State Representation and Termination for Recursive Reasoning Systems

TL;DR

This paper introduces a framework for representing and terminating recursive reasoning systems using epistemic state graphs and the order-gap metric, providing conditions for effective iteration stopping.

Contribution

It formalizes the reasoning state representation and termination criteria, offering a local condition for when further reasoning is unlikely to improve results.

Findings

Defines the epistemic state graph for reasoning states

Introduces the order-gap metric to assess iteration usefulness

Provides a necessary and sufficient condition for non-degenerate order-gap near fixed points

Abstract

Recursive reasoning systems alternate between acquiring new evidence and refining an accumulated understanding. Two design choices are typically left implicit: how to represent the evolving reasoning state, and when to stop iterating. This paper addresses both. We represent the reasoning state as an epistemic state graph encoding extracted claims, evidential relations, open questions, and confidence weights. We define the order-gap as the distance between the states reached by expand-then-consolidate versus consolidate-then-expand; a small order-gap suggests that the two orderings agree and further iteration is unlikely to help. Our main result gives a necessary and sufficient condition for the linearised order-gap to be non-degenerate near the fixed point, showing when the criterion is informative rather than algebraically vacuous. This is a local condition, not a global convergence guarantee. We apply the framework to recursive reasoning systems and sketch its application to agent loops, tree-of-thought reasoning, theorem proving, and continual learning.