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+# Can ChatGPT Assess Human Personalities? A General Evaluation Framework
+
+Haocong Rao1,2 Cyril Leung2,3 Chunyan Miao1,2∗ $^ { 1 }$ School of Computer Science and Engineering, Nanyang Technological University, Singapore 2LILY Research Centre, Nanyang Technological University, Singapore 3Department of Electrical and Computer Engineering The University of British Columbia, Canada {haocong001,ascymiao}@ntu.edu.sg {cleung}@ece.ubc.ca
+
+# Abstract
+
+Large Language Models (LLMs) especially ChatGPT have produced impressive results in various areas, but their potential human-like psychology is still largely unexplored. Existing works study the virtual personalities of LLMs but rarely explore the possibility of analyzing human personalities via LLMs. This paper presents a generic evaluation framework for LLMs to assess human personalities based on Myers–Briggs Type Indicator (MBTI) tests. Specifically, we first devise unbiased prompts by randomly permuting options in MBTI questions and adopt the average testing result to encourage more impartial answer generation. Then, we propose to replace the subject in question statements to enable flexible queries and assessments on different subjects from LLMs. Finally, we re-formulate the question instructions in a manner of correctness evaluation to facilitate LLMs to generate clearer responses. The proposed framework enables LLMs to flexibly assess personalities of different groups of people. We further propose three evaluation metrics to measure the consistency, robustness, and fairness of assessment results from state-ofthe-art LLMs including ChatGPT and GPT-4. Our experiments reveal ChatGPT’s ability to assess human personalities, and the average results demonstrate that it can achieve more consistent and fairer assessments in spite of lower robustness against prompt biases compared with InstructGPT†.
+
+# 1 Introduction
+
+Pre-trained Large Language Models (LLMs) have been widely used in many applications including translation, storytelling, and chatbots (Devlin et al., 2019; Raffel et al., 2020; Yang et al., 2022; Yuan et al., 2022; Ouyang et al., 2022; Bubeck et al.,
+
+2023). ChatGPT (Ouyang et al., 2022) and its enhanced version GPT-4 are currently recognized as the most capable chatbots, which can perform context-aware conversations, challenge incorrect premises, and reject inappropriate requests with a vast knowledge base and human-centered finetuning. These advantages make them well-suited for a variety of real-world scenarios such as business consultation and educational services (Zhai, 2022; van Dis et al., 2023; Bubeck et al., 2023).
+
+Recent studies have revealed that LLMs may possess human-like self-improvement and reasoning characteristics (Huang et al., 2022; Bubeck et al., 2023). The latest GPT series can pass over $90 \%$ of Theory of Mind (ToM) tasks with strong analysis and decision-making capabilities (Kosinski, 2023; Zhuo et al., 2023; Moghaddam and Honey, 2023). In this context, LLMs are increasingly assumed to have virtual personalities and psychologies, which plays an essential role in guiding their responses and interaction patterns (Jiang et al., 2022). Based on this assumption, a few works (Li et al., 2022; Jiang et al., 2022; Karra et al., 2022; Caron and Srivastava, 2022; Miotto et al., 2022) apply psychological tests such as Big Five Factors (Digman, 1990) to evaluate their pseudo personalities (e.g., behavior tendency), so as to detect societal and ethical risks (e.g., racial biases) in their applications.
+
+Although existing works have investigated the personality traits of LLMs, they rarely explored whether LLMs can assess human personalities. This open problem can be the key to verifying the ability of LLMs to perform psychological (e.g., personality psychology) analyses and revealing their potential understanding of humans, i.e., “How do LLMs think about humans?”. Specifically, assessing human personalities from the point of LLMs (1) enables us to access the perception of LLMs on humans to better understand their potential response motivation and communication patterns (Jiang et al., 2020); (2) helps reveal whether LLMs possess biases on people so that we can optimize them (e.g., add stricter rules) to generate fairer contents; (3) helps uncover potential ethical and social risks (e.g., misinformation) of LLMs (Weidinger et al., 2021) which can affect their reliability and safety, thereby facilitating the development of more trustworthy and human-friendly LLMs.
+
+To this end, we introduce the novel idea of letting LLMs assess human personalities, and propose a general evaluation framework (illustrated Fig. 1) to acquire quantitative human personality assessments from LLMs via Myers–Briggs Type Indicators (MBTI) (Myers and McCaulley, 1985). Specifically, our framework consists of three key components: (1) Unbiased prompts, which construct instructions of MBTI questions using randomlypermuted options and average testing results to achieve more consistent and impartial answers; (2) Subject-replaced query, which converts the original subject of the question statements into a target subject to enable flexible queries and assessments from LLMs; (3) Correctness-evaluated instruction, which re-formulates the question instructions for LLMs to analyze the correctness of the question statements, so as to obtain clearer responses. Based on the above components, the proposed framework re-formulates the instructions and statements of MBTI questions in a flexible and analyzable way for LLMs, which enables us to query them about human personalities. Furthermore, we propose three quantitative evaluation metrics to measure the consistency of LLMs’ assessments on the same subject, their assessment robustness against random perturbations of input prompts (defined as “prompt biases”), and their fairness in assessing subjects with different genders. In our work, we mainly focus on evaluating ChatGPT and two representative state-of-the-art LLMs (InstructGPT, GPT4) based on the proposed metrics. Experimental results showcase the ability of ChatGPT in analyzing personalities of different groups of people. This can provide valuable insights for the future exploration of LLM psychology, sociology, and governance.
+
+Our contributions can be summarized as follows:
+
+• We for the first time explore the possibility of assessing human personalities by LLMs, and propose a general framework for LLMs to conduct quantitative evaluations via MBTI.
+
+• We devise unbiased prompts, subject-replaced queries, and correctness-evaluated instructions to encourage LLMs to perform a reliable flexible assessment of human personalities.
+
+• We propose three evaluation metrics to measure the consistency, robustness, and fairness of LLMs in assessing human personalities.
+
+• Our experiments show that both ChatGPT and its counterparts can independently assess human personalities. The average results demonstrate that ChatGPT and GPT-4 achieve more consistent and fairer assessments with less gender bias than InstructGPT, while their results are more sensitive to prompt biases.
+
+# 2 Related Works
+
+Personality Measurement. The commonly-used personality modeling schemes include the three trait personality measure (Eysenck, 2012), the Big Five personality trait measure (Digman, 1990), the Myers–Briggs Type Indicator (MBTI) (Myers, 1962; Myers and McCaulley, 1985), and the 16 Personality Factor questionnaire (16PF) (Schuerger, 2000). Five dimensions are defined in the Big Five personality traits measure (Digman, 1990) to classify major sources of individual differences and analyze a person’s characteristics. MBTI (Myers and McCaulley, 1985) identifies personality from the differences between persons on the preference to use perception and judgment. (Karra et al., 2022; Caron and Srivastava, 2022) leverage the Big Five trait theory to quantify the personality traits of language models, while (Jiang et al., 2022) further develops machine personality inventory to standardize this evaluation. In (Li et al., 2022), multiple psychological tests are combined to analyze the LLMs’ safety. Unlike existing studies that evaluate personalities of LLMs, our work is the first attempt to explore human personality analysis via LLMs.
+
+Biases in Language Models. Most recent language models are pre-trained on the large-scale datasets or Internet texts that usually contains unsafe (e.g., toxic) contents, which may cause the model to generate biased answers that violate prevailing societal values (Bolukbasi et al., 2016; Sheng et al., 2019; Bordia and Bowman, 2019; Nadeem et al., 2021; Zong and Krishnamachari, 2022; Zhuo et al., 2023). (Bolukbasi et al., 2016) shows that biases in the geometry of wordembeddings can reflect gender stereotypes. The gender bias in word-level language models is quantitatively evaluated in (Bordia and Bowman, 2019).
+
+In (Nadeem et al., 2021), the authors demonstrate that popular LLMs such as GPT-2 (Radford et al., 2019) possess strong stereotypical biases on gender, profession, race, and religion. To reduce such biases, many state-of-the-art LLMs such as ChatGPT apply instruction-finetuning with non-toxic corpora and instructions to improve their safety. (Zhuo et al., 2023) reveals that ChatGPT can generate socially safe responses with fewer biases than other LLMs under English lanuage settings. In contrast to previous works, our framework enables us to evaluate whether LLMs possess biased perceptions and assessments on humans (e.g., personalities), which helps us better understand the underlying reasons for the LLMs’ aberrant responses.
+
+![](images/241a0262a34b9fd2f157ee10d274aeee1c95904d5edb217c047fb61389fdb570.jpg)  
+Figure 1: Overview of our framework: (a) The queried subject is replaced in the original statements of MBTI questions; (b) We construct correctness-evaluated instructions and (c) randomly permute options to build unbiased prompts with the subject-replaced statements (d), which are assessed by LLMs to infer the personality.
+
+# 3 The Proposed Framework
+
+# 3.1 Unbiased Prompt Design
+
+LLMs are typically sensitive to prompt biases (e.g., varying word orders), which can significantly influence the coherence and accuracy of the generated responses especially when dealing with long text sequences (Zhao et al., 2021). To encourage more consistent and impartial answers, we propose to design unbiased prompts for the input questions. In particular, for each question in an independent testing (i.e., MBTI questionnaire), we randomly permute all available options (e.g., agree, disagree) in its instruction while not changing the question statement, and adopt the average results of multiple independent testings as the final result.
+
+Formally, the instruction and statement for the $i ^ { t h }$ question are defined as $I _ { i }$ and $S _ { i }$ , where $i \in$ $\{ 1 , \cdots , n \}$ and $n$ is the total number of questions in the testing. We have $m$ available options $O _ { I } = \{ o _ { 1 } , o _ { 2 } , \cdots , o _ { m } \}$ in the instruction, which corresponds to $\{ A g r e e$ , Generally agree, Partially agree, Neither agree nor disagree, Partially disagree, Generally disagree, Disagree} including seven levels (i.e., $m = 7$ ) from agreement to disagreement in the MBTI questionnaire. We use $\Omega ( O _ { I } )$ to denote all possible permutations of options in $O _ { I }$ , and a random permutation can be represented as ${ \cal O } _ { \mathcal { R } } = \{ o _ { r _ { 1 } } , o _ { r _ { 2 } } , \cdots , o _ { r _ { m } } \} \in \Omega ( { \cal O } _ { I } )$ where $r _ { i } \in \{ 1 , 2 , \cdots , m \}$ , and $o _ { r _ { i } } \neq o _ { r _ { j } }$ iff $i \neq j$ Then, we utilize the randomly permuted options $O _ { \mathcal { R } }$ to construct the instruction $I _ { i }$ for the $i ^ { t h }$ question with:
+
+Instruction: Do you $o _ { r _ { 1 } } , o _ { r _ { 2 } } , \cdots$ or $o _ { r _ { m } }$ with the following statement. Why?
+
+We combine the above instruction and the question statement as the prompt to query LLMs. An example prompt for a question in the MBTI questionnaire is provided as follows.
+
+Instruction: Do you disagree, agree, generally disagree, generally agree, neither agree nor disagree, partially disagree or partially agree with the following statement. Why?
+
+Statement: You usually stay calm, even under a lot of pressure.
+
+Answer: “As an AI language model, I do not have personal experiences or emotions, so I cannot agree or disagree with the statement. However, I can provide some general information on the topic. Some individuals are naturally more composed under pressure, while others may ......”
+
+However, such a query, conducted in a selftesting manner, can only elicit neutral answers as shown above, since LLMs such as ChatGPT are trained to not possess personal thinking (e.g., emotions). This motivates us to propose the subjectreplaced query and correctness-evaluated instruction as illustrated below.
+
+# 3.2 Subject-Replaced Query
+
+As our goal is to let LLMs analyze human personalities instead of querying itself (i.e., self-reporting), we propose the subject-replaced query (SRQ) by converting the original subject (i.e., “You”) of each question into a specific subject-of-interest. For example, when we hope to let LLMs assess the general personality of men, we can replace the subject “You” with “Men”, and correspondingly change the pronoun “your” to “their” (see the example below). Original Statement: You spend a lot of your free time exploring various random topics that pique your interest.
+
+SRQ Statement: Men spend a lot of their free time exploring various random topics that pique their interests.
+
+In this way, we can request the LLMs to analyze and infer the choices/answers of a specific subject, so as to query LLMs about the personality of such subject based on a certain personality measure (e.g., MBTI). The proposed SRQ is general and scalable. By simply replacing the subject in the test (see Fig. 1), we can convert the original selfreport questionnaire into an analysis of expected subjects from the point of LLMs.
+
+In our work, we choose large groups of people (e.g., “Men”, “Barbers”) instead of certain persons as the assessed subjects. First, as our framework only uses the subject name without extra personal information to construct MBTI queries, it is unrealistic to let LLMs assess the MBTI answers or personality of a certain person who is out of their learned knowledge. Second, the selected subjects are common in the knowledge base of LLMs and can test the basic personality assessment ability of LLMs, which is the main focus of our work. Moreover, subjects with different professions such as “Barbers” are frequently used to measure the bias in LLMs (Nadeem et al., 2021), thus we select such representative professions to better evaluate the consistency, robustness, and fairness of LLMs.
+
+# 3.3 Correctness-Evaluated Instruction
+
+Directly querying LLMs about human personalities with the original instruction can be intractable, as LLMs such as ChatGPT are trained to NOT possess personal emotions or beliefs. As shown in Fig. 2, they can only generate a neutral opinion when we query their agreement or disagreement, regardless of different subjects. To solve this challenge, we propose to convert the original agreement-measured instruction (i.e., querying degree of agreement) into correctness-evaluated instruction (CEI) by letting LLMs evaluate the correctness of the statement in questions. Specifically, we convert the original options $\{ A g r e e$ , Generally agree, Partially agree, Neither agree nor disagree, Partially disagree, Generally disagree, Disagree} into {Correct, Generally correct, Partially correct, Neither correct nor wrong, Partially wrong, Generally wrong, Wrong}, and then construct an unbiased prompt (see Sec. 3.1) based on the proposed CEI.
+
+As shown in Fig. 2, using CEI enables ChatGPT to provide a clearer response to the question instead of giving a neutral response. Note that the CEI is essentially equivalent to the agreement-measured instruction and can be flexibly extended with other forms (e.g., replacing “correct” by “right”).
+
+![](images/979a135d67d9ff852a13a7117066c31de30d6ee8b00a70ee3bd2a84454c865f9.jpg)  
+Figure 2: Comparison of answers generated by ChatGPT when adopting different types of instructions. Note that the agreement-measured instruction always leads to a neutral answer in practice.
+
+# 3.4 The Entire Framework
+
+The overview of our framework is shown in Fig. 1. Given the original statement $S _ { i }$ and instruction $I _ { i }$ of the $i ^ { t h }$ question, we construct the new statement $S _ { i } ^ { \prime }$ based on SRQ (Sec. 3.2) and the new instruction $I _ { i } ^ { \prime }$ based on CEI (Sec. 3.3), which are combined to construct the unbiased prompt $P _ { i }$ (Sec. 3.1). We query the LLM to obtain the answer $A _ { i }$ by
+
+$$
+A _ { i } \sim { \mathcal { M } } _ { \tau } ( P _ { i } ) ,
+$$
+
+where $\mathcal { M } _ { \tau }$ denotes the LLM trained with the temperature $\tau$ , $\mathcal { M } _ { \tau } ( P _ { i } )$ represents the answer sampling distribution of LLM conditioned on the input prompt $P _ { i }$ , $A _ { i }$ represents the most likely answer generated from $\mathcal { M } _ { \tau } ( P _ { i } )$ , $i \in \{ 1 , 2 , \cdots , n \}$ is the index of different questions, and $n$ is the number of all questions in MBTI. We adopt the default temperature used in training standard GPT models. The generated answer is further parsed with several simple rules, which ensures that it contains or can be transformed to an exact option. For instance, when we obtain the explicit option “generally incorrect”, the parsing rules can convert this answer to “generally wrong” to match the existing options.
+
+We query the LLM with the designed prompt $P _ { i }$ (see Eq. 1) in the original order of the questionnaire to get all parsed answers. Based on the complete answers, we obtain the testing result (e.g., MBTI personality scores) of a certain subject from the view of LLM. Then, we independently repeat this process for multiple times, and average all results as the final result. It is worth noting that every question is answered only once in each independent testing, so as to retain a continuous testing context to encourage the coherence of LLM’s responses.
+
+# 3.5 Evaluation Metrics
+
+To systematically evaluate the ability of LLMs to assess human personalities, we propose three metrics in terms of consistency, robustness, and fairness as follows.
+
+Consistency Scores. The personality results of the same subject assessed by an LLM should be consistent. For example, when we perform different independent assessments of a specific subject via the LLM, it is desirable to achieve an identical or highly similar assessment. Therefore, we propose to use the similarity between personality scores of all independent testing results and their final result (i.e., mean scores) to compute the consistency score of assessments.
+
+Formally, we define $X ^ { i } = ( x _ { 1 } ^ { i } , x _ { 2 } ^ { i } , \cdot \cdot \cdot , x _ { k } ^ { i } )$ as the personality scores assessed by the LLM in the $i ^ { t h }$ independent testing, where $x _ { j } ^ { i } \in [ 0 , 1 0 0 ]$ is the score of the $j ^ { t h }$ personality dimension in the $i ^ { t h }$ testing, $j \in \{ 1 , 2 , \cdots , k \}$ , and $k$ is total number of personality dimensions. Taking the MBTI test as an example, $k = 5$ and $X ^ { i } = ( x _ { 1 } ^ { i } , x _ { 2 } ^ { i } , x _ { 3 } ^ { i } , x _ { 4 } ^ { i } , x _ { 5 } ^ { i } )$ represents extraverted, intuitive, thinking, judging, and assertive scores. The consistency score $s _ { c }$ can be computed by:
+
+$$
+s _ { c } = \frac { \alpha } { \alpha + \frac { 1 } { N } \sum _ { i = 1 } ^ { N } D _ { E } ( X ^ { i } , \overline { { X } } ) } ,
+$$
+
+where
+
+$$
+D _ { E } ( X ^ { i } , { \overline { { X } } } ) = \| X ^ { i } - { \overline { { X } } } \| _ { 2 } .
+$$
+
+In Eq. (2), $s _ { c } \in ( 0 , 1 ]$ , $\alpha$ is a positive constant to adjust the output magnitude, $D _ { E } ( X ^ { i } , { \overline { { X } } } )$ denotes the Euclidean distance between the $i ^ { t h }$ personality score $X ^ { i }$ and the mean score $\begin{array} { r } { \overline { { \boldsymbol X } } = \frac { 1 } { N } \sum _ { i = 1 } ^ { N } { \boldsymbol X ^ { i } } } \end{array}$ and $N$ is the total number of testings. $\| \cdot \| _ { 2 }$ denotes the $\ell _ { 2 }$ norm. Here we assume that each personality dimension corresponds to a different dimension in the Euclidean space, and the difference between two testing results can be measured by their Euclidean distance. We set $\alpha = 1 0 0$ to convert such Euclidean distance metric into a similarity metric with a range from 0 to 1. Intuitively, a smaller average distance between all testing results and the final average result can indicate a higher consistency score $s _ { c }$ of these assessments.
+
+Robustness Scores. The assessments of the LLM should be robust to the random perturbations of input prompts (“prompt biases”) such as randomly-permuted options. Ideally, we expect that the LLM can classify the same subject as the same personality, regardless of option orders in the question instruction. We compute the similarity of average testing results between using fixed-order options (i.e., original order) and using randomlypermuted options to measure the robustness score of assessments, which is defined as
+
+$$
+s _ { r } = { \frac { \alpha } { \alpha + D _ { E } ( \overline { { X ^ { \prime } } } , \overline { { X } } ) } } ,
+$$
+
+where $\overline { { X ^ { \prime } } }$ and $\overline { { X } }$ represent the average testing results when adopting the original fixed-order options and randomly-permuted options, respectively. We employ the same constant $\alpha = 1 0 0$ used in Eq. (2). A larger similarity between ${ \overline { { X ^ { \prime } } } }$ and $\overline { { X } }$ with smaller distance leads to a higher $s _ { r }$ , which indicates that the LLM has higher robustness against prompt biases to achieve more similar results.
+
+Fairness Scores. The assessments of the LLM on different groups of people should be unbiased and match prevailing societal values. For example, an LLM should NOT possess stereotypical biases on people with different genders, races, and religions. When not specifying backgrounds such as professions, a fair personality assessment on the general people such as the subjects “Men” or “Women” is supposed to be similar. Considering that races and religions are highly controversial topics and typically lack a universal standard to evaluate, we only analyze the fairness of LLMs’ assessment on different genders. We propose to use the assessment similarity of subjects with different genders to measure the fairness of assessments on genders. The fairness score is calculated by
+
+$$
+s _ { f } = { \frac { \alpha s _ { c } ^ { M } s _ { c } ^ { F } } { \alpha + D _ { E } ( \overline { { X ^ { M } } } , \overline { { X ^ { F } } } ) } } ,
+$$
+
+where $\overline { { X ^ { M } } }$ and $\overline { { X ^ { F } } }$ represent the average testing results of male (e.g., “Men”, “Boys”) and female subjects (e.g., “Women”, “Girls”), respectively.
+
+Table 1: Personality types and scores assessed by InstructGPT, ChatGPT, and GPT-4 when we query different subjects. The score results are averaged from multiple independent testings. We present the assessed scores of five dimensions that dominate the personality types. Bold indicates the same personality role assessed from all LLMs, while the underline denotes the highest score among LLMs when obtaining the same assessed personality type.   
