Beyond Theorem Proving: Formulation, Framework and Benchmark for Formal Problem-Solving

TL;DR

This paper introduces a formal framework for problem-solving using theorem proving environments, creating benchmarks and verification methods to improve process transparency and alignment with human reasoning.

Contribution

It proposes a new formal problem-solving framework, FPS, and its deductive variant D-FPS, along with benchmarks and a verification method, RPE, to advance AI problem-solving research.

Findings

Frameworks are proven to be sound and complete.

Constructed three new problem-solving benchmarks.

Baseline models solve up to 27.47% of benchmark problems.

Abstract

As a seemingly self-explanatory task, problem-solving has been a significant component of science and engineering. However, a general yet concrete formulation of problem-solving itself is missing. With the recent development of AI-based problem-solving agents, the demand for process-level verifiability is rapidly increasing yet underexplored. To fill these gaps, we present a principled formulation of problem-solving as a deterministic Markov decision process; a novel framework, FPS (Formal Problem-Solving), which utilizes existing FTP (formal theorem proving) environments to perform process-verified problem-solving; and D-FPS (Deductive FPS), decoupling solving and answer verification for better human-alignment. The expressiveness, soundness and completeness of the frameworks are proven. We construct three benchmarks on problem-solving: FormalMath500, a formalization of a subset of the MATH500 benchmark; MiniF2F-Solving and PutnamBench-Solving, adaptations of FTP benchmarks MiniF2F and PutnamBench. For faithful, interpretable, and human-aligned evaluation, we propose RPE (Restricted Propositional Equivalence), a symbolic approach to determine the correctness of answers by formal verification. We evaluate four prevalent FTP models and two prompting methods as baselines, solving at most 23.77% of FormalMath500, 27.47% of MiniF2F-Solving, and 0.31% of PutnamBench-Solving.