIXED-SAT remains IIF-complete even when we restrict ourself to binary mixed CSP with an empty K. This results furthermore offers a “canonical” lI,P-complete problem to the CSP community. The problem of deciding strong consistency is easily shown to be in Cc, but com- pleteness is still to be proven in the general case. 3and remains so when F is restricted to 3-CNF (Stockmeyer, 1977). Constraint Satisfaction 177 In practice, the complexity of MIXED-SAT may be reason- able if we consider that K should usually drastically reduce the number of possible worlds. Lastly, when K = 0 (or equivalently Pass(P) = L), parameters are independent (since their possible values are not constrained). This as- sumption has a drastic effect on the complexity of strong consistency: Proposition 2 Under the parameter independence assump- tion, deciding strong consistency in a binary mixed CSP is NP-complete. Sketch of proof: membership in NP is straightforward (the de- cision d is a polynomial length certificate which is checkable in polynomial time since Pass(P) = L). The completeness comes from the fact that the consistency of any binary CSP can be re- duced to the strong consistency of the same CSP, considered as a mixed CSP with an empty parameter set A. Searching for a conditional decision The high complexity of finding a solution of a mixed CSP leads us to look for an algorithm which gives an optimal conditional solution if we let it run until it stops naturally, or otherwise gives an approximate (non optimal) conditional decision (the longer we let it run, the closer it gets to op- timality). Intuitively, the algorithm incrementally builds a list of decisions which eventually covers a superset of all good worlds. Repeatedly, we pick a new decision d (to be added to the list) that covers at least one possible world among those which are not yet covered, we compute the set R of worlds (possible or not) that this decision covers and we subtract this set from the set of worlds which haven’t yet been covered. In order to easily compute worlds cov- ered by d, we assume that any constraint of C involves at most one parameter. Therefore, Coversp (d) forms a Carte- sian product of subsets of the parameter domains and can be computed in polynomial time in binary mixed CSP. The successive subtractions of these Cartesian products is per- formed using a technique recently proposed in (Freuder and Hubbe, 1995) , called subdomain subproblem extraction. We recah this technique before we describe our algorithm. Sub-domain extraction We define an environment E as a set of worlds of the form El x . . . x Z,, with V’k, lk E Lk. An example of environment is (Xi E {a, c), X2 E {b, c, d}), also written {a, c} x {b, c, d) or [ tc>] when there is no ambiguity on the order of parameters. The set of all possible environments is obviously a lattice (equipped with the inclusion order), whose top is the set of all possible parameter assignments (in the example, [ $$]) and whose bottom is the empty set. Given two environments E and R, sub-domain subprob- Zem extraction technique (Freuder and Hubbe, 1995) decom- poses E into a set of disjoint sub-environments Dec( E, R) such that all worlds of E belong either to R or to one of the sub-environments of the decomposition. The decom- position is unique if an ordering on the variables is fixed. WhenEnR = 0 we let Dec(E, R) = (E); and when E C R,Dec(E, R) = 0. Thus, this decomposition is ac- tually a subtraction since we end up with a list of disjoint environments which cover exactly the worlds of E \ R. Namely: Proposition 3 (Freuder and Hubbe 1995) w E E and w $Z R e there exists a unique F E Dec(E, R) such thatw E F The algorithm Start LD:= W; {list of decisions with the env. covered} Env:= {& x .a. x Lp}; {list of env. not yet covered by LD} Bad:= 0; {list of non coverable env.} Repeat E := an environment from Env; {possible heuristics} If (A, L, K: U E) is consistent (1) {E contains possible worlds} then If (A U X, L x D, K U C U E) is inconsistent (2) {E contains only non-coverable worlds} then Bad :=Bad UE else s := a solution of (A U X, L x D, K: U C U E); d := projection of s on the decision variables; {d covers at least one possible world of E} R := Caversp (d) {unary constraints on par-am.