Construction of local structure maps for cellular automata

Abstract

The paper formalizes and extends the idea of local structure approximation for cellular automata originally proposed by Gutowitz et. al. We start with a review of the construction of a probability measure on the set of bi-infinite strings over a finite alphabet of $N$ symbols. We then demonstrate that for a shift-invariant probability measure, probabilities of all blocks of length up to $k$ can be expressed by $(N-1)N^{k-1}$ linearly independent block probabilities. Two choices of these independent blocks are discussed in detail, one in which we choose the longest possible blocks ("long form") and one in which we choose the shortest possible blocks ("short form"). We then proceed to review the method which allows to approximate probabilities of blocks longer than $k$ by blocks of length $k$ or less. This approximation, known as Bayesian extension or Markov measure, is then used to construct approximate orbits of shift-invariant probability measures under the action of probabilistic or deterministic cellular automaton. We show that the aforementioned approximate orbit is completely determined by an $(N-1)N^{k-1}$-dimensional map. When the short form of block probabilities is used, this map takes particularly simple form, often revealing important features of a particular cellular automaton.