Finite Observations, Infinite Behaviour: bicategorical semantics for stateful monoidal processes

2026-07-04

AI Summary

This work addresses the problem of characterizing behavioral equivalence for time-varying stateful processes based solely on finite input-output observations, without relying on inaccessible internal states. To this end, it introduces a novel structure called the “discard double category,” which provides a unified framework for modeling partial, nondeterministic, probabilistic, and quantum processes, and constructs a functorial semantics into the free feedback category. By leveraging preordered enriched monoidal categories and closed relations over compact Hausdorff spaces, the paper establishes a categorical compactness theorem. This framework not only subsumes Willems’ behavioral theory of linear time-invariant systems as a special case but also offers a unified, internal-state-free foundation for behavioral semantics across a broad spectrum of process types.

Abstract

Time-dependent processes are often described by machines with an internal state which is updated as time evolves. An external observer cannot see this state and learns about a process only through finite observations of its inputs and outputs, each of which imposes a constraint on the trajectories the process can exhibit. We introduce a semantic construction in which two stateful processes have the same behaviour when they have the same constraints, as determined by finite observations, independent of their internal state. The construction is defined over any preorder-enriched monoidal category with a compatible notion of discarding, which we call a discard bicategory, capturing partial, non-deterministic, probabilistic, and quantum processes. The resulting category of behaviours provides a functorial semantics for free feedback categories in the sense of Katis, Sabadini, and Walters. For non-deterministic systems, we prove a categorified compactness theorem: every compatible family of finite observations between compact Hausdorff spaces extends uniquely and functorially to an infinite closed relation. Restricted to affine relations over finite fields, the compactness theorem recovers Willems' notion of behaviour for linear time-invariant systems.

Problem

• stateful processes
• finite observations
• behavioural equivalence
• monoidal categories
• compactness theorem

Innovation

• bicategorical semantics
• stateful processes
• finite observations
• compactness theorem
• discard bicategory