text stringlengths 128 2.05k |
|---|
We can also examine how fast the community of agents arrives at a steady state. Figure 14 shows that at short timescales ( [MATH] ), better convergence may be achieved allowing only unhedged assertions. In a more extreme case, figure 15 shows that for [MATH] , better performance on the ALO metric is only achieved after... |
DISCUSSION These results show that, in a model of language development across a population, hedged assertions can improve both the level of convergence to shared language as measured by average pairwise difference between label sets (APD) and, to an extent, the discriminatory power of individuals’ label sets, as measur... |
The improved performance against the two metrics is tempered by the fact that the speed at which the steady state is achieved is somewhat slower than when using simply unhedged assertions. However, the improvement in APD is seen relatively quickly at [MATH] , soon after the unhedged model has reached its steady state. ... |
If the speed of development of shared categories is not important, the two types of hedges would be useful in different types of situation, depending whether convergence or discriminatory power is more important. This might be dependent on, for example, the structure of the underlying environment. In the current simula... |
If speed is important, using contraction hedges can still improve levels of convergence in a relatively short timeframe. There are many parameters in the simulation that bear further investigation. The distribution of objects in the environment, as mentioned above, is likely to have an effect on performance again the t... |
CONCLUSIONS We have investigated the utility of hedged assertions in the development of a shared language, and shown that allowing agents to make hedged assertions improves the ability to develop common categories in two distinct ways. Firstly, using contraction hedges, i.e. words like ‘very’, allows improved levels of... |
# Source: arxiv 1602.07455 # Title: Automatically Proving Mathematical Theorems with Evolutionary Algorithms and Proof Assistants # Sections: all # Downloaded: 2026-03-03T01:58:08.251670+00:00 |
Automatically Proving Mathematical Theorems with Evolutionary Algorithms and Proof Assistants Abstract Mathematical theorems are human knowledge able to be accumulated in the form of symbolic representation, and proving theorems has been considered intelligent behavior. Based on the BHK interpretation and the Curry-How... |
Index Terms: Evolutionary algorithm, proof assistant, Coq, automatic theorem proving. Introduction Human knowledge has been accumulated in various forms. Among them are theorems in mathematics that are presented in a precise fashion and can be applied to innumerous domains. The importance of mathematics and its influen... |
Before the proposal of the link between logic and computation, the principle of Propositions as Types, logic and computation were previously considered two separate fields |
. Based on the BHK interpretation and the Curry-Howard isomorphism , (functional) programming languages , including Haskell , and proof assistants, such as Coq |
, HOL , Isabelle , and LEGO have been developed. Due to the transformation from logic to computation, proofs to mathematical theorems can then be expressed as programs and verified computationally. |
Mathematical theorem proving has been considered intelligent behavior . Even for today, while a Go computer program, called AlphaGo , able to beat the human Go champion of Europe, Fan Hui |
, by five games to zero and Lee Sedol , who was ranked world #2, by four games to one exists, the ability of computers to fully automatically prove mathematical theorems still seems quite limited. It is probably because most of the effort made on proof assistants aims at automated proof checking, which ensures the corr... |
Evolutionary algorithms , as stochastic methods, have been known for their flexility and versatility. They have been successfully applied to handle a variety of scientific and engineering problems in numerous disciplines. Hence, in this study, we would like to make our first attempt to link evolutionary algorithms, as ... |
for fitness evaluation. The preliminary results indicate the feasibility and demonstrate that this direction of research is promising. |
The remainder of the paper is organized as follows. Section II introduces the background regarding the primary goal of the present work. Section III describes in detail the ad-hoc attempt we adopt to investigate the feasibility of automatically proving mathematical theorems. The preliminary results obtained in our expe... |
II Backound The goal of this study is to make an attempt to automatically prove mathematical theorems, and the present work is done by linking two well developed fundations, proof assistants and evolutionary algorithms. In this section, the background regarding the primiary goal of this study is presented. |
First of all, automated theorem proving is a term quite closely related to this study. Although automated theorem proving and the research line of this study may share the same ultimate goal, the ways to progress towards the ultimate goal are actually entirely different, especially that evolutionary algorithms, as stoc... |
The term automated theorem proving appeared in the 1950s as the most developed field within automated reasoning and was applied in 1956 to Logic Theory Machine |
