CogAtom: From Cognitive Atoms to Olympiad-level Mathematical Reasoning in Large Language Models

2025-09-21

AI Summary

Large language models (LLMs) still face significant challenges in multi-step mathematical reasoning and abstract concept integration, while the scarcity of high-quality, Olympiad-level mathematical problems hinders the development of test-time scaling techniques. To address these limitations, we propose the “Cognitive Atom” framework, which formalizes problem generation as the selection, recombination, and evolution of elementary reasoning units. Our approach integrates graph-structured cognitive atom extraction, diversity-oriented stochastic walk exploration, constraint-satisfying recombination, and logical consistency verification—enabling controllable reasoning-path generation and precise difficulty calibration. Evaluated against state-of-the-art baselines, our generated problems achieve superior accuracy, deeper reasoning depth, and greater structural diversity; they reach AIME-level difficulty and exhibit broader problem-type coverage. Consequently, the framework substantially enhances the scale, quality, and scalability of mathematical reasoning datasets.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Cognitive Modeling & Cognitive Systems: Conceptual Inference and Reasoning
• Machine Learning: Large Multimodal Models (LMMs)

Application Category

• Graph Algorithms and Modeling for the Web: Foundation models and LLMs for Web-related graphs
• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

Mathematical reasoning poses significant challenges for Large Language Models (LLMs) due to its demand for multi-step reasoning and abstract conceptual integration. While recent test-time scaling techniques rely heavily on high-quality, challenging problems, the scarcity of Olympiad-level math problems remains a bottleneck. We introduce CogAtom, a novel cognitive atom-based framework for synthesizing mathematically rigorous and cognitively diverse problems. Unlike prior approaches, CogAtom models problem construction as a process of selecting and recombining fundamental reasoning units, cognitive atoms, extracted from human-authored solutions. A diversity-promoting random walk algorithm enables exploration of the cognitive atom space, while a constraint-based recombination mechanism ensures logical soundness and structural validity. The combinatorial nature of the graph structure provides a near-infinite space of reasoning paths, and the walk algorithm systematically explores this space to achieve large-scale synthesis of high-quality problems; meanwhile, by controlling the number of cognitive atoms, we can precisely adjust problem difficulty, ensuring diversity, scalability, and controllability of the generated problems. Experimental results demonstrate that CogAtom outperforms existing methods in accuracy, reasoning depth, and diversity, generating problems that closely match the difficulty of AIME while exceeding it in structural variation. Our work offers a cognitively grounded pathway toward scalable, high-quality math problem generation.Our code is publicly available at https://github.com/Icarus-1111/CogAtom.

Problem

• Generating Olympiad-level math problems with controlled difficulty
• Overcoming scarcity of high-quality challenging mathematical problems
• Ensuring logical soundness and diversity in problem generation

Innovation

• Cognitive atom-based framework for problem synthesis
• Diversity-promoting random walk explores reasoning paths
• Constraint-based recombination ensures logical soundness