Causal dynamics of discrete manifolds

TL;DR

This paper generalizes Cellular Automata to model the evolution of discrete manifolds over time, establishing formal theorems and graph-based characterizations, with decidability results in low dimensions.

Contribution

It introduces a formal framework for discrete manifolds evolving via Cellular Automata, including graph-based characterizations and decidability results in certain dimensions.

Findings

Formalization of discrete manifold evolution

Graph-based characterization of manifolds

Decidability in dimensions less than 4

Abstract

We extend Cellular Automata to time-varying discrete geometries. In other words we formalize, and prove theorems about, the intuitive idea of a discrete manifold which evolves in time, subject to two natural constraints: the evolution does not propagate information too fast; and it acts everywhere the same. For this purpose we develop a correspondence between complexes and labeled graphs. In particular we reformulate the properties that characterize discrete manifolds amongst complexes, solely in terms of graphs. In dimensions $n<4$, over bounded-star graphs, it is decidable whether a Cellular Automaton maps discrete manifolds into discrete manifolds.