Boolean derivatives and computation of cellular automata

TL;DR

This paper introduces Boolean derivatives and their use in computing cellular automata, providing more efficient algorithms and compact representations for Boolean functions, especially in totalistic cases.

Contribution

It develops a framework for Boolