Some Applications of Information Theory to Cellular Automata

In this paper general deterministic one-dimensional cellular automata are identified with mappings of the unit interval into itself. This allows the machinery of dynamical systems analysis to be employed. The emphasis of the paper, however, is on applications of existing concepts and techniques of information theory to these automata. A basic paper by W.M. Conner is utilized to obtain equality of the capacity and HausdorlT dimension of each line of the automata, and existence of limiting values of these quantities is established. Assuming a probability measure on the initial line that is stationary and ergodic for the shift, a consistent ergodic theory is derived for any finite or infinite collection of lines of the automata. A body of related work by Russian authors on probabilistic automata is briefly examined. Important questions about the existence and properties of limiting distributions remain unresolved.