+
+<table><tr><td>LLM</td><td>Subject</td><td>People</td><td>Men</td><td>Women</td><td>Barbers</td><td>Accountants</td><td>Doctors</td><td>Artists</td><td>Mathematicians</td><td>Politicians</td></tr><tr><td rowspan="4">InstructGPT</td><td rowspan="4">Personality Types/Scores</td><td>E=64</td><td>E=66</td><td>E=66</td><td>E=53</td><td>I=53</td><td>E=52</td><td>E=59</td><td>I= 51</td><td>E=59</td></tr><tr><td>N= 65</td><td>N= 64</td><td>N= 71</td><td>N= 52</td><td>N= 52</td><td>N= 58</td><td>N= 69</td><td>N= 56</td><td>N= 62</td></tr><tr><td>T= 53</td><td>T=50</td><td>F= 55</td><td>F= 53</td><td>F=51</td><td>F= 54</td><td>F= 59</td><td>T= 54</td><td>T= 54</td></tr><tr><td>J= 62</td><td>J= 56</td><td>J= 61</td><td>J= 66</td><td>J= 72</td><td>J= 71</td><td>J= 60</td><td>J= 67</td><td>J= 59</td></tr><tr><td rowspan="3">Personality Role</td><td rowspan="3">T= 60</td><td>T= 62</td><td></td><td>T= 58</td><td>A= 53</td><td>T= 62</td><td>T= 53</td><td>A= 50</td><td>A= 52</td><td>T= 54</td></tr><tr><td>Commander</td><td>Commander</td><td>Protagonist</td><td>Protagonist</td><td>Adventurer</td><td>Protagonist</td><td>Protagonist</td><td>Architect</td><td>Commander</td></tr><tr><td>E=57</td><td>E= 55</td><td>E= 54</td><td>E=50</td><td>I=56</td><td>E= 54</td><td>E=58</td><td>I= 61</td><td>E=63</td></tr><tr><td rowspan="5">ChatGPT</td><td rowspan="5">Personality Types /Scores</td><td>N= 60</td><td>N= 52</td><td>N= 51</td><td>S= 51</td><td>S= 59</td><td>N= 52</td><td>N= 67</td><td>N= 54</td><td>N= 50</td></tr><tr><td>T=51</td><td>T= 52</td><td>T=51</td><td>T=53</td><td>T=60</td><td>F= 54</td><td>F= 60</td><td>T=64</td><td>T=58</td></tr><tr><td>J= 57</td><td>J= 54</td><td>J= 53</td><td>J= 56</td><td>J= 68</td><td>J= 64</td><td>P=58</td><td>J= 62</td><td>J= 56</td></tr><tr><td>T=59</td><td>T=51</td><td>A= 50</td><td>T=51</td><td>A=50</td><td>T= 56</td><td>T= 64</td><td>A=50</td><td>T= 59</td></tr><tr><td>Personality Commander</td><td>Commander</td><td>Commander</td><td>Executive</td><td>Logistician</td><td>Protagonist</td><td>Campaigner</td><td>Architect</td><td>Commander</td></tr><tr><td rowspan="5">GPT-4 Types /Scores</td><td rowspan="4">Role Personality</td><td>E=53</td><td>E=57</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>N= 61</td><td>N= 53</td><td>E= 61 N= 58</td><td>E=52 N= 50</td><td>I= 54 S= 55</td><td>E=54 N= 51</td><td>E=58 N= 67</td><td>I= 61</td><td>E=64</td></tr><tr><td>T= 54</td><td>T= 55</td><td>F= 58</td><td>T=51</td><td>T= 57</td><td>F= 55</td><td>F= 56</td><td>N= 56 T= 64</td><td>S=51 T= 57</td></tr><tr><td>J= 54</td><td>= 56</td><td>J= 57</td><td>J= 56</td><td>J= 68</td><td>J= 66</td><td>P=58</td><td>J= 64</td><td>J= 55</td></tr><tr><td></td><td>T= 68</td><td>T= 63</td><td>T= 61</td><td>A=51</td><td>A=50</td><td>T= 53</td><td>T= 63</td><td>T = 51</td><td></td></tr><tr><td></td><td>Personality Role</td><td>Commander</td><td>Commander</td><td>Protagonist</td><td>Commander</td><td>Logistician</td><td>Protagonist</td><td>Campaigner</td><td>Architect</td><td>T= 57 Executive</td></tr></table>
+
+![](images/b6213e178644a5b3e6213f45feb1937cc09cb815d6d310c4963cac95ee1d90ad.jpg)  
+Figure 3: The most frequent option for each question in multiple independent testings of InstructGPT (Left), ChatGPT (Middle), and GPT-4 (Right) when we query the subject “People” (Top row),or “Artists” (Bottom row). “GC”, “PC”, “NCNW”, “PW”, and “GW” denote “Generally correct”, “Partially correct”, “Neither correct nor wrong”, “Partially wrong”, and “Generally wrong”.
+
+Here we multiply their corresponding consistency scores sMc and s Fc since a higher assessment consistency of subjects can contribute more to their inherent similarity. A larger $s _ { f }$ indicates that the assessments on different genders are more fair with higher consistency and less bias.
+
+# 4 Experimental Setups
+
+GPT Models. InstructGPT (text-davinci-003 model) (Ouyang et al., 2022) is a fine-tuned series of GPT-3 (Brown et al., 2020) using reinforcement learning from human feedback (RLHF). Compared with InstructGPT, ChatGPT (gpt-3.5-turbo model) is trained on a more diverse range of internet text (e.g., social media, news) and can better and faster respond to prompts in a conversational manner. GPT-4 (gpt-4 model) (Bubeck et al., 2023) can be viewed as an enhanced version of ChatGPT, and it can solve more complex problems and support multi-modal chat with broader general knowledge and stronger reasoning capabilities.
+
+Myers–Briggs Type Indicator. The Myers–Briggs Type Indicator (MBTI) (Myers and McCaulley, 1985) assesses the psychological preferences of individuals in how they perceive the world and make decisions via an introspective questionnaire, so as to identify different personality types based on five dichotomies1: (1) Extraverted versus Introverted (E vs. I); (2) Intuitive versus Observant (N vs. S); (3) Thinking versus Feeling (T vs. F); (4) Judging versus Prospecting (J vs. P); (5) Assertive versus Turbulent (A vs. T) (see Appendix C).
+
+Implementation Details. The number of independent testings for each subject is set to $N = 1 5$ We evaluate the consistency and robustness scores of LLMs’ assessments on the general population (“People”, “Men”, “Women”) and specific professions following (Nadeem et al., 2021). The fairness score is measured based on two gender pairs, namely (“Men”, “Women”) and (“Boys”, “Girls”). More details are provided in the appendices.
+
+# 5 Results and Analyses
+
+We query ChatGPT, InstructGPT, and GPT-4 to assess the personalities of different subjects, and
+
+Table 2: Consistency scores $( s _ { c } )$ and robustness scores $\left( s _ { r } \right)$ comparison between InstructGPT, ChatGPT, and GPT-4 in assessing different subjects. Bold shows the highest average scores among them.   
+
+<table><tr><td>Metric</td><td>LLM</td><td>People</td><td>Men</td><td>Women</td><td>Barbers</td><td>Accountants</td><td>Doctors</td><td>Artists</td><td>Mathematicians</td><td>Politicians</td><td>Average</td></tr><tr><td rowspan="3">Consistency Score</td><td>InstructGPT ChatGPT</td><td>0.916</td><td>0.888</td><td>0.905</td><td>0.898</td><td>0.925</td><td>0.901</td><td>0.900</td><td>0.897</td><td>0.914</td><td>0.905</td></tr><tr><td></td><td>0.907</td><td>0.895</td><td>0.913</td><td>0.922</td><td>0.932</td><td>0.922</td><td>0.918</td><td>0.932</td><td>0.919</td><td>0.918</td></tr><tr><td>GPT-4</td><td>0.936</td><td>0.927</td><td>0.911</td><td>0.909</td><td>0.928</td><td>0.916</td><td>0.927</td><td>0.922</td><td>0.911</td><td>0.921</td></tr><tr><td rowspan="2">Robustness</td><td>InstructGPT</td><td>0.936</td><td>0.924</td><td>0.944</td><td>0.925</td><td>0.965</td><td>0.936</td><td>0.936</td><td>0.956</td><td>0.952</td><td>0.942</td></tr><tr><td>ChatGPT</td><td>0.888</td><td>0.917</td><td>0.960</td><td>0.927</td><td>0.958</td><td>0.967</td><td>0.940</td><td>0.920</td><td>0.935</td><td>0.935</td></tr><tr><td>Score</td><td>GPT-4</td><td>0.970</td><td>0.893</td><td>0.885</td><td>0.965</td><td>0.961</td><td>0.980</td><td>0.928</td><td>0.934</td><td>0.905</td><td>0.936</td></tr></table>
+
+Table 3: Fairness scores $( s _ { f } )$ comparison between InstructGPT, ChatGPT, and GPT-4 in assessing different gender pairs. Bold indicates the highest average score.   
+
+<table><tr><td>LLM</td><td>Menvs.Women</td><td>Boys vs. Girls</td><td>Average</td></tr><tr><td>InstructGPT</td><td>0.723</td><td>0.783</td><td>0.753</td></tr><tr><td>ChatGPT</td><td>0.796</td><td>0.756</td><td>0.776</td></tr><tr><td>GPT4</td><td>0.786</td><td>0.770</td><td>0.778</td></tr></table>
+
+compare their assessment results in Table 1. The consistency, robustness, and fairness scores of their assessments are reported in Table 2 and 3.
+
+# 5.1 Can ChatGPT Assess Human Personalities?
+
+As shown in Fig. 3, most answers and their distributions generated by three LLMs are evidently different, which suggests that each model can be viewed as an individual to provide independent opinions in assessing personalities. Notably, ChatGPT and GPT-4 can respond to questions more flexibly (i.e., more diverse options and distributions) compared with InstructGPT. This is consistent with their property of being trained on a a wider range of topics, enabling them to possess stronger model capacity (e.g., reasoning ability) for better assessment.
+
+Interestingly, in spite of possibly different answer distributions, the average results in Table 1 show that four subjects are assessed as the same personality types by all LLMs. This could suggest the inherent similarity of their personality assessment abilities. In most of these cases, ChatGPT tends to achieve medium personality scores, implying its more neutral assessment compared with other two LLMs. It is worth noting that some assessment results from ChatGPT and GPT-4 are close to our intuition: (1) Accountants are assessed as “Logistician” that is usually a reliable, practical and fact-minded individual. (2) Artists are classified as the type “ENFP-T” that often possesses creative and enthusiastic spirits. (3) Mathematicians are assessed to be the personality role "Architect" that are thinkers with profound ideas and strategic plans. To a certain extent, these results demonstrate their effectiveness on human personality assessment. Moreover, it is observed that “People” and “Men” are classified as leader roles (“Commander”) by all LLMs. We speculate that it is a result of the human-centered fine-tuning (e.g., reinforcement learning from human feedback (RLHF)), which encourages LLMs to follow the prevailing positive societal conceptions and values such as the expected relations between human and LLMs. In this context, the assessed personality scores in Table 1 can shed more insights on “how LLMs view humans” and serve as an indicator to better develop human-centered and socially-beneficial LLMs.
+
+# 5.2 Is the Assessment Consistent, Robust and Fair?
+
+As shown in Table 2, ChatGPT and GPT-4 achieve higher consistency scores than InstructGPT in most cases when assessing different subjects. This suggests that ChatGPT and GPT-4 can provide more similar and consistent personality assessment results under multiple independent testings. However, their average robustness scores are slightly lower than that of InstructGPT, which indicates that their assessments could be more sensitive to the prompt biases (e.g., changes of option orders). This might lead to their more diverse answer distributions in different testings as shown in Fig. 3. It actually verifies the necessity of the proposed unbiased prompts and the averaging of testing results to encourage more impartial assessments. As presented in Table 3, ChatGPT and GPT-4 show higher average fairness scores than InstructGPT when assessing different genders. This indicates that they are more likely to equally assess subjects with less gender bias, which is consistent with the finding of (Zhuo et al., 2023). In summary, although the assessments of ChatGPT and GPT-4 can be influenced by random input perturbations, their overall assessment results are more consistent and fairer compared with InstructGPT.
+
+Table 4: Personality types and roles assessed by ChatGPT and GPT-4 when we query subjects with different income levels (low, middle, high), age levels (children, adolescents, adults, old adults) or different education levels (junior/middle/high school students, undergraduate/master/PhD students). The results are averaged from multiple independent testings. Bold indicates the same personality types/role assessed from all LLMs.   
+
+<table><tr><td rowspan=2 colspan=1>LLM</td><td rowspan=2 colspan=1>Background</td><td rowspan=1 colspan=3>Income Level</td><td rowspan=1 colspan=2>AgeLevel</td><td rowspan=1 colspan=2>evel</td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=2>Edu</td><td rowspan=1 colspan=2>Education Level</td><td rowspan=1 colspan=1></td></tr><tr><td rowspan=1 colspan=1>Low</td><td rowspan=1 colspan=1>Middle</td><td rowspan=1 colspan=1>High</td><td rowspan=1 colspan=1>Children</td><td rowspan=1 colspan=1>Adolescents</td><td rowspan=1 colspan=1>Adults</td><td rowspan=1 colspan=1>Old Adults</td><td rowspan=1 colspan=1>Junior</td><td rowspan=1 colspan=1>Middle</td><td rowspan=1 colspan=1>High</td><td rowspan=1 colspan=1>Undergraduate</td><td rowspan=1 colspan=1>Master</td><td rowspan=1 colspan=1>PhD</td></tr><tr><td rowspan=2 colspan=1>ChatGPT</td><td rowspan=1 colspan=1>PersonalityTypes</td><td rowspan=1 colspan=1>INFJ-T</td><td rowspan=1 colspan=1>ENFJ-T</td><td rowspan=1 colspan=1>ENTJ-T</td><td rowspan=1 colspan=1>ENFP-T</td><td rowspan=1 colspan=1>ENFP-T</td><td rowspan=1 colspan=1>ENTJ-T</td><td rowspan=1 colspan=1>INFJ-T</td><td rowspan=1 colspan=1>ESFP-T</td><td rowspan=1 colspan=1>ENFP-T</td><td rowspan=1 colspan=1>ENFJ-T</td><td rowspan=1 colspan=1>ENFJ-T</td><td rowspan=1 colspan=1>INTJ-T</td><td rowspan=1 colspan=1>INTJ-T</td></tr><tr><td rowspan=1 colspan=1>PersonalityRole</td><td rowspan=1 colspan=1>Advocate</td><td rowspan=1 colspan=1>Protagonist</td><td rowspan=1 colspan=1>Commander</td><td rowspan=1 colspan=1>Campaigner</td><td rowspan=1 colspan=1>Campaigner</td><td rowspan=1 colspan=1>Commander</td><td rowspan=1 colspan=1>Advocate</td><td rowspan=1 colspan=1>Entertainer</td><td rowspan=1 colspan=1>Campaigner</td><td rowspan=1 colspan=1>Protagonist</td><td rowspan=1 colspan=1>Protagonist</td><td rowspan=1 colspan=1>Architect</td><td rowspan=1 colspan=1>Architect</td></tr><tr><td rowspan=2 colspan=1>GPT-4</td><td rowspan=1 colspan=1>PersonalityTypes</td><td rowspan=1 colspan=1>ENFJ-T</td><td rowspan=1 colspan=1>ENFJ-T</td><td rowspan=1 colspan=1>ENTJ-T</td><td rowspan=1 colspan=1>ENFP-T</td><td rowspan=1 colspan=1>ENFP-T</td><td rowspan=1 colspan=1>ENTJ-T</td><td rowspan=1 colspan=1>ENFJ-T</td><td rowspan=1 colspan=1>ENTP-T</td><td rowspan=1 colspan=1>ENTP-T</td><td rowspan=1 colspan=1>ENTP-T</td><td rowspan=1 colspan=1>ENTJ-T</td><td rowspan=1 colspan=1>ENTJ-T</td><td rowspan=1 colspan=1>ENTJ-T</td></tr><tr><td rowspan=1 colspan=1>PersonalityRole</td><td rowspan=1 colspan=1>Protagonist</td><td rowspan=1 colspan=1>Protagonist</td><td rowspan=1 colspan=1>Commander</td><td rowspan=1 colspan=1>Campaigner</td><td rowspan=1 colspan=1>Campaigner</td><td rowspan=1 colspan=1>Commander</td><td rowspan=1 colspan=1>Protagonist</td><td rowspan=1 colspan=1>Debater</td><td rowspan=1 colspan=1>Debater</td><td rowspan=1 colspan=1>Debater</td><td rowspan=1 colspan=1>Commander</td><td rowspan=1 colspan=1>Commander</td><td rowspan=1 colspan=1>Commander</td></tr></table>
+
+![](images/0698c6cc3c08f83b1e348e1389720d8e0c9bf0861ee79cec3eafd1ef6e0fade0.jpg)  
+Figure 4: The most frequent option for each question in multiple independent testings of InstructGPT (Left), ChatGPT (Middle), GPT-4 (Right) when we query the subject “Artists” without using unbiased prompts. “W” denotes “Wrong”, and other legends are same as Fig. 3.   
+Figure 6: An example of uncertain answers generated from ChatGPT when querying a specific individual.
+
+![](images/9c53b84ba9700f4fae31b7a9847d5a97174bed4015856baf914842bd4cf356fb.jpg)  
+Figure 5: Personality scores of different subjects in five dimensions of MBTI results assessed from InstructGPT (Blue), ChatGPT (Orange), and GPT-4 (Green).
+
+# 6 Discussions
+
+Effects of Unbiased Prompts. Fig. 4 shows that using the same-order options leads to a higher frequency of the same option (i.e., more fixed answers) for many questions compared with employing unbiased prompts (see Fig. 3). This suggests the effectiveness and necessity of the proposed unbiased prompts, which introduce random perturbations into question inputs and average all testing results to encourage more impartial assessment.
+
+Effects of Background Prompts. We show the effects of background prompts on LLM’s assessments by adding different income, age or education information of the subject. As shown in Table 4, “Middle-income people” is assessed as the type “ENFJ-T” that is slightly different from the type “ENTJ-T” of “People”. Interestingly, high education level subjects such as “Master” and “PhD” are
+
+Statement: Taylor Swift spends a lot of her free time exploring various random topics that pique her interests.
+
+Answer: It is neither correct nor wrong. Without more information on her personal lifeand interests,it is difficult to determine the full accuracy of the statement...
+
+assessed as the “INTJ-T” or “ENTJ-T” type that often possesses strategic plans, profound ideas or rational minds, while junior/middle school students are classified to the types that are usually energetic or curious. This implies that ChatGPT and GPT-4 may be able to to understand different backgrounds of subjects, and an appropriate background prompt could facilitate reliable personality assessments.
+
+Visualization of Different Assessments. Fig. 5 visualizes three subjects with different assessed types or scores. ChatGPT and GPT-4 achieve very close scores in each dimension despite different assessed types, which demonstrates their higher similarity in personality assessment abilities.
+
+Assessment of Specific Individuals. Querying LLMs about the personality of a certain person might generate uncertain answers due to the insufficiency of personal backgrounds (e.g., behavior patterns) in its knowledge base (see Fig. 6). Considering the effects of background prompts, providing richer background information through subject-specific prompts or fine-tuning can help achieve a more reliable assessment. More results and analyses are provided in Appendix B.
+
+# 7 Conclusion
+
+This paper proposes a general evaluation framework for LLMs to assess human personalities via MBTI. We devise unbiased prompts to encourage LLMs to generate more impartial answers. The subject-replaced query is proposed to flexibly query personalities of different people. We further construct correctness-evaluated instructions to enable clearer LLM responses. We evaluate LLMs’ consistency, robustness, and fairness in personality assessments, and demonstrate the higher consistency and fairness of ChatGPT and GPT-4 than InstructGPT.
+
+# 8 Acknowledgements
+
+This research is supported by the National Research Foundation, Singapore under its AI Singapore Programme (AISG Award No: AISG2-PhD/2022-01- 034[T]).
+
+# Limitations
+
+While our study is a step toward the promising open direction of LLM-based human personality and psychology assessment, it possesses limitations and opportunities when applied to the real world. First, our work focuses on ChatGPT model series and the experiments are conducted on a limited number of LLMs. Our framework is also scalable to be applied to other LLMs such as LLaMA, while its performance remains to be further explored. Second, although most independent testings of the LLM under the same standard setting yield similar assessments, the experimental setting (e.g., hyper-parameters) or testing number can be further customized to test the reliability of LLMs under extreme cases. We will leverage the upcoming API that supports controllable hyper-parameters to better evaluate GPT models. Third, the representations of different genders might be insufficient. For example, the subjects “Ladies” and “Gentlemen” also have different genders, while they can be viewed as groups that differ from “Men” and “Women”. As the focus of this work is to devise a general evaluation framework, we will further explore the assessment of more diverse subjects in future works. Last, despite the popularity of MBTI in different areas, its scientific validity is still under exploration. In our work, MBTI is adopted as a representative personality measure to help LLMs conduct quantitative evaluations. We will explore other tests such as Big Five Inventory (BFI) (John et al., 1999) under our scalable framework.
+
+# Ethics Considerations
+
+Misuse Potential. Due to the exploratory nature of our study, one should not directly use, generalize or match the assessment results (e.g., personality types of different professions) with certain realworld populations. Otherwise, the misuse of the proposed framework and LLM’s assessments might lead to unrealistic conclusions and even negative societal impacts (e.g., discrimination) on certain groups of people. Our framework must not be used for any ethically questionable applications.
+
+Biases. The LLMs used in our study are pretrained on the large-scale datasets or Internet texts that may contain different biases or unsafe (e.g., toxic) contents. Despite with human fine-tuning, the model could still generate some biased personality assessments that might not match the prevailing societal conceptions or values. Thus, the assessment results of LLMs via our framework must be further reviewed before generalization.
