State Representation and Termination for Recursive Reasoning Systems
Authors: Debashis Guha, Amritendu Mukherjee, Sanjay Kukreja et al.
Summary
This paper tackles two design questions for recursive reasoning systems (systems that alternate between gathering evidence and refining understanding): how to represent the evolving state and when to stop iterating. The authors propose an epistemic state graph encoding claims, evidence, open questions, and confidence weights, and introduce the "order-gap"—the difference between expand-then-consolidate versus consolidate-then-expand. A small order-gap suggests the two orders agree and further iteration won't help. They provide a necessary and sufficient condition for the linearized order-gap to be informative (non-degenerate) near a fixed point, and sketch applications to agent loops, tree-of-thought, theorem proving, and continual learning.
Main takeaways:
- Recursive reasoning systems need a state representation (epistemic state graph) and a stopping criterion (order-gap).
- The order-gap measures whether expand-then-consolidate ≈ consolidate-then-expand; small gap means stop iterating.
- A formal condition tells you when the order-gap is informative versus algebraically vacuous near the fixed point (local, not global).
- Framework applies to agent loops, tree-of-thought reasoning, theorem proving, and continual learning.
- Addresses the "when to stop" problem in iterative reasoning without relying on heuristics or fixed iteration counts.
Relevance
Not directly related to my persona/marker implantation or conditional behavior work. This is about stopping criteria and state representation for iterative reasoning systems. Only tangentially relevant if I ever wanted to model persona installation as an iterative refinement process with a formal stopping rule, but that's not my current focus.
Abstract
arXiv:2605.06690v1 Announce Type: new 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.