State Representation and Termination for Recursive Reasoning Systems

Researchers present a formal framework for recursive reasoning systems that addresses two critical design challenges: how to represent evolving reasoning states and when to terminate iteration. The paper introduces an epistemic state graph representation and proposes the 'order-gap' metric as a stopping criterion, with theoretical guarantees for when this criterion provides meaningful guidance.

This research addresses fundamental architectural questions in recursive reasoning systems—a category encompassing tree-of-thought prompting, iterative theorem proving, and agent loop design. The paper's contribution centers on formalizing what has previously been implicit: how to track reasoning progress and recognize convergence. The epistemic state graph approach encodes not just claims but also evidential relationships, open questions, and confidence weights, providing richer state representation than typical token sequences or embedding vectors. The order-gap metric measures whether expand-then-consolidate and consolidate-then-expand orderings yield similar results; a small gap suggests diminishing returns from further iteration. This matters because recursive systems currently lack principled stopping conditions, often relying on fixed iteration counts or arbitrary heuristics. The theoretical contribution establishes when the linearized order-gap criterion becomes non-degenerate near fixed points—essentially determining when the stopping metric is genuinely informative versus mathematically trivial. While the paper acknowledges this is a local condition rather than a global convergence guarantee, it provides the first rigorous framework for this class of problems. For AI practitioners, this research enables more efficient recursive systems by reducing wasted computation on fruitless iterations. The framework applies across multiple domains: agentic systems could terminate execution more intelligently, theorem provers could stop searching when refinement becomes counterproductive, and continual learning systems could recognize when additional evidence integration is unhelpful. The work bridges theoretical computer science and practical AI engineering, offering both mathematical rigor and practical applicability.

• →The paper formalizes state representation for recursive reasoning through epistemic state graphs encoding claims, evidence, questions, and confidence weights.
• →The order-gap metric provides a principled stopping criterion by measuring convergence between different consolidation orderings.
• →Theoretical conditions ensure the stopping criterion is informative rather than algebraically degenerate near convergence.
• →Framework applies to tree-of-thought, agent loops, theorem proving, and continual learning systems.
• →Addresses practical efficiency problem of knowing when recursive refinement yields diminishing returns.