README.md

---
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** (_**F**ormal **P**roblem-**S**olving_), utilizing FTP (formal theorem proving) environments to perform process-verified problem-solving;
- **D-FPS** (_**D**eductive **FPS**_), decoupling solving and answer verification for better human-alignment;
- **RPE** (_**R**estricted **P**ropositional **E**quivalence_), 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., `open`s. 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).