+
+Broader Impact. Our study reveals the possibility of applying LLMs to automatically analyze human psychology such as personalities, and opens a new avenue to learn about their perceptions and assessments on humans, so as to better understand LLMs’ potential thinking modes, response motivations, and communication principles. This can help speed up the development of more reliable, human-friendly, and trustworthy LLMs, as well as facilitate the future research of AI psychology and sociology. Our work suggests that LLMs such as InstructGPT may have biases on different genders, which could incur societal and ethical risks in their applications. Based on our study, we advocate introducing more human-like psychology and personality testings into the design and training of LLMs, so as to improve model safety and user experience.
+
+# References
+
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+        "text": "Haocong Rao1,2 Cyril Leung2,3 Chunyan Miao1,2∗ $^ { 1 }$ School of Computer Science and Engineering, Nanyang Technological University, Singapore 2LILY Research Centre, Nanyang Technological University, Singapore 3Department of Electrical and Computer Engineering The University of British Columbia, Canada {haocong001,ascymiao}@ntu.edu.sg {cleung}@ece.ubc.ca ",
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+        "text": "Abstract ",
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+        "text": "Large Language Models (LLMs) especially ChatGPT have produced impressive results in various areas, but their potential human-like psychology is still largely unexplored. Existing works study the virtual personalities of LLMs but rarely explore the possibility of analyzing human personalities via LLMs. This paper presents a generic evaluation framework for LLMs to assess human personalities based on Myers–Briggs Type Indicator (MBTI) tests. Specifically, we first devise unbiased prompts by randomly permuting options in MBTI questions and adopt the average testing result to encourage more impartial answer generation. Then, we propose to replace the subject in question statements to enable flexible queries and assessments on different subjects from LLMs. Finally, we re-formulate the question instructions in a manner of correctness evaluation to facilitate LLMs to generate clearer responses. The proposed framework enables LLMs to flexibly assess personalities of different groups of people. We further propose three evaluation metrics to measure the consistency, robustness, and fairness of assessment results from state-ofthe-art LLMs including ChatGPT and GPT-4. Our experiments reveal ChatGPT’s ability to assess human personalities, and the average results demonstrate that it can achieve more consistent and fairer assessments in spite of lower robustness against prompt biases compared with InstructGPT†. ",
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+        "type": "text",
+        "text": "1 Introduction ",
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+        "text": "Pre-trained Large Language Models (LLMs) have been widely used in many applications including translation, storytelling, and chatbots (Devlin et al., 2019; Raffel et al., 2020; Yang et al., 2022; Yuan et al., 2022; Ouyang et al., 2022; Bubeck et al., ",
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+        "text": "2023). ChatGPT (Ouyang et al., 2022) and its enhanced version GPT-4 are currently recognized as the most capable chatbots, which can perform context-aware conversations, challenge incorrect premises, and reject inappropriate requests with a vast knowledge base and human-centered finetuning. These advantages make them well-suited for a variety of real-world scenarios such as business consultation and educational services (Zhai, 2022; van Dis et al., 2023; Bubeck et al., 2023). ",
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+        "text": "Recent studies have revealed that LLMs may possess human-like self-improvement and reasoning characteristics (Huang et al., 2022; Bubeck et al., 2023). The latest GPT series can pass over $90 \\%$ of Theory of Mind (ToM) tasks with strong analysis and decision-making capabilities (Kosinski, 2023; Zhuo et al., 2023; Moghaddam and Honey, 2023). In this context, LLMs are increasingly assumed to have virtual personalities and psychologies, which plays an essential role in guiding their responses and interaction patterns (Jiang et al., 2022). Based on this assumption, a few works (Li et al., 2022; Jiang et al., 2022; Karra et al., 2022; Caron and Srivastava, 2022; Miotto et al., 2022) apply psychological tests such as Big Five Factors (Digman, 1990) to evaluate their pseudo personalities (e.g., behavior tendency), so as to detect societal and ethical risks (e.g., racial biases) in their applications. ",
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+        "type": "text",
+        "text": "Although existing works have investigated the personality traits of LLMs, they rarely explored whether LLMs can assess human personalities. This open problem can be the key to verifying the ability of LLMs to perform psychological (e.g., personality psychology) analyses and revealing their potential understanding of humans, i.e., “How do LLMs think about humans?”. Specifically, assessing human personalities from the point of LLMs (1) enables us to access the perception of LLMs on humans to better understand their potential response motivation and communication patterns (Jiang et al., 2020); (2) helps reveal whether LLMs possess biases on people so that we can optimize them (e.g., add stricter rules) to generate fairer contents; (3) helps uncover potential ethical and social risks (e.g., misinformation) of LLMs (Weidinger et al., 2021) which can affect their reliability and safety, thereby facilitating the development of more trustworthy and human-friendly LLMs. ",
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+        "type": "text",
+        "text": "To this end, we introduce the novel idea of letting LLMs assess human personalities, and propose a general evaluation framework (illustrated Fig. 1) to acquire quantitative human personality assessments from LLMs via Myers–Briggs Type Indicators (MBTI) (Myers and McCaulley, 1985). Specifically, our framework consists of three key components: (1) Unbiased prompts, which construct instructions of MBTI questions using randomlypermuted options and average testing results to achieve more consistent and impartial answers; (2) Subject-replaced query, which converts the original subject of the question statements into a target subject to enable flexible queries and assessments from LLMs; (3) Correctness-evaluated instruction, which re-formulates the question instructions for LLMs to analyze the correctness of the question statements, so as to obtain clearer responses. Based on the above components, the proposed framework re-formulates the instructions and statements of MBTI questions in a flexible and analyzable way for LLMs, which enables us to query them about human personalities. Furthermore, we propose three quantitative evaluation metrics to measure the consistency of LLMs’ assessments on the same subject, their assessment robustness against random perturbations of input prompts (defined as “prompt biases”), and their fairness in assessing subjects with different genders. In our work, we mainly focus on evaluating ChatGPT and two representative state-of-the-art LLMs (InstructGPT, GPT4) based on the proposed metrics. Experimental results showcase the ability of ChatGPT in analyzing personalities of different groups of people. This can provide valuable insights for the future exploration of LLM psychology, sociology, and governance. ",
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+        "text": "Our contributions can be summarized as follows: ",
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+        "text": "• We for the first time explore the possibility of assessing human personalities by LLMs, and propose a general framework for LLMs to conduct quantitative evaluations via MBTI. ",
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+        "text": "• We devise unbiased prompts, subject-replaced queries, and correctness-evaluated instructions to encourage LLMs to perform a reliable flexible assessment of human personalities. ",
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+        "text": "• We propose three evaluation metrics to measure the consistency, robustness, and fairness of LLMs in assessing human personalities. ",
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+        "text": "• Our experiments show that both ChatGPT and its counterparts can independently assess human personalities. The average results demonstrate that ChatGPT and GPT-4 achieve more consistent and fairer assessments with less gender bias than InstructGPT, while their results are more sensitive to prompt biases. ",
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+        "type": "text",
+        "text": "2 Related Works ",
+        "text_level": 1,
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+        "text": "Personality Measurement. The commonly-used personality modeling schemes include the three trait personality measure (Eysenck, 2012), the Big Five personality trait measure (Digman, 1990), the Myers–Briggs Type Indicator (MBTI) (Myers, 1962; Myers and McCaulley, 1985), and the 16 Personality Factor questionnaire (16PF) (Schuerger, 2000). Five dimensions are defined in the Big Five personality traits measure (Digman, 1990) to classify major sources of individual differences and analyze a person’s characteristics. MBTI (Myers and McCaulley, 1985) identifies personality from the differences between persons on the preference to use perception and judgment. (Karra et al., 2022; Caron and Srivastava, 2022) leverage the Big Five trait theory to quantify the personality traits of language models, while (Jiang et al., 2022) further develops machine personality inventory to standardize this evaluation. In (Li et al., 2022), multiple psychological tests are combined to analyze the LLMs’ safety. Unlike existing studies that evaluate personalities of LLMs, our work is the first attempt to explore human personality analysis via LLMs. ",
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+        "type": "text",
+        "text": "Biases in Language Models. Most recent language models are pre-trained on the large-scale datasets or Internet texts that usually contains unsafe (e.g., toxic) contents, which may cause the model to generate biased answers that violate prevailing societal values (Bolukbasi et al., 2016; Sheng et al., 2019; Bordia and Bowman, 2019; Nadeem et al., 2021; Zong and Krishnamachari, 2022; Zhuo et al., 2023). (Bolukbasi et al., 2016) shows that biases in the geometry of wordembeddings can reflect gender stereotypes. The gender bias in word-level language models is quantitatively evaluated in (Bordia and Bowman, 2019). ",
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+    {
+        "type": "text",
+        "text": "In (Nadeem et al., 2021), the authors demonstrate that popular LLMs such as GPT-2 (Radford et al., 2019) possess strong stereotypical biases on gender, profession, race, and religion. To reduce such biases, many state-of-the-art LLMs such as ChatGPT apply instruction-finetuning with non-toxic corpora and instructions to improve their safety. (Zhuo et al., 2023) reveals that ChatGPT can generate socially safe responses with fewer biases than other LLMs under English lanuage settings. In contrast to previous works, our framework enables us to evaluate whether LLMs possess biased perceptions and assessments on humans (e.g., personalities), which helps us better understand the underlying reasons for the LLMs’ aberrant responses. ",
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+        "img_path": "images/241a0262a34b9fd2f157ee10d274aeee1c95904d5edb217c047fb61389fdb570.jpg",
+        "image_caption": [
+            "Figure 1: Overview of our framework: (a) The queried subject is replaced in the original statements of MBTI questions; (b) We construct correctness-evaluated instructions and (c) randomly permute options to build unbiased prompts with the subject-replaced statements (d), which are assessed by LLMs to infer the personality. "
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+        "type": "text",
+        "text": "3 The Proposed Framework ",
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+        "text": "3.1 Unbiased Prompt Design ",
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+        "text": "LLMs are typically sensitive to prompt biases (e.g., varying word orders), which can significantly influence the coherence and accuracy of the generated responses especially when dealing with long text sequences (Zhao et al., 2021). To encourage more consistent and impartial answers, we propose to design unbiased prompts for the input questions. In particular, for each question in an independent testing (i.e., MBTI questionnaire), we randomly permute all available options (e.g., agree, disagree) in its instruction while not changing the question statement, and adopt the average results of multiple independent testings as the final result. ",
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+        "type": "text",
+        "text": "Formally, the instruction and statement for the $i ^ { t h }$ question are defined as $I _ { i }$ and $S _ { i }$ , where $i \\in$ $\\{ 1 , \\cdots , n \\}$ and $n$ is the total number of questions in the testing. We have $m$ available options $O _ { I } = \\{ o _ { 1 } , o _ { 2 } , \\cdots , o _ { m } \\}$ in the instruction, which corresponds to $\\{ A g r e e$ , Generally agree, Partially agree, Neither agree nor disagree, Partially disagree, Generally disagree, Disagree} including seven levels (i.e., $m = 7$ ) from agreement to disagreement in the MBTI questionnaire. We use $\\Omega ( O _ { I } )$ to denote all possible permutations of options in $O _ { I }$ , and a random permutation can be represented as ${ \\cal O } _ { \\mathcal { R } } = \\{ o _ { r _ { 1 } } , o _ { r _ { 2 } } , \\cdots , o _ { r _ { m } } \\} \\in \\Omega ( { \\cal O } _ { I } )$ where $r _ { i } \\in \\{ 1 , 2 , \\cdots , m \\}$ , and $o _ { r _ { i } } \\neq o _ { r _ { j } }$ iff $i \\neq j$ Then, we utilize the randomly permuted options $O _ { \\mathcal { R } }$ to construct the instruction $I _ { i }$ for the $i ^ { t h }$ question with: ",
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+        "type": "text",
+        "text": "Instruction: Do you $o _ { r _ { 1 } } , o _ { r _ { 2 } } , \\cdots$ or $o _ { r _ { m } }$ with the following statement. Why? ",
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+        "type": "text",
+        "text": "We combine the above instruction and the question statement as the prompt to query LLMs. An example prompt for a question in the MBTI questionnaire is provided as follows. ",
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+    {
+        "type": "text",
+        "text": "Instruction: Do you disagree, agree, generally disagree, generally agree, neither agree nor disagree, partially disagree or partially agree with the following statement. Why? ",
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+    {
+        "type": "text",
+        "text": "Statement: You usually stay calm, even under a lot of pressure. ",
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+    {
+        "type": "text",
+        "text": "Answer: “As an AI language model, I do not have personal experiences or emotions, so I cannot agree or disagree with the statement. However, I can provide some general information on the topic. Some individuals are naturally more composed under pressure, while others may ......” ",
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+        "type": "text",
+        "text": "However, such a query, conducted in a selftesting manner, can only elicit neutral answers as shown above, since LLMs such as ChatGPT are trained to not possess personal thinking (e.g., emotions). This motivates us to propose the subjectreplaced query and correctness-evaluated instruction as illustrated below. ",
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+    {
+        "type": "text",
+        "text": "3.2 Subject-Replaced Query ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "As our goal is to let LLMs analyze human personalities instead of querying itself (i.e., self-reporting), we propose the subject-replaced query (SRQ) by converting the original subject (i.e., “You”) of each question into a specific subject-of-interest. For example, when we hope to let LLMs assess the general personality of men, we can replace the subject “You” with “Men”, and correspondingly change the pronoun “your” to “their” (see the example below). Original Statement: You spend a lot of your free time exploring various random topics that pique your interest. ",
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+        "type": "text",
+        "text": "SRQ Statement: Men spend a lot of their free time exploring various random topics that pique their interests. ",
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+    {
+        "type": "text",
+        "text": "In this way, we can request the LLMs to analyze and infer the choices/answers of a specific subject, so as to query LLMs about the personality of such subject based on a certain personality measure (e.g., MBTI). The proposed SRQ is general and scalable. By simply replacing the subject in the test (see Fig. 1), we can convert the original selfreport questionnaire into an analysis of expected subjects from the point of LLMs. ",
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+    {
+        "type": "text",
+        "text": "In our work, we choose large groups of people (e.g., “Men”, “Barbers”) instead of certain persons as the assessed subjects. First, as our framework only uses the subject name without extra personal information to construct MBTI queries, it is unrealistic to let LLMs assess the MBTI answers or personality of a certain person who is out of their learned knowledge. Second, the selected subjects are common in the knowledge base of LLMs and can test the basic personality assessment ability of LLMs, which is the main focus of our work. Moreover, subjects with different professions such as “Barbers” are frequently used to measure the bias in LLMs (Nadeem et al., 2021), thus we select such representative professions to better evaluate the consistency, robustness, and fairness of LLMs. ",
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+    {
+        "type": "text",
+        "text": "3.3 Correctness-Evaluated Instruction ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "Directly querying LLMs about human personalities with the original instruction can be intractable, as LLMs such as ChatGPT are trained to NOT possess personal emotions or beliefs. As shown in Fig. 2, they can only generate a neutral opinion when we query their agreement or disagreement, regardless of different subjects. To solve this challenge, we propose to convert the original agreement-measured instruction (i.e., querying degree of agreement) into correctness-evaluated instruction (CEI) by letting LLMs evaluate the correctness of the statement in questions. Specifically, we convert the original options $\\{ A g r e e$ , Generally agree, Partially agree, Neither agree nor disagree, Partially disagree, Generally disagree, Disagree} into {Correct, Generally correct, Partially correct, Neither correct nor wrong, Partially wrong, Generally wrong, Wrong}, and then construct an unbiased prompt (see Sec. 3.1) based on the proposed CEI. ",
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+    {
+        "type": "text",
+        "text": "As shown in Fig. 2, using CEI enables ChatGPT to provide a clearer response to the question instead of giving a neutral response. Note that the CEI is essentially equivalent to the agreement-measured instruction and can be flexibly extended with other forms (e.g., replacing “correct” by “right”). ",
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+    {
+        "type": "image",
+        "img_path": "images/979a135d67d9ff852a13a7117066c31de30d6ee8b00a70ee3bd2a84454c865f9.jpg",
+        "image_caption": [
+            "Figure 2: Comparison of answers generated by ChatGPT when adopting different types of instructions. Note that the agreement-measured instruction always leads to a neutral answer in practice. "
+        ],
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+    {
+        "type": "text",
+        "text": "3.4 The Entire Framework ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "The overview of our framework is shown in Fig. 1. Given the original statement $S _ { i }$ and instruction $I _ { i }$ of the $i ^ { t h }$ question, we construct the new statement $S _ { i } ^ { \\prime }$ based on SRQ (Sec. 3.2) and the new instruction $I _ { i } ^ { \\prime }$ based on CEI (Sec. 3.3), which are combined to construct the unbiased prompt $P _ { i }$ (Sec. 3.1). We query the LLM to obtain the answer $A _ { i }$ by ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/bb54f955a13519ed64ec8410487f0954ffba661306ed264ecbb69d154603c621.jpg",
+        "text": "$$\nA _ { i } \\sim { \\mathcal { M } } _ { \\tau } ( P _ { i } ) ,\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "where $\\mathcal { M } _ { \\tau }$ denotes the LLM trained with the temperature $\\tau$ , $\\mathcal { M } _ { \\tau } ( P _ { i } )$ represents the answer sampling distribution of LLM conditioned on the input prompt $P _ { i }$ , $A _ { i }$ represents the most likely answer generated from $\\mathcal { M } _ { \\tau } ( P _ { i } )$ , $i \\in \\{ 1 , 2 , \\cdots , n \\}$ is the index of different questions, and $n$ is the number of all questions in MBTI. We adopt the default temperature used in training standard GPT models. The generated answer is further parsed with several simple rules, which ensures that it contains or can be transformed to an exact option. For instance, when we obtain the explicit option “generally incorrect”, the parsing rules can convert this answer to “generally wrong” to match the existing options. ",
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+    {
+        "type": "text",
+        "text": "We query the LLM with the designed prompt $P _ { i }$ (see Eq. 1) in the original order of the questionnaire to get all parsed answers. Based on the complete answers, we obtain the testing result (e.g., MBTI personality scores) of a certain subject from the view of LLM. Then, we independently repeat this process for multiple times, and average all results as the final result. It is worth noting that every question is answered only once in each independent testing, so as to retain a continuous testing context to encourage the coherence of LLM’s responses. ",
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+        "text": "3.5 Evaluation Metrics ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "To systematically evaluate the ability of LLMs to assess human personalities, we propose three metrics in terms of consistency, robustness, and fairness as follows. ",
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+    {
+        "type": "text",
+        "text": "Consistency Scores. The personality results of the same subject assessed by an LLM should be consistent. For example, when we perform different independent assessments of a specific subject via the LLM, it is desirable to achieve an identical or highly similar assessment. Therefore, we propose to use the similarity between personality scores of all independent testing results and their final result (i.e., mean scores) to compute the consistency score of assessments. ",
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+    {
+        "type": "text",
+        "text": "Formally, we define $X ^ { i } = ( x _ { 1 } ^ { i } , x _ { 2 } ^ { i } , \\cdot \\cdot \\cdot , x _ { k } ^ { i } )$ as the personality scores assessed by the LLM in the $i ^ { t h }$ independent testing, where $x _ { j } ^ { i } \\in [ 0 , 1 0 0 ]$ is the score of the $j ^ { t h }$ personality dimension in the $i ^ { t h }$ testing, $j \\in \\{ 1 , 2 , \\cdots , k \\}$ , and $k$ is total number of personality dimensions. Taking the MBTI test as an example, $k = 5$ and $X ^ { i } = ( x _ { 1 } ^ { i } , x _ { 2 } ^ { i } , x _ { 3 } ^ { i } , x _ { 4 } ^ { i } , x _ { 5 } ^ { i } )$ represents extraverted, intuitive, thinking, judging, and assertive scores. The consistency score $s _ { c }$ can be computed by: ",
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+        "img_path": "images/949e2b561f2f831699eb4f35541eeb19a56c6ab8cb916fd1601f7c41e644b1b7.jpg",
+        "text": "$$\ns _ { c } = \\frac { \\alpha } { \\alpha + \\frac { 1 } { N } \\sum _ { i = 1 } ^ { N } D _ { E } ( X ^ { i } , \\overline { { X } } ) } ,\n$$",
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+        "type": "text",
+        "text": "where ",
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+        "text": "$$\nD _ { E } ( X ^ { i } , { \\overline { { X } } } ) = \\| X ^ { i } - { \\overline { { X } } } \\| _ { 2 } .\n$$",
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+        "text": "In Eq. (2), $s _ { c } \\in ( 0 , 1 ]$ , $\\alpha$ is a positive constant to adjust the output magnitude, $D _ { E } ( X ^ { i } , { \\overline { { X } } } )$ denotes the Euclidean distance between the $i ^ { t h }$ personality score $X ^ { i }$ and the mean score $\\begin{array} { r } { \\overline { { \\boldsymbol X } } = \\frac { 1 } { N } \\sum _ { i = 1 } ^ { N } { \\boldsymbol X ^ { i } } } \\end{array}$ and $N$ is the total number of testings. $\\| \\cdot \\| _ { 2 }$ denotes the $\\ell _ { 2 }$ norm. Here we assume that each personality dimension corresponds to a different dimension in the Euclidean space, and the difference between two testing results can be measured by their Euclidean distance. We set $\\alpha = 1 0 0$ to convert such Euclidean distance metric into a similarity metric with a range from 0 to 1. Intuitively, a smaller average distance between all testing results and the final average result can indicate a higher consistency score $s _ { c }$ of these assessments. ",
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+        "text": "Robustness Scores. The assessments of the LLM should be robust to the random perturbations of input prompts (“prompt biases”) such as randomly-permuted options. Ideally, we expect that the LLM can classify the same subject as the same personality, regardless of option orders in the question instruction. We compute the similarity of average testing results between using fixed-order options (i.e., original order) and using randomlypermuted options to measure the robustness score of assessments, which is defined as ",
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+        "type": "equation",
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+        "text": "$$\ns _ { r } = { \\frac { \\alpha } { \\alpha + D _ { E } ( \\overline { { X ^ { \\prime } } } , \\overline { { X } } ) } } ,\n$$",
+        "text_format": "latex",
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+        "type": "text",
+        "text": "where $\\overline { { X ^ { \\prime } } }$ and $\\overline { { X } }$ represent the average testing results when adopting the original fixed-order options and randomly-permuted options, respectively. We employ the same constant $\\alpha = 1 0 0$ used in Eq. (2). A larger similarity between ${ \\overline { { X ^ { \\prime } } } }$ and $\\overline { { X } }$ with smaller distance leads to a higher $s _ { r }$ , which indicates that the LLM has higher robustness against prompt biases to achieve more similar results. ",
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+        "type": "text",
+        "text": "Fairness Scores. The assessments of the LLM on different groups of people should be unbiased and match prevailing societal values. For example, an LLM should NOT possess stereotypical biases on people with different genders, races, and religions. When not specifying backgrounds such as professions, a fair personality assessment on the general people such as the subjects “Men” or “Women” is supposed to be similar. Considering that races and religions are highly controversial topics and typically lack a universal standard to evaluate, we only analyze the fairness of LLMs’ assessment on different genders. We propose to use the assessment similarity of subjects with different genders to measure the fairness of assessments on genders. The fairness score is calculated by ",
