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purewhite42
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formal_problem_solving_main

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Text Generation
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json
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1K - 10K
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metadata
license: apache-2.0
task_categories:
  - text-generation
pretty_name: Formal Problem-Solving
size_categories:
  - 100K<n<1M
configs:
  - config_name: default
    data_files:
      - split: formal_math500
        path: formal_math500_format_unified.jsonl
      - split: minif2f_solving
        path: minif2f_solving_format_unified.jsonl
      - split: putnam_solving
        path: putnam_solving_format_unified.jsonl

Dataset Card for Formal Problem-Solving

This benchmark is part of the official implementation of Beyond Theorem Proving: Formulation, Framework and Benchmark for Formal Problem-Solving.

Our research focuses on:

  1. What is problem-solving?
  2. Beyond proving known targets, how can process-verified problem-solving be conducted inside existing formal theorem proving (FTP) environments?

Contribution

  • A principled formulation of problem-solving as a deterministic Markov decision process;
  • FPS (Formal Problem-Solving), utilizing FTP (formal theorem proving) environments to perform process-verified problem-solving;
  • D-FPS (Deductive FPS), decoupling solving and answer verification for better human-alignment;
  • RPE (Restricted Propositional Equivalence), a symbolic approach to determine the correctness of answers by formal verification;
  • Three benchmarks on problem-solving: FormalMath500, MiniF2F-Solving and PutnamBench-Solving.

Benchmark Details

  • FormalMath500 is a formalized subset of the prevalent MATH500 benchmark[5,6], including 387 data points:

    • 123 about Algebra
    • 92 about Intermediate Algebra
    • 62 about Number Theory
    • 65 about Prealgebra
    • 45 about Precalculus

  • MiniF2F-Solving is a refactored subset of MiniF2F[7], containing in 375 data points with:

    • 30 from AIME
    • 140 from MATH-Algebra
    • 82 from AMC
    • 3 from IMO
    • 120 from MATH-Number Theory

  • PutnamBench-Solving is a refactored subset of PutnamBench[8], containing 324 data points with:

    • 9 about Abstract Algebra
    • 138 about Algebra
    • 122 about Analysis
    • 14 about Combinatorics
    • 28 about Geometry
    • 25 about Linear Algebra
    • 49 about Number Theory
    • 8 about Probability
    • 4 about Set Theory

Direct Use

  • Formal Problem-Solving (FPS): Given a formal problem, generate a formal solution. The formal solution should solve all goals and provide a direct answer.

  • Deductive Formal Problem-Solving (D-FPS): Given a formal problem, generate a forward solution and, optionally, a backward proof. The forward solution should use deductive reasoning to derive a direct answer and prove its completeness. The backward proof should prove the answer's soundness.

  • Formal Theorem Proving (FTP): Given a formal problem and its ground-truth answer, generate a formal proof to prove the ground-truth's correctness.

Dataset Structure

Each problem contains the following fields:

  • informal_problem: The problem in natural language (including LaTeX).
  • informal_answer: The ground-truth answer in natural language (including LaTeX).
  • informal_solution: A step-by-step solution in natural language (including LaTeX).
  • header: Code that should be executed before initializing the formal problem, e.g., opens. If null, open BigOperators Real Nat Topology should be used. (Unified in *_format_unified.jsonl)
  • intros: Independent variables $V$ and hypotheses $\Phi$. $V={v_i}{i=1}^n$ is the set of variables independent to the queriable $a$. $\Phi = {\phi_i}{i=1}^p$ is the set of propositions that depend on $V$ (whose all free variables are included in $V$), consisting of conditions that can be used to deduce the answer.
  • outros: Conclusions $\Psi = {\psi_i}_{i=1}^q$ is the set of propositions which depend on $V \cup {a}$, consisting of conclusions that should be satisfied.
  • formal_answer: The ground-truth answer in formal language (Lean 4).
  • formal_answer_type: The type of the ground-truth answer in formal language (Lean 4).
  • metainfo: Meta-information of the problem. (Removed in *_format_unified.jsonl)

References

[1] Moura, Leonardo de, and Sebastian Ullrich. "The Lean 4 theorem prover and programming language." Automated Deduction–CADE 28: 28th International Conference on Automated Deduction, Virtual Event, July 12–15, 2021, Proceedings 28. Springer International Publishing, 2021.

[2] Community, Mathlib . "The Lean mathematical library.", 10.1145/3372885.3373824. 2019.

[3] Limperg, Jannis, and Asta Halkjær From. "Aesop: White-box best-first proof search for Lean." Proceedings of the 12th ACM SIGPLAN International Conference on Certified Programs and Proofs. 2023.

[4] Aniva, Leni, et al. "Pantograph: A Machine-to-Machine Interaction Interface for Advanced Theorem Proving, High Level Reasoning, and Data Extraction in Lean 4." arXiv preprint arXiv:2410.16429 (2024).

[5] Lightman, Hunter, et al. "Let's verify step by step." The Twelfth International Conference on Learning Representations. 2023.

[6] Hendrycks, Dan, et al. "Measuring mathematical problem solving with the math dataset." arXiv preprint arXiv:2103.03874 (2021).

[7] Zheng, Kunhao, Jesse Michael Han, and Stanislas Polu. "Minif2f: a cross-system benchmark for formal olympiad-level mathematics." arXiv preprint arXiv:2109.00110 (2021).

[8] Tsoukalas, George, et al. "Putnambench: Evaluating neural theorem-provers on the putnam mathematical competition." arXiv preprint arXiv:2407.11214 (2024).