Finite fields
Information dynamics of cellular automata I: an algebraic study
Fundamenta Informaticae - Special issue on cellular automata
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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 Xq–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 cX and let GcX be the set of polynomials appearing in cX. Our problem is to ask how much information of X is contained by GcX. For GcX we define the degree of completenessλ(GcX) = logq|〈GcX 〉|, where 〈GcX 〉 is the subring of Q[X] generated by GcX and investigate its relation to the degree of degeneracym(cX) introduced before. We note here that m(cX)=q−|V(GcX)|, where |V(GcX)| is the cardinality of the value set of GcX. Then, we prove that λ(GcX) and in turn that λ(GcX)+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.