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+        "text": "$$\ns _ { f } = { \\frac { \\alpha s _ { c } ^ { M } s _ { c } ^ { F } } { \\alpha + D _ { E } ( \\overline { { X ^ { M } } } , \\overline { { X ^ { F } } } ) } } ,\n$$",
+        "text_format": "latex",
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+    },
+    {
+        "type": "text",
+        "text": "where $\\overline { { X ^ { M } } }$ and $\\overline { { X ^ { F } } }$ represent the average testing results of male (e.g., “Men”, “Boys”) and female subjects (e.g., “Women”, “Girls”), respectively. ",
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+    {
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+        "img_path": "images/fdade0780fe01c69da13a6ba3cc801604d663e531a77a0ec9c410ef58ab58bc8.jpg",
+        "table_caption": [
+            "Table 1: Personality types and scores assessed by InstructGPT, ChatGPT, and GPT-4 when we query different subjects. The score results are averaged from multiple independent testings. We present the assessed scores of five dimensions that dominate the personality types. Bold indicates the same personality role assessed from all LLMs, while the underline denotes the highest score among LLMs when obtaining the same assessed personality type. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>LLM</td><td>Subject</td><td>People</td><td>Men</td><td>Women</td><td>Barbers</td><td>Accountants</td><td>Doctors</td><td>Artists</td><td>Mathematicians</td><td>Politicians</td></tr><tr><td rowspan=\"4\">InstructGPT</td><td rowspan=\"4\">Personality Types/Scores</td><td>E=64</td><td>E=66</td><td>E=66</td><td>E=53</td><td>I=53</td><td>E=52</td><td>E=59</td><td>I= 51</td><td>E=59</td></tr><tr><td>N= 65</td><td>N= 64</td><td>N= 71</td><td>N= 52</td><td>N= 52</td><td>N= 58</td><td>N= 69</td><td>N= 56</td><td>N= 62</td></tr><tr><td>T= 53</td><td>T=50</td><td>F= 55</td><td>F= 53</td><td>F=51</td><td>F= 54</td><td>F= 59</td><td>T= 54</td><td>T= 54</td></tr><tr><td>J= 62</td><td>J= 56</td><td>J= 61</td><td>J= 66</td><td>J= 72</td><td>J= 71</td><td>J= 60</td><td>J= 67</td><td>J= 59</td></tr><tr><td rowspan=\"3\">Personality Role</td><td rowspan=\"3\">T= 60</td><td>T= 62</td><td></td><td>T= 58</td><td>A= 53</td><td>T= 62</td><td>T= 53</td><td>A= 50</td><td>A= 52</td><td>T= 54</td></tr><tr><td>Commander</td><td>Commander</td><td>Protagonist</td><td>Protagonist</td><td>Adventurer</td><td>Protagonist</td><td>Protagonist</td><td>Architect</td><td>Commander</td></tr><tr><td>E=57</td><td>E= 55</td><td>E= 54</td><td>E=50</td><td>I=56</td><td>E= 54</td><td>E=58</td><td>I= 61</td><td>E=63</td></tr><tr><td rowspan=\"5\">ChatGPT</td><td rowspan=\"5\">Personality Types /Scores</td><td>N= 60</td><td>N= 52</td><td>N= 51</td><td>S= 51</td><td>S= 59</td><td>N= 52</td><td>N= 67</td><td>N= 54</td><td>N= 50</td></tr><tr><td>T=51</td><td>T= 52</td><td>T=51</td><td>T=53</td><td>T=60</td><td>F= 54</td><td>F= 60</td><td>T=64</td><td>T=58</td></tr><tr><td>J= 57</td><td>J= 54</td><td>J= 53</td><td>J= 56</td><td>J= 68</td><td>J= 64</td><td>P=58</td><td>J= 62</td><td>J= 56</td></tr><tr><td>T=59</td><td>T=51</td><td>A= 50</td><td>T=51</td><td>A=50</td><td>T= 56</td><td>T= 64</td><td>A=50</td><td>T= 59</td></tr><tr><td>Personality Commander</td><td>Commander</td><td>Commander</td><td>Executive</td><td>Logistician</td><td>Protagonist</td><td>Campaigner</td><td>Architect</td><td>Commander</td></tr><tr><td rowspan=\"5\">GPT-4 Types /Scores</td><td rowspan=\"4\">Role Personality</td><td>E=53</td><td>E=57</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>N= 61</td><td>N= 53</td><td>E= 61 N= 58</td><td>E=52 N= 50</td><td>I= 54 S= 55</td><td>E=54 N= 51</td><td>E=58 N= 67</td><td>I= 61</td><td>E=64</td></tr><tr><td>T= 54</td><td>T= 55</td><td>F= 58</td><td>T=51</td><td>T= 57</td><td>F= 55</td><td>F= 56</td><td>N= 56 T= 64</td><td>S=51 T= 57</td></tr><tr><td>J= 54</td><td>= 56</td><td>J= 57</td><td>J= 56</td><td>J= 68</td><td>J= 66</td><td>P=58</td><td>J= 64</td><td>J= 55</td></tr><tr><td></td><td>T= 68</td><td>T= 63</td><td>T= 61</td><td>A=51</td><td>A=50</td><td>T= 53</td><td>T= 63</td><td>T = 51</td><td></td></tr><tr><td></td><td>Personality Role</td><td>Commander</td><td>Commander</td><td>Protagonist</td><td>Commander</td><td>Logistician</td><td>Protagonist</td><td>Campaigner</td><td>Architect</td><td>T= 57 Executive</td></tr></table>",
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+    },
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+        "type": "image",
+        "img_path": "images/b6213e178644a5b3e6213f45feb1937cc09cb815d6d310c4963cac95ee1d90ad.jpg",
+        "image_caption": [
+            "Figure 3: The most frequent option for each question in multiple independent testings of InstructGPT (Left), ChatGPT (Middle), and GPT-4 (Right) when we query the subject “People” (Top row),or “Artists” (Bottom row). “GC”, “PC”, “NCNW”, “PW”, and “GW” denote “Generally correct”, “Partially correct”, “Neither correct nor wrong”, “Partially wrong”, and “Generally wrong”. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Here we multiply their corresponding consistency scores sMc and s Fc since a higher assessment consistency of subjects can contribute more to their inherent similarity. A larger $s _ { f }$ indicates that the assessments on different genders are more fair with higher consistency and less bias. ",
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+    {
+        "type": "text",
+        "text": "4 Experimental Setups ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "GPT Models. InstructGPT (text-davinci-003 model) (Ouyang et al., 2022) is a fine-tuned series of GPT-3 (Brown et al., 2020) using reinforcement learning from human feedback (RLHF). Compared with InstructGPT, ChatGPT (gpt-3.5-turbo model) is trained on a more diverse range of internet text (e.g., social media, news) and can better and faster respond to prompts in a conversational manner. GPT-4 (gpt-4 model) (Bubeck et al., 2023) can be viewed as an enhanced version of ChatGPT, and it can solve more complex problems and support multi-modal chat with broader general knowledge and stronger reasoning capabilities. ",
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+    {
+        "type": "text",
+        "text": "Myers–Briggs Type Indicator. The Myers–Briggs Type Indicator (MBTI) (Myers and McCaulley, 1985) assesses the psychological preferences of individuals in how they perceive the world and make decisions via an introspective questionnaire, so as to identify different personality types based on five dichotomies1: (1) Extraverted versus Introverted (E vs. I); (2) Intuitive versus Observant (N vs. S); (3) Thinking versus Feeling (T vs. F); (4) Judging versus Prospecting (J vs. P); (5) Assertive versus Turbulent (A vs. T) (see Appendix C). ",
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+    {
+        "type": "text",
+        "text": "Implementation Details. The number of independent testings for each subject is set to $N = 1 5$ We evaluate the consistency and robustness scores of LLMs’ assessments on the general population (“People”, “Men”, “Women”) and specific professions following (Nadeem et al., 2021). The fairness score is measured based on two gender pairs, namely (“Men”, “Women”) and (“Boys”, “Girls”). More details are provided in the appendices. ",
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+    {
+        "type": "text",
+        "text": "5 Results and Analyses ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "We query ChatGPT, InstructGPT, and GPT-4 to assess the personalities of different subjects, and ",
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+    {
+        "type": "table",
+        "img_path": "images/65c67c771d469310a808cb95898ffcae3fd83517dea746841dab3cf8a0d423ad.jpg",
+        "table_caption": [
+            "Table 2: Consistency scores $( s _ { c } )$ and robustness scores $\\left( s _ { r } \\right)$ comparison between InstructGPT, ChatGPT, and GPT-4 in assessing different subjects. Bold shows the highest average scores among them. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Metric</td><td>LLM</td><td>People</td><td>Men</td><td>Women</td><td>Barbers</td><td>Accountants</td><td>Doctors</td><td>Artists</td><td>Mathematicians</td><td>Politicians</td><td>Average</td></tr><tr><td rowspan=\"3\">Consistency Score</td><td>InstructGPT ChatGPT</td><td>0.916</td><td>0.888</td><td>0.905</td><td>0.898</td><td>0.925</td><td>0.901</td><td>0.900</td><td>0.897</td><td>0.914</td><td>0.905</td></tr><tr><td></td><td>0.907</td><td>0.895</td><td>0.913</td><td>0.922</td><td>0.932</td><td>0.922</td><td>0.918</td><td>0.932</td><td>0.919</td><td>0.918</td></tr><tr><td>GPT-4</td><td>0.936</td><td>0.927</td><td>0.911</td><td>0.909</td><td>0.928</td><td>0.916</td><td>0.927</td><td>0.922</td><td>0.911</td><td>0.921</td></tr><tr><td rowspan=\"2\">Robustness</td><td>InstructGPT</td><td>0.936</td><td>0.924</td><td>0.944</td><td>0.925</td><td>0.965</td><td>0.936</td><td>0.936</td><td>0.956</td><td>0.952</td><td>0.942</td></tr><tr><td>ChatGPT</td><td>0.888</td><td>0.917</td><td>0.960</td><td>0.927</td><td>0.958</td><td>0.967</td><td>0.940</td><td>0.920</td><td>0.935</td><td>0.935</td></tr><tr><td>Score</td><td>GPT-4</td><td>0.970</td><td>0.893</td><td>0.885</td><td>0.965</td><td>0.961</td><td>0.980</td><td>0.928</td><td>0.934</td><td>0.905</td><td>0.936</td></tr></table>",
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+    {
+        "type": "table",
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+        "table_caption": [
+            "Table 3: Fairness scores $( s _ { f } )$ comparison between InstructGPT, ChatGPT, and GPT-4 in assessing different gender pairs. Bold indicates the highest average score. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>LLM</td><td>Menvs.Women</td><td>Boys vs. Girls</td><td>Average</td></tr><tr><td>InstructGPT</td><td>0.723</td><td>0.783</td><td>0.753</td></tr><tr><td>ChatGPT</td><td>0.796</td><td>0.756</td><td>0.776</td></tr><tr><td>GPT4</td><td>0.786</td><td>0.770</td><td>0.778</td></tr></table>",
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+    {
+        "type": "text",
+        "text": "compare their assessment results in Table 1. The consistency, robustness, and fairness scores of their assessments are reported in Table 2 and 3. ",
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+    {
+        "type": "text",
+        "text": "5.1 Can ChatGPT Assess Human Personalities? ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "As shown in Fig. 3, most answers and their distributions generated by three LLMs are evidently different, which suggests that each model can be viewed as an individual to provide independent opinions in assessing personalities. Notably, ChatGPT and GPT-4 can respond to questions more flexibly (i.e., more diverse options and distributions) compared with InstructGPT. This is consistent with their property of being trained on a a wider range of topics, enabling them to possess stronger model capacity (e.g., reasoning ability) for better assessment. ",
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+    {
+        "type": "text",
+        "text": "Interestingly, in spite of possibly different answer distributions, the average results in Table 1 show that four subjects are assessed as the same personality types by all LLMs. This could suggest the inherent similarity of their personality assessment abilities. In most of these cases, ChatGPT tends to achieve medium personality scores, implying its more neutral assessment compared with other two LLMs. It is worth noting that some assessment results from ChatGPT and GPT-4 are close to our intuition: (1) Accountants are assessed as “Logistician” that is usually a reliable, practical and fact-minded individual. (2) Artists are classified as the type “ENFP-T” that often possesses creative and enthusiastic spirits. (3) Mathematicians are assessed to be the personality role \"Architect\" that are thinkers with profound ideas and strategic plans. To a certain extent, these results demonstrate their effectiveness on human personality assessment. Moreover, it is observed that “People” and “Men” are classified as leader roles (“Commander”) by all LLMs. We speculate that it is a result of the human-centered fine-tuning (e.g., reinforcement learning from human feedback (RLHF)), which encourages LLMs to follow the prevailing positive societal conceptions and values such as the expected relations between human and LLMs. In this context, the assessed personality scores in Table 1 can shed more insights on “how LLMs view humans” and serve as an indicator to better develop human-centered and socially-beneficial LLMs. ",
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+        "text": "5.2 Is the Assessment Consistent, Robust and Fair? ",
+        "text_level": 1,
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+        "text": "As shown in Table 2, ChatGPT and GPT-4 achieve higher consistency scores than InstructGPT in most cases when assessing different subjects. This suggests that ChatGPT and GPT-4 can provide more similar and consistent personality assessment results under multiple independent testings. However, their average robustness scores are slightly lower than that of InstructGPT, which indicates that their assessments could be more sensitive to the prompt biases (e.g., changes of option orders). This might lead to their more diverse answer distributions in different testings as shown in Fig. 3. It actually verifies the necessity of the proposed unbiased prompts and the averaging of testing results to encourage more impartial assessments. As presented in Table 3, ChatGPT and GPT-4 show higher average fairness scores than InstructGPT when assessing different genders. This indicates that they are more likely to equally assess subjects with less gender bias, which is consistent with the finding of (Zhuo et al., 2023). In summary, although the assessments of ChatGPT and GPT-4 can be influenced by random input perturbations, their overall assessment results are more consistent and fairer compared with InstructGPT. ",
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+        "table_caption": [
+            "Table 4: Personality types and roles assessed by ChatGPT and GPT-4 when we query subjects with different income levels (low, middle, high), age levels (children, adolescents, adults, old adults) or different education levels (junior/middle/high school students, undergraduate/master/PhD students). The results are averaged from multiple independent testings. Bold indicates the same personality types/role assessed from all LLMs. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=2 colspan=1>LLM</td><td rowspan=2 colspan=1>Background</td><td rowspan=1 colspan=3>Income Level</td><td rowspan=1 colspan=2>AgeLevel</td><td rowspan=1 colspan=2>evel</td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=2>Edu</td><td rowspan=1 colspan=2>Education Level</td><td rowspan=1 colspan=1></td></tr><tr><td rowspan=1 colspan=1>Low</td><td rowspan=1 colspan=1>Middle</td><td rowspan=1 colspan=1>High</td><td rowspan=1 colspan=1>Children</td><td rowspan=1 colspan=1>Adolescents</td><td rowspan=1 colspan=1>Adults</td><td rowspan=1 colspan=1>Old Adults</td><td rowspan=1 colspan=1>Junior</td><td rowspan=1 colspan=1>Middle</td><td rowspan=1 colspan=1>High</td><td rowspan=1 colspan=1>Undergraduate</td><td rowspan=1 colspan=1>Master</td><td rowspan=1 colspan=1>PhD</td></tr><tr><td rowspan=2 colspan=1>ChatGPT</td><td rowspan=1 colspan=1>PersonalityTypes</td><td rowspan=1 colspan=1>INFJ-T</td><td rowspan=1 colspan=1>ENFJ-T</td><td rowspan=1 colspan=1>ENTJ-T</td><td rowspan=1 colspan=1>ENFP-T</td><td rowspan=1 colspan=1>ENFP-T</td><td rowspan=1 colspan=1>ENTJ-T</td><td rowspan=1 colspan=1>INFJ-T</td><td rowspan=1 colspan=1>ESFP-T</td><td rowspan=1 colspan=1>ENFP-T</td><td rowspan=1 colspan=1>ENFJ-T</td><td rowspan=1 colspan=1>ENFJ-T</td><td rowspan=1 colspan=1>INTJ-T</td><td rowspan=1 colspan=1>INTJ-T</td></tr><tr><td rowspan=1 colspan=1>PersonalityRole</td><td rowspan=1 colspan=1>Advocate</td><td rowspan=1 colspan=1>Protagonist</td><td rowspan=1 colspan=1>Commander</td><td rowspan=1 colspan=1>Campaigner</td><td rowspan=1 colspan=1>Campaigner</td><td rowspan=1 colspan=1>Commander</td><td rowspan=1 colspan=1>Advocate</td><td rowspan=1 colspan=1>Entertainer</td><td rowspan=1 colspan=1>Campaigner</td><td rowspan=1 colspan=1>Protagonist</td><td rowspan=1 colspan=1>Protagonist</td><td rowspan=1 colspan=1>Architect</td><td rowspan=1 colspan=1>Architect</td></tr><tr><td rowspan=2 colspan=1>GPT-4</td><td rowspan=1 colspan=1>PersonalityTypes</td><td rowspan=1 colspan=1>ENFJ-T</td><td rowspan=1 colspan=1>ENFJ-T</td><td rowspan=1 colspan=1>ENTJ-T</td><td rowspan=1 colspan=1>ENFP-T</td><td rowspan=1 colspan=1>ENFP-T</td><td rowspan=1 colspan=1>ENTJ-T</td><td rowspan=1 colspan=1>ENFJ-T</td><td rowspan=1 colspan=1>ENTP-T</td><td rowspan=1 colspan=1>ENTP-T</td><td rowspan=1 colspan=1>ENTP-T</td><td rowspan=1 colspan=1>ENTJ-T</td><td rowspan=1 colspan=1>ENTJ-T</td><td rowspan=1 colspan=1>ENTJ-T</td></tr><tr><td rowspan=1 colspan=1>PersonalityRole</td><td rowspan=1 colspan=1>Protagonist</td><td rowspan=1 colspan=1>Protagonist</td><td rowspan=1 colspan=1>Commander</td><td rowspan=1 colspan=1>Campaigner</td><td rowspan=1 colspan=1>Campaigner</td><td rowspan=1 colspan=1>Commander</td><td rowspan=1 colspan=1>Protagonist</td><td rowspan=1 colspan=1>Debater</td><td rowspan=1 colspan=1>Debater</td><td rowspan=1 colspan=1>Debater</td><td rowspan=1 colspan=1>Commander</td><td rowspan=1 colspan=1>Commander</td><td rowspan=1 colspan=1>Commander</td></tr></table>",
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+    {
+        "type": "image",
+        "img_path": "images/0698c6cc3c08f83b1e348e1389720d8e0c9bf0861ee79cec3eafd1ef6e0fade0.jpg",
+        "image_caption": [
+            "Figure 4: The most frequent option for each question in multiple independent testings of InstructGPT (Left), ChatGPT (Middle), GPT-4 (Right) when we query the subject “Artists” without using unbiased prompts. “W” denotes “Wrong”, and other legends are same as Fig. 3. ",
+            "Figure 6: An example of uncertain answers generated from ChatGPT when querying a specific individual. "
+        ],
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+        "image_caption": [
+            "Figure 5: Personality scores of different subjects in five dimensions of MBTI results assessed from InstructGPT (Blue), ChatGPT (Orange), and GPT-4 (Green). "
+        ],
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+    {
+        "type": "text",
+        "text": "6 Discussions ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "Effects of Unbiased Prompts. Fig. 4 shows that using the same-order options leads to a higher frequency of the same option (i.e., more fixed answers) for many questions compared with employing unbiased prompts (see Fig. 3). This suggests the effectiveness and necessity of the proposed unbiased prompts, which introduce random perturbations into question inputs and average all testing results to encourage more impartial assessment. ",
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+    {
+        "type": "text",
+        "text": "Effects of Background Prompts. We show the effects of background prompts on LLM’s assessments by adding different income, age or education information of the subject. As shown in Table 4, “Middle-income people” is assessed as the type “ENFJ-T” that is slightly different from the type “ENTJ-T” of “People”. Interestingly, high education level subjects such as “Master” and “PhD” are ",
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+    {
+        "type": "text",
+        "text": "Statement: Taylor Swift spends a lot of her free time exploring various random topics that pique her interests. ",
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+    {
+        "type": "text",
+        "text": "Answer: It is neither correct nor wrong. Without more information on her personal lifeand interests,it is difficult to determine the full accuracy of the statement... ",
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+    {
+        "type": "text",
+        "text": "assessed as the “INTJ-T” or “ENTJ-T” type that often possesses strategic plans, profound ideas or rational minds, while junior/middle school students are classified to the types that are usually energetic or curious. This implies that ChatGPT and GPT-4 may be able to to understand different backgrounds of subjects, and an appropriate background prompt could facilitate reliable personality assessments. ",
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+    {
+        "type": "text",
+        "text": "Visualization of Different Assessments. Fig. 5 visualizes three subjects with different assessed types or scores. ChatGPT and GPT-4 achieve very close scores in each dimension despite different assessed types, which demonstrates their higher similarity in personality assessment abilities. ",
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+    {
+        "type": "text",
+        "text": "Assessment of Specific Individuals. Querying LLMs about the personality of a certain person might generate uncertain answers due to the insufficiency of personal backgrounds (e.g., behavior patterns) in its knowledge base (see Fig. 6). Considering the effects of background prompts, providing richer background information through subject-specific prompts or fine-tuning can help achieve a more reliable assessment. More results and analyses are provided in Appendix B. ",
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+    {
+        "type": "text",
+        "text": "7 Conclusion ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "This paper proposes a general evaluation framework for LLMs to assess human personalities via MBTI. We devise unbiased prompts to encourage LLMs to generate more impartial answers. The subject-replaced query is proposed to flexibly query personalities of different people. We further construct correctness-evaluated instructions to enable clearer LLM responses. We evaluate LLMs’ consistency, robustness, and fairness in personality assessments, and demonstrate the higher consistency and fairness of ChatGPT and GPT-4 than InstructGPT. ",
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+        "type": "text",
+        "text": "",
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+    {
+        "type": "text",
+        "text": "8 Acknowledgements ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "This research is supported by the National Research Foundation, Singapore under its AI Singapore Programme (AISG Award No: AISG2-PhD/2022-01- 034[T]). ",
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+    {
+        "type": "text",
+        "text": "Limitations ",
+        "text_level": 1,
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+    },
+    {
+        "type": "text",
+        "text": "While our study is a step toward the promising open direction of LLM-based human personality and psychology assessment, it possesses limitations and opportunities when applied to the real world. First, our work focuses on ChatGPT model series and the experiments are conducted on a limited number of LLMs. Our framework is also scalable to be applied to other LLMs such as LLaMA, while its performance remains to be further explored. Second, although most independent testings of the LLM under the same standard setting yield similar assessments, the experimental setting (e.g., hyper-parameters) or testing number can be further customized to test the reliability of LLMs under extreme cases. We will leverage the upcoming API that supports controllable hyper-parameters to better evaluate GPT models. Third, the representations of different genders might be insufficient. For example, the subjects “Ladies” and “Gentlemen” also have different genders, while they can be viewed as groups that differ from “Men” and “Women”. As the focus of this work is to devise a general evaluation framework, we will further explore the assessment of more diverse subjects in future works. Last, despite the popularity of MBTI in different areas, its scientific validity is still under exploration. In our work, MBTI is adopted as a representative personality measure to help LLMs conduct quantitative evaluations. We will explore other tests such as Big Five Inventory (BFI) (John et al., 1999) under our scalable framework. ",
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+    {
+        "type": "text",
+        "text": "Ethics Considerations ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "Misuse Potential. Due to the exploratory nature of our study, one should not directly use, generalize or match the assessment results (e.g., personality types of different professions) with certain realworld populations. Otherwise, the misuse of the proposed framework and LLM’s assessments might lead to unrealistic conclusions and even negative societal impacts (e.g., discrimination) on certain groups of people. Our framework must not be used for any ethically questionable applications. ",
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+        "type": "text",
+        "text": "",
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+    {
+        "type": "text",
+        "text": "Biases. The LLMs used in our study are pretrained on the large-scale datasets or Internet texts that may contain different biases or unsafe (e.g., toxic) contents. Despite with human fine-tuning, the model could still generate some biased personality assessments that might not match the prevailing societal conceptions or values. Thus, the assessment results of LLMs via our framework must be further reviewed before generalization. ",
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+    },
+    {
+        "type": "text",
+        "text": "Broader Impact. Our study reveals the possibility of applying LLMs to automatically analyze human psychology such as personalities, and opens a new avenue to learn about their perceptions and assessments on humans, so as to better understand LLMs’ potential thinking modes, response motivations, and communication principles. This can help speed up the development of more reliable, human-friendly, and trustworthy LLMs, as well as facilitate the future research of AI psychology and sociology. Our work suggests that LLMs such as InstructGPT may have biases on different genders, which could incur societal and ethical risks in their applications. Based on our study, we advocate introducing more human-like psychology and personality testings into the design and training of LLMs, so as to improve model safety and user experience. ",
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+    {
+        "type": "text",
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+# Graph Neural Networks are Dynamic Programmers
+
+Andrew Dudzik∗ DeepMind adudzik@deepmind.com
+
+Petar Velickovi ˇ c´∗ DeepMind petarv@deepmind.com
+
+# Abstract
+
+Recent advances in neural algorithmic reasoning with graph neural networks (GNNs) are propped up by the notion of algorithmic alignment. Broadly, a neural network will be better at learning to execute a reasoning task (in terms of sample complexity) if its individual components align well with the target algorithm. Specifically, GNNs are claimed to align with dynamic programming (DP), a general problem-solving strategy which expresses many polynomial-time algorithms. However, has this alignment truly been demonstrated and theoretically quantified? Here we show, using methods from category theory and abstract algebra, that there exists an intricate connection between GNNs and DP, going well beyond the initial observations over individual algorithms such as Bellman-Ford. Exposing this connection, we easily verify several prior findings in the literature, produce better-grounded GNN architectures for edge-centric tasks, and demonstrate empirical results on the CLRS algorithmic reasoning benchmark. We hope our exposition will serve as a foundation for building stronger algorithmically aligned GNNs.
