The Myhill-Nerode Theorem for Bounded Interaction: Canonical Abstractions via Agent-Bounded Indistinguishability
The paper addresses state-space explosion in partially observable environments by formalizing a bounded-interaction analogue of the Myhill-Nerode theorem for finite POMDPs. Its core insight is that two observation histories are equivalent if no bounded finite-state controller can distinguish them via closed-loop interaction, inducing a canonical quotient that is minimal and unique for that observer capacity. This yields a principled separation between exact decision sufficiency for observation-measurable objectives and approximate bounds for latent-state rewards, with the canonical object strictly requiring clock-aware probe families.
The paper delivers a rigorous structural theory establishing that bounded probe families induce canonical minimal quotients on POMDPs. The Bounded-Interaction Myhill-Nerode theorem (Theorem 4.4) rigorously establishes well-definedness, soundness, universality, minimality, and uniqueness, while Theorem 4.6 proves exact sufficiency for agent-accessible objectives $\mathcal{G}_{\mathrm{obs}}$. The work is methodologically exemplary in explicitly separating theorem-facing validation (Tier I: clock-aware exact) from operational surrogacy (Tier II/III), though this strict separation means the scalable operational results inherit guarantees only when an explicit cross-family measurement $\delta_{\mathrm{clk}}$ is available.
The closed-loop Wasserstein pseudometric construction (Definition 3.1) and the probe-exact quotient POMDP are mathematically sound. The witness boundary is crisply delineated: Theorem 4.9 proves deterministic clock-aware FSCs suffice as witnesses, while Proposition 4.10 provides an explicit $m=1$, $T=3$ counterexample where deterministic stationary FSCs fail. The experimental tier structure (Table 1) honestly distinguishes exhaustive clock-aware validation ($Q_{m,T}^{\mathrm{clk}}$) from operational deterministic-stationary surrogacy ($Q_{m,T}^{\mathrm{op}}$), with Tier I and Tier II results directly supporting the theorem claims.
The observation-Lipschitz value bound $|V_M^{\pi} - V_{\widetilde{M}}^{\pi}| \leq L_R T \varepsilon$ (Theorem 3.6) is vacuous for latent-state rewards when $L_R$ is large; the paper explicitly admits that for Tiger's standard reward ($L_R = 110$), "the bound is explicitly vacuous" despite being structurally correct. Exact computation of the quotient remains exponential in the horizon $T$, and while the paper honestly frames this limitation, it restricts practical applicability. The operational experiments at scale (Tier III) explicitly disclaim theorem validation, meaning the paper's central scalable claims rest on approximation heuristics without the same guarantees as the clock-aware exact objects.
The evidence strongly supports structural claims regarding the clock-aware quotient: Table 3 verifies exact preservation of observation-measurable objectives (Theorem 4.6), while Table 2 validates the operational-to-clock-aware gap on small exact cases. However, cross-family value transfer certificates are only available when the gap $\delta_{\mathrm{clk}}$ is explicitly measured, and the paper notes that "heavier operational stress tests are archived separately in the appendix and artifact package" rather than treated as core theorem validation. Comparisons to bisimulation metrics and predictive state representations (Section 2) accurately position the work as distinctively closed-loop and capacity-parameterized.
Algorithmic specifications are detailed (Algorithms 1-2) and code is available at a linked GitHub repository, supporting reproduction. The paper provides explicit certificates ($\delta_S = 0$) for exact-for-family operational cases (Table 5) and bootstrap confidence intervals for sampling-based estimates (Table 6), enhancing reproducibility for certified strata. However, heavier Tier III archival stress tests are explicitly excluded from the core verification target, and exact computation requires exhaustive enumeration over controller spaces exponential in the horizon $T$, which may be computationally prohibitive for independent reproduction of the full theorem-aligned experiments at larger scales.
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.
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