q-VFCA: q-state Vector-valued Fuzzy Cellular Automata
Yuki Nishida, Sennosuke Watanabe, Akiko Fukuda and Yoshihide Watanabe
Elementary fuzzy Cellular Automata (CA) are known as continuous counterpart of elementary CA, which are 2-state CA, via the polynomial representation of local rules. In this paper, we first develop a new fuzzification methodology for q-state CA. It is based on the vector representation of q-state CA, that is, the q-states are assigned to the standard basis vectors of the q-dimensional real space and the local rule can be expressed by a tuple of q polynomials. Then, the q-state vector-valued fuzzy CA are defined by expanding the set of the states to the convex hull of the standard basis vectors in the q-dimensional real space. The vector representation of states enables us to enumerate the number-conserving rules of 3-state vector-valued fuzzy CA in a systematic way.
Keywords: Cellular automata, fuzzy cellular automata, vector-valued cellular automata, conservation law, number-conserving rule, periodic boundary condition