Response Curves and Preimage Sequences of Two-Dimensional Cellular Automata

TL;DR

This paper investigates response curves for binary two-dimensional cellular automata with L-shaped neighborhoods, using preimage sets to compute the density of ones over iterations, revealing properties of surjective and permutive rules.

Contribution

It introduces a method to calculate response curves for a broad class of cellular automata using preimage sets, including explicit density calculations for certain initial conditions.

Findings

Response curves can be computed for many rules using preimage sets.

All surjective rules with L-shaped neighborhoods are permutive.

Rules with explicitly computable preimage curves are either identity or shift emulators.

Abstract

We consider the problem of finding response curves for a class of binary two-dimensional cellular automata with $L$-shaped neighbourhood. We show that the dependence of the density of ones after an arbitrary number of iterations, on the initial density of ones, can be calculated for a fairly large number of rules by considering preimage sets. We provide several examples and a summary of all known results. We consider a special case of initial density equal to 0.5 for other rules and compute explicitly the density of ones after $n$ iterations of the rule. This analysis includes surjective rules, which in the case of $L$-shaped neighbourhood are all found to be permutive. We conclude with the observation that all rules for which preimage curves can be computed explicitly are either finite or asymptotic emulators of identity or shift.