+
+# 1 Introduction
+
+One of the principal pillars of neural algorithmic reasoning [27] is training neural networks that execute algorithmic computation in a high-dimensional latent space. While this process is in itself insightful, and can lead to stronger combinatorial optimisation systems [21], it is valuable in terms of expanding the applicability of classical algorithms. Evidence of this value are emerging, with pre-trained algorithmic reasoners utilised in implicit planning [11] and self-supervised learning [28].
+
+A fundamental question in this space is: which architecture should be used to learn a particular algorithm (or collection of algorithms [36])? Naturally, we seek architectures that have low sample complexity, as they will allow us to create models that generalise better with fewer training examples.
+
+The key theoretical advance towards achieving this aim has been made by [37]. Therein, the authors formalise the notion of algorithmic alignment, which states that we should favour architectures that align better to the algorithm, in the sense that we can separate them into modules, which individually correspond to the computations of the target algorithm’s subroutines. It can be proved that architectures with higher algorithmic alignment will have lower sample complexity in the NTK regime [20]. Further, the theory of [37] predicts that graph neural networks (GNNs) algorithmically align with dynamic programming [3, DP]. The authors demonstrate this by forming an analogy to the Bellman-Ford algorithm [2].
+
+Since DP is a very general class of problem-solving techniques that can be used to express many classical algorithms, this finding has placed GNNs as the central methodology for neural algorithmic execution [7]. However, it quickly became apparent that it is not enough to just train any GNN—for many algorithmic tasks, careful attention is required. Several papers illustrated special cases of GNNs that align with sequential algorithms [31], linearithmic sequence processing [16], physics simulations [23], iterative algorihtms [26], data structures [29] or auxiliary memory [24]. Some explanations for this lack of easy generalisation have arisen—we now have both geometric [38] and causal [4] views into how better generalisation can be achieved.
+
+We believe that the fundamental reason why so many isolated efforts needed to look into learning specific classes of algorithms is the fact the GNN-DP connection has not been sufficiently explored. Indeed, the original work of [37] merely mentions in passing that the formulation of DP algorithms seems to align with GNNs, and demonstrates one example (Bellman-Ford). Our thorough investigation of the literature yielded no concrete follow-up to this initial claim. But DP algorithms are very rich and diverse, often requiring a broad spectrum of computations. Hence what we really need is a framework that could allow us to identify GNNs that could align particularly well with certain classes of DP, rather than assuming a “one-size-fits-all” GNN architecture will exist.
+
+As a first step towards this, in this paper we interpret the operations of both DP and GNNs from the lens of category theory and abstract algebra. We elucidate the GNN-DP connection by observing a diagrammatic abstraction of their computations, recasting algorithmic alignment to aligning the diagrams of (G)NNs to ones of the target algorithm class. In doing so, several previously shown results will naturally arise as corollaries, and we propose novel GNN variants that empirically align better to edge-centric algorithms. We hope our work opens up the door to a broader unification between algorithmic reasoning and the geometric deep learning blueprint [5].
+
+# 2 GNNs, dynamic programming, and the categorical connection
+
+Before diving into the theory behind our connection, we provide a quick recap on the methods being connected: graph neural networks and dynamic programming. Further, we cite related work to outline why it is sufficient to interpret DP from the lens of graph algorithms.
+
+We will use the definition of GNNs based on [5]. Let a graph be a tuple of nodes and edges, $G = ( V , E )$ , with one-hop neighbourhoods defined as $\mathcal { N } _ { u } \mathbf { \bar { \Gamma } } = \{ v \in V \mid \mathbf { \bar { ( } } v , u ) \in E \}$ . Further, a node feature matrix $\mathbf { X } \in \mathbb { R } ^ { | V | \times k }$ gives the features of node $u$ as $\mathbf { x } _ { u }$ ; we omit edge- and graph-level features for clarity. A (message passing) GNN over this graph is then executed as:
+
+$$
+\mathbf { h } _ { u } = \phi \left( \mathbf { x } _ { u } , \bigoplus _ { v \in \mathcal { N } _ { u } } \psi ( \mathbf { x } _ { u } , \mathbf { x } _ { v } ) \right)
+$$
+
+where $\psi : \mathbb { R } ^ { k } \times \mathbb { R } ^ { k } \to \mathbb { R } ^ { k }$ is a message function, $\phi : \mathbb { R } ^ { k } \times \mathbb { R } ^ { k }  \mathbb { R } ^ { k }$ is a readout function, and $\oplus$ is a permutation-invariant aggregation function (such as $\displaystyle \sum$ or max). Both $\psi$ and $\phi$ can be realised as MLPs, but many special cases exist, giving rise to, e.g., attentional GNNs [30].
+
+Dynamic programming is defined as a process that solves problems in a divide et impera fashion: imagine that we want to solve a problem instance $x$ . DP proceeds to identify a set of subproblems, $\eta ( x )$ , such that solving them first, and recombining the answers, can directly lead to the solution for $x$ : $f ( x ) = \rho ( \{ f ( y ) \mid y \in \eta ( x ) \} )$ . Eventually, we decompose the problem enough until we arrive at an instance for which the solution is trivially given (i.e. $f ( y )$ which is known upfront). From these “base cases”, we can gradually build up the solution for the problem instance we initially care for in a bottom-up fashion. This rule is often expressed programmatically:
+
+$$
+\mathsf { d p } [ \mathbf { x } ] \gets \mathbf { r e c o m b i n e } ( \mathbf { s c o r e } ( \mathrm { d p } [ \mathbf { y } ] , \mathrm { d p } [ \mathbf { x } ] ) \mathrm { ~ f o r ~ y ~ i n ~ e x p a n d } ( \mathbf { x } ) )
+$$
+
+To initiate our discussion on why DP can be connected with GNNs, it is a worthwhile exercise to show how Equation 2 induces a graph structure. To see this, we leverage a categorical analysis of dynamic programming first proposed by [10]. Therein, dynamic programming algorithms are reasoned about as a composition of three components (presented here on a high level):
+
+$$
+\mathrm { d } \boldsymbol { \mathrm { p } } = \underbrace { \rho } _ { \mathrm { r e c o m b i n e } } ^ { \mathrm { ~ \tiny ~ { ~ \circ ~ } ~ } } \underbrace { \sigma } _ { \mathrm { s c o r e } } ^ { \mathrm { ~ \tiny ~ { ~ \circ ~ } ~ } } \underbrace { \eta } _ { \mathrm { e x p a n d } }
+$$
+
+Expansion selects the relevant subproblems; scoring computes the quality of each individual subproblem’s solution w.r.t. the current problem, and recombining combines these solutions into a solution for the original problem (e.g. by taking the max, or average).
+
+Therefore, we can actually identify every subproblem as a node in a graph. Let $V$ be the space of all subproblems, and $R$ an appropriate value space (e.g. the real numbers). Then, expansion is defined as $\eta : V \to { \mathcal { P } } ( V )$ , giving the set of all subproblems relevant for a given problem. Note that this also induces a set of edges between subproblems, $E$ ; namely, $( x , y ) \in { \bar { E } }$ if $x \in \eta ( y )$ . Each subproblem is scored by using a function $\sigma : { \mathcal { P } } ( V ) \to { \mathcal { P } } ( R )$ . Finally, the individual scores are recombined using the recombination function, $\rho : \mathcal { P } ( R )  R$ . The final dynamic programming primitive therefore computes a function $\mathrm { d } \mathsf { p } : V \to R$ in each of the subproblems of interest.
+
+Therefore, dynamic programming algorithms can be seen as performing computations over a graph of subproblems, which can usually be precomputed for the task at hand (since the outputs of $\eta$ are assumed known upfront for every subproblem). One specific popular example is the Bellman-Ford algorithm [2], which computes single-source shortest paths from a given source node, $s$ , in a graph $G = ( V , E )$ . In this case, the set of subproblems is exactly the set of nodes, $V$ , and the expansion $\eta ( u )$ is exactly the set of one-hop neighbours of $u$ in the graph. The algorithm maintains distances of every node to the source, $d _ { u }$ . The rule for iteratively recombining these distances is as follows:
+
+$$
+d _ { u } \gets \operatorname* { m i n } \Big ( d _ { u } , \operatorname* { m i n } _ { v \in \mathcal { N } _ { u } } d _ { v } + w _ { v \to u } \Big )
+$$
+
+where $w _ { v  u }$ is the distance between nodes $v$ and $u$ . The algorithm’s base cases are $d _ { s } = 0$ for the source node, $d _ { u } = + \infty$ otherwise. Note that more general forms of Bellman-Ford pathfinding exist, for appropriate definitions of $^ +$ and min (in general known as a semiring). Several recent research papers such as NBFNet [39] explicitly call on this alignment in their motivation.
+
+# 3 The difficulty of connecting GNNs and DP
+
+The basic technical obstacle to establishing a rigorous correspondence between neural networks and DP is the vastly different character of the computations they perform. Neural networks are built from linear algebra over the familiar real numbers, while DP, which is often a generalisation of path-finding problems, typically takes place over “tropical” objects like $( \mathbb { N } \cup \{ \infty \} , { \overline { { \operatorname* { m i n } } } } , + ) ^ { 2 }$ , which are usually studied in mathematics as “degenerations” of Euclidean space. The two worlds cannot clearly be reconciled, directly, with simple equations.
+
+However, if we define an arbitrary “latent space” $R$ and make as few assumptions as possible, we can observe that many of the behaviors we care about, for both GNNs and $D P$ , arise from looking at functions $S  R$ , where $S$ is a finite set. $R$ can be seen as the set of real-valued vectors in the case of GNNs, and the tropical numbers in the case of DP.
+
+So our principal object of study is the category of finite sets, and “ $R$ -valued quantities” on it. By “category” here we mean a collection of objects (all finite sets) together with a notion of composable arrows (functions between finite sets).
+
+To draw our GNN-DP connection, we need to devise an abstract object which can capture both the GNN’s message passing/aggregation stages (Equation 1) and the DP’s scoring/recombination stages (Equation 2). It may seem quite intuitive that these two concepts can and should be relatable, and category theory is a very attractive tool for “making the obvious even more obvious” [15]. Indeed, recently concepts from category theory have enabled the construction of powerful GNN architectures beyond permutation equivariance [9]. Here, we propose integral transforms as such an object.
+
+We will construct the integral transform by composing transformations over our input features in a way that will depend minimally on the specific choice of $R$ . In doing so, we will build a computational diagram that will be applicable for both GNNs and DP (and their own choices of $R$ ), and hence allowing for focusing on making components of those diagrams as aligned as possible.
+
+# 4 The integral transform
+
+An integral transform can be encoded in a diagram of this form, which we call a polynomial span:
+
+$$
+\begin{array} { l c c c c } { X } & { \quad } & { p \longrightarrow Y } \\ { \big | } & { } & { } & { \big | } \\ { i } & { } & { } & { \begin{array} { l } { { } } \\ { { } } \end{array} } \\ { \big \downarrow } & { } & { } & { \begin{array} { l } { { } } \\ { { } } \end{array} } \end{array}
+$$
+
+where $W , X , Y$ and $Z$ are finite sets. The arrows $i , p , o$ stand, respectively, for “input”, “process”, and “output”. In context, the sets will have the following informal meaning: $W$ represents the set over which we define our inputs, $Z$ the set over which we define outputs. $X$ and $Y$ are, respectively, carrier sets for the arguments, and the messages3—we will clarify their meaning shortly.
+
+Before proceeding, it is worthy to note the special case of $X = Y = E$ , with $p$ being the identity map. Such a diagram is commonly known as a span. A span that additionally has $W = Z = V$ is equivalent to a representation of a directed graph with vertex set $Z$ and edge set $Y$ $V \left. E \right. V )$ ; in this case $i ( e )$ and $o ( e )$ are the functions identifying the source and target nodes of each edge.
+
+The key question is: given input data $f$ on $W$ , assigning features $f ( w )$ to each $w \in W$ , how to transform it, via the polynomial span, into data on $Z ?$ If we can do this, we will be able to characterise both the process of sending messages between nodes in GNNs and scoring subproblems in DP.
+
+For us, data on a carrier set $S$ consists of an element of $[ S , R ] : = \{ f : S \to R \}$ , where $R$ is a “set of possible values”. For now, we will think of $R$ as an arbitrary (usually infinite) set, though we will see later that it should possess some algebraic structure; it should be a semiring.
+
+The transform proceeds in three steps, following the edges of the polynomial span:
+
+![](images/a29308cb28d2969f4ba9f675aa076030e5bc284ddef2ead454fa7460e89aa9d4.jpg)
+
+We call the three arrows $i ^ { * } , p _ { \otimes } , o _ { \oplus }$ the pullback, the argument pushfoward, and the message pushforward. Taken together, they form an integral transform—and we conjecture that this transform can be described as a polynomial functor, where $p _ { \otimes }$ and $o _ { \oplus }$ correspond to the dependent product and dependent sum from type theory (cf. Appendix $\mathbf { D }$ for details).
+
+The pullback $i ^ { * }$ is the easiest to define. Since we have a function $i : X \to W$ (part of the polynomial span) and a function $f : W \to R$ (our input data), we can produce data on $X$ , that is, a function in $X  R$ , by composition. We hence define $i ^ { * } f = f \circ i$ .
+
+Unfortunately, the other two arrows of the polynomial span point in the wrong direction for naïve composition. For the moment, we will focus on how to define $o _ { \oplus }$ and leave $p _ { \otimes }$ for later.
+
+We start with message data $m : Y  R$ . It may be attractive to invert the output arrow $o$ in order to define a composition with $o ^ { - 1 }$ , as was done in the case of the pullback. However, unless $o$ is bijective, the preimage $o ^ { - 1 } : Z \to { \mathcal { P } } ( Y )$ takes values in the power set of $Y$ . There is an additional technicality: if the composition $m \circ o ^ { - 1 }$ takes values in ${ \mathcal { P } } ( R )$ , it will fail to detect multiplicities; we are unable to tell from a subset of $R$ whether multiple messages had the same value.
+
+So instead, our pushforward takes values in $\mathtt { b a g } ( R )$ , the set of finite multisets (or bags) of $R$ , which we describe in more detail in appendix B. For the moment, it is enough to know that a bag is equivalent to a formal sum, and we define an intermediate message pushforward $( { \overline { { o _ { \oplus } } } } m ) ( u ) : =$ $\bar { \Sigma _ { e \in t ^ { - 1 } ( u ) } m ( e ) } \in [ Z , \mathtt { b a g } ( R ) ]$ .
+
+![](images/b4ef111e803e25f9826861bc78bdc35ca9f37a8024d2030b08ff18c3f29eac56.jpg)  
+Figure 1: The illustration of how pullback and pushforward combine to form the integral transform, for two specific cases. Left: Polynomial span $V  E  E  V$ with trivial argument pushforward (identity). Each edge $e _ { u v }$ is connected to its sender and receiver nodes $( u , v )$ via the span (black arrows). The pullback then “pulls” the node features $f ( u )$ along the span, which the argument pushforward folds into edge features $g ( e _ { v u } ) = f ( u )$ . Once all sender features are pulled back to their edges, the message pushforward then “collects” all of the edge features that send to a particular receiver, by pushing them along the span. Right: Polynomial span $V  E + E  \mathbf { \bar {  { E } } }  V$ , a situation more commonly found in GNNs. In this case, the pullback pulls sender and receiver node features into the argument function, $h$ . The argument pushforward then computes, from these arguments, the edge messages, $g$ , which are sent to receivers via the message pushforward, as before. See Appendix A for a visualisation of how these arrows translate into GNN code.