} Add (R, d) to LD; Env:= UFEEnVDeC(F, R) (3) end {else} end {else} Until Env = 0 (or interruption by the user) End {LD covers Good(P)} Several steps of the algorithm deserve some comments: (1) The test of consistency of (A, L, K u E) may seem superfluous - since its role is only to distinguish bad environments from impossible environments, and in any case no new decision will be added-but is necessary* if we care about about the consistency of P (since Lemma 4 will not hold any longer if this test is removed). (l)+(2) The two sequences of CSP (A, L,K u E) and (A U X, L x D, K U C U E), repeatedly solved by the algorithm, define dynamic CSP (Dechter et al., 1988) and their resolution may be improved by any techniques developed to solve such problems. (3) Once a new R is built, it is subtracted from all the awaiting environments to avoid some redundant compu- tations. This furthermore guarantees that the computation will stop when the current list LD defines a solution. (3) The update of the list of environments Env, may be performed lazily to avoid memory consumption. Lastly, the algorithm considers an environment as impossi- ble only when all of its worlds are impossible; consequently, there will be environments in LD and Bad containing im- possible worlds. This is not a problem, since (i) these 4However, this test can be performed during the resolution (2) of (A U X, L x D, K U C U E) by the backtrack-based algorithm simply by imposing that the parameters be instantiated first. As soon as a consistent labeling of the A is found, any vertical ordering heuristics may apply. 178 Constraint Satisfaction impossible worlds will never be observed; the important point is that any possible, coverable world is covered by a decision, and any possible, bad world is in the Bad list; and (ii) any environment added to the Bad list contains at least a possible world, which ensures that at the end of the algorithm, P is consistent @Bad = 0. 1. 2. 3. 4. 5. 6. 7. unning the algorithm on the example Env = {[$":]};LD = M;Bad= 0 E = [;$I; dl = (3,1,2); R = {b, d} x {a, b}; Dec([:;“,:l> [:I) = tr:;1> LZlh Env = tc”,l~ La; LD = ccr:1, (3,1,2)>h E = [ !;I; d2 = (1,2,3); R = {a, b} x {a, b, d}; %Lbcd,l~ r,sP,l) = tr% m; ~f4LabcCdl Eil> = Kl, LAdlh Env = tr:17 r,“,1, [:I, L&bccdlh LD = c<r3 (3,1,2)), ([:$I, (1,2,3))h E = [ “,I; d3 = (1,3,3); R = {a, b} x {a, c); Dec([El, [~f3=Dec([~l, [::1)=0; ~4,d,l7 rzl) =c& ~4[abCcdl’ [::I> = Uabccdlh Env= tr,“,1, Lbccdlh LD=N:I> c41> 2% &gJ (1>2,3)), ([::I7 (19 39 3))) E = [ ,“,I; K: U C U E inconsistent (but K U E consistent). Bad = tr,“,1> E = [=bCcd]; & = (2,2,3); R = (a, c} x (a, b, d); ~4[,Ll7 Lx& = Klh Env = Klh LD = {<[:I, (3,1,2)), (l:;d]' (L2,3)), ([::I, (L3, W, ([::g’ (29 27 3))) E = [:I; K: U E inconsistent. Env = 0. STOP Finally, the last value of LD is {([if], (3,1,2)), ([Qabbd], (1,2,3)), ([:!I, (L3,3)L ([~~.Jl’ C&T WI, and Bad = {[,“,Ih This al- lows us to build a conditional decision where each world w is mapped to the decision S(W) = dj associated to the first item (E, dj) such that E contains w (shown on the table below). This search of a decision for a given world could be made more effi- cient via the use of a discrimination tree for example. The worlds marked with a “*” are actually impossible but the list LD assigns them a decision, or labels them bad. Correctness of the algorithm It is based on the following properties; the proofs (omitted for length considerations) appear in (Fargier et al., 1996). Proposition 4 At any point of the algorithm, if E E Bad then E n Pass(P) C Bad(Q) and E n Pass(P) # 0. Proposition 5 Atanypointofthe algorithm, if(E, d) E LD then VW E E, (w, d) satisfies C. Proposition 6 (Correctness of the algorithm) Zf run qui- etly, the algorithm stops, the jnal value of LD dejnes an optimal conditional decision for Q and the jinal value of Bad contains Bad(Q), Sketch of proof: to prove the termination, we prove that LJE~Q~E decreases strictly at each iteration (using Proposition 3). The second part of the theorem relies on the following invariant, verified at any iteration: VW E Poss( P), 3( E, d) E LD such that w E E or 3!E E Env such that w E E or 3!E E Bad such that w E E, The invariant is proved by induction on the algorithm (we omit the proof for length considerations). Applying the invariant to the end of the last iteration gives VW E Pass(P), either 3( E, d) in LD such that w E E, or 3E E Bad such that w E E. Together with Propositions 4 and 5, this achieves proving the theorem. Since the set of possible worlds covered by LD grows monotonically as the algorithm runs (and so is Bad) it is possible to use the algorithm as an “anytime” algorithm. Actually, LD tends to a cover of Good(Q) and Bad tends to a subset of Bad(Q) . Proposition 7 If the algorithm runs until it stops, Q is con- sistent ifl Bad # 0. The proof follows easily from Propositions 6 and 4. Qb- viously, if Bad = 0 and 1 LDI = 1 then Q is strongly consistent. The converse is not true; more generally, the list LD may not be minima15. Partial