Cellular automata in operational probabilistic theories

TL;DR

This paper develops a framework for cellular automata within operational probabilistic theories, introducing concepts like composition, causality, homogeneity, and locality, and establishing their mathematical properties and relations.

Contribution

It extends cellular automata theory to operational probabilistic frameworks, defining new concepts and proving foundational lemmas for infinite systems and Cayley graph structures.

Findings

Defined composition of infinite elementary systems.

Introduced and analyzed causal influence and signalling.

Proved a general wrapping lemma connecting cellular automata on Cayley graphs.

Abstract

The theory of cellular automata in operational probabilistic theories is developed. We start introducing the composition of infinitely many elementary systems, and then use this notion to define update rules for such infinite composite systems. The notion of causal influence is introduced, and its relation with the usual property of signalling is discussed. We then introduce homogeneity, namely the property of an update rule to evolve every system in the same way, and prove that systems evolving by a homogeneous rule always correspond to vertices of a Cayley graph. Next, we define the notion of locality for update rules. Cellular automata are then defined as homogeneous and local update rules. Finally, we prove a general version of the wrapping lemma, that connects CA on different Cayley graphs sharing some small-scale structure of neighbourhoods.