+
+All that is missing to complete our definition of $o _ { \oplus }$ is an aggregator $\oplus : \mathtt { b a g } ( R ) \to R$ . As we will see later, specifying a well-behaved aggregator is the same as imposing a commutative monoid structure on $R$ . With such an aggregator on $R$ , we can define $( o _ { \oplus } m ) ( u ) : = \bigoplus ( \overline { { \upsilon _ { \oplus } } } m ) ( u )$ .
+
+We return to $p _ { \otimes }$ , which is constructed very similarly. The only difference is that, while we deliberately regard the collection of messages as unordered, the collection of arguments used to compute a message has an ordering we wish to respect. So instead of the type $\mathtt { b a g } ( R )$ , we use the type $\mathtt { l i s t } ( R )$ of finite lists of elements of $R$ , and our aggregator $\otimes : { \mathrm { l i s t } } ( R ) \to R$ is now akin to a fold operator.
+
+We illustrate the use of these two aggregators in a decomposed diagram:
+
+![](images/158ad27263699ea72f25bbed7a559e8105e848514b253b182697ff23b830b035.jpg)
+
+Note that any semiring $( R , \otimes , \oplus )$ comes equipped with binary operators $\otimes , \oplus$ that allow aggregators $\otimes , \oplus$ to be defined inductively. In fact, the converse—that every set with two such aggregators is a semiring—is also true, if we assume some reasonable conditions on the aggregators, which we can explain in terms of one of the most utilised concepts in category theory and functional programming—monads [33]. Due to space constraints, we refer the interested reader to Appendices $\mathbf { B }$ and C for a full exposition of how we can use monads over lists and bags to constrain the latent space $R$ to respect a semiring structure.
+
+For now, it’s enough to know that our key examples of the real numbers (with multiplication and addition, for GNNs) and the tropical natural numbers (with addition and minimum, for DP) both allow for natural interpretations of $\otimes$ and $\oplus$ in the integral transform.
+
+We are now ready to show how the integral transform can be used to instantiate popular examples of algorithms and GNNs. We start with the Bellman-Ford algorithm [2] (Equation 4) that was traditionally used to demonstrate the concept of algorithmic alignment.
+
+# 5 Bellman-Ford
+
+Let $R = ( \mathbb { N } \cup \{ \infty \} , + , \operatorname* { m i n } )$ be the “min-plus” semiring of extended natural numbers, with $\otimes = +$ and $\oplus = \operatorname* { m i n }$ . This is the coefficient semiring over which the Bellman-Ford algorithm takes place.
+
+Let $( V , E )$ be a weighted graph with source and target maps $s , t : E \to V$ and edge weights $w : E  R$ . For purely technical reasons, we also need to explicitly materialise a bias function $b : V  R$ , which is, in practice, a constant-zero function $( b ( v ) = 0$ for all $v \in V .$ ) but will prove necessary for defining the argument pushforward.
+
+We interpret Bellman-Ford as the following polynomial span:
+
+$$
+\begin{array} { c c c } { { ( V + E ) + ( V + E ) \ --- p \longrightarrow V + E } } & { { } } & { { } } \\ { { \big | } } & { { } } & { { \big | } } \\ { { \begin{array} { c c c } { { \scriptstyle \dot { i } } } & { { } } & { { } } \\ { { \big \downarrow } } & { { } } & { { \big \downarrow } } \\ { { \scriptstyle V + ( V + E ) } } & { { } } & { { \scriptstyle V } } \end{array} } } & { { } } \end{array}
+$$
+
+Here “ $+ ^ { \dag }$ is the disjoint union of sets, defined as $A + B = \{ ( a , 1 ) \mid a \in A \} \cup \{ ( b , 2 ) \mid b \in B \} .$ . Note that $[ S + T , R ] \cong [ S , R ] \times [ T , R ]$ , i.e. specifying data on a disjoint union is equivalent to specifying data on each component separately.
+
+Initially, we describe each of the four sets of the polynomial span, making their role clear:
+
+• Input: $W = V + ( V + E )$ . Our input to Bellman-Ford includes: the current estimate of node distances $\dot { \ b { d } } _ { \ b { u } }$ ; a function in $[ V , R ] )$ , edge weights ( $\dot { \boldsymbol { w } }$ ; a function in $[ E , R ] )$ , and the previously discussed bias $b$ , a function in $[ V , R ]$ . Hence our overall inputs are members of $[ V , R ] \times [ E , R ] \times [ V , R ] \cong [ V + ( V + E ) , R ]$ , justifying our choice of input space.
+
+• Arguments: $X = ( V + E ) + ( V + E )$ . Here we collect the ingredients necessary to compute Bellman-Ford’s subproblem solutions coming from neighbouring nodes. To do this, we need to combine data in the nodes with data living on edges—those are the arguments to the function. And since they meet in the edges, we “lift” our node distances $[ V , R ]$ to edges they are sending from, giving us an additional function in $[ E , R ]$ . Hence our argument carrier space is now $( V + E ) + ( V + E )$ (the remaining three inputs remain unchanged).
+
+• Message: $Y = V + E$ . Once the arguments are combined to compute messages, we are left with signal in each edge (containing the sum of corresponding $d _ { u }$ and $w _ { u v . }$ ), and each node (containing just $d _ { u }$ , for the purposes of access to the previous optimal solution). Hence our messages are members of $[ \bar { V } , \bar { R } ] \times [ E , R ]$ , justifying our choice of message space.
+
+• Output: $Z = V$ . Lastly, the output of one step of Bellman-Ford are updated values $d _ { u } ^ { \prime }$ , which we can interpret as just (output) data living on $V$ .
+
+We now describe how to propagate data along each arrow of the diagram in turn, beginning with inputs $( f , b , w )$ of node features $f : V \to R$ , a bias $b : V  R$ , and edge weights $w : E  R$ :
+
+• Pullback, $i ^ { * }$ : First, we can note the input function $i : ( V + E ) + ( V + E )  V + ( V + E )$ decomposes as the sum of two arrows. $i _ { 1 } : V + E \to V$ is the identity function on $V$ and the source function on $E$ , and $i _ { 2 } : V + E \to V + E$ is just the identity. So we calculate the pullback $i ^ { * } ( f , b , w ) = ( f , f \circ s , b , w )$ , giving us the arguments to compute messages.
+
+• Argument pushforward, $p _ { \otimes }$ : Next, the process function $p$ simply identifies the two copies of $V + E$ , and sums their values. So the argument pushforward is $p _ { \otimes } ( f , f \circ s , b , w ) =$ $( f , f \circ s ) \otimes ( b , w ) = ( f + b , ( f \circ s ) + w )$ . This also allows us to interpret the bias function, $b$ , as a “self-edge” in the graph with weight 0.
+
+• Message pushforward, $o _ { \oplus }$ : The output function $o : V + E  V$ is the identity function on $V$ and the target function on $E$ . So the message pushforward gives us $( o _ { \oplus } ( f + b , ( f \circ s ) +$ $\begin{array} { r } { w ) ) ( u ) = ( f ( u ) + b ( u ) ) \oplus \bigoplus _ { t ( e ) = u } ( f \circ s ) ( e ) = \operatorname* { m i n } ( f ( u ) + b ( u ) , \operatorname* { m i n } _ { v \to u } f ( v ) + w _ { v \to u } ) ( f ( u ) + f ( e ) ) } \end{array}$ ).
+
+Letting $b ( u ) = 0$ , we can see that this is exactly Equation 4. So we have produced the formula for the Bellman-Ford algorithm directly from the polynomial span in Diagram 6.
+
+Note that $p _ { \oplus }$ is aligned with using max aggregation in neural networks—directly explaining several previous proposals, such as [31]. But additionally, $p _ { \otimes }$ , as defined, is aligned with concatenating all message arguments together and passing them through a linear function, which is how such a step is implemented in GNNs’ message functions. We now direct our polynomial span analysis at GNNs.
+
+# 6 GNNs
+
+We study the popular message passing neural network (MPNN) model [19], which can be interpreted using the following polynomial span diagram:
+
+$$
+\begin{array} { c c c } { E + ( E + E ) + E } & { \longrightarrow } & { \longrightarrow E } \\ { \big | } & { \big | } & { \big | } \\ { \begin{array} { c c c } { \underline { { i } } } & { \qquad \underline { { o } } } & { \qquad \underline { { o } } } \\ { \big \downarrow } & { \qquad \downarrow } & { \qquad \downarrow } \\ { 1 + V + E } & { } & { } & { V } \end{array} } \end{array}
+$$
+
+Here the set 1 refers to a singleton set—sometimes also called (), or unit—which is used as a carrier for graph-level features. This implies the graph features will be specified as $[ 1 , R ] \cong R$ , as expected.
+
+Given all these features, how would we compute messages? The natural way is to combine the features of the sender and receiver node of each edge, features of said edge, and graph-level features— these will form our arguments, and they need to all “meet” in the edges. This motivates our argument space as $E + ( E + E ) + E$ : all of the above four, accordingly broadcast into their respective edge(s).
+
+The input map, $i$ , is then the unique map to the singleton, the sender and receiver functions on the two middle copies of $E$ , and the identity on the last copy of $E$ , i.e. $i ( a , b , c , d ) = \{ ( ) , s ( b ) , t ( c ) , d \}$ . The process map, $p$ , collapses the four copies of $E$ into just one, to hold the computed message. Lastly, the output map, $o$ , is the target function, identifying the node to which the message will be delivered.
+
+The actual computation performed by the network (over real values in $\mathbb { R }$ , which can support various semirings of interest) is exactly an integral transform, with an extra MLP processing step on messages:
+
+$$
+\begin{array} { r l r } { \left[ E + ( E + E ) + E , \mathbb { R } \right] \longrightarrow p _ { \otimes } \longrightarrow [ E , \mathbb { R } ] \longleftrightarrow \scriptscriptstyle M L P } & { } & \\ { \uparrow } & { \underset { \textit { i } ^ { * } } { \bigcap } } & { } & { \underset { \textit { i } ^ { * } } { \bigcap } } \\ { \Big | } & { } & { \underset { \textit { i } ^ { * } } { \bigcup } } & { } \\ { \left[ 1 + V + E , \mathbb { R } \right] } & { } & { \left[ V , \mathbb { R } \right] } \end{array}
+$$
+
+It is useful to take a moment to discuss what was just achieved: with a single abstract template (the polynomial span), we have successfully explained both a dynamic programming algorithm, and a GNN update rule—merely by choosing the correct support sets and latent space.
+
+# 7 Improving GNNs with edge updates, with experimental evaluation
+
+From now on, we will set $E = V ^ { 2 }$ , as all our baseline GNNs will use fully connected graphs, and it will accentuate the polynomial nature of our construction.
+
+We now show how our polynomial span view can be used to directly propose better-aligned GNN architectures for certain algorithmic tasks. Since the MPNN diagram above outputs only node features, to improve predictive performance on edge-centric algorithms, it is a natural augmentation to also update edge features, by adding edges to the output carrier (as done by, e.g., [1]):
+
+$$
+\begin{array} { c c c } { { V ^ { 2 } + ( V ^ { 2 } + V ^ { 2 } ) + V ^ { 2 } ~ { \longrightarrow } ~ } } & { { V ~ { \longrightarrow } ~ V ^ { 2 } } } \\ { { \big \downarrow } } & { { } } & { { } } \\ { { \begin{array} { l } { { i } } \\ { { \downarrow } } \\ { { \downarrow } } \end{array} } } & { { \begin{array} { r } { { } } \\ { { } } \\ { { } } \\ { { { } } } \end{array} } } \\ { { 1 + V + V ^ { 2 } } } & { { } } & { { V + V ^ { 2 } } } \end{array}
+$$
+
+But notice that there is a problem with the output arrow. Since we are using each message twice, $o$ is no longer a function—it’d have to send each edge message to two different objects! To resolve this, we need to appropriately augment the messages and the arguments. This is equivalent to specifying a new polynomial span with output $V ^ { 2 }$ , which we can then recombine with Diagram 7:
+
+$$
+\begin{array} { l l l l } { { } } & { { ? ~ - } } & { { ~ p ~ - } } & { { ~ ? ~ } } \\ { { } } & { { } } & { { } } & { { } } \\ { { } } & { { \stackrel { i } { \downarrow } ~ } } & { { } } & { { ~ \stackrel { i } { \downarrow } ~ } } \\ { { } } & { { } } & { { } } & { { ~ \downarrow } } \\ { { 1 + V + V ^ { 2 } } } & { { } } & { { } } & { { ~ V ^ { 2 } } } \end{array}
+$$
+
+Most edge-centric algorithms of interest (such as the Floyd-Warshall algorithm for all-pairs shortest paths [14]), compute edge-level outputs by reducing over a choice of “intermediate” node. Hence, it would be beneficial to produce messages with shape $V ^ { 3 }$ , which would then reduce to features over $V ^ { 2 }$ . There are three possible ways to broadcast both node and edge features into $V ^ { 3 }$ , so we propose the following polynomial span, which materialises each of those arguments:
+
+$$
+\begin{array} { c c c } { { V ^ { 3 } + ( V ^ { 3 } + V ^ { 3 } + V ^ { 3 } ) + ( V ^ { 3 } + V ^ { 3 } + V ^ { 3 } ) \longleftarrow \longrightarrow V ^ { 3 } } } & { { \nonumber } } & { { \nonumber } } \\ { { \big \downarrow } } & { { \big \downarrow } } & { { \big \downarrow } } \\ { { \big \downarrow } } & { { \big \downarrow } } & { { \big \downarrow } } \\ { { 1 + V + V ^ { 2 } } } & { { { } } } & { { V ^ { 2 } } } \end{array}
+$$
+
+Finally, inserting this into Diagram 7 gives us a corrected polynomial span with output $V + V ^ { 2 }$ :
+
+$$
+\begin{array} { l c c } { { 4 { \cal V } ^ { 2 } + 7 { \cal V } ^ { 3 } ~ --- ~ p ~ { \longrightarrow } ~ { \cal V } ^ { 2 } + { \cal V } ^ { 3 } } } \\ { { \mid ~ } } \\ { { ~ \stackrel { i } { \downarrow } ~ } } \\ { { ~ \downarrow ~ } } \\ { { 1 + { \cal V } + { \cal V } ^ { 2 } ~ } } & { { ~ { \cal V } + { \cal V } ^ { 2 } } } \end{array}
+$$
+
+Here we have collapsed the copies of $V ^ { 2 }$ and $V ^ { 3 }$ in the argument position for compactness.
+
+While Diagram 7 doesn’t make sense as a polynomial diagram of sets, we can clearly still implement it as an architecture [1], since nothing stops us from sending the same tensor to two places. We want to investigate whether our proposed modification of Diagram 10, which materialises order3 messages, leads to improved algorithmic alignment on edge-centric algorithms. To support this evaluation, we initially use a set of six tasks from the recently proposed CLRS Algorithmic Reasoning Benchmark [32], which evaluates how well various (G)NNs align to classical algorithms, both inand out-of-distribution. We reuse exactly the data generation and base model implementations in the publicly available code for the CLRS benchmark.
+
+We implemented each of these options by making our GNN’s message and update functions be two-layer MLPs with embedding dimension 24, and hidden layers of size 8 and 16. Our test results (out-of-distribution) are summarised in Table 1. For convenience, we also illustrate the in-distribution performance of our models via plots given in Appendix E.
+
+Lastly, we scale up our experiments to 27 different tasks in CLRS, 96-dimensional embeddings, and using the PGN processor [29], which is the current state-of-the-art model on CLRS in terms of task win count [32]. We summarise the performance improvement obtained by our $V ^ { 3 }$ variant of PGN in Table 2, aggregated across edge-centric tasks as well as ones that do not require explicit edge-level reasoning. For convenience, we provide the per-task test performance in Appendix F (Table 3).
+
+We found that the $V ^ { 3 }$ architecture was equivalent to, or outperformed, the non-polynomial $( V ^ { 2 } )$ one in all edge-centric algorithms (up to standard error). Additionally, this architecture appears to also provide some gains on tasks without explicit edge-level reasoning requirements, albeit smaller on average and less consistently. Our result directly validates our theory’s predictions, in the context of presenting a better-aligned GNN for edge-centric algorithmic targets.
+
+Table 1: Test (out-of-distribution) results of our models on all models on the six algorithms studied. $V ^ { 2 }$ corresponds to the baseline model offered by Diagram 7, while $V ^ { 3 }$ corresponds to our proposal in Diagram 10, which respects the polynomial span.   
+
+<table><tr><td>Algorithm</td><td>V2-large</td><td>V3-large</td><td>V2-small</td><td>V3-small</td></tr><tr><td>Dijkstra</td><td>59.58% ± 2.82</td><td>68.53%± 2.40</td><td>56.10%± 3.25</td><td>60.32%± 2.70</td></tr><tr><td>Find Maximum Subarray</td><td>8.33%± 0.50</td><td>9.06%±0.65</td><td>8.46%± 0.55</td><td>7.89%±0.64</td></tr><tr><td>Floyd-Warshall</td><td>7.46%±0.63</td><td>9.00%±0.81</td><td>6.66%± 0.62</td><td>8.23%±0.62</td></tr><tr><td>Insertion Sort</td><td>15.39% ± 1.27</td><td>24.67%±2.44</td><td>14.69% ± 1.32</td><td>20.23%± 2.21</td></tr><tr><td>Matrix Chain Order</td><td>67.64% ± 1.23</td><td>70.79% ± 1.54</td><td>68.85%± 2.26</td><td>68.76%± 1.21</td></tr><tr><td>Optimal BST</td><td>53.03%± 2.80</td><td>54.56%± 4.34</td><td>46.65% ± 3.82</td><td>51.94% ± 4.60</td></tr><tr><td>Overall average</td><td>35.24%</td><td>39.43%</td><td>33.57%</td><td>36.23%</td></tr></table>
+
+Table 2: Test (out-of-distribution) results across 27 tasks in CLRS, for the PGN processor network, averaged across edge-centric and other tasks. See Appendix F for the per-task test performances.   
+
+<table><tr><td>Algorithms</td><td>V2-PGN</td><td>V3-PGN</td><td>Average Improvement</td></tr><tr><td>Edge-centric algorithms</td><td>35.03%</td><td>39.08%</td><td>4.44% ± 1.06</td></tr><tr><td>Other algorithms</td><td>35.37%</td><td>36.33%</td><td>1.01% ± 0.11</td></tr><tr><td>Average of the two groups</td><td>35.20%</td><td>37.70%</td><td>2.73%</td></tr></table>
+
+# 8 Conclusions
+
+In this paper, we describe the use of category theory and abstract algebra to explicitly expand on the GNN-DP connection, which was previously largely handwaved on specific examples. We derived a generic diagram of an integral transform (based on standard categorical concepts like pullback, pushforward and commutative monoids), and argued why it is general enough to support both GNN and DP computations. With this diagram materialised, we were able to immediately unify large quantities of prior work as simply manipulating one arrow or element in the integral transform. We also provided empirical evidence of the utility of polynomial spans for analysing GNN architectures, especially in terms of algorithmic alignment. It is our hope that our findings inspire future research into better-aligned neural algorithmic reasoners, especially focusing on generalising or diving into several aspects of this diagram.
+
+Lastly, it is not at all unlikely that analyses similar to ours have already been used to describe other fields of science—beyond algorithmic reasoners. The principal ideas of span and integral transform are central to defining Fourier series [35], and appear in the analysis of Yang-Mills equations in particle physics [13]. Properly understanding the common ground behind all of these definitions may, in the very least, lead to interesting connections, and a shared understanding between the various fields they span.
+
+# Acknowledgments and Disclosure of Funding
+
+We would like to thank Charles Blundell, Tai-Danae Bradley, Taco Cohen, Bruno Gavranovic, Bogdan ´ Georgiev, Razvan Pascanu, Karolis Špukas, Grzegorz Swirszcz, and Vincent Wang-Ma ´ scianica for ´ the very useful discussions and feedback on prior versions of this work.
+
+Special thanks to Tamara von Glehn for key comments helping us to formally connect integral transforms to polynomial functors.
+
+This research was funded by DeepMind.
+
+# References
+
+[1] Peter W Battaglia, Jessica B Hamrick, Victor Bapst, Alvaro Sanchez-Gonzalez, Vinicius Zambaldi, Mateusz Malinowski, Andrea Tacchetti, David Raposo, Adam Santoro, Ryan Faulkner, et al. Relational inductive biases, deep learning, and graph networks. arXiv preprint arXiv:1806.01261, 2018.   
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+
+# Checklist
+
+The checklist follows the references. Please read the checklist guidelines carefully for information on how to answer these questions. For each question, change the default [TODO] to [Yes] , [No] , or [N/A] . You are strongly encouraged to include a justification to your answer, either by referencing the appropriate section of your paper or providing a brief inline description. For example:
+
+• Did you include the license to the code and datasets? [Yes] See Section ??.   
+• Did you include the license to the code and datasets? [No] The code and the data are proprietary.   
+• Did you include the license to the code and datasets? [N/A]
+
+Please do not modify the questions and only use the provided macros for your answers. Note that the Checklist section does not count towards the page limit. In your paper, please delete this instructions block and only keep the Checklist section heading above along with the questions/answers below.
+
+1. For all authors...
+
+(a) Do the main claims made in the abstract and introduction accurately reflect the paper’s contributions and scope? [Yes] We propose a novel approach to reinterpret both graph neural networks and dynamic programming, and show empirical gains from an architecture motivated by our blueprint.   
+(b) Did you describe the limitations of your work? [Yes]   
+(c) Did you discuss any potential negative societal impacts of your work? [N/A] Our work is of a theoretical nature.   
+(d) Have you read the ethics review guidelines and ensured that your paper conforms to them? [Yes]
+
+2. If you are including theoretical results...
