Mathematical Basis of Cellular Automata, Introduction to 1819

Mathematical Basis of CellularAutomata, Introduction to

ANDREW ADAMATZKYUniversity of the West of England, Bristol, UK

A cellular automaton is a discrete universe with discretetime, discrete space and discrete states. Cells of the uni-verse are arranged into regular structures called lattices orarrays. Each cell takes a finite number of states and up-dates its states in a discrete time, depending on the statesof its neighbors, and all cells update their states in parallel.Cellular automata are mathematical models of massivelyparallel computing; computational models of spatially ex-tended non-linear physical, biological, chemical and socialsystems; and primary tools for studying large-scale com-plex systems.

Cellular automata are ubiquitous; they are objects oftheoretical study and also tools of applied modeling in sci-ence and engineering. Purely for ease of representation,one can roughly split articles in this section into threegroups: cellular automata theory, cellular automata mod-els of computation and cellular automata models of nat-ural phenomena. Many topics, however, belong to severalgroups.

We recommend that the reader begin with articleson history and modern analysis of classifying cellular au-tomata based on internal characteristics of their cell-statetransition functions, development of cellular automataconfigurations in space and time, and decidability of thecellular automata (see � Identification of Cellular Au-tomata). Studies in dynamical behavior are essential inprogressing cellular automata theory. They include topo-logical dynamics, for example, in relation to symbolic dy-namics, surjectivity, and permutations (see � Topologi-cal Dynamics of Cellular Automata); chaos, entropy anddecidability of cellular automata behavior (see � ChaoticBehavior of Cellular Automata), and insights into cellularautomata as dynamical systems with invariant measures(see � Ergodic Theory of Cellular Automata).

Self-reproducing patterns and gliders are amongst themost remarkable features of cellular automata. Certain cel-lular automata can reproduce configurations of cell-states,for example, the von Neumann universal constructor, andthus can be used in designs of self-replicating hardware(see � Self-Replication and Cellular Automata). Glidersare translating oscillators, or traveling patterns, of non-quiescent states, for example, gliders in Conway’s Game ofLife. Gliders are particularly fascinating in two- and three-dimensional spaces (see � Gliders in Cellular Automata).

Historically, an orthogonal lattice was the main sub-strate for cellular automata implementation. In the lastdecade the limit was lifted and nowadays you can findcellular automata on non-orthogonal lattices and tilings(see � Cellular Automata in Triangular, Pentagonaland Hexagonal Tessellations), non-Euclidean geometries,such as hyperbolic spaces (see � Cellular Automatain Hyperbolic Spaces) and various topological spaces(see � Dynamics of Cellular Automata in Non-compactSpaces).

Typically, a cell neighborhood is fixed during cellularautomaton development, and a cell updates its state de-pending on current states of its neighbors. But even inthis very basic setup, the space-time dynamics of cellu-lar automata are incredibly complex, as can be observedfrom analysis of the simplest one-dimensional automatawhere a transition rule applied to the sum of two statesis equal to the sum of its actions on the two states sepa-rately (see � Additive Cellular Automata). The automatadynamics becomes much richer if we allow the topologyof the cell neighborhood to be updated dynamically dur-ing automaton development (see � Structurally DynamicCellular Automata) or also allow a cell’s state to becomedependent on the cells’ previous states (see � Cellular Au-tomata with Memory). Talking about non-standard cell-transition rules, we must mention cellular automata withinjective global functions, where every configuration hasexactly one preceding configuration (see � Reversible Cel-lular Automata), and also cellular automata with quan-tum-bit cell-states and cell-transition functions incited byprinciples of quantum mechanics (see � Quantum Cellu-lar Automata).

The reader’s initial excursion into the theory of cellu-lar automata themselves can conclude in reading about de-cision problems of cellular automata expressed in terms offilling the plane using tiles with colored edges (see � TilingProblem and Undecidability in Cellular Automata) andabout algebraic properties of cellular automata transfor-mations, such as group representation of the Garden ofEden theorem and matrix representation of cellular au-tomata (see � Cellular Automata and Groups).

Firing squad synchronization is the oldest problem ofcellular automaton computation: all cells of a one-dimen-sional cellular automaton are quiescent apart from one cellin the firing state; we wish to design minimal cell-statetransition rules enabling all other cells to assume the fir-ing state at the same time (see � Firing Squad Synchro-nization Problem in Cellular Automata). Universality ofcellular automata is another classical issue. Two kinds ofuniversality are of most importance: computation univer-sality, that is, an ability to compute any computable func-

1820 Mathematical Basis of Cellular Automata, Introduction to

tion or implement a functionally complete logical system,and intrinsic, or simulation universality, such as an abil-ity to simulate any cellular automaton (see � Cellular Au-tomata, Universality of).

Readers can familiarize themselves with basics of spaceand time complexity cellular-automata parallel computingin (see � Cellular Automata as Models of Parallel Com-putation). The knowledge will then be extended by mea-sures of complexity, parallels between cellular automataand dynamics systems, Kolmogorov complexity of cellu-lar automata (see � Algorithmic Complexity and CellularAutomata) and studies of cellular automata as acceptors offormal languages (see � Cellular Automata and LanguageTheory). As demonstrated in (see � Evolving Cellular Au-tomata) cellular automata can be evolved to perform diffi-cult computational tasks.

Cellular automata models of natural systems such ascell differentiation, road traffic, reaction-diffusion, and ex-citable media (see � Cellular Automata Modeling of Phys-ical Systems) are ideal candidates for studying all im-portant phenomena of pattern growth (see � GrowthPhenomena in Cellular Automata); for studying trans-formation of a system’s state from one phase to another(see � Phase Transitions in Cellular Automata), and forstudying the ability of a system to be attracted to the stateswhere boundary between the system’s phases is indistin-guishable (see � Self-organized Criticality and CellularAutomata). Cellular automata models can be designed, inprinciple, by reconstructing cell-state transition rules ofcellular automata from snapshots of space-time dynamicsof the system we wish to simulate (see � Identification ofCellular Automata).