Cellular Automata and Kan Extensions

TL;DR

This paper explores how cellular automata can be extended using Kan extensions from category theory, revealing new connections between computer science and category theory without requiring prior categorical knowledge.

Contribution

It formalizes cellular automaton extensions via Kan extensions, linking automata theory with category theory and offering a new perspective on the Curtis-Hedlung theorem.

Findings

Categorical framework for automaton extension

New insights into Curtis-Hedlung theorem

Bridges between computer science and category theory

Abstract

In this paper, we formalize precisely the sense in which the application of cellular automaton to partial configuration is a natural extension of its local transition function through the categorical notion of Kan extension. In fact, the two possible ways to do such an extension and the ingredients involved in their definition are related through Kan extensions in many ways. These relations provide additional links between computer science and category theory, and also give a new point of view on the famous Curtis-Hedlung theorem of cellular automata from the extended topological point of view provided by category theory. These relations provide additional links between computer science and category theory. No prior knowledge of category theory is assumed.