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