Information Dynamics of Cellular Automata I -An Algebraic Study

Information dynamics of cellular automata(CA) is studied using polynomials over finite fields. The information about the uncertainty of cell states is expressed by an indeterminate X called information variable and its dynamics is investigated by extending CA to CA[X] whose cell states are polynomials in X. For the global configuration of extended CA[X], new notions of completeness and degeneracy are defined and their dynamical properties are investigated. A theorem is proved that completeness equals non-degeneracy. With respect to the reversibility, we prove that a CA is reversible, if and only if its extension CA[X] preserves the set of complete configurations. Information dynamics of finite CAs and linear CAs are treated in the separate sections. Decision problems are also referred.