The Myhill-Nerode Theorem for Bounded Interaction: Canonical Abstractions via Agent-Bounded Indistinguishability

Persistent communication limits force a multi-agent system to decide which agents may coordinate throughout a rollout. Current proximity alone is insufficient: separated agents may interact later, whereas a large pair reward may remain unreachable until it is heavily discounted. We introduce Reachability-Certified Subteam Decomposition (RCSD) for finite multi-agent Markov decision processes with factorized physical dynamics, finite-range ordered pair rewards, and almost-sure motion bounds. RCSD combines a speed-limit lower bound on pairwise contact time with a reward envelope to form a current-state affinity. For any capacity-valid persistent partition, the sum of cut affinities bounds the reward-deletion error of every unchanged stationary Markov state-feedback policy. A product of team-optimal policies for the resulting cut MDP incurs at most twice this certificate in regret against the centralized optimum. Both bounds are worst-case tight. On a controlled five-agent family, RCSD-Exact reduces aggregate normalized execution regret by 56.0%, 28.8%, and 25.3% relative to uniform, distance-only, and envelope-only partitions. A separate stochastic two-dimensional study finds no bound violation over 384 exact-partition and 1,440 restricted-controller evaluations. Exact four-agent evidence favors RCSD over uniform and distance-only grouping; raw evidence for current contact is borderline and envelope-only is unresolved. Across balanced 8-20-agent strata, controller-library utility is mixed: pointwise paired intervals favor RCSD over distance and current contact, include zero for uniform, and favor envelope-only and Value-MIP over RCSD. Partition construction remains subsecond in median up to 100 agents; this last result does not include affinity formation or MDP planning.

When two agents of different computational capacities interact with the same environment, they need not compress a common semantic alphabet differently; they can induce different semantic alphabets altogether. We show that the quotient POMDP $Q_{m,T}(M)$ - the unique coarsest abstraction consistent with an agent's capacity - serves as a capacity-derived semantic space for any bounded agent, and that communication between heterogeneous agents exhibits a sharp structural phase transition. Below a critical rate $R_{\text{crit}}$ determined by the quotient mismatch, intent-preserving communication is structurally impossible. In the supported one-way memoryless regime, classical side-information coding then yields exponential decay above the induced benchmark. Classical coding theorems tell you the rate once the source alphabet is fixed; our contribution is to derive that alphabet from bounded interaction itself. Concretely, we prove: (1) a fixed-$\varepsilon$ structural phase-transition theorem whose lower bound is fully general on the common-history quotient comparison; (2) a one-way Wyner-Ziv benchmark identification on quotient alphabets, with exact converse, exact operational equality for memoryless quotient sources, and an ergodic long-run bridge via explicit mixing bounds; (3) an asymptotic one-way converse in the shrinking-distortion regime $\varepsilon = O(1/T)$, proved from the message stream and decoder side information; and (4) alignment traversal bounds enabling compositional communication through intermediate capacity levels. Experiments on eight POMDP environments (including RockSample(4,4)) illustrate the phase transition, a structured-policy benchmark shows the one-way rate can drop by up to $19\times$ relative to the counting bound, and a shrinking-distortion sweep matches the regime of the asymptotic converse.

An agent acting under partial observability must retain a recursively updateable statistic of history that restores the Markov property, but the smallest such statistic is generally unknown. We characterize this minimal Markov sufficient statistic for holonomy-cover decision processes, a structured POMDP class in which the visible dynamics are Markov and every realized visible transition applies a fixed permutation to a hidden mode. In particular, we construct the stable quotient, the coarsest observation-wise abstraction preserving one-step rewards and quotient successors, and prove that the pair of the current observation and stable class forms an exact finite Markov state. When the current class is correctly initialized, exact class tracking requires exactly the minimal memory symbols, in the sense that under reachability and pairwise decision separation at a maximizing observation, no arbitrary finite-memory controller can use fewer. Under resettable diagnostics, nearest-prototype class inference has exponentially decaying error, and a calibrate-then-restart reduction transfers finite-MDP guarantees to the recovered state. The results enable Holonomy Memory Reinforcement Learning. It represents memory by the current stable class, updates it through ordered edge transports, identifies local class coordinates when diagnostics are available, and applies a standard finite-MDP RL backbone after synchronization. Experiments recover an exact compression from raw states to quotient states and achieve perfect paired-order accuracy with three decision-time memory states, matching the quotient oracle and outperforming the non-oracle baselines.