Completeness and Degeneracy in Information Dynamics of Cellular Automata

Abstract

This paper addresses an algebraic problem which arises from our study on the information dynamics of cellular automata (CA). The state set of a cell is assumed to be a polynomial ring Q[X] modulo X ^q–X over a finite field GF(q), where X is the indeterminate called the information variable. When a CA starts with an initial configuration containing a cell with state X, the information of X is transmitted to neighboring cells by cellular computation. In such a computation, every cell of a global configuration takes a polynomial in Q[X]. Generally denote such a configuration by c _X and let G _cX be the set of polynomials appearing in c _X . Our problem is to ask how much information of X is contained by G _cX . For G _cX we define the degree of completenessλ(G _cX ) = log _q |〈G _cX 〉|, where 〈G _cX 〉 is the subring of Q[X] generated by G _cX and investigate its relation to the degree of degeneracym(c _X ) introduced before. We note here that m(cX) = q − |V(G _cX )|, where |V(G _cX )| is the cardinality of the value set of G _cX . Then, we prove that λ(G _cX ) and in turn that λ(G _cX ) + m(cX) = q. This result suggests that the computation of the size of subrings is reduced to that of the value size, which is executed much easier than subring generation.