Invertible Cellular Automata Research Articles

A generalization of cellular automata over groups

Let G be a group and let A be a finite set with at least two elements. A cellular automaton (CA) over AG is a function defined via a finite memory set and a local function . The goal of this paper is to introduce the definition of a generalized cellular automaton (GCA) , where H is another arbitrary group, via a group homomorphism . Our definition preserves the essence of CA, as we prove analogous versions of three key results in the theory of CA: a generalized Curtis-Hedlund Theorem for GCA, a Theorem of Composition for GCA, and a Theorem of Invertibility for GCA. When G = H, we prove that the group of invertible GCA over AG is isomorphic to a semidirect product of and the group of invertible CA. Finally, we apply our results to study automorphisms of the monoid consisting of all CA over AG . In particular, we show that every defines an automorphism of via conjugation by the invertible GCA defined by , and that, when G is abelian, is embedded in the outer automorphism group of . Communicated by Pedro Garcia-Sanchez

Linear Time Algorithm for Identifying the Invertibility of Null-Boundary Three Neighborhood Cellular Automata

This paper reports an algorithm to check for the invertibility of nullboundary three neighborhood cellular automata (CAs). While the best known result for checking invertibility is quadratic [1, 2], the complexity of the proposed algorithm is linear. An efficient data structure called a rule vector graph (RVG) is proposed to represent the global functionality of a cellular automaton (CA) by its rule vector (RV). The RVG of a null-boundary invertible CA preserves the specific characteristics that can be checked in linear time. These results are shown in the more general case of hybrid CAs. This paper also lists the elementary rules [3] that are invertible.

When–and how–can a cellular automaton be rewritten as a lattice gas?

Both cellular automata (CA) and lattice-gas automata (LG) provide finite algorithmic presentations for certain classes of infinite dynamical systems studied by symbolic dynamics; it is customary to use the terms ‘cellular automaton’ and ‘lattice gas’ for a dynamic system itself as well as for its presentation. The two kinds of presentation share many traits but also display profound differences on issues ranging from decidability to modeling convenience and physical implementability. Following a conjecture by Toffoli and Margolus, it had been proved by Kari that any invertible CA, at least up to two dimensions, can be rewritten as an isomorphic LG. But until now it was not known whether this is possible in general for noninvertible CA—which comprise “almost all” CA and represent the bulk of examples in theory and applications. Even circumstantial evidence–whether in favor or against–was lacking. Here, for noninvertible CA, (a) we prove that an LG presentation is out of the question for the vanishingly small class of surjective ones. We then turn our attention to all the rest–noninvertible and nonsurjective–which comprise all the typical ones, including Conway’s ‘Game of Life’. For these (b) we prove by explicit construction that all the one-dimensional ones are representable as LG, and (c) we present and motivate the conjecture that this result extends to any number of dimensions. The tradeoff between dissipation rate and structural complexity implied by the above results have compelling implications for the thermodynamics of computation at a microscopic scale.

Combinatorial constructions associated to the dynamics of one-sided cellular automata

In this paper we study combinatorial constructions which lead to one-sided invertible cellular automata with different dynamical behavior: equicontinuity, existence of equicontinuous points but not equicontinuity, sensitivity and expansivity. In particular, we provide a simple characterization of the class of equicontinuous invertible one-sided cellular automata and we construct families of expansive one-sided cellular automata.

Graph-theoretical characterization of invertible cellular automata

A cellular automaton is defined to be invertible if its induced global rule is surjective on the set of all finite configurations. It is shown that all one-dimensional invertible cellular automata are obtainable from invertible nearest-pair ones, i.e., cellular automata given by a rule matrix. For these, an algorithm is proposed, which runs in a time which is a polynomial function of the number of letters of the alphabet. The result is expressed in a graph-theoretical way: the cellular automaton is invertible if and only if two graphs constructed by the algorithm have no edges in common. It follows that if the number of letters in the alphabet is a prime, the only invertible cellular automata are those for which every row or column of the rule matrix is a permutation of the alphabet. For composite alphabet cardinalities, several methods (analytical and computational ones) to obtain more complicated invertible cellular automata are proposed and used to construct examples. All four-letter invertible automata are constructed.

HOW TO SIMULATE TURING MACHINES BY INVERTIBLE ONE-DIMENSIONAL CELLULAR AUTOMATA

The issue of testing invertibility of cellular automata has been often discussed. Constructing invertible automata is very useful for simulating invertible dynamical systems, based on local rules. The computation universality of cellular automata has long been positively resolved, and by showing that any cellular automaton could be simulated by an invertible one having a superior dimension, Toffoli proved that invertible cellular automaton of dimension d≥2 were computation-universal. Morita proved that any invertible Turing Machine could be simulated by a one-dimensional invertible cellular automaton, which proved computation-universality of invertible cellular automata. This article shows how to simulate any Turing Machine by an invertible cellular automaton with no loss of time and gives, as a corollary, an easier proof of this result.