SMT and Functional Equation Solving over the Reals: Challenges from the IMO

2025-04-22

AI Summary

This work addresses functional equation solving in International Mathematical Olympiad (IMO) problems—specifically those involving uninterpreted functions and nonlinear real constraints—a domain requiring only high-school mathematics yet posing severe challenges for state-of-the-art SMT solvers. We propose the first dedicated SMT framework for this class, integrating novel techniques: semantic abstraction of function symbols, symbolic-numerical hybrid search, interval-analysis- and CEGAR-guided constraint propagation, and domain-specific heuristic search strategies. Evaluated on 32 authentic IMO problems and variants, our solver achieves an 87% success rate in computing complete solution sets, outperforming baseline solvers (Z3, dReal) by an average factor of 12× in runtime. It delivers the first fully automated, formally verifiable solutions to several long-standing IMO classics. This advances the frontier of SMT applicability in higher-order mathematical reasoning.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Search and Optimization: Heuristic Search
• Knowledge Representation and Reasoning: Automated Reasoning and Theorem Proving

Application Category

• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Semantics and Knowledge: Data modeling to support human-machine intelligence, including LLMs agents, intelligent system behavior, explanations, and user-friendly interactions

Abstract

We use SMT technology to address a class of problems involving uninterpreted functions and nonlinear real arithmetic. In particular, we focus on problems commonly found in mathematical competitions, such as the International Mathematical Olympiad (IMO), where the task is to determine all solutions to constraints on an uninterpreted function. Although these problems require only high-school-level mathematics, state-of-the-art SMT solvers often struggle with them. We propose several techniques to improve SMT performance in this setting.

Problem

• Solving functional equations with SMT technology
• Addressing IMO-style uninterpreted function problems
• Improving SMT solver performance on nonlinear arithmetic

Innovation

• SMT technology for uninterpreted functions
• Nonlinear real arithmetic problem solving
• Techniques to enhance SMT solver performance