Inverse Design of Finite-State Channels by Dynamic Measure Matching

Inspired by the optimization approach to biology, we formulate an inverse problem for communication with memory: construct a finite-state channel for which a given signaling behavior supports capacity-achieving codes. Dynamic measure matching is introduced to turn a directed-information converse into conditions on the channel dynamics. These conditions lead to a new notion of dynamic surprise: the log ratio of physical to test output probabilities with a continuation correction for future signaling opportunities. The matching inequalities bound its conditional mean. For fixed architecture, test law, and potential, channel row synthesis under convex physical restrictions is convex. For every signaling constraint represented by a finite irreducible constrained graph of positive entropy, we construct a channel law under which undesired inputs remain available and behaviorally distinct while incurring a strict information penalty. A noisy refractory family has an exact capacity plateau, followed by an interval in which Shannon and zero-error feedback capacities differ. Explicit zero-error codes attain capacity in the graph construction and on the plateau. In this family, a Gibbs choice maximizes the penalty and makes dynamic surprise equal its conditional mean on every transition, yielding an exact trajectory-wise likelihood identity, a finite-block bound on successful decoding with premature requests, and an exponential strong converse. An exact gap decomposition into action residuals and output-law divergence constrains every reliable capacity-approaching code. Within the strict plateau, positive discrimination exponents for unused dynamics require a positive rate loss, while sparse probing permits consistent estimation at asymptotic capacity. A chemical channel illustrates how premature stimulation can alter future responsiveness while leaving the immediate output law unchanged.