, a deduction system for the propositional logic which adopts a heuristic approach to emulate human reasoning. It was the first program designed to prove mathematical theorems. Another field of automated reasoning with less automation but more pragmatic actions is called interactive theorem proving , in which proof ass... |
have been developed. As software tools, proof assistants incorporate automated reasoning techniques to make logical decisions on mathematical theorems. |
Proof assistants, such as Coq , HOL , Isabelle , and LEGO have been designed to interact with users for developing formal proofs and to verify the logical validity of proofs. Among them is Coq, an interactive theorem prover developed by Inria |
that allows users to express mathematical assertions by specifying various strategies called tactics . It works based on the theory of the Calculus of Inductive Construction, a derivative of the calculus of constructions. Coq also provides a formal language to write mathematical definitions, executable algorithms and t... |
The other foundation is evolutionary algorithms , which have been successfully applied to resolve issues of many different natures in a host of domains. Evolutionary algorithms are known for their flexibility and versatility because their ability to handle black-box optimization makes it possible, even easy, to interfa... |
III Ad-Hoc Attempt First of all, the source code developed in this study, the supporting materials required for conducting the experiments, and our experimental results are open source in the repository of nclab |
on GitHub . It should be viable and relatively easy for practitioners and interested readers to replicate the experimental results described later in this article. |
Our present ad-hoc attempt, as suggested by the title of this paper, consists of two major components: an evolutionary algorithm for finding proofs and the proof assistant for verifying proofs. For part of the proof assistant, we employ Coq |
in this study. The reason to employ Coq is quite straightforward that according to the entry of proof assistants in Wikipedia, Coq supports most features, including higher-order logic and dependent types, which are most likely necessary if research along this line is continually pursued in the future. |
Proof development in Coq is done through a sequence of tactics to interact with the current goals shown in the Coq interface. These tactics convert a goal of a proof into subgoals by implementing backward reasoning from conclusions to premises. For example, in order to prove [MATH] [MATH] and [MATH] must be both proven... |
Because a deep structure of three-layered correspondence between logic and computation exists: propositions as types, proofs as programs, and simplification of proofs as evaluation of programs, Coq as a proof assistant has been developed based on this correspondence in order to provide the capability of checking the va... |
In some sense, we can consider that what Coq does is to provide an execution environment in which programs are written in the programming language defined by Coq. The tactics for composing proofs in Coq can be considered as instructions, such as those of x86 CPUs |
or ARM CPUs . Thus, a proof written as a sequence of tactics in Coq can be viewed as a program composed of a sequence of (CPU) instructions. If the “Coq CPU” can successfully execute a given sequence of instructions, the proof expressed as this sequence of tactics is then verified to be logically valid. It is exactly w... |
” indicates. Thanks to the subtle and strong correspondence between proofs and programs, finding proofs can now be regarded as writing programs. In the realm of evolutionary computation, Genetic Programming |
is a well known and established branch which aims at automatically generating computer programs fulfilling given specifications. Since Coq tactics are similar to CPU instructions, among the variants of Genetic Programming is Linear Genetic Programming |
which is in particular designed for handling programs in the form of instruction sequences. Algorithm 1 Flow of the adopted evolutionary algorithm |
1: procedure Ad-Hoc_EA [MATH] [MATH] 2: [MATH] 3: [MATH] 4: [MATH] 5: repeat 6: [MATH] 7: [MATH] 8: [MATH] 9: repeat 10: [MATH] 11: |
[MATH] 12: [MATH] 13: if [MATH] then 14: [MATH] 15: end if 16: [MATH] 17: [MATH] 18: until [MATH] 19: [MATH] [MATH] Generational model |
20: [MATH] 21: [MATH] 22: until [MATH] 23: end procedure While techniques developed for Linear Genetic Programming may be a good choice to perform the role of generating CPU instruction sequences, i.e., searching for proofs, we employ a more rudimentary evolutionary approach in this study, because the goal is to make a... |
Algorithm 2 Initialization of the population 1: Initialized population; 2: procedure Initialization [MATH] 3: [MATH] 4: [MATH] 5: |
repeat 6: [MATH] 7: [MATH] 8: [MATH] 9: repeat 10: [MATH] 11: [MATH] 12: [MATH] 13: until [MATH] 14: [MATH] 15: [MATH] 16: until |
[MATH] 17: return [MATH] 18: end procedure As a consequence, in this study, we adopt a simple, GP-like evolutionary algorithm of which the pseudo code is outlined in Algorithm to work with Coq using the standard library and a “tactic base” consisting of 153 tactics, which is available for download from our GitHub repos... |
