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Learning to Disprove: Formal Counterexample Generation with Large Language Models

arXiv:2603.19514v1 Announce Type: new Abstract: Mathematical reasoning demands two critical, complementary skills: constructing rigorous proofs for true statements and discovering counterexamples that disprove false ones. However, current AI efforts in mathematics focus almost exclusively on proof construction, often neglecting the equally important task of finding counterexamples. In this paper, we address this gap by fine-tuning large language models (LLMs) to reason about and generate counterexamples. We formalize this task as formal counterexample generation, which requires LLMs not only to propose candidate counterexamples but also to produce formal proofs that can be automatically verified in the Lean 4 theorem prover. To enable effective learning, we introduce a symbolic mutation strategy that synthesizes diverse training data by systematically extracting theorems and discarding selected hypotheses, thereby producing diverse counterexample instances. Together with curated datas

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Zenan Li, Zhaoyu Li, Kaiyu Yang, Xiaoxing Ma, Zhendong Su · · 1 min read · 95 views
#cs.AI

arXiv:2603.19514v1 Announce Type: new Abstract: Mathematical reasoning demands two critical, complementary skills: constructing rigorous proofs for true statements and discovering counterexamples that disprove false ones. However, current AI efforts in mathematics focus almost exclusively on proof construction, often neglecting the equally important task of finding counterexamples. In this paper, we address this gap by fine-tuning large language models (LLMs) to reason about and generate counterexamples. We formalize this task as formal counterexample generation, which requires LLMs not only to propose candidate counterexamples but also to produce formal proofs that can be automatically verified in the Lean 4 theorem prover. To enable effective learning, we introduce a symbolic mutation strategy that synthesizes diverse training data by systematically extracting theorems and discarding selected hypotheses, thereby producing diverse counterexample instances. Together with curated datasets, this strategy enables a multi-reward expert iteration framework that substantially enhances both the effectiveness and efficiency of training LLMs for counterexample generation and theorem proving. Experiments on three newly collected benchmarks validate the advantages of our approach, showing that the mutation strategy and training framework yield significant performance gains.

Executive Summary

This article proposes a novel approach to augmenting large language models (LLMs) with the ability to generate formal counterexamples. By fine-tuning LLMs with a symbolic mutation strategy and multi-reward expert iteration framework, the authors demonstrate significant performance gains in counterexample generation and theorem proving. The approach addresses a critical gap in AI efforts in mathematics, where proof construction has traditionally received more attention. The experiments on three newly collected benchmarks validate the effectiveness of the proposed method, showcasing its potential in augmenting mathematical reasoning capabilities of LLMs.

Key Points

  • ▸ The article introduces formal counterexample generation, a task that requires LLMs to propose candidate counterexamples and produce formal proofs.
  • ▸ The authors propose a symbolic mutation strategy to synthesize diverse training data and a multi-reward expert iteration framework for effective learning.
  • ▸ Experiments on three benchmarks demonstrate significant performance gains in counterexample generation and theorem proving.

Merits

Strength in Addressing a Critical Gap

The article effectively addresses the gap in AI efforts in mathematics, where proof construction has traditionally received more attention than counterexample generation.

Improvements in Performance

The proposed method demonstrates significant performance gains in counterexample generation and theorem proving, as validated by experiments on three benchmarks.

Potential in Augmenting Mathematical Reasoning

The approach has the potential to augment the mathematical reasoning capabilities of LLMs, enabling them to reason more effectively about mathematical statements.

Demerits

Limited Generalizability

The proposed method may not generalize well to other domains or tasks beyond formal counterexample generation and theorem proving.

Dependence on Curated Datasets

The effectiveness of the approach relies on the availability of curated datasets, which may be limited or biased.

Potential for Overfitting

The multi-reward expert iteration framework may lead to overfitting if not properly regularized.

Expert Commentary

The article presents a novel and promising approach to augmenting LLMs with the ability to generate formal counterexamples. The proposed method addresses a critical gap in AI efforts in mathematics and demonstrates significant performance gains in counterexample generation and theorem proving. However, the limitations of the approach, such as limited generalizability and dependence on curated datasets, must be carefully considered. Furthermore, the potential for overfitting highlights the need for proper regularization. Nonetheless, the article's findings have significant implications for various domains and policymakers, underscoring the importance of addressing the gap in AI efforts in mathematics.

Recommendations

  • ✓ Future research should focus on exploring the generalizability of the proposed method to other domains and tasks.
  • ✓ The development of more robust and diversified training datasets is essential to further improve the performance of LLMs in formal counterexample generation and theorem proving.

Sources

Original: arXiv - cs.AI

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