Non-Archimedean Valued and p-Adic Valued Fuzzy Cellular Automata
The negation of Archimedes’ axiom allows to postulate infinitesimals and infinitely large integers and, as a result, to consider non-wellfounded phenomena. In this paper I introduce the non-Archimedean and p-adic logical multiple-validities in the framework of logical cellular automata. Notice that in logical cellular automata the set of states can be considered as the set of truth values and the local transition function as a many-valued logical function. The non-Archimedean and p-adic multiple-validities are regarded as a validity that runs either the unit interval [0 1 ] of hypernumbers or the ring of p-adic integers. Using non-Archimedean and p-adic valued logics, we can describe the evolution of a logical cellular automaton. Non-Archimedean fuzzy cellular automata are defined as infinite Cartesian products of fuzzy cellular automata and p-adic valued cellular automata are defined as infinite Cartesian products of p-valued cellular automata.