[Paper Review] Automatically Proving Mathematical Theorems with Evolutionary Algorithms and Proof Assistants

Li-An Yang, Jui-Pin Liu+2 more|arXiv (Cornell University)|Feb 24, 2016

Evolutionary Algorithms and Applications16 references3 citations

TL;DR

This paper proposes a novel framework that combines evolutionary algorithms with proof assistants—specifically Coq—to automatically generate and verify formal proofs of mathematical theorems. By encoding proof strategies as integer sequences evolved via a simple genetic algorithm and evaluating them in Coq, the approach successfully proved ten non-trivial theorems that could not be solved by Coq’s built-in 'auto' tactic, demonstrating the feasibility of automated theorem proving through evolutionary search.

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-Howard isomorphism, proof assistants, software capable of interacting with human for constructing formal proofs, have been developed in the past several decades. Since proofs can be considered and expressed as programs, proof assistants simplify and verify a proof by computationally evaluating the program corresponding to the proof. Thanks to the transformation from logic to computation, it is now possible to generate or search for formal proofs directly in the realm of computation. Evolutionary algorithms, known to be flexible and versatile, have been successfully applied to handle a variety of scientific and engineering problems in numerous disciplines for also several decades. Examining the feasibility of establishing the link between evolutionary algorithms, as the program generator, and proof assistants, as the proof verifier, in order to automatically find formal proofs to a given logic sentence is the primary goal of this study. In the article, we describe in detail our first, ad-hoc attempt to fully automatically prove theorems as well as the preliminary results. Ten simple theorems from various branches of mathematics were proven, and most of these theorems cannot be proven by using the tactic auto alone in Coq, the adopted proof assistant. The implication and potential influence of this study are discussed, and the developed source code with the obtained experimental results are released as open source.

Motivation & Objective

• To investigate the feasibility of using evolutionary algorithms as proof generators and proof assistants as proof verifiers in automated theorem proving.
• To explore whether stochastic, global search methods like evolutionary algorithms can effectively explore the space of formal proofs in Coq.
• To assess whether this hybrid approach can prove theorems that are not solvable by standard automated tactics like 'auto' in Coq.
• To lay the foundation for future systems capable of crowdsourced, automated mathematical knowledge accumulation through computational proof search.

Proposed method

• An evolutionary algorithm generates integer sequences representing potential tactic sequences in Coq.
• Each integer sequence is interpreted as a proof script and evaluated for correctness by the Coq proof assistant.
• Fitness evaluation is performed by Coq’s type checker, which verifies whether the generated proof script successfully proves the target theorem.
• The algorithm uses a simple genetic programming approach without advanced mechanisms, focusing on proof validity rather than proof length or efficiency.
• The framework leverages the Curry-Howard isomorphism, treating proofs as programs and validating them via program evaluation in Coq.
• The system is designed to be extensible, with future enhancements planned such as runtime status inspection and integration of Monte Carlo tree search.

Experimental results

Research questions

• RQ1 Can evolutionary algorithms effectively explore the space of formal proofs in a proof assistant like Coq?
• RQ2 Can this approach automatically prove theorems that are not solvable by standard automated tactics such as 'auto' in Coq?
• RQ3 What is the feasibility of using a simple evolutionary algorithm to generate correct proofs when combined with a proof assistant’s verification capability?
• RQ4 How can proof assistants be used to guide evolutionary search through real-time feedback on proof state?
• RQ5 What are the potential long-term implications of such a framework for the automated accumulation of mathematical knowledge?

Key findings

• The framework successfully proved ten theorems from various branches of mathematics that could not be proven using Coq’s 'auto' tactic alone.
• The evolutionary algorithm generated valid proof scripts that were correctly verified by Coq, demonstrating the viability of the approach.
• Proofs were generated without prior knowledge of the solution space, indicating that the evolutionary process could discover non-trivial proof paths.
• The results suggest that evolutionary algorithms can serve as effective search mechanisms in formal proof space, especially when combined with the correctness guarantees of proof assistants.
• The approach opens a path toward automated, crowdsourced mathematical knowledge accumulation through computational proof search.
• The authors conclude that the integration of evolutionary algorithms with proof assistants is a promising direction for future research in automated theorem proving.