Nondeterministic Behaviours in Double Categorical Systems Theory

TL;DR

This paper develops a double categorical framework to model nondeterministic behaviors in systems, using monoidal semi double categories, semimodules, and Markov categories with conditionals for compositional trajectory representation.

Contribution

It introduces a novel double categorical approach to nondeterminism, extending existing theories with new structures for systems and trajectories.

Findings

Constructed monoidal semi double categories of interfaces

Defined trajectories using conditional products in Markov categories

Represented nondeterministic systems via Markov maps

Abstract

In this paper, we build double theories capturing the idea of nondeterministic behaviors and trajectories. Following Libkind and Myers' Double Operadic Theory of Systems, we construct monoidal semi double categories of interfaces, along with what we call semimodules of systems, in the case of Moore machines, working with Markov categories with conditionals to handle nondeterminism. We use conditional products in these Markov categories to define trajectories in a compositional way, and represent nondeterministic systems using Markov maps; channels between systems are assumed to be deterministic.