Communication Complexity and Intrinsic Universality in Cellular Automata

The notions of universality and completeness are central in the theories of computation and computational complexity. However, proving lower bounds and necessary conditions remains hard in most of the cases. In this article, we introduce necessary conditions for a cellular automaton to be "universal", according to a precise notion of simulation, related both to the dynamics of cellular automata and to their computational power. This notion of simulation relies on simple operations of space-time rescaling and it is intrinsic to the model of cellular automata. Intrinsinc universality, the derived notion, is stronger than Turing universality, but more uniform, and easier to define and study. Our approach builds upon the notion of communication complexity, which was primarily designed to study parallel programs, and thus is, as we show in this article, particulary well suited to the study of cellular automata: it allowed to show, by studying natural problems on the dynamics of cellular automata, that several classes of cellular automata, as well as many natural (elementary) examples, could not be intrinsically universal

Bulking II: Classifications of Cellular Automata

This paper is the second part of a series of two papers dealing with bulking: a way to define quasi-order on cellular automata by comparing space-time diagrams up to rescaling. In the present paper, we introduce three notions of simulation between cellular automata and study the quasi-order structures induced by these simulation relations on the whole set of cellular automata. Various aspects of these quasi-orders are considered (induced equivalence relations, maximum elements, induced orders, etc) providing several formal tools allowing to classify cellular automata

Algebraic Methods for Finite Linear Cellular Automata

PhDCellular automata are a simple class of extended dynamical systems which have been much studied in recent years. Linear cellular automata are the class of cellular automata most amenable to algebraic analytic treatments, algebraic techniques are used to study finite linear cellular automata and also finite linear cellular automata with external inputs. General results are developed for state alphabet a finite commutative ring and a notion of qualitative dynamical similarity is introduced for those systems consisting of a fixed linear cellular automata rule but with distinct time independent inputs. Sufficient conditions for qualitative dynamical similarity are obtained in the general case. Exact results are obtained for the case of state alphabet a finite field, including new results for finite linear cellular automata without inputs and a complete description of the behaviour of the corresponding system with time independent inputs. Necessary and sufficient conditions for qualitative dynamical similarity in this case are given. Results for the hitherto untreated case of state alphabet the integers modulo pk, p prime and k>1, are obtained from those for the finite field case by the technique of idempotent lifting. These two cases suffice for the treatment of the general case of st, ),t e alphabet the integers modulo any positive integer m>1, in particular a necessary and sufficient condition for qualitatively similar dynamics in the presence of time independent inputs is given for this case. The extension of the results for time independent inputs to the case of periodic and eventually periodic inputs is treated and the generalisation of the techniques developed to higher dimensional linear cellular automata is discussed

An Experimental Study of Robustness to Asynchronism for Elementary Cellular Automata

Cellular Automata (CA) are a class of discrete dynamical systems that have been widely used to model complex systems in which the dynamics is specified at local cell-scale. Classically, CA are run on a regular lattice and with perfect synchronicity. However, these two assumptions have little chance to truthfully represent what happens at the microscopic scale for physical, biological or social systems. One may thus wonder whether CA do keep their behavior when submitted to small perturbations of synchronicity. This work focuses on the study of one-dimensional (1D) asynchronous CA with two states and nearest-neighbors. We define what we mean by ``the behavior of CA is robust to asynchronism'' using a statistical approach with macroscopic parameters. and we present an experimental protocol aimed at finding which are the robust 1D elementary CA. To conclude, we examine how the results exposed can be used as a guideline for the research of suitable models according to robustness criteria.Comment: Version : Feb 13th, 2004, submitted to Complex System

Multiband linear cellular automata and endomorphisms of algebraic vector groups

We propose a correspondence between certain multiband linear cellular automata - models of computation widely used in the description of physical phenomena - and endomorphisms of certain algebraic unipotent groups over finite fields. The correspondence is based on the construction of a universal element specialising to a normal generator for any finite field. We use this correspondence to deduce new results concerning the temporal dynamics of such automata, using our prior, purely algebraic, study of the endomorphism ring of vector groups. These produce 'for free' a formula for the number of fixed points of the $n$-iterate in terms of the $p$-adic valuation of $n$, a dichotomy for the Artin-Mazur dynamical zeta function, and an asymptotic formula for the number of periodic orbits. Since multiband linear cellular automata simulate higher order linear automata (in which states depend on finitely many prior temporal states, not just the direct predecessor), the results apply equally well to that class.Comment: 11 page

A Survey of Cellular Automata: Types, Dynamics, Non-uniformity and Applications

Cellular automata (CAs) are dynamical systems which exhibit complex global behavior from simple local interaction and computation. Since the inception of cellular automaton (CA) by von Neumann in 1950s, it has attracted the attention of several researchers over various backgrounds and fields for modelling different physical, natural as well as real-life phenomena. Classically, CAs are uniform. However, non-uniformity has also been introduced in update pattern, lattice structure, neighborhood dependency and local rule. In this survey, we tour to the various types of CAs introduced till date, the different characterization tools, the global behaviors of CAs, like universality, reversibility, dynamics etc. Special attention is given to non-uniformity in CAs and especially to non-uniform elementary CAs, which have been very useful in solving several real-life problems.Comment: 43 pages; Under review in Natural Computin

Convolution equations on lattices: periodic solutions with values in a prime characteristic field

These notes are inspired by the theory of cellular automata. A linear cellular automaton on a lattice of finite rank or on a toric grid is a discrete dinamical system generated by a convolution operator with kernel concentrated in the nearest neighborhood of the origin. In the present paper we deal with general convolution operators. We propose an approach via harmonic analysis which works over a field of positive characteristic. It occurs that a standard spectral problem for a convolution operator is equivalent to counting points on an associate algebraic hypersurface in a torus according to the torsion orders of their coordinates.Comment: 30 pages, a new editio

On Sloane's persistence problem

We investigate the so-called persistence problem of Sloane, exploiting connections with the dynamics of circle maps and the ergodic theory of $\mathbb{Z}^d$ actions. We also formulate a conjecture concerning the asymptotic distribution of digits in long products of finitely many primes whose truth would, in particular, solve the persistence problem. The heuristics that we propose to complement our numerical studies can be thought in terms of a simple model in statistical mechanics.Comment: 5 figure