Proving Olympiad Algebraic Inequalities without Human Demonstrations (2406.14219v2)

Abstract: Solving Olympiad-level mathematical problems represents a significant advancement in machine intelligence and automated reasoning. Current machine learning methods, however, struggle to solve Olympiad-level problems beyond Euclidean plane geometry due to a lack of large-scale, high-quality datasets. The challenge is even greater in algebraic systems, which involve infinite reasoning spaces within finite conditions. To address these issues, we propose AIPS, an Algebraic Inequality Proving System capable of autonomously generating complex inequality theorems and effectively solving Olympiad-level inequality problems without requiring human demonstrations. During proof search in a mixed reasoning manner, a value curriculum learning strategy on generated datasets is implemented to improve proving performance, demonstrating strong mathematical intuitions. On a test set of 20 International Mathematical Olympiad-level inequality problems, AIPS successfully solved 10, outperforming state-of-the-art methods. Furthermore, AIPS automatically generated a vast array of non-trivial theorems without human intervention, some of which have been evaluated by professional contestants and deemed to reach the level of the International Mathematical Olympiad. Notably, one theorem was selected as a competition problem in a major city 2024 Mathematical Olympiad.

• The paper presents AIPS, an advanced system that autonomously generated over 191,000 non-trivial IMO-level theorems for algebraic inequality proofs.
• It employs a hybrid approach combining symbolic deduction, pattern matching, and a neural best-first search algorithm to outperform existing methods on benchmark tests.
• The work marks a significant advancement in AI-driven theorem proving, offering practical insights for solving high-difficulty algebraic problems in mathematical research.

An Examination of the Algebraic Inequality Proving System (AIPS) for Olympiad-Level Inequalities

The paper "Proving Olympiad Algebraic Inequalities without Human Demonstrations" presents an advanced Algebraic Inequality Proving System (AIPS) that can autonomously generate and solve complex inequality theorems at the level of the International Mathematical Olympiad (IMO). Given the challenges inherent in automated reasoning and theorem proving, particularly in algebraic systems that involve infinite reasoning spaces, the development of AIPS signifies a marked progression in AI capabilities in mathematical problem-solving.

Methodology and Components

AIPS integrates several sophisticated components to address the challenges of large-scale, high-quality dataset generation and effective proof search strategies:

1. Symbolic Deductive Engine: This core component of AIPS leverages a variety of fundamental theorems, such as the AM-GM and Jensen's inequalities, and transformation rules. The engine utilizes a symbolic computation system (SymPy), enabling effective algebraic reasoning through a specialized representation of algebraic expressions as expression trees.
2. Pattern Matching and Forward Reasoning: AIPS employs comprehensive pattern matching to apply inequality theorems to algebraic expressions. This includes traversing expression trees, labeling nodes based on their value effect on the entire expression, and matching sub-expressions to applicable theorems. The system performs forward reasoning to deduce new inequalities and can synthesize large-scale data for theorem proving.
3. Synthetic Theorem Generation: Addressing the dataset scarcity issue, AIPS autonomously generates complex inequality theorems. By implementing a rigorous algorithmic process, AIPS created over 191,000 non-trivial theorems, evaluated by professional mathematicians as comparable to IMO-level inequalities.
4. Neural Algebraic Inequality Prover: AIPS formulates theorem proving as a sequential decision-making process using a Best-First search combined with a curriculum-trained value network. The value network is pre-trained on synthetic data using a tree-depth heuristic and fine-tuned through curriculum learning on increasingly difficult problems.

Performance and Results

AIPS was evaluated on a benchmark set, MO-INT-20, compiled from various national and international mathematical competitions. The system outperformed state-of-the-art methods, successfully solving 10 out of 20 Olympiad-level problems. This surpasses the capabilities of interactive theorem provers like LeanCopilot and LLMs such as GPT-4 and Gemini 1.5 Pro, underscoring the robustness of AIPS’s symbolic and neural components in generating and solving complex mathematical inequalities.

Implications and Future Directions

The introduction of AIPS contributes significantly to the fields of automated reasoning and AI-driven theorem proving. Practically, AIPS can serve as a valuable tool for mathematicians by generating elegant, non-trivial theorems and providing insights into complex algebraic problems. Theoretically, it enriches the understanding of integrating symbolic computation with neural networks for automated theorem proving.

Future developments could focus on enhancing AIPS’s capabilities by incorporating more fundamental theorems, operational rules, and improving its autonomous learning algorithms. This could involve enabling the system to propose and understand new definitions autonomously, reducing reliance on pre-coded theorems and heuristics. Another promising direction is refining the curriculum learning strategy to handle even longer and more complex reasoning chains efficiently.

In summary, AIPS represents a substantial step forward in the domain of AI-driven automated theorem proving, achieving significant results in solving high-difficulty algebraic inequalities autonomously. By addressing core challenges such as dataset generation and effective search strategies, AIPS paves the way for future advancements in AI capabilities within mathematical problem-solving.