CSP (Freuder, 1989) also handles simultaneously a family of CSPs (representing the possible relaxations of an overconstrained CSP) but the original motivation of this framework is not to handle decision under incomplete knowledge, and therefore lacks the distinction between con- trollable and uncontrollable variables. In (Helzerman and Harper, 1994) an efficient algorithm is proposed for en- forcing arc-consistency simultaneously in a family of CSPs sharing some variables and constraints; such a family is re- lated to our set of candidates, but again, this approach does not enable decision under incomplete knowledge. Now, in the field of knowledge representation (Boutilier, 1994) pro- posed a logical, qualitative basis for decision theory, dis- tinguishing like us between controllable and uncontrollable propositions, but without the notion of conditional decision. Poole’s independent choice logic (Poole, 1995) also distin- guishes between the nature’s choice space and the agent’s choice space, in order to build strategies which are similar to our conditional decisions; it also includes the representation of probabilities, utilities and knowledge-gathering actions. In this paper we proposed an anytime algorithm for com- puting conditional decisions under the full observability as- sumption. Further work will consist first in implementing and testing our algorithm; for this we think of exploiting 5For instance, it may be the case that the last decision added to the list covers all possible worlds but that many decisions have been added to the list before. Of course, one could think of checking afterwards whether some decisions are redundant, but this could be very costly. Constraint Satisfaction 179 results from the area of dynamic constraint satisfaction (Dechter et al., 1988), where several techniques are pro- posed in order to solve a sequence of CSP that differs only in some constraints more efficiently than by naively solving each CSP one after the other. Furthermore, the compu- tation of a compact representation of conditional decisions could also probably be performed using Finite Automata (as (Vempaty, 1992) for sets of solutions) or Binary Decision Diagrams (Bryant, 1992). It is clear that, at this level of complexity, algorithmic issues may be crucial. Since our contribution consists in extending the CSP framework in order to deal with decision problems under incomplete knowledge (namely by distinguishing between controllable and uncontrollable variables), it can be consid- ered as a first step to embed decision theory into constraint satisfaction; other important steps in this direction would consist in considering probabilities, utilities and sequences of decisions. A first step towards handling probabilistic knowledge in constraint satisfaction has been proposed in (Fargier et al., 1995), where the uncertainty on the values of parameters is encoded by a given probability distribu- tion instead of a set of constraints. Utility functions would enable us to represent flexible goals and it should not be hard to embed them in our framework; in the constraints of C, an extra field is added to each tuple, namely the util- ity of the corresponding assignment, as in (Schiex et al., 1995, Bistarelli et al., 1995). Extending our framework to sequences of decisions is significantly harder. The partition of the variables is not sufficient: not only decisions are in- fluenced by parameters, but the value of some parameters may also be influenced by earlier decisions; for this we may structure decision variables and parameters in a directed network, in the same spirit as influence diagrams (Howard and Matheson, 1984)6 , a link from Xi to xj meaning that the allowed values of the decision variable xj depend on X;, and a link from xi to Xj meaning that the decision of assigning a value to xj has or may have some effects on Xj . An interesting potential application of mixed constraint satisfaction and its possible extensions is planning under uncertainty. Clearly, this topic needs the notion of control- lability, and would gain a lot in being encoded in an exten- sion of the CSP framework - since it could benefit from the numerous advances on the resolution of CSPs. The similar- ity between our conditional decisions and conditional plans is clear, at least for a single action. For handling sequences of actions, mixed CSPs have to be extended as evoked in the previous paragraph. Non-deterministic effects of ac- tions may be encoded by using extra parameters. Lastly, many recent approaches to planning make use of decision theory; clearly, extending mixed CSP with a larger part of