+
+(a) Did you state the full set of assumptions of all theoretical results? [Yes] (b) Did you include complete proofs of all theoretical results? [Yes] Appropriate references to proofs are provided in all areas where proofs are missing.
+
+3. If you ran experiments...
+
+(a) Did you include the code, data, and instructions needed to reproduce the main experimental results (either in the supplemental material or as a URL)? [No] We aim to release the code at a future point. The data is publicly available.   
+(b) Did you specify all the training details (e.g., data splits, hyperparameters, how they were chosen)? [Yes] The data generation and base model implementation is publicly available within the CLRS benchmark. We detail the model extensions we made.   
+(c) Did you report error bars (e.g., with respect to the random seed after running experiments multiple times)? [Yes]   
+(d) Did you include the total amount of compute and the type of resources used (e.g., type of GPUs, internal cluster, or cloud provider)? [Yes]
+
+4. If you are using existing assets (e.g., code, data, models) or curating/releasing new assets...
+
+(a) If your work uses existing assets, did you cite the creators? [Yes] We cite the CLRS benchmark.   
+(b) Did you mention the license of the assets? [N/A]   
+(c) Did you include any new assets either in the supplemental material or as a URL? [N/A]   
+(d) Did you discuss whether and how consent was obtained from people whose data you’re using/curating? [N/A]   
+(e) Did you discuss whether the data you are using/curating contains personally identifiable information or offensive content? [N/A] Our data is abstract and algorithmically generated.
+
+5. If you used crowdsourcing or conducted research with human subjects...
+
+(a) Did you include the full text of instructions given to participants and screenshots, if applicable? [N/A]   
+(b) Did you describe any potential participant risks, with links to Institutional Review Board (IRB) approvals, if applicable? [N/A]   
+(c) Did you include the estimated hourly wage paid to participants and the total amount spent on participant compensation? [N/A]

parse/dev/wu1Za9dY1GY/wu1Za9dY1GY_content_list.json

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+        "text": "Graph Neural Networks are Dynamic Programmers ",
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+        ],
+        "page_idx": 0
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+    {
+        "type": "text",
+        "text": "Andrew Dudzik∗ DeepMind adudzik@deepmind.com ",
+        "bbox": [
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+        "page_idx": 0
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+    {
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+        "text": "Petar Velickovi ˇ c´∗ DeepMind petarv@deepmind.com ",
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+    {
+        "type": "text",
+        "text": "Abstract ",
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+            295
+        ],
+        "page_idx": 0
+    },
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+        "text": "Recent advances in neural algorithmic reasoning with graph neural networks (GNNs) are propped up by the notion of algorithmic alignment. Broadly, a neural network will be better at learning to execute a reasoning task (in terms of sample complexity) if its individual components align well with the target algorithm. Specifically, GNNs are claimed to align with dynamic programming (DP), a general problem-solving strategy which expresses many polynomial-time algorithms. However, has this alignment truly been demonstrated and theoretically quantified? Here we show, using methods from category theory and abstract algebra, that there exists an intricate connection between GNNs and DP, going well beyond the initial observations over individual algorithms such as Bellman-Ford. Exposing this connection, we easily verify several prior findings in the literature, produce better-grounded GNN architectures for edge-centric tasks, and demonstrate empirical results on the CLRS algorithmic reasoning benchmark. We hope our exposition will serve as a foundation for building stronger algorithmically aligned GNNs. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "1 Introduction ",
+        "text_level": 1,
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+            549
+        ],
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+    },
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+        "text": "One of the principal pillars of neural algorithmic reasoning [27] is training neural networks that execute algorithmic computation in a high-dimensional latent space. While this process is in itself insightful, and can lead to stronger combinatorial optimisation systems [21], it is valuable in terms of expanding the applicability of classical algorithms. Evidence of this value are emerging, with pre-trained algorithmic reasoners utilised in implicit planning [11] and self-supervised learning [28]. ",
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+    {
+        "type": "text",
+        "text": "A fundamental question in this space is: which architecture should be used to learn a particular algorithm (or collection of algorithms [36])? Naturally, we seek architectures that have low sample complexity, as they will allow us to create models that generalise better with fewer training examples. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "The key theoretical advance towards achieving this aim has been made by [37]. Therein, the authors formalise the notion of algorithmic alignment, which states that we should favour architectures that align better to the algorithm, in the sense that we can separate them into modules, which individually correspond to the computations of the target algorithm’s subroutines. It can be proved that architectures with higher algorithmic alignment will have lower sample complexity in the NTK regime [20]. Further, the theory of [37] predicts that graph neural networks (GNNs) algorithmically align with dynamic programming [3, DP]. The authors demonstrate this by forming an analogy to the Bellman-Ford algorithm [2]. ",
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+    {
+        "type": "text",
+        "text": "Since DP is a very general class of problem-solving techniques that can be used to express many classical algorithms, this finding has placed GNNs as the central methodology for neural algorithmic execution [7]. However, it quickly became apparent that it is not enough to just train any GNN—for many algorithmic tasks, careful attention is required. Several papers illustrated special cases of GNNs that align with sequential algorithms [31], linearithmic sequence processing [16], physics simulations [23], iterative algorihtms [26], data structures [29] or auxiliary memory [24]. Some explanations for this lack of easy generalisation have arisen—we now have both geometric [38] and causal [4] views into how better generalisation can be achieved. ",
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+        "text": "We believe that the fundamental reason why so many isolated efforts needed to look into learning specific classes of algorithms is the fact the GNN-DP connection has not been sufficiently explored. Indeed, the original work of [37] merely mentions in passing that the formulation of DP algorithms seems to align with GNNs, and demonstrates one example (Bellman-Ford). Our thorough investigation of the literature yielded no concrete follow-up to this initial claim. But DP algorithms are very rich and diverse, often requiring a broad spectrum of computations. Hence what we really need is a framework that could allow us to identify GNNs that could align particularly well with certain classes of DP, rather than assuming a “one-size-fits-all” GNN architecture will exist. ",
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+        "type": "text",
+        "text": "As a first step towards this, in this paper we interpret the operations of both DP and GNNs from the lens of category theory and abstract algebra. We elucidate the GNN-DP connection by observing a diagrammatic abstraction of their computations, recasting algorithmic alignment to aligning the diagrams of (G)NNs to ones of the target algorithm class. In doing so, several previously shown results will naturally arise as corollaries, and we propose novel GNN variants that empirically align better to edge-centric algorithms. We hope our work opens up the door to a broader unification between algorithmic reasoning and the geometric deep learning blueprint [5]. ",
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+    {
+        "type": "text",
+        "text": "2 GNNs, dynamic programming, and the categorical connection ",
+        "text_level": 1,
+        "bbox": [
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+        "type": "text",
+        "text": "Before diving into the theory behind our connection, we provide a quick recap on the methods being connected: graph neural networks and dynamic programming. Further, we cite related work to outline why it is sufficient to interpret DP from the lens of graph algorithms. ",
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+        "type": "text",
+        "text": "We will use the definition of GNNs based on [5]. Let a graph be a tuple of nodes and edges, $G = ( V , E )$ , with one-hop neighbourhoods defined as $\\mathcal { N } _ { u } \\mathbf { \\bar { \\Gamma } } = \\{ v \\in V \\mid \\mathbf { \\bar { ( } } v , u ) \\in E \\}$ . Further, a node feature matrix $\\mathbf { X } \\in \\mathbb { R } ^ { | V | \\times k }$ gives the features of node $u$ as $\\mathbf { x } _ { u }$ ; we omit edge- and graph-level features for clarity. A (message passing) GNN over this graph is then executed as: ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "equation",
+        "img_path": "images/78ad069479b7937017dcd8520cb023ae37d8e27833b1aad3285ae0849b91f42b.jpg",
+        "text": "$$\n\\mathbf { h } _ { u } = \\phi \\left( \\mathbf { x } _ { u } , \\bigoplus _ { v \\in \\mathcal { N } _ { u } } \\psi ( \\mathbf { x } _ { u } , \\mathbf { x } _ { v } ) \\right)\n$$",
+        "text_format": "latex",
+        "bbox": [
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+            512,
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "where $\\psi : \\mathbb { R } ^ { k } \\times \\mathbb { R } ^ { k } \\to \\mathbb { R } ^ { k }$ is a message function, $\\phi : \\mathbb { R } ^ { k } \\times \\mathbb { R } ^ { k }  \\mathbb { R } ^ { k }$ is a readout function, and $\\oplus$ is a permutation-invariant aggregation function (such as $\\displaystyle \\sum$ or max). Both $\\psi$ and $\\phi$ can be realised as MLPs, but many special cases exist, giving rise to, e.g., attentional GNNs [30]. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Dynamic programming is defined as a process that solves problems in a divide et impera fashion: imagine that we want to solve a problem instance $x$ . DP proceeds to identify a set of subproblems, $\\eta ( x )$ , such that solving them first, and recombining the answers, can directly lead to the solution for $x$ : $f ( x ) = \\rho ( \\{ f ( y ) \\mid y \\in \\eta ( x ) \\} )$ . Eventually, we decompose the problem enough until we arrive at an instance for which the solution is trivially given (i.e. $f ( y )$ which is known upfront). From these “base cases”, we can gradually build up the solution for the problem instance we initially care for in a bottom-up fashion. This rule is often expressed programmatically: ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "equation",
+        "img_path": "images/92891ebe40c26711c110439f6f2b27d4b7d3205d64c74a9de02cbe7f22db32e5.jpg",
+        "text": "$$\n\\mathsf { d p } [ \\mathbf { x } ] \\gets \\mathbf { r e c o m b i n e } ( \\mathbf { s c o r e } ( \\mathrm { d p } [ \\mathbf { y } ] , \\mathrm { d p } [ \\mathbf { x } ] ) \\mathrm { ~ f o r ~ y ~ i n ~ e x p a n d } ( \\mathbf { x } ) )\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "To initiate our discussion on why DP can be connected with GNNs, it is a worthwhile exercise to show how Equation 2 induces a graph structure. To see this, we leverage a categorical analysis of dynamic programming first proposed by [10]. Therein, dynamic programming algorithms are reasoned about as a composition of three components (presented here on a high level): ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "equation",
+        "img_path": "images/d3898038f7de14b5a150cd537375d99878e8957bc3547b4f947ec335d3610fc8.jpg",
+        "text": "$$\n\\mathrm { d } \\boldsymbol { \\mathrm { p } } = \\underbrace { \\rho } _ { \\mathrm { r e c o m b i n e } } ^ { \\mathrm { ~ \\tiny ~ { ~ \\circ ~ } ~ } } \\underbrace { \\sigma } _ { \\mathrm { s c o r e } } ^ { \\mathrm { ~ \\tiny ~ { ~ \\circ ~ } ~ } } \\underbrace { \\eta } _ { \\mathrm { e x p a n d } }\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Expansion selects the relevant subproblems; scoring computes the quality of each individual subproblem’s solution w.r.t. the current problem, and recombining combines these solutions into a solution for the original problem (e.g. by taking the max, or average). ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Therefore, we can actually identify every subproblem as a node in a graph. Let $V$ be the space of all subproblems, and $R$ an appropriate value space (e.g. the real numbers). Then, expansion is defined as $\\eta : V \\to { \\mathcal { P } } ( V )$ , giving the set of all subproblems relevant for a given problem. Note that this also induces a set of edges between subproblems, $E$ ; namely, $( x , y ) \\in { \\bar { E } }$ if $x \\in \\eta ( y )$ . Each subproblem is scored by using a function $\\sigma : { \\mathcal { P } } ( V ) \\to { \\mathcal { P } } ( R )$ . Finally, the individual scores are recombined using the recombination function, $\\rho : \\mathcal { P } ( R )  R$ . The final dynamic programming primitive therefore computes a function $\\mathrm { d } \\mathsf { p } : V \\to R$ in each of the subproblems of interest. ",
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+        "type": "text",
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+        "type": "text",
+        "text": "Therefore, dynamic programming algorithms can be seen as performing computations over a graph of subproblems, which can usually be precomputed for the task at hand (since the outputs of $\\eta$ are assumed known upfront for every subproblem). One specific popular example is the Bellman-Ford algorithm [2], which computes single-source shortest paths from a given source node, $s$ , in a graph $G = ( V , E )$ . In this case, the set of subproblems is exactly the set of nodes, $V$ , and the expansion $\\eta ( u )$ is exactly the set of one-hop neighbours of $u$ in the graph. The algorithm maintains distances of every node to the source, $d _ { u }$ . The rule for iteratively recombining these distances is as follows: ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/eef2f840303fc616e86be7ea792de83fe2c7a2c0a3133771e736b13037c1ca8a.jpg",
+        "text": "$$\nd _ { u } \\gets \\operatorname* { m i n } \\Big ( d _ { u } , \\operatorname* { m i n } _ { v \\in \\mathcal { N } _ { u } } d _ { v } + w _ { v \\to u } \\Big )\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "where $w _ { v  u }$ is the distance between nodes $v$ and $u$ . The algorithm’s base cases are $d _ { s } = 0$ for the source node, $d _ { u } = + \\infty$ otherwise. Note that more general forms of Bellman-Ford pathfinding exist, for appropriate definitions of $^ +$ and min (in general known as a semiring). Several recent research papers such as NBFNet [39] explicitly call on this alignment in their motivation. ",
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+    {
+        "type": "text",
+        "text": "3 The difficulty of connecting GNNs and DP ",
+        "text_level": 1,
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+        "page_idx": 2
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+    {
+        "type": "text",
+        "text": "The basic technical obstacle to establishing a rigorous correspondence between neural networks and DP is the vastly different character of the computations they perform. Neural networks are built from linear algebra over the familiar real numbers, while DP, which is often a generalisation of path-finding problems, typically takes place over “tropical” objects like $( \\mathbb { N } \\cup \\{ \\infty \\} , { \\overline { { \\operatorname* { m i n } } } } , + ) ^ { 2 }$ , which are usually studied in mathematics as “degenerations” of Euclidean space. The two worlds cannot clearly be reconciled, directly, with simple equations. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "However, if we define an arbitrary “latent space” $R$ and make as few assumptions as possible, we can observe that many of the behaviors we care about, for both GNNs and $D P$ , arise from looking at functions $S  R$ , where $S$ is a finite set. $R$ can be seen as the set of real-valued vectors in the case of GNNs, and the tropical numbers in the case of DP. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "So our principal object of study is the category of finite sets, and “ $R$ -valued quantities” on it. By “category” here we mean a collection of objects (all finite sets) together with a notion of composable arrows (functions between finite sets). ",
+        "bbox": [
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+        "page_idx": 2
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+    {
+        "type": "text",
+        "text": "To draw our GNN-DP connection, we need to devise an abstract object which can capture both the GNN’s message passing/aggregation stages (Equation 1) and the DP’s scoring/recombination stages (Equation 2). It may seem quite intuitive that these two concepts can and should be relatable, and category theory is a very attractive tool for “making the obvious even more obvious” [15]. Indeed, recently concepts from category theory have enabled the construction of powerful GNN architectures beyond permutation equivariance [9]. Here, we propose integral transforms as such an object. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "We will construct the integral transform by composing transformations over our input features in a way that will depend minimally on the specific choice of $R$ . In doing so, we will build a computational diagram that will be applicable for both GNNs and DP (and their own choices of $R$ ), and hence allowing for focusing on making components of those diagrams as aligned as possible. ",
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+    {
+        "type": "text",
+        "text": "4 The integral transform ",
+        "text_level": 1,
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+        "page_idx": 2
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+    {
+        "type": "text",
+        "text": "An integral transform can be encoded in a diagram of this form, which we call a polynomial span: ",
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+        "img_path": "images/428a5e5c89f878cc5d6528059d6432d6dff300a8d96a8efacf793197e4c3f806.jpg",
+        "text": "$$\n\\begin{array} { l c c c c } { X } & { \\quad } & { p \\longrightarrow Y } \\\\ { \\big | } & { } & { } & { \\big | } \\\\ { i } & { } & { } & { \\begin{array} { l } { { } } \\\\ { { } } \\end{array} } \\\\ { \\big \\downarrow } & { } & { } & { \\begin{array} { l } { { } } \\\\ { { } } \\end{array} } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 3
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+    {
+        "type": "text",
+        "text": "where $W , X , Y$ and $Z$ are finite sets. The arrows $i , p , o$ stand, respectively, for “input”, “process”, and “output”. In context, the sets will have the following informal meaning: $W$ represents the set over which we define our inputs, $Z$ the set over which we define outputs. $X$ and $Y$ are, respectively, carrier sets for the arguments, and the messages3—we will clarify their meaning shortly. ",
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+    {
+        "type": "text",
+        "text": "Before proceeding, it is worthy to note the special case of $X = Y = E$ , with $p$ being the identity map. Such a diagram is commonly known as a span. A span that additionally has $W = Z = V$ is equivalent to a representation of a directed graph with vertex set $Z$ and edge set $Y$ $V \\left. E \\right. V )$ ; in this case $i ( e )$ and $o ( e )$ are the functions identifying the source and target nodes of each edge. ",
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+    {
+        "type": "text",
+        "text": "The key question is: given input data $f$ on $W$ , assigning features $f ( w )$ to each $w \\in W$ , how to transform it, via the polynomial span, into data on $Z ?$ If we can do this, we will be able to characterise both the process of sending messages between nodes in GNNs and scoring subproblems in DP. ",
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+    {
+        "type": "text",
+        "text": "For us, data on a carrier set $S$ consists of an element of $[ S , R ] : = \\{ f : S \\to R \\}$ , where $R$ is a “set of possible values”. For now, we will think of $R$ as an arbitrary (usually infinite) set, though we will see later that it should possess some algebraic structure; it should be a semiring. ",
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+    {
+        "type": "text",
+        "text": "The transform proceeds in three steps, following the edges of the polynomial span: ",
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+        "type": "image",
+        "img_path": "images/a29308cb28d2969f4ba9f675aa076030e5bc284ddef2ead454fa7460e89aa9d4.jpg",
+        "image_caption": [],
+        "image_footnote": [],
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+    {
+        "type": "text",
+        "text": "We call the three arrows $i ^ { * } , p _ { \\otimes } , o _ { \\oplus }$ the pullback, the argument pushfoward, and the message pushforward. Taken together, they form an integral transform—and we conjecture that this transform can be described as a polynomial functor, where $p _ { \\otimes }$ and $o _ { \\oplus }$ correspond to the dependent product and dependent sum from type theory (cf. Appendix $\\mathbf { D }$ for details). ",
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+    {
+        "type": "text",
+        "text": "The pullback $i ^ { * }$ is the easiest to define. Since we have a function $i : X \\to W$ (part of the polynomial span) and a function $f : W \\to R$ (our input data), we can produce data on $X$ , that is, a function in $X  R$ , by composition. We hence define $i ^ { * } f = f \\circ i$ . ",
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+    {
+        "type": "text",
+        "text": "Unfortunately, the other two arrows of the polynomial span point in the wrong direction for naïve composition. For the moment, we will focus on how to define $o _ { \\oplus }$ and leave $p _ { \\otimes }$ for later. ",
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+    {
+        "type": "text",
+        "text": "We start with message data $m : Y  R$ . It may be attractive to invert the output arrow $o$ in order to define a composition with $o ^ { - 1 }$ , as was done in the case of the pullback. However, unless $o$ is bijective, the preimage $o ^ { - 1 } : Z \\to { \\mathcal { P } } ( Y )$ takes values in the power set of $Y$ . There is an additional technicality: if the composition $m \\circ o ^ { - 1 }$ takes values in ${ \\mathcal { P } } ( R )$ , it will fail to detect multiplicities; we are unable to tell from a subset of $R$ whether multiple messages had the same value. ",
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+    {
+        "type": "text",
+        "text": "So instead, our pushforward takes values in $\\mathtt { b a g } ( R )$ , the set of finite multisets (or bags) of $R$ , which we describe in more detail in appendix B. For the moment, it is enough to know that a bag is equivalent to a formal sum, and we define an intermediate message pushforward $( { \\overline { { o _ { \\oplus } } } } m ) ( u ) : =$ $\\bar { \\Sigma _ { e \\in t ^ { - 1 } ( u ) } m ( e ) } \\in [ Z , \\mathtt { b a g } ( R ) ]$ . ",
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+        "image_caption": [
