On the Equivalence of Cellular Automata and the Tile Assembly Model

TL;DR

This paper establishes a formal equivalence between cellular automata and the abstract Tile Assembly Model by defining simulation relations and demonstrating how each model can simulate the other in 2D systems.

Contribution

It introduces formal notions of simulation between CA and aTAM, and constructs specific systems that can simulate any system of the other model, showing their fundamental computational equivalence.

Findings

A nondeterministic CA can simulate any aTAM system.

An aTAM tile set can simulate any nondeterministic CA with finite initial configuration.

The models are computationally equivalent through defined simulation relations.

Abstract

In this paper, we explore relationships between two models of systems which are governed by only the local interactions of large collections of simple components: cellular automata (CA) and the abstract Tile Assembly Model (aTAM). While sharing several similarities, the models have fundamental differences, most notably the dynamic nature of CA (in which every cell location is allowed to change state an infinite number of times) versus the static nature of the aTAM (in which tiles are static components that can never change or be removed once they attach to a growing assembly). We work with 2-dimensional systems in both models, and for our results we first define what it means for CA systems to simulate aTAM systems, and then for aTAM systems to simulate CA systems. We use notions of simulate which are similar to those used in the study of intrinsic universality since they are in some sense strict, but also intuitively natural notions of simulation. We then demonstrate a particular nondeterministic CA which can be configured so that it can simulate any arbitrary aTAM system, and finally an aTAM tile set which can be configured so that it can be used to simulate any arbitrary nondeterministic CA system which begins with a finite initial configuration.