[EQUATION] Then, the chromosome representation adopted in this work is a variable-length integer sequence because the number of tactics needed to complete a proof may vary in length. For example, if there are chromosomes [MATH] and [MATH] |
[EQUATION] these two chromosomes respectively correspond to the following Coq proofs: Individual [MATH] Individual [MATH] Proof. |
Proof. intros intros [MATH] [MATH] [MATH] [MATH] [MATH] [MATH] [MATH] [MATH] Qed. [MATH] [MATH] [MATH] Qed. The tactic intros is not included in the adopted tactic base and is inserted in the front of each individual. |
The procedure for initializing the population is given in Algorithm . Currently, the tactic base contains 153 tactics, and thus, [MATH] is 152. We use [MATH] and [MATH] in all the conducted experiments. |
Parental selection, crossover, and mutation are given in Algorithms and , respectively. For a given population, we select two individuals among the top 50% as parents to go through crossover and, probably, mutation to generate one offspring. The individuals ranked top 50% have an equal probability to be selected as par... |
Algorithm 3 Parental selection 1: One selected parent; 2: procedure SelectOneParent [MATH] 3: [MATH] 4: [MATH] The 0th is the highest. |
5: return the individual with the [MATH] th highest fitness; 6: end procedure Algorithm 4 Crossover 1: One offspring 2: procedure |
Crossover [MATH] 3: [MATH] 4: [MATH] 5: return [MATH] 6: end procedure Algorithm 5 Mutation 1: One mutated invidual; 2: procedure |
Mutation [MATH] 3: [MATH] 4: [MATH] 5: return [MATH] 6: end procedure Finally, the evaluation procedure invokes Coq to evaluate individuals in the population and is shown in Algorithm . As aforementioned, this study is a proof-of-principle one to check the viability of linking evolutionary algorithms and proof assistan... |
In order to use the number of tactics as fitness, some limitation is required to avoid meaningless conditions. For example, the tactic simpl may be repeated forever, and tactics regarding the commutative laws can be repeated forever by pairs. For this kind of cases, we give a list of unrepeatable tactics without affect... |
Algorithm 6 Evaluation 1: procedure Evaluation [MATH] 2: [MATH] 3: repeat 4: Invoke Coq on individual [MATH] 5: [MATH] the number of passed tactics; |
6: if Error: No such unproven subgoal then 7: Assign ( [MATH] ) as fitness to individual [MATH] 8: else 9: Assign [MATH] as fitness to individual [MATH] |
10: end if 11: [MATH] 12: until [MATH] 13: end procedure In the source code available from our GitHub repository, we also pick some individuals with the top scores for further examination by invoking Coq on them in a tactic-by-tactic manner. If a complete proof is found, it is archived for our research purpose and noth... |
IV Preliminary Results With the adopted evolutionary algorithm and our ad-hoc attempt presented in section III , we have successfully proven ten theorems of different branches in mathematics automatically. As aforementioned, the proofs found by the introduced approach for the following theorems are also available as su... |
Arithmetic Theorem n_le_k : forall n k, n = 0 [MATH] [MATH] k. Theorem plus_n_0 : forall n, n + 0 = 0. Theorem n_1_n : forall n, n ^ 1 = n. |
Logic Lemma solving_by_eapply : forall(P Q : nat [MATH] Prop), (forall n k, Q k [MATH] P n) [MATH] Q 1 [MATH] P 2. Theorem andb_prop : forall n k, |
andb n k = true [MATH] n = true [MATH] k = true. Theorem andb_true_elim2 : forall n k : bool, andb n k = true [MATH] k = true. Theorem andb_true_intro : forall n k, |
n = true [MATH] k = true [MATH] andb n k = true. Parity Theorem ev_minus2 : forall n, ev n [MATH] ev (pred (pred n)). Theorem SSev_even : forall n, ev (S (S n)) [MATH] ev n. |
Theorem silly_prob (forall n, evenb n = true [MATH] oddb (S n) = true) [MATH] oddb 4 = true. Note that the tactic auto is not included in the tactic base, and more importantly, most of the theorems listed above cannot be proven by using only auto in Coq. Interested readers may further investigate into Coq and check out... |
The obtained, preliminary results are quite within our expectation. We were able to prove a number of simple theorems as listed above, while we failed to do the same on relatively advanced theorems. Simple theorems are more likely to be proven in several steps with different sequences of tactics. For a concrete example... |
[EQUATION] According to our experimental results, more than thirty different valid sequences of Coq tactics have been found in one hundred generations with 153 tactics in the tactic base. As aforementioned, the size of population, i.e., the number of individuals, is 1000, and in one successful run, the first complete p... |
Despite the fact that proven theorems in these preliminary results seem quite simple, even straightforward in human eyes, the potential for research and development on linking the two domains, evolutionary algorithms and proof assistants, can clearly be identified. These ten proven theorems are from various branches of... |
It can also be justified that finding the proofs to the above theorems is hardly by chance. The capability of automatically proving mathematical theorems does exist in a system linking together evolutionary algorithms and proof assistants. Taking the theorem of Equation ( ) and 153 Coq tactics for example, one has bare... |
