Wu's Method can Boost Symbolic AI to Rival Silver Medalists and AlphaGeometry to Outperform Gold Medalists at IMO Geometry (2404.06405v2)

Abstract: Proving geometric theorems constitutes a haLLMark of visual reasoning combining both intuitive and logical skills. Therefore, automated theorem proving of Olympiad-level geometry problems is considered a notable milestone in human-level automated reasoning. The introduction of AlphaGeometry, a neuro-symbolic model trained with 100 million synthetic samples, marked a major breakthrough. It solved 25 of 30 International Mathematical Olympiad (IMO) problems whereas the reported baseline based on Wu's method solved only ten. In this note, we revisit the IMO-AG-30 Challenge introduced with AlphaGeometry, and find that Wu's method is surprisingly strong. Wu's method alone can solve 15 problems, and some of them are not solved by any of the other methods. This leads to two key findings: (i) Combining Wu's method with the classic synthetic methods of deductive databases and angle, ratio, and distance chasing solves 21 out of 30 methods by just using a CPU-only laptop with a time limit of 5 minutes per problem. Essentially, this classic method solves just 4 problems less than AlphaGeometry and establishes the first fully symbolic baseline strong enough to rival the performance of an IMO silver medalist. (ii) Wu's method even solves 2 of the 5 problems that AlphaGeometry failed to solve. Thus, by combining AlphaGeometry with Wu's method we set a new state-of-the-art for automated theorem proving on IMO-AG-30, solving 27 out of 30 problems, the first AI method which outperforms an IMO gold medalist.

• The paper demonstrates that integrating Wu's method with synthetic techniques solves 27 out of 30 IMO problems, setting a new state-of-the-art benchmark.
• It details a novel synergy of algebraic and deductive approaches that enables performance comparable to IMO silver medalists on standard hardware.
• It highlights the potential of fully symbolic theorem proving systems to rival neuro-symbolic models, paving the way for advanced research in automated geometry.

Revisiting Automated Theorem Proving for IMO Geometry with Enhanced Wu's Method

Introduction

Automated theorem proving for geometry, particularly at the Olympiad level, represents a complex challenge that tests the boundaries of artificial intelligence in the field of intellectual and intuitive reasoning. The International Mathematical Olympiad (IMO) problems serve as a benchmark for evaluating the abilities of automated theorem proving systems. This paper focuses on an in-depth examination of Wu's method applied to the challenging IMO-AG-30 problem set, previously assessed by the introduction of AlphaGeometry, a neuro-symbolic model. Our findings underscore the surprising efficacy of Wu's method, presenting a combination of traditional synthetic methodologies and advanced neuro-symbolic approaches that surpass previous capabilities in automated geometry theorem proving.

Key Findings and Methodological Insights

A revisitation of the IMO-AG-30 Challenge, originally intended for assessment using AlphaGeometry, reveals some notable accomplishments and insights concerning automated theorem proving techniques, especially regarding Wu's method:

• Performance of Wu's Method: Contrary to initial evaluations, Wu's method alone demonstrates a strong performance by solving 15 of the 30 problems. This includes problems that were intractable for other theorem proving methodologies, thus exposing the underrecognized potential of algebraic approaches in solving complex geometry problems.
• Synergistic Approach: The novel combination of Wu's method with classic synthetic methods – deductive databases (DD) and angle, ratio, and distance chasing (AR) – establishes a formidable approach, solving 21 of the 30 problems. This combined technique, referred to as Wu&DD+AR, operates efficiently on a CPU-only laptop, achieving results comparable to an IMO silver medalist's performance and only slightly trailing behind AlphaGeometry's achievements.
• Setting New Benchmarks: The integration of Wu's method with AlphaGeometry (Wu&AG) introduces a groundbreaking achievement in automated theorem proving, solving 27 out of the 30 IMO-AG-30 problems. This marks the first instance of an AI system outperforming an IMO gold medalist, thereby setting a new state-of-the-art standard for the field.

Practical and Theoretical Implications

The remarkable performance of algebraic methods, particularly Wu's method, in this evaluation carries significant implications for both the practical application and theoretical understanding of automated theorem proving in geometry:

• Fully Symbolic Baseline: The establishment of Wu&DD+AR as a fully symbolic baseline that closely rivals human-level performance emphasizes the untapped potential of combining algebraic and synthetic methods. This approach not only advances the capabilities of automated theorem proving systems but also aligns closely with human problem-solving techniques, presenting a more interpretable model compared to purely neuro-symbolic models.
• Complementarity of Methods: The success of combining Wu's method with AlphaGeometry demonstrates the complementary nature of algebraic and synthetic methods. This synergy suggests that the most effective pathway forward may involve a holistic integration of diverse proving techniques, leveraging the strengths of each to tackle the multifaceted challenges posed by IMO geometry problems.

Future Directions

Despite the success of the presented approaches, the field of automated theorem proving for geometry remains ripe for further exploration and advancement:

• Improving Computational Tools: The current limitations of computational tools, especially in terms of constructions available and optimization, highlight the need for continued development and enhancement of software implementations. Such improvements could potentially unlock even greater achievements in automated theorem proving, leveraging the full theoretical capabilities of methods like Wu's.
• Developing More Challenging Benchmarks: The relative ease with which modern computational solvers have addressed IMO geometry puzzles suggests the necessity for new benchmarks. These would ideally push the boundaries of automated theorem proving further, testing the systems' abilities to engage in deeper and more complex problem-solving, akin to the "middlegame" of chess, rather than the comparatively constrained "endgame" scenarios presented by current problems.

Conclusion

This paper's exploration of Wu's method in the context of IMO-AG-30 has revealed both the robust capabilities of traditional algebraic approaches and the potential for synergistic combinations with synthetic and neuro-symbolic methodologies. As automated theorem proving continues to evolve, the insights garnered from this research will undoubtedly contribute to the development of more advanced, efficient, and holistic systems capable of tackling the myriad challenges posed by mathematical problem-solving at the highest levels.