decision theory goes in this direction; our long-term goal is thus to provide a constraint satisfaction based framework to decision-theoretic planning. References Craig Boutilier, Toward a logic for qualitative decision theory, Proc. of KR’94,75-86. R. E. Bryant, Symbolic Boolean Manipulation with Or- dered Binary-Decision Diagrams, ACM Computing Sur- veys, 24,3, 1992,293-3 18. Rina Dechter and Avi Dechter, Belief Maintenance in Dy- namic Constraint Networks, Proc. of AAAI’88, 37-42, St. Paul, MN. Helene Fargier, Jerome Lang and Thomas Schiex, Con- straint satisfaction and decision under uncertainty, Tech. Report IRIT-96-06, Universite Paul Sabatier (Toulouse, France), Feb. 1996. Helene Fargier, Jerome Lang, Roger Martin-Clouaire and Thomas Schiex, A constraint satisfaction framework for decision under uncertainty, Proc. of Uncertainty in AI’95, 167-174. Eugene Freuder, Partial constraint satisfaction, Proceed- ings of IJCAI’89,278-283. Eugene Freuder and Paul Hubbe, Extracting constraint sat- isfaction subproblems, Proc. of ZJCAZ’9.5, Montreal. Randall A. Helzerman and Mary P. Harper, An approach to multiply segmented constraint satisfaction problems, Proc. AAAZ’94,350-355. R.A. Howard and J.E. Matheson, Influence Diagrams, In- fluence Diagrams, in R.A. Howard and J.E. Matheson, eds., The Principles and Applications of Decision Analy- sis, vol.2 (1984), 720-761. Alan K. Mackworth, Consistency in networks of relations, Artificial Intelligence, 8, 1977,99-l 18. David Poole, Exploiting the rule structure for decision making within the independent choice logic, Proc. of Un- certainty in AI’95,454-463. S. Bistarelli, F. Rossi and U. Montanari, Constraint solv- ing over Semi-rings, Proceedings of NCAZ’95, Montreal, Canada, 624-630. T. Schiex, H. Fargier and G. Verfaillie, Valued Constraint Satisfaction Problems: hard and easy problems, Proc. of ZJCAI’95, Montreal, Canada, 63 l-637. L. J. Stockmeyer, The polynomial-time hierarchy, Theo- retical Computer Science, 3, 1977, l-22. N. Rao Vempaty, Solving Constraint Satisfaction Problems Using Finite State Automata, Proceedings of AAAI’92, 453-458. ‘Note that the multi-stage extension of mixed CSP would differ from influence diagrams in the representational and computational basis of our approach, which is, namely, constraint satisfaction; in particular, certain kinds of cycles are allowed in our approach, contrarily to the acyclicity requirement for influence diagrams. 180 Constraint Satisfaction | 1996 | 25 |
1,899 | trolling State-S ractio ayesia &act) Chao-Lin Liu Artificial Intelligence Laboratory, University of Michigan 1101 Beal Avenue Ann Arbor, Michigan 48105 chaolin @umich.edu Many applications require computational systems to re- spond to queries by a particular deadline. Failure to meet the deadlines may render the returned solution useless. Moreover, the deadlines of such time-critical applications are often uncertain at system design time. Anytime algo- rithms have been suggested to cope with these challenges by trading the quality of the solutions for the reactiveness of the systems at run time [ 13. We have introduced an any- time evaluation algorithm [2] for a formalism commonly used in uncertain reasoning: Bayesian networks. Empirical results indicate that approximations of good quality can be obtained within a much shorter time than would be re- quired to directly evaluate the networks by exact algo- rithms. Also the quality of the approximation improves with the allocated computational time on average. Approximations of the desired solutions may be acquired by evaluating abstract versions of the given Bayesian net- works, where the abstract networks ignore some of the dis- tinctions between states of the variables, called abstracted variables. Reduction of the state spaces can dramatically cut down the computational time, at the expense of the quality of the solutions. When there is time for more com- putation, the evaluation algorithm recovers some of the ignored distinctions, and gets another approximation by evaluating the refined network. The algorithm iteratively evaluates refined abstract networks until stopped. The structure of the Bayesian net- work has a special bearing on the quality of approximations K , defined based on the divergence between the true and the approximated distributions. Notice that larger K means worse quality. Specifically, we have shown that the quality of the approximated distribution of a set of vari- ables that d-separates another set of variables from the ab- stracted variables is not better than that of the d-separated set. D-separation is a graphical property that implies con- ditional