BOOLEAN DERIVATIVES AND COMPUTATION OF CELLULAR AUTOMATA

Abstract

The derivatives of a Boolean function are defined up to any order. The Taylor and MacLaurin expansions of a Boolean function are thus obtained. The last corresponds to the ring sum expansion (RSE) of a Boolean function, and is a more compact form than the usual canonical disjunctive form. For totalistic functions the RSE allows the saving of a large number of Boolean operations. The algorithm has natural applications to the simulations of cellular automata using the multi-site coding technique. Several already published algorithms are analyzed, and expressions with fewer terms are generally found.