Selfsimilarity, Simulation and Spacetime Symmetries

TL;DR

This paper investigates the intrinsic simulation capabilities of cellular automata, especially linear and surjective ones, introducing a new necessary condition and proof techniques to determine simulation possibilities and limitations.

Contribution

It presents a novel necessary condition for CA simulation, particularly for linear surjective CA, and develops methods to analyze their simulation potential.

Findings

A linear reversible CA cannot simulate the identity.

Some linear CA are 'time-asymmetric' and cannot simulate their inverse.

The necessary condition can be heuristically checked via space-time diagrams.

Abstract

We study intrinsic simulations between cellular automata and introduce a new necessary condition for a CA to simulate another one. Although expressed for general CA, this condition is targeted towards surjective CA and especially linear ones. Following the approach introduced by the first author in an earlier paper, we develop proof techniques to tell whether some linear CA can simulate another linear CA. Besides rigorous proofs, the necessary condition for the simulation to occur can be heuristically checked via simple observations of typical space-time diagrams generated from finite configurations. As an illustration, we give an example of linear reversible CA which cannot simulate the identity and which is 'time-asymmetric', i.e. which can neither simulate its own inverse, nor the mirror of its own inverse.