independence. Namely when variables in set X d- separate variables in set Y from the abstracted variables A given the instantiated variables, Y is conditionally inde- pendent of A. Consider the Bayesian network shown above. If we instantiate variable B3 and abstract BI, then the qualities of the approximated marginal distributions of other variables have the relationship: K, 2 K,, 2 K, . We are searching for the control me&ods 1.0 * 0.8 of the state-space 1 o6 abstraction, aiming ’ 3 at improving per- & 0 0.4 0.2 formance profiles of z 0.0 the algorithm. In 1 2 3 4 5 6 7 8 9 10 II 12 13 14 IS Iteration particular, we are exploring score functions for selecting the aggregated states to refine. The score functions compute the goodness of the aggregated states based on factors that are local to the aggregated states. It is hoped that the scores be highly correlated with the actual improvement of the approxima- tion. Among the functions being investigated, the REMB function based on the relative entropy of the Markov boundary of the abstracted variable has demonstrated the best average performance in experiments. This is illustrated by the closeness of the REMB curve to the “Best” curve in the chart shown above. The “Best” curve is achieved by an ideal function that always lead to correct selection of the aggregated state to refine. Such a perfect capability is achieved in part by the evaluating the original network, and is not a practically achievable feature. We have established a theorem that addresses the relation- ship between state-space abstraction and the quality of ap- proximations. We also have limited, empirical results re- garding control of state-space abstraction using a variety of heuristics. Future work will consider the usage of value of information in score functions. The goal is to establish strategies for controlling the state-space abstraction such that the algorithm provides the best performance possible. References [l] Boddy, M., and T. L. Dean. 1994. Deliberation sched- uling for problem solving in time-constrained environ- ments. Artificial Intelligence 67: 245-285. [2] Wellman, M. I?., and C.-L. Liu. 1994. State-space ab- straction for anytime evaluation of probabilistic networks. Proceedings of the Tenth Conference on Uncertainty in Artificial Intelligence, pp. 567-574. * This work was supported in part by AFOSR Grant F49620-94- l-0027. Student Abstracts 1395 From: AAAI-96 Proceedings. Copyright © 1996, AAAI (www.aaai.org). All rights reserved. | 1996 | 250 |
Subsets and Splits
SQL Console for Seed42Lab/AI-paper-crawl
Finds papers discussing interpretability and explainability in machine learning from after 2010, offering insight into emerging areas of research focus.
Interpretability Papers Since 2011
Reveals papers from the AAAI dataset after 2010 that discuss interpretability or explainability, highlighting key research in these areas.
SQL Console for Seed42Lab/AI-paper-crawl
Searches for papers related to interpretability and explainability in NIPS proceedings after 2010, providing a filtered dataset for further analysis of research trends.
AI Papers on Interpretability
Finds papers discussing interpretability or explainability published after 2010, providing insight into recent trends in research focus.
ICML Papers on Interpretability
Retrieves papers from the ICML dataset after 2010 that mention interpretability or explainability, offering insights into trends in model transparency research.
ICLR Papers on Interpretability
Retrieves papers from the ICLR dataset that discuss interpretability or explainability, focusing on those published after 2010, providing insights into evolving research trends in these areas.
ICCV Papers on Interpretability
Finds papers from the ICCV dataset published after 2010 that discuss interpretability or explainability, providing insight into trends in research focus.
EMNLP Papers on Interpretability
Retrieves papers related to interpretability and explainability published after 2010, providing a focused look at research trends in these areas.
ECCV Papers on Interpretability
The query retrieves papers from the ECCV dataset related to interpretability and explainability published after 2010, providing insights into recent trends in these research areas.
CVPR Papers on Interpretability
Retrieves papers from the CVPR dataset published after 2010 that mention 'interpretability' or 'explainability', providing insights into the focus on these topics over time.
AI Papers on Interpretability
Retrieves papers from the ACL dataset that discuss interpretability or explainability, providing insights into research focus in these areas.