Efficient Test-Time Optimization for Multi-Agent Proof Autoformalization

2026-07-13

AI Summary

This work addresses the challenge of automatically translating multi-step mathematical proofs from natural language into formal languages by proposing ToMap, a multi-agent framework. ToMap employs a decomposer-formalizer-prover pipeline and identifies the decomposer as the critical bottleneck, innovatively concentrating test-time computation on its iterative refinement. By integrating formal verification feedback, semantic proof scoring, and a GEPA-inspired Pareto-front guidance mechanism, the framework jointly enhances syntactic correctness and semantic fidelity. Evaluated on ProofFlowBench, ToMap outperforms the previous state-of-the-art method by 19.0%, with most performance gains achieved within just a few iterations while simultaneously reducing overall test-time computational overhead.

Abstract

Full-proof autoformalization bridges extensive mathematical proofs in natural language with formally validated reasoning, offering a pathway to elevate the ceiling of verifiable mathematical reasoning. Unlike statement-level formalization, proof autoformalization is a long-horizon challenge requiring coordination of claims, contexts, and dependencies across many proof steps, yet has only recently come under focused study. Current approaches either rely on costly model training or apply excessive, unguided repair at inference time. To this end, we introduce ToMap, a multi-agent framework that structures proof autoformalization as a Decomposer-Formalizer-Prover pipeline with efficient test-time optimization guided by formal verification and semantic rubrics for proof quality. Rather than distributing test-time compute across all agents, we perform bottleneck analysis and identify the Decomposer as the critical bottleneck: the quality of its atomic, self-contained proof units directly determines whether downstream agents can successfully formalize and prove each step. ToMap therefore treats the Formalizer and Prover as downstream executors and efficiently focuses test-time compute on Decomposer refinement. This refinement follows a loop inspired by GEPA, evolving prompts over candidate decompositions and using formal verification progress together with semantic proof rubrics to define a Pareto frontier that guides the next decomposition update. Experiments on ProofFlowBench show that ToMap improves over the best previous method by 19.0% when evaluated by both syntactic correctness and semantic faithfulness, while requiring lower test-time cost. Scaling analysis shows that most gains emerge within a few iterations of decomposition evolution, guiding test-time budget selection.

Problem

• proof autoformalization
• test-time optimization
• multi-agent
• formal verification
• decomposition

Innovation

• proof autoformalization
• multi-agent framework
• test-time optimization
• formal verification
• prompt evolution