Class Of Cellular Automata Research Articles

Cellular automaton fluids — A review

This paper is basically a review of cellular automaton fluids, which are the class of cellular automata used for describing fluids. Cellular automaton fluids are discrete analogues of molecular dynamics in which the particles have discrete velocities and move on the sites of a lattice according to some rule of evolution. We restrict ourselves mainly to two-dimensional fluids, but make some comments regarding models for fluids in three dimensions. Analytical as well as numerical simulation results, including ours on the wake behind a cylinder, are discussed for the two-dimensional cellular automaton (ca) models. We also discuss briefly some issues which need resolution before theca models can be used for practical simulations.

Local structure theory for cellular automata

The mean-field theory for cellular automata (Wolfram, and Schulman and Seiden) is generalized to the local structure theory. The local structure theory is a sequence of finitely-parameterized models of the statistical features of a cellular automaton's evolution. The nth model in the sequence takes into account correlations in terms of the probability of blocks of n states. A class of measures, the n-block measures, is introduced.The local structure operator of order n maps n-block measures to n-block measures in a manner which reflects the cellular automaton map on blocks of states. The fixed points of the map on measures approximate the invariant measures of the cellular automaton. The ability of the local structure theory to model evolution from uncorrelated initial distributions is studied. The theory gives exact results in simple cases. In more complex cases, Monte Carlo numerical experiments suggest that an accurate statistical portrait of cellular automaton evolution is obtained. The invariant measures of a cellular automaton and the stability of these measures may be obtained from the local structure theory. The local structure theory appears to be a powerful method for characterization and classification of cellular automata. Nearest neighbor cellular automata with two states per cell are studied using this method.

Global properties of cellular automata

Cellular automata are discrete mathematical systems that generate diverse, often complicated, behavior using simple deterministic rules. Analysis of the local structure of these rules makes possible a description of the global properties of the associated automata. A class of cellular automata that generate infinitely many aperiodic temporal sequences is defined, as is the set of rules for which inverses exist. Necessary and sufficient conditions are derived characterizing the classes of “nearest-neighbor” rules for which arbitrary finite initial conditions (i) evolve to a homogeneous state; (ii) generate at least one constant temporal sequence.

Computation theory of cellular automata

Self-organizing behaviour in cellular automata is discussed as a computational process. Formal language theory is used to extend dynamical systems theory descriptions of cellular automata. The sets of configurations generated after a finite number of time steps of cellular automaton evolution are shown to form regular languages. Many examples are given. The sizes of the minimal grammars for these languages provide measures of the complexities of the sets. This complexity is usually found to be non-decreasing with time. The limit sets generated by some classes of cellular automata correspond to regular languages. For other classes of cellular automata they appear to correspond to more complicated languages. Many properties of these sets are then formally non-computable. It is suggested that such undecidability is common in these and other dynamical systems.

Triangle cellular automata

This paper introduces a special class of cellular automata, called triangle cellular automata, which accept as input strings of length a power of two. In particular, we study the special case in which state information can only move upward through a complete binary tree of finite-state automata. It is shown that this class of cellular automata can accept various string languages in O(log n) time, where n is the length of the input string defining the initial states of the leaf vertices in the tree. Extensions to two dimensions, defining a pyramid cellular automaton, are also given.