Clock systems for stochastic and non-deterministic categorical systems theories

TL;DR

This paper demonstrates that stochastic and non-deterministic categorical systems theories possess clock systems, which facilitate well-behaved behavior functors, extending to generalized notions of non-linear time.

Contribution

It proves the existence of clock systems in stochastic and non-deterministic systems theories, a previously unknown result, and explores their implications for behavior modeling.

Findings

Stochastic systems have clock systems.

Non-deterministic systems have clock systems.

Clock systems relate to generalized non-linear time behaviors.

Abstract

One of the characteristic features of categorical systems theory is that the behavior of systems can be characterized by certain morphisms into them. In other words, behaviors form a representable covariant functor to Set. And more generally, in the compositional setting, behaviors form a representable double functor to Span. Clock systems are convenient because behavior functors represented by clock systems are automatically well-behaved. It was previously not known whether stochastic and non-deterministic systems theories have clock systems. In this paper, we show that indeed they do have clock systems. Moreover, the clock systems for non-deterministic systems point to generalized notions of behavior for non-linear time.