[EQUATION] If there are actually 100 valid proofs of different sequences of which the length is 5 or shorter than 5, the probability is still about the magnitude of [MATH] or [MATH] . On the other hand, it took around only 8000 evaluations to “guess it right,” which costs little with modern computational facilities. |
In summary, ten simple mathematical theorems from various branches of mathematics, most of them cannot be proven by the current, limited automatic mechanism of proof assistants alone, were proven by the proposed ad-hoc approach with a reasonable cost of computational resource. The feasibility of automatically generatin... |
Discussion In order to assess the feasibility of automatically proving mathematical theorems, an ad-hoc attempt, composed of a straightforward evolutionary algorithm and the direct use of Coq, was described in section III . The preliminary results presented in section IV provide the evidence that the line of research o... |
As mentioned in section IV , the proposed ad-hoc approach failed to prove relatively advanced theorems which require proofs of longer tactic sequences. Although it is reasonable and acceptable in the present work, efforts on both sides need to be done to develop an automatic theorem proving framework for practical use. |
Firstly, on the side of proof assistants, in the ad-hoc attempt, we basically use the number of passed tactics and its calculation as fitness. In addition to forbidding some tactics to repeat, we may consider different fitness assignment for compound usage of certain tactics or variable fitness weights on tactics for s... |
On the side of the search methodology, since the feasibility to use evolutionary algorithms to search for formal proofs has been preliminarily identified, techniques from Linear Genetic Programming can then be considered and put into action. In addition to Linear Genetic Programming, other variants of Genetic Programmi... |
, or even techniques commonly used in Genetic Programming, such as automatically defined function (ADF) and automatically defined macro (ADM), can be integrated into the evolutionary proof-search engine to extract “modularized” segments of proofs in a sense which may yet be understandable for the time being. |
Further into the realm of randomized search methods, given the recent success in Go playing with computer programs , Monte Carlo tree search |
may also be considered as the search mechanism. Although a great amount of efforts will be required for its adoption and adaptation in the theorem proving framework, Monte Carlo tree search is a promising methd alternative to the techniques of Genetic Programming. |
From the obtained the preliminary results, we can easily find that even an automatic theorem proving framework as simple as our ad-hoc attempt is capable of finding alternative proofs, composed of different sequences of tactics, to a given theorem. Figures to show some of these alternative proofs found in our experimen... |
These tactic sequences verified by Coq might not be totally different from the viewpoint of concepts in mathematics, while they still differ from each other with the distinct adoption of logical strategies expressed by the tactic sequences. Taking the proofs in Figure as an example, one can easily discriminate these pr... |
# Source: arxiv 1604.07110 # Title: Divergent Cumulative Cultural Evolution # Sections: all # Downloaded: 2026-03-03T02:00:03.418592+00:00 |
Divergent Cumulative Cultural Evolution Abstract Divergent cumulative cultural evolution occurs when the cultural evolutionary trajectory diverges from the biological evolutionary trajectory. We consider the conditions under which divergent cumulative cultural evolution can occur. We hypothesize that two conditions are... |
Introduction Social learning is a form of learning that arises from social situatedness (Lindblom and Ziemke,, 2003 and is characterized by agents interacting with one another in order to learn. Social learning can accelerate learning beyond that of individual learning strategies (see Marriott and Chebib, ( 2014 ); Mar... |
Genetic evolution and cultural evolution are parallel processes that optimize information in a population. It is common to consider the interaction between these parallel processes. Two effects have been well discussed with respect to learning: the hiding effect is when learning shields genetics from selection pressure... |
Another way we can compare genetic and cultural evolution is according to the direction of the evolutionary trajectory. It is possible for the cultural evolutionary trajectory to diverge from the biological evolutionary trajectory. In particular, this means that the culture may evolve in directions that are neutral or ... |
A simple non-human example of divergent evolutionary trajectories is sexual selection. Females could select for traits that correlate with fitness. In this case the culture and the genetic evolution agree. However, females could select for traits that do not correlate with fitness or correlate negatively with fitness. ... |
Some human cultures may frustrate the reproductive or survival capabilities of some of their members (usually for the apparent benefit of the culture). For instance, a Catholic priest will abstain from reproducing according to the rule of his culture. Also, a Samurai might kill himself out of shame for failing to meet ... |
