ReForm: Reflective Autoformalization with Prospective Bounded Sequence Optimization

TL;DR

ReForm introduces a reflective autoformalization approach with iterative self-correction and semantic validation, significantly improving the accuracy of translating natural language mathematics into formal statements.

Contribution

The paper presents ReForm, a novel reflective autoformalization method with Prospective Bounded Sequence Optimization for better semantic fidelity.

Findings

ReForm outperforms baselines by 22.6 percentage points on four benchmarks.

Introduces ConsistencyCheck, a benchmark with 859 expert-annotated items.

Human experts still make semantic errors in up to 38.5% of cases.

Abstract

Autoformalization, which translates natural language mathematics into machine-verifiable formal statements, is critical for using formal mathematical reasoning to solve math problems stated in natural language. While Large Language Models can generate syntactically correct formal statements, they often fail to preserve the original problem's semantic intent. This limitation arises from the LLM approaches' treating autoformalization as a simplistic translation task which lacks mechanisms for self-reflection and iterative refinement that human experts naturally employ. To address these issues, we propose ReForm, a Reflective Autoformalization method that tightly integrates semantic consistency evaluation into the autoformalization process. This enables the model to iteratively generate formal statements, assess its semantic fidelity, and self-correct identified errors through progressive refinement. To effectively train this reflective model, we introduce Prospective Bounded Sequence Optimization (PBSO), which employs different rewards at different sequence positions to ensure that the model develops both accurate autoformalization and correct semantic validations, preventing superficial critiques that would undermine the purpose of reflection. Extensive experiments across four autoformalization benchmarks demonstrate that ReForm achieves an average improvement of 22.6 percentage points over the strongest baselines. To further ensure evaluation reliability, we introduce ConsistencyCheck, a benchmark of 859 expert-annotated items that not only validates LLMs as judges but also reveals that autoformalization is inherently difficult: even human experts produce semantic errors in up to 38.5% of cases.