OMEGA: Can LLMs Reason Outside the Box in Math? Evaluating Exploratory, Compositional, and Transformative Generalization

TL;DR

This paper introduces OMEGA, a diverse benchmark to evaluate large language models' ability to generalize in mathematics through exploratory, compositional, and transformative reasoning, revealing current limitations and guiding future improvements.

Contribution

The paper presents OMEGA, a novel benchmark for assessing out-of-distribution mathematical reasoning in LLMs across three creativity-inspired axes, with detailed analysis of model performance and limitations.

Findings

LLMs' performance drops with increasing problem complexity.

Fine-tuning improves exploratory reasoning but not compositional or transformative reasoning.

Transformative reasoning remains a significant challenge for current LLMs.

Abstract

Recent large-scale language models (LLMs) with long Chain-of-Thought reasoning-such as DeepSeek-R1-have achieved impressive results on Olympiad-level mathematics benchmarks. However, they often rely on a narrow set of strategies and struggle with problems that require a novel way of thinking. To systematically investigate these limitations, we introduce OMEGA-Out-of-distribution Math Problems Evaluation with 3 Generalization Axes-a controlled yet diverse benchmark designed to evaluate three axes of out-of-distribution generalization, inspired by Boden's typology of creativity: (1) Exploratory-applying known problem solving skills to more complex instances within the same problem domain; (2) Compositional-combining distinct reasoning skills, previously learned in isolation, to solve novel problems that require integrating these skills in new and coherent ways; and (3) Transformative-adopting novel, often unconventional strategies by moving beyond familiar approaches to solve problems more effectively. OMEGA consists of programmatically generated training-test pairs derived from templated problem generators across geometry, number theory, algebra, combinatorics, logic, and puzzles, with solutions verified using symbolic, numerical, or graphical methods. We evaluate frontier (or top-tier) LLMs and observe sharp performance degradation as problem complexity increases. Moreover, we fine-tune the Qwen-series models across all generalization settings and observe notable improvements in exploratory generalization, while compositional generalization remains limited and transformative reasoning shows little to no improvement. By isolating and quantifying these fine-grained failures, OMEGA lays the groundwork for advancing LLMs toward genuine mathematical creativity beyond mechanical proficiency.