Reversible space–time simulation of cellular automata

Abstract

The goal of this paper is to design a reversible d-dimensional cellular automaton which is capable of simulating the behavior of any given d-dimensional cellular automaton over any given configuration (even infinite) with respect to a well suited notion of simulation we introduce. We generalize a problem which was originally addressed in a paper by Toffoli in 1977. He asked whether a d-dimensional reversible cellular automaton could simulate d-dimensional cellular automata. In the same paper he proved that there exists a (d+1)-dimensional reversible cellular automaton which can simulate a given d-dimensional cellular automaton. To prove our result, we use as an intermediate model partition cellular automata defined by Morita et al. in 1989.