Local structure theory for cellular automata

Abstract

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.