DAG-Math: Graph-Guided Mathematical Reasoning in LLMs

Large Language Models (LLMs) demonstrate strong performance on mathematical problems when prompted with Chain-of-Thought (CoT), yet it remains unclear whether this success stems from search, rote procedures, or rule-consistent reasoning. To address this, we propose modeling CoT as a certain rule-based stochastic process over directed acyclic graphs (DAGs), where nodes represent intermediate derivation states and edges encode rule applications. Within this framework, we introduce **logical closeness**, a metric that quantifies how well a model’s CoT trajectory (i.e., the LLM's output) adheres to the DAG structure, providing evaluation beyond classical PASS@$k$ metrics. Building on this, we introduce the *DAG-MATH* CoT format and construct a benchmark that guides LLMs to generate CoT trajectories in this format, thereby enabling the evaluation of their reasoning ability under our framework. Across standard mathematical reasoning datasets, our analysis uncovers statistically significant differences in reasoning fidelity among representative LLM families—even when PASS@$k$ is comparable—highlighting gaps between final-answer accuracy and rule-consistent derivation. Our framework provides a balance between free-form CoT and formal proofs systems, offering actionable diagnostics for LLMs reasoning evaluation.