Commutativity of Composition of some Elementary Cellular Automata with Essentially 2-Neighborhood Local Functions over Monoids

Abstract

The local function of a cellular automaton with binary states can be expressed by a formula in propositional logic. The inverse function of a local function of any reversible cellular automation can also be expressed as a propositional logic formula, and using it as a local function then, we can define the cellular automation. The multiplication of these formulae in propositional logic is defined using the action of the cell space as a dynamical system and yields the local function of the composition of two cellular automata.In this study, we deal with logical formulae on a commutative monoid as local functions of elementary cellular automata. We focus on essentially 2-neighborhood local functions and the logical symbol "Implication" on commutative monoids. We discuss the commutativity of multiplication of the formulae and show some conditions for formulae to satisfy the commutativity of the composition of elementary cellular automata.