Lean Meets Theoretical Computer Science: Scalable Synthesis of Theorem Proving Challenges in Formal-Informal Pairs
Pith reviewed 2026-05-21 22:17 UTC · model grok-4.3
The pith
Theoretical computer science supplies a scalable source of verified formal-informal theorem proving challenges in Lean4.
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
Core claim
By leveraging algorithmic definitions from theoretical computer science, specifically Busy Beaver problems and Mixed Boolean Arithmetic problems, it is possible to automatically synthesize arbitrarily many theorem-proof pairs that include parallel formal specifications in Lean4 and informal specifications in Markdown, enabling a scalable pipeline for generating verified proof challenges.
What carries the argument
The synthesis framework that automatically translates algorithmic definitions into parallel formal Lean4 statements and informal Markdown descriptions while supporting automatic verification of correctness.
If this is right
- This creates a scalable pipeline for generating verified proof challenges without relying on manual curation.
- Frontier models exhibit substantial gaps in automated theorem proving success rates across the two domains.
- Long-form proof generation remains difficult even for problems that are computationally easy to verify.
- TCS domains provide a valuable source for advancing automated reasoning research.
Where Pith is reading between the lines
- Extending the synthesis method to additional areas of theoretical computer science could increase the variety of available challenges.
- Models could be fine-tuned on the generated formal-informal pairs to target improvements in long-form reasoning.
- The same pipeline might be adapted to produce benchmarks that isolate specific weaknesses in current theorem provers.
- This approach may support research that connects informal mathematical reasoning with formal verification in AI systems.
Load-bearing premise
The algorithmic definitions of Busy Beaver and Mixed Boolean Arithmetic problems can be translated into Lean4 formal statements whose correctness is automatically verifiable and whose informal counterparts remain faithful to the original mathematical intent.
What would settle it
Generating a large batch of problems and finding that many Lean4 statements fail to capture the intended mathematical properties or that the informal Markdown versions deviate from the formal ones would show the translation step does not work as claimed.
read the original abstract
Formal theorem proving (FTP) has emerged as a critical foundation for evaluating the reasoning capabilities of large language models, enabling automated verification of mathematical proofs at scale. However, progress has been constrained by limited datasets due to the high cost of manual curation and the scarcity of challenging problems with verified formal-informal correspondences. We propose leveraging theoretical computer science (TCS) as a scalable source of rigorous proof problems, where algorithmic definitions enable automated generation of arbitrarily many challenging theorem-proof pairs. We demonstrate this approach on two TCS domains: Busy Beaver problems, which involve proving bounds on Turing machine halting behavior, and Mixed Boolean Arithmetic problems, which combine logical and arithmetic reasoning. Our framework automatically synthesizes problems with parallel formal (Lean4) and informal (Markdown) specifications, creating a scalable pipeline for generating verified proof challenges. Evaluation on frontier models reveals substantial gaps in automated theorem proving: while DeepSeekProver-V2-671B achieves 57.5\% success on Busy Beaver problems, it manages only 12\% on Mixed Boolean Arithmetic problems. These results highlight the difficulty of long-form proof generation even for problems that are computationally easy to verify, demonstrating the value of TCS domains for advancing automated reasoning research.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper proposes a scalable framework that automatically synthesizes theorem-proving challenges as paired formal (Lean4) and informal (Markdown) specifications drawn from algorithmic definitions in theoretical computer science, specifically Busy Beaver halting bounds and Mixed Boolean Arithmetic expressions. It reports concrete evaluation results on frontier models, including a 57.5% success rate for DeepSeekProver-V2-671B on Busy Beaver problems versus 12% on Mixed Boolean Arithmetic problems, to illustrate gaps in long-form automated proof generation.
Significance. If the formal-informal pairs preserve semantic fidelity, the work supplies a reproducible, arbitrarily scalable source of verified proof challenges that are computationally easy to check yet difficult for current LLMs, directly addressing the scarcity of high-quality datasets for automated theorem proving research. The reported performance differentials on two distinct TCS domains provide falsifiable, quantitative evidence of reasoning limitations.
major comments (1)
- The central claim that the synthesized problems test genuine reasoning difficulty in the stated TCS domains requires that the Lean4 encodings of Busy Beaver transition tables and Mixed Boolean Arithmetic operators remain faithful to the informal specifications. The manuscript describes the synthesis pipeline only at a high level (abstract and framework overview) and provides neither the code-to-formal mapping details, equivalence lemmas, nor post-generation semantic audits. Without these, the 57.5% versus 12% gap cannot be unambiguously attributed to proof-generation difficulty rather than encoding artifacts.
Simulated Author's Rebuttal
We thank the referee for their constructive feedback on the importance of establishing semantic fidelity in our synthesized formal-informal pairs. We address the major comment below and will revise the manuscript to incorporate additional details on the encoding process.
read point-by-point responses
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Referee: The central claim that the synthesized problems test genuine reasoning difficulty in the stated TCS domains requires that the Lean4 encodings of Busy Beaver transition tables and Mixed Boolean Arithmetic operators remain faithful to the informal specifications. The manuscript describes the synthesis pipeline only at a high level (abstract and framework overview) and provides neither the code-to-formal mapping details, equivalence lemmas, nor post-generation semantic audits. Without these, the 57.5% versus 12% gap cannot be unambiguously attributed to proof-generation difficulty rather than encoding artifacts.
Authors: We agree that explicit documentation of the encoding faithfulness is necessary to support attribution of the observed performance gap to reasoning difficulty. The synthesis pipeline generates Lean4 specifications through a direct, deterministic translation from the same algorithmic definitions used to produce the informal Markdown statements, ensuring semantic correspondence by construction. For Busy Beaver problems, transition tables are encoded as Lean4 inductive definitions of finite-state machines that replicate the standard 5-tuple Turing machine specification, with the halting predicate defined to match the informal bound exactly. For Mixed Boolean Arithmetic, expressions are mapped to Lean's BitVec and Bool primitives with operator-for-operator correspondence. While the current manuscript emphasizes the high-level framework, we will revise it to include a dedicated subsection detailing these mappings, selected equivalence lemmas proving that formal statements entail the informal specifications, and a description of our post-generation audit procedure (random sampling with cross-verification against reference implementations). These additions will be placed in the main text or appendix of the revised manuscript. revision: yes
Circularity Check
No significant circularity in synthesis pipeline or model evaluations
full rationale
The paper introduces a new framework for automatically generating formal-informal theorem pairs from TCS domains (Busy Beaver and Mixed Boolean Arithmetic) and evaluates frontier models on the resulting problems. The reported success rates (57.5% on Busy Beaver, 12% on Mixed Boolean Arithmetic) are obtained by testing external models on newly synthesized instances whose correctness is claimed to be automatically verifiable in Lean4. No equations, fitted parameters, or derivations reduce outputs to inputs by construction; there are no self-definitional steps, predictions that are statistically forced from the same data, or load-bearing self-citations that justify the central claims. The derivation chain is self-contained against external benchmarks and model testing.
Axiom & Free-Parameter Ledger
axioms (1)
- domain assumption Busy Beaver and Mixed Boolean Arithmetic problems admit faithful formalizations in Lean4 that preserve the original computational or logical intent.
discussion (0)
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