State Representation and Termination for Recursive Reasoning Systems
For researchers designing recursive reasoning systems (e.g., agent loops, tree-of-thought reasoning), this work offers a formal framework for state representation and termination, though the result is local and not a global convergence guarantee.
The paper introduces a state representation for recursive reasoning systems using an epistemic state graph and proposes an order-gap criterion to determine when to stop iterating. The main result provides a necessary and sufficient condition for the order-gap to be non-degenerate near the fixed point, indicating when the stopping criterion is informative.
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.