A Categorical Outlook on Cellular Automata

TL;DR

This paper applies category theory, specifically comonads, to model cellular automata, providing a new perspective that recovers classical results like the Curtis-Hedlund theorem.

Contribution

It introduces a categorical framework using comonads to analyze cellular automata, linking local behaviors with coKleisli maps and generalizing existing theorems.

Findings

Categorical modeling of cellular automata using comonads.

Recovery of the Curtis-Hedlund theorem within this framework.

Establishment of equivalences between local behaviors and coKleisli maps.

Abstract

In programming language semantics, it has proved to be fruitful to analyze context-dependent notions of computation, e.g., dataflow computation and attribute grammars, using comonads. We explore the viability and value of similar modeling of cellular automata. We identify local behaviors of cellular automata with coKleisli maps of the exponent comonad on the category of uniform spaces and uniformly continuous functions and exploit this equivalence to conclude some standard results about cellular automata as instances of basic category-theoretic generalities. In particular, we recover Ceccherini-Silberstein and Coornaert's version of the Curtis-Hedlund theorem.