An Algebraic Characterization of Fuzzy Cellular Automata

It is well known that, in the case of a one dimensional two-neighbor situation, there are 256 fuzzy cellular automata obtained by the fuzzi-fication of the disjunctive normal form in classical (Wolfram) boolean cellular automata using classical aristotelian logic. Starting from a single polynomial in three variables we find its invariants under affine linear transformations and show that among these there are precisely eight equivalent polynomials whose span over Z(2) generates a vector space of dimension 8 over the field Z(2) containing 256 distinct elements. Its elements are, in fact, the transition (local) rules of the fuzzy cellular automata studied by previous authors. Our result allows for an alternate characterization of such fuzzy cellular automata and permits their generalization to arbitrary number of variables over general (finite or infinite) fields thus bypassing the need for a disjunctive normal form approach.