An extreme and rare case of cultural divergence would be a case where every individual of the culture engages in detrimental cultural practices. This would include cases of mass abstinence or mass suicide. Mass abstinence was a cultural belief of the Shakers (in the 1770s-1780s). Some mass suicides are caused for fear ... |
We believe divergent cultural evolution requires at least two properties. First, genetic and cultural information must be stored in separate information stores. This rules out models with horizontal transfer of genetic material. Second, horizontal transfer of cultural information occurs between individuals of the same ... |
We believe these are necessary conditions and we believe they are probably not sufficient conditions. It is difficult to test this hypothesis since our implementation has many other implicit conditions that may play an important role. Our experiment is designed to test whether these conditions (plus implicit others) ca... |
In Marriott and Chebib, ( 2014 we demonstrated a simple instance of divergent genetic and cultural evolution in a population with these properties. The experiment involved a simple optimization problem, asexual agents, and no spatial environment. Agents in that experiment showed accelerated optimization and divergence ... |
We have reproduced this experiment in a virtual environment representing real space. Our agents engage in sexual reproduction and are subject to natural selection. Our first experiments in this environment involved simple agents with no learning capabilities Marriott and Chebib, 2015a, Marriott and Chebib, 2015b, . We ... |
Divergent Cultural Evolution In Marriott and Chebib, ( 2014 we implemented a simple proof of concept and demonstration of divergent cumulative cultural evolution. Agents in our model engage in all three modes of adaptation: phylogenetic, ontogenetic, and sociogenetic. Phylogenetic adaptation is adaptation by genetic ev... |
The dual inheritance model (see Fig. ) describes how these three modes of adaptation interact to create the agent Marriott and Chebib, 2016b, . Genetic information is inert over the lifetime of the agent in our model. It is transmitted vertically (from parent to child) during reproductive events and is responsible for ... |
As agents are units of selection in our simulation natural selection occurs on the lifetime behavior of an agent (i.e. its phenotype). This behavior is determined by an interaction of an agent’s genome, memome and environment. As a result both genetic and memetic information is important in determining if an agent live... |
There are two important ways that divergence can occur between genetic and cultural evolution. It is common that evolutionary trajectories in both the genetic and cultural realm are aligned. This is common when they are both trying to optimize a behavior. In these cases it is expected that cultural optimization of the ... |
The second type of divergence occurs when genetic selection pressures and cultural selection pressures are contrary. For instance, sexual reproduction is favored by genetic selection but suppressed in many (human and non-human) cultures. We call this divergence under competitive selection pressures |
We believe that both types of divergence require the properties stated above. That is, genetic and cultural information must be separate and cultural information must be transmitted horizontally. We will test our implementation for both types of divergence. |
Model We have improved on our proof of concept by placing our agents in an environment in which they compete for resources and are subjected to a form of natural selection (i.e. compete for mates) instead of artificial selection (i.e. face a fitness function). |
Our agents live in a random geometric network of resource sites (Penrose,, 2003 . Random geometric networks are an approximation of two dimensional physical space. At each site agents can spend time gathering the resources available at that site. Sites in our current model have one, two or three resources available to ... |
Agents have a simple metabolism in which resources are converted into energy. Energy is used to move around the environment, gather resources, and perform actions like breeding, learning and social learning. Additional small daily energy penalties are administered for idle activity, old age, and length of genome (only ... |
Genome As mentioned above, in the dual inheritance model an agent’s genome is inert during its lifetime and therefore is not adaptive nor directly active in behavior selection. The primary purpose of its genome is to spread genetic information in reproductive events. The secondary purpose of its genome is to produce an... |
A genome of an agent represents a path of resources sites in the random geometric network. At each site on this path is also encoded possible behaviors for an agent at that site. That is, a genome represents a single long path through the network and the actions an agent might take at each site. |
We call each site in this path a gene in the genome. A typical gene consists of three parts. A gathering component encodes a strategy for gathering resources at the respective site. A non-gathering component encodes the energy spent on non-gathering actions like breeding, learning and socializing. Energy in our model c... |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.