The Myhill-Nerode Theorem for Bounded Interaction: Canonical Abstractions via Agent-Bounded Indistinguishability
This provides a theoretical framework for capacity-limited observers in reinforcement learning, though it appears incremental as an extension of existing concepts to bounded interaction.
The paper formalizes a canonical quotient for finite POMDPs based on bounded agent indistinguishability, analogous to the Myhill-Nerode theorem, and validates it on Tiger and GridWorld environments.
Any capacity-limited observer induces a canonical quotient on its environment: two situations that no bounded agent can distinguish are, for that agent, the same. We formalise this for finite POMDPs. A fixed probe family of finite-state controllers induces a closed-loop Wasserstein pseudometric on observation histories and a probe-exact quotient merging histories that no controller in the family can distinguish. The quotient is canonical, minimal, and unique-a bounded-interaction analogue of the Myhill-Nerode theorem. For clock-aware probes, it is exactly decision-sufficient for objectives that depend only on the agent's observations and actions; for latent-state rewards, we use an observation-Lipschitz approximation bound. The main theorem object is the clock-aware quotient; scalable deterministic-stationary experiments study a tractable coarsening with gap measured on small exact cases and explored empirically at larger scale. We validate theorem-level claims on Tiger and GridWorld. We also report operational case studies on Tiger, GridWorld, and RockSample as exploratory diagnostics of approximation behavior and runtime, not as theorem-facing evidence when no exact cross-family certificate is available; heavier stress tests are archived in the appendix and artifact package.