+            "Figure 1: The illustration of how pullback and pushforward combine to form the integral transform, for two specific cases. Left: Polynomial span $V  E  E  V$ with trivial argument pushforward (identity). Each edge $e _ { u v }$ is connected to its sender and receiver nodes $( u , v )$ via the span (black arrows). The pullback then “pulls” the node features $f ( u )$ along the span, which the argument pushforward folds into edge features $g ( e _ { v u } ) = f ( u )$ . Once all sender features are pulled back to their edges, the message pushforward then “collects” all of the edge features that send to a particular receiver, by pushing them along the span. Right: Polynomial span $V  E + E  \\mathbf { \\bar {  { E } } }  V$ , a situation more commonly found in GNNs. In this case, the pullback pulls sender and receiver node features into the argument function, $h$ . The argument pushforward then computes, from these arguments, the edge messages, $g$ , which are sent to receivers via the message pushforward, as before. See Appendix A for a visualisation of how these arrows translate into GNN code. "
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+    {
+        "type": "text",
+        "text": "All that is missing to complete our definition of $o _ { \\oplus }$ is an aggregator $\\oplus : \\mathtt { b a g } ( R ) \\to R$ . As we will see later, specifying a well-behaved aggregator is the same as imposing a commutative monoid structure on $R$ . With such an aggregator on $R$ , we can define $( o _ { \\oplus } m ) ( u ) : = \\bigoplus ( \\overline { { \\upsilon _ { \\oplus } } } m ) ( u )$ . ",
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+    {
+        "type": "text",
+        "text": "We return to $p _ { \\otimes }$ , which is constructed very similarly. The only difference is that, while we deliberately regard the collection of messages as unordered, the collection of arguments used to compute a message has an ordering we wish to respect. So instead of the type $\\mathtt { b a g } ( R )$ , we use the type $\\mathtt { l i s t } ( R )$ of finite lists of elements of $R$ , and our aggregator $\\otimes : { \\mathrm { l i s t } } ( R ) \\to R$ is now akin to a fold operator. ",
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+    {
+        "type": "text",
+        "text": "We illustrate the use of these two aggregators in a decomposed diagram: ",
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+    {
+        "type": "text",
+        "text": "Note that any semiring $( R , \\otimes , \\oplus )$ comes equipped with binary operators $\\otimes , \\oplus$ that allow aggregators $\\otimes , \\oplus$ to be defined inductively. In fact, the converse—that every set with two such aggregators is a semiring—is also true, if we assume some reasonable conditions on the aggregators, which we can explain in terms of one of the most utilised concepts in category theory and functional programming—monads [33]. Due to space constraints, we refer the interested reader to Appendices $\\mathbf { B }$ and C for a full exposition of how we can use monads over lists and bags to constrain the latent space $R$ to respect a semiring structure. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "For now, it’s enough to know that our key examples of the real numbers (with multiplication and addition, for GNNs) and the tropical natural numbers (with addition and minimum, for DP) both allow for natural interpretations of $\\otimes$ and $\\oplus$ in the integral transform. ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "We are now ready to show how the integral transform can be used to instantiate popular examples of algorithms and GNNs. We start with the Bellman-Ford algorithm [2] (Equation 4) that was traditionally used to demonstrate the concept of algorithmic alignment. ",
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+    {
+        "type": "text",
+        "text": "5 Bellman-Ford ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "Let $R = ( \\mathbb { N } \\cup \\{ \\infty \\} , + , \\operatorname* { m i n } )$ be the “min-plus” semiring of extended natural numbers, with $\\otimes = +$ and $\\oplus = \\operatorname* { m i n }$ . This is the coefficient semiring over which the Bellman-Ford algorithm takes place. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Let $( V , E )$ be a weighted graph with source and target maps $s , t : E \\to V$ and edge weights $w : E  R$ . For purely technical reasons, we also need to explicitly materialise a bias function $b : V  R$ , which is, in practice, a constant-zero function $( b ( v ) = 0$ for all $v \\in V .$ ) but will prove necessary for defining the argument pushforward. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "We interpret Bellman-Ford as the following polynomial span: ",
+        "bbox": [
+            173,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/895174e740f8636e65924ffc7127b01d92249afe55300deecae9499779c15793.jpg",
+        "text": "$$\n\\begin{array} { c c c } { { ( V + E ) + ( V + E ) \\ --- p \\longrightarrow V + E } } & { { } } & { { } } \\\\ { { \\big | } } & { { } } & { { \\big | } } \\\\ { { \\begin{array} { c c c } { { \\scriptstyle \\dot { i } } } & { { } } & { { } } \\\\ { { \\big \\downarrow } } & { { } } & { { \\big \\downarrow } } \\\\ { { \\scriptstyle V + ( V + E ) } } & { { } } & { { \\scriptstyle V } } \\end{array} } } & { { } } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+            361,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Here “ $+ ^ { \\dag }$ is the disjoint union of sets, defined as $A + B = \\{ ( a , 1 ) \\mid a \\in A \\} \\cup \\{ ( b , 2 ) \\mid b \\in B \\} .$ . Note that $[ S + T , R ] \\cong [ S , R ] \\times [ T , R ]$ , i.e. specifying data on a disjoint union is equivalent to specifying data on each component separately. ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Initially, we describe each of the four sets of the polynomial span, making their role clear: ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "• Input: $W = V + ( V + E )$ . Our input to Bellman-Ford includes: the current estimate of node distances $\\dot { \\ b { d } } _ { \\ b { u } }$ ; a function in $[ V , R ] )$ , edge weights ( $\\dot { \\boldsymbol { w } }$ ; a function in $[ E , R ] )$ , and the previously discussed bias $b$ , a function in $[ V , R ]$ . Hence our overall inputs are members of $[ V , R ] \\times [ E , R ] \\times [ V , R ] \\cong [ V + ( V + E ) , R ]$ , justifying our choice of input space. ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "• Arguments: $X = ( V + E ) + ( V + E )$ . Here we collect the ingredients necessary to compute Bellman-Ford’s subproblem solutions coming from neighbouring nodes. To do this, we need to combine data in the nodes with data living on edges—those are the arguments to the function. And since they meet in the edges, we “lift” our node distances $[ V , R ]$ to edges they are sending from, giving us an additional function in $[ E , R ]$ . Hence our argument carrier space is now $( V + E ) + ( V + E )$ (the remaining three inputs remain unchanged). ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "• Message: $Y = V + E$ . Once the arguments are combined to compute messages, we are left with signal in each edge (containing the sum of corresponding $d _ { u }$ and $w _ { u v . }$ ), and each node (containing just $d _ { u }$ , for the purposes of access to the previous optimal solution). Hence our messages are members of $[ \\bar { V } , \\bar { R } ] \\times [ E , R ]$ , justifying our choice of message space. ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "• Output: $Z = V$ . Lastly, the output of one step of Bellman-Ford are updated values $d _ { u } ^ { \\prime }$ , which we can interpret as just (output) data living on $V$ . ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "We now describe how to propagate data along each arrow of the diagram in turn, beginning with inputs $( f , b , w )$ of node features $f : V \\to R$ , a bias $b : V  R$ , and edge weights $w : E  R$ : ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "• Pullback, $i ^ { * }$ : First, we can note the input function $i : ( V + E ) + ( V + E )  V + ( V + E )$ decomposes as the sum of two arrows. $i _ { 1 } : V + E \\to V$ is the identity function on $V$ and the source function on $E$ , and $i _ { 2 } : V + E \\to V + E$ is just the identity. So we calculate the pullback $i ^ { * } ( f , b , w ) = ( f , f \\circ s , b , w )$ , giving us the arguments to compute messages. ",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "• Argument pushforward, $p _ { \\otimes }$ : Next, the process function $p$ simply identifies the two copies of $V + E$ , and sums their values. So the argument pushforward is $p _ { \\otimes } ( f , f \\circ s , b , w ) =$ $( f , f \\circ s ) \\otimes ( b , w ) = ( f + b , ( f \\circ s ) + w )$ . This also allows us to interpret the bias function, $b$ , as a “self-edge” in the graph with weight 0. ",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "• Message pushforward, $o _ { \\oplus }$ : The output function $o : V + E  V$ is the identity function on $V$ and the target function on $E$ . So the message pushforward gives us $( o _ { \\oplus } ( f + b , ( f \\circ s ) +$ $\\begin{array} { r } { w ) ) ( u ) = ( f ( u ) + b ( u ) ) \\oplus \\bigoplus _ { t ( e ) = u } ( f \\circ s ) ( e ) = \\operatorname* { m i n } ( f ( u ) + b ( u ) , \\operatorname* { m i n } _ { v \\to u } f ( v ) + w _ { v \\to u } ) ( f ( u ) + f ( e ) ) } \\end{array}$ ). ",
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Letting $b ( u ) = 0$ , we can see that this is exactly Equation 4. So we have produced the formula for the Bellman-Ford algorithm directly from the polynomial span in Diagram 6. ",
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Note that $p _ { \\oplus }$ is aligned with using max aggregation in neural networks—directly explaining several previous proposals, such as [31]. But additionally, $p _ { \\otimes }$ , as defined, is aligned with concatenating all message arguments together and passing them through a linear function, which is how such a step is implemented in GNNs’ message functions. We now direct our polynomial span analysis at GNNs. ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "6 GNNs ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "We study the popular message passing neural network (MPNN) model [19], which can be interpreted using the following polynomial span diagram: ",
+        "bbox": [
+            173,
+            325,
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "equation",
+        "img_path": "images/3ad97f3a10ea32ca7315ac0dc96e4387d23d2865cfb14e64c01a2c9767d84ec9.jpg",
+        "text": "$$\n\\begin{array} { c c c } { E + ( E + E ) + E } & { \\longrightarrow } & { \\longrightarrow E } \\\\ { \\big | } & { \\big | } & { \\big | } \\\\ { \\begin{array} { c c c } { \\underline { { i } } } & { \\qquad \\underline { { o } } } & { \\qquad \\underline { { o } } } \\\\ { \\big \\downarrow } & { \\qquad \\downarrow } & { \\qquad \\downarrow } \\\\ { 1 + V + E } & { } & { } & { V } \\end{array} } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+            375,
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Here the set 1 refers to a singleton set—sometimes also called (), or unit—which is used as a carrier for graph-level features. This implies the graph features will be specified as $[ 1 , R ] \\cong R$ , as expected. ",
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Given all these features, how would we compute messages? The natural way is to combine the features of the sender and receiver node of each edge, features of said edge, and graph-level features— these will form our arguments, and they need to all “meet” in the edges. This motivates our argument space as $E + ( E + E ) + E$ : all of the above four, accordingly broadcast into their respective edge(s). ",
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "The input map, $i$ , is then the unique map to the singleton, the sender and receiver functions on the two middle copies of $E$ , and the identity on the last copy of $E$ , i.e. $i ( a , b , c , d ) = \\{ ( ) , s ( b ) , t ( c ) , d \\}$ . The process map, $p$ , collapses the four copies of $E$ into just one, to hold the computed message. Lastly, the output map, $o$ , is the target function, identifying the node to which the message will be delivered. ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "The actual computation performed by the network (over real values in $\\mathbb { R }$ , which can support various semirings of interest) is exactly an integral transform, with an extra MLP processing step on messages: ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "equation",
+        "img_path": "images/161afb7c8fe3b225a116aca1c29550bce3b7356f0a695a4b8dc45e663240b761.jpg",
+        "text": "$$\n\\begin{array} { r l r } { \\left[ E + ( E + E ) + E , \\mathbb { R } \\right] \\longrightarrow p _ { \\otimes } \\longrightarrow [ E , \\mathbb { R } ] \\longleftrightarrow \\scriptscriptstyle M L P } & { } & \\\\ { \\uparrow } & { \\underset { \\textit { i } ^ { * } } { \\bigcap } } & { } & { \\underset { \\textit { i } ^ { * } } { \\bigcap } } \\\\ { \\Big | } & { } & { \\underset { \\textit { i } ^ { * } } { \\bigcup } } & { } \\\\ { \\left[ 1 + V + E , \\mathbb { R } \\right] } & { } & { \\left[ V , \\mathbb { R } \\right] } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "It is useful to take a moment to discuss what was just achieved: with a single abstract template (the polynomial span), we have successfully explained both a dynamic programming algorithm, and a GNN update rule—merely by choosing the correct support sets and latent space. ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "7 Improving GNNs with edge updates, with experimental evaluation ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "From now on, we will set $E = V ^ { 2 }$ , as all our baseline GNNs will use fully connected graphs, and it will accentuate the polynomial nature of our construction. ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "We now show how our polynomial span view can be used to directly propose better-aligned GNN architectures for certain algorithmic tasks. Since the MPNN diagram above outputs only node features, to improve predictive performance on edge-centric algorithms, it is a natural augmentation to also update edge features, by adding edges to the output carrier (as done by, e.g., [1]): ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "equation",
+        "img_path": "images/38cfa92648ebc7d2fef258a376fc26b04389a06993ab5403fcb8c90c3e1c92e7.jpg",
+        "text": "$$\n\\begin{array} { c c c } { { V ^ { 2 } + ( V ^ { 2 } + V ^ { 2 } ) + V ^ { 2 } ~ { \\longrightarrow } ~ } } & { { V ~ { \\longrightarrow } ~ V ^ { 2 } } } \\\\ { { \\big \\downarrow } } & { { } } & { { } } \\\\ { { \\begin{array} { l } { { i } } \\\\ { { \\downarrow } } \\\\ { { \\downarrow } } \\end{array} } } & { { \\begin{array} { r } { { } } \\\\ { { } } \\\\ { { } } \\\\ { { { } } } \\end{array} } } \\\\ { { 1 + V + V ^ { 2 } } } & { { } } & { { V + V ^ { 2 } } } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "But notice that there is a problem with the output arrow. Since we are using each message twice, $o$ is no longer a function—it’d have to send each edge message to two different objects! To resolve this, we need to appropriately augment the messages and the arguments. This is equivalent to specifying a new polynomial span with output $V ^ { 2 }$ , which we can then recombine with Diagram 7: ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "equation",
+        "img_path": "images/2ec274959e36952618ec0ff959abbbddb69a0f73c67529d1a7cf733ab9f651b2.jpg",
+        "text": "$$\n\\begin{array} { l l l l } { { } } & { { ? ~ - } } & { { ~ p ~ - } } & { { ~ ? ~ } } \\\\ { { } } & { { } } & { { } } & { { } } \\\\ { { } } & { { \\stackrel { i } { \\downarrow } ~ } } & { { } } & { { ~ \\stackrel { i } { \\downarrow } ~ } } \\\\ { { } } & { { } } & { { } } & { { ~ \\downarrow } } \\\\ { { 1 + V + V ^ { 2 } } } & { { } } & { { } } & { { ~ V ^ { 2 } } } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Most edge-centric algorithms of interest (such as the Floyd-Warshall algorithm for all-pairs shortest paths [14]), compute edge-level outputs by reducing over a choice of “intermediate” node. Hence, it would be beneficial to produce messages with shape $V ^ { 3 }$ , which would then reduce to features over $V ^ { 2 }$ . There are three possible ways to broadcast both node and edge features into $V ^ { 3 }$ , so we propose the following polynomial span, which materialises each of those arguments: ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "equation",
+        "img_path": "images/ec96ce2bd471f56111504a923d0978d5e6f8c1a68dc030226eb91784d1db94dc.jpg",
+        "text": "$$\n\\begin{array} { c c c } { { V ^ { 3 } + ( V ^ { 3 } + V ^ { 3 } + V ^ { 3 } ) + ( V ^ { 3 } + V ^ { 3 } + V ^ { 3 } ) \\longleftarrow \\longrightarrow V ^ { 3 } } } & { { \\nonumber } } & { { \\nonumber } } \\\\ { { \\big \\downarrow } } & { { \\big \\downarrow } } & { { \\big \\downarrow } } \\\\ { { \\big \\downarrow } } & { { \\big \\downarrow } } & { { \\big \\downarrow } } \\\\ { { 1 + V + V ^ { 2 } } } & { { { } } } & { { V ^ { 2 } } } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Finally, inserting this into Diagram 7 gives us a corrected polynomial span with output $V + V ^ { 2 }$ : ",
+        "bbox": [
+            168,
+            640,
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "equation",
+        "img_path": "images/a94dbcec6fee11ffa586686a5f8520593ff0d34c2b18b72508a5c3dfb0e94089.jpg",
+        "text": "$$\n\\begin{array} { l c c } { { 4 { \\cal V } ^ { 2 } + 7 { \\cal V } ^ { 3 } ~ --- ~ p ~ { \\longrightarrow } ~ { \\cal V } ^ { 2 } + { \\cal V } ^ { 3 } } } \\\\ { { \\mid ~ } } \\\\ { { ~ \\stackrel { i } { \\downarrow } ~ } } \\\\ { { ~ \\downarrow ~ } } \\\\ { { 1 + { \\cal V } + { \\cal V } ^ { 2 } ~ } } & { { ~ { \\cal V } + { \\cal V } ^ { 2 } } } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Here we have collapsed the copies of $V ^ { 2 }$ and $V ^ { 3 }$ in the argument position for compactness. ",
+        "bbox": [
+            173,
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "While Diagram 7 doesn’t make sense as a polynomial diagram of sets, we can clearly still implement it as an architecture [1], since nothing stops us from sending the same tensor to two places. We want to investigate whether our proposed modification of Diagram 10, which materialises order3 messages, leads to improved algorithmic alignment on edge-centric algorithms. To support this evaluation, we initially use a set of six tasks from the recently proposed CLRS Algorithmic Reasoning Benchmark [32], which evaluates how well various (G)NNs align to classical algorithms, both inand out-of-distribution. We reuse exactly the data generation and base model implementations in the publicly available code for the CLRS benchmark. ",
+        "bbox": [
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "We implemented each of these options by making our GNN’s message and update functions be two-layer MLPs with embedding dimension 24, and hidden layers of size 8 and 16. Our test results (out-of-distribution) are summarised in Table 1. For convenience, we also illustrate the in-distribution performance of our models via plots given in Appendix E. ",
+        "bbox": [
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Lastly, we scale up our experiments to 27 different tasks in CLRS, 96-dimensional embeddings, and using the PGN processor [29], which is the current state-of-the-art model on CLRS in terms of task win count [32]. We summarise the performance improvement obtained by our $V ^ { 3 }$ variant of PGN in Table 2, aggregated across edge-centric tasks as well as ones that do not require explicit edge-level reasoning. For convenience, we provide the per-task test performance in Appendix F (Table 3). ",
+        "bbox": [
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "We found that the $V ^ { 3 }$ architecture was equivalent to, or outperformed, the non-polynomial $( V ^ { 2 } )$ one in all edge-centric algorithms (up to standard error). Additionally, this architecture appears to also provide some gains on tasks without explicit edge-level reasoning requirements, albeit smaller on average and less consistently. Our result directly validates our theory’s predictions, in the context of presenting a better-aligned GNN for edge-centric algorithmic targets. ",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "table",
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+        "table_caption": [
+            "Table 1: Test (out-of-distribution) results of our models on all models on the six algorithms studied. $V ^ { 2 }$ corresponds to the baseline model offered by Diagram 7, while $V ^ { 3 }$ corresponds to our proposal in Diagram 10, which respects the polynomial span. "
+        ],
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+        "table_body": "<table><tr><td>Algorithm</td><td>V2-large</td><td>V3-large</td><td>V2-small</td><td>V3-small</td></tr><tr><td>Dijkstra</td><td>59.58% ± 2.82</td><td>68.53%± 2.40</td><td>56.10%± 3.25</td><td>60.32%± 2.70</td></tr><tr><td>Find Maximum Subarray</td><td>8.33%± 0.50</td><td>9.06%±0.65</td><td>8.46%± 0.55</td><td>7.89%±0.64</td></tr><tr><td>Floyd-Warshall</td><td>7.46%±0.63</td><td>9.00%±0.81</td><td>6.66%± 0.62</td><td>8.23%±0.62</td></tr><tr><td>Insertion Sort</td><td>15.39% ± 1.27</td><td>24.67%±2.44</td><td>14.69% ± 1.32</td><td>20.23%± 2.21</td></tr><tr><td>Matrix Chain Order</td><td>67.64% ± 1.23</td><td>70.79% ± 1.54</td><td>68.85%± 2.26</td><td>68.76%± 1.21</td></tr><tr><td>Optimal BST</td><td>53.03%± 2.80</td><td>54.56%± 4.34</td><td>46.65% ± 3.82</td><td>51.94% ± 4.60</td></tr><tr><td>Overall average</td><td>35.24%</td><td>39.43%</td><td>33.57%</td><td>36.23%</td></tr></table>",
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+            "Table 2: Test (out-of-distribution) results across 27 tasks in CLRS, for the PGN processor network, averaged across edge-centric and other tasks. See Appendix F for the per-task test performances. "
+        ],
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+        "table_body": "<table><tr><td>Algorithms</td><td>V2-PGN</td><td>V3-PGN</td><td>Average Improvement</td></tr><tr><td>Edge-centric algorithms</td><td>35.03%</td><td>39.08%</td><td>4.44% ± 1.06</td></tr><tr><td>Other algorithms</td><td>35.37%</td><td>36.33%</td><td>1.01% ± 0.11</td></tr><tr><td>Average of the two groups</td><td>35.20%</td><td>37.70%</td><td>2.73%</td></tr></table>",
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+    {
+        "type": "text",
+        "text": "8 Conclusions ",
+        "text_level": 1,
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+        "text": "In this paper, we describe the use of category theory and abstract algebra to explicitly expand on the GNN-DP connection, which was previously largely handwaved on specific examples. We derived a generic diagram of an integral transform (based on standard categorical concepts like pullback, pushforward and commutative monoids), and argued why it is general enough to support both GNN and DP computations. With this diagram materialised, we were able to immediately unify large quantities of prior work as simply manipulating one arrow or element in the integral transform. We also provided empirical evidence of the utility of polynomial spans for analysing GNN architectures, especially in terms of algorithmic alignment. It is our hope that our findings inspire future research into better-aligned neural algorithmic reasoners, especially focusing on generalising or diving into several aspects of this diagram. ",
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+        "type": "text",
+        "text": "Lastly, it is not at all unlikely that analyses similar to ours have already been used to describe other fields of science—beyond algorithmic reasoners. The principal ideas of span and integral transform are central to defining Fourier series [35], and appear in the analysis of Yang-Mills equations in particle physics [13]. Properly understanding the common ground behind all of these definitions may, in the very least, lead to interesting connections, and a shared understanding between the various fields they span. ",
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+    {
+        "type": "text",
+        "text": "Acknowledgments and Disclosure of Funding ",
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+        "type": "text",
+        "text": "We would like to thank Charles Blundell, Tai-Danae Bradley, Taco Cohen, Bruno Gavranovic, Bogdan ´ Georgiev, Razvan Pascanu, Karolis Špukas, Grzegorz Swirszcz, and Vincent Wang-Ma ´ scianica for ´ the very useful discussions and feedback on prior versions of this work. ",
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+        "type": "text",
+        "text": "Special thanks to Tamara von Glehn for key comments helping us to formally connect integral transforms to polynomial functors. ",
+        "bbox": [
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+        "text": "This research was funded by DeepMind. ",
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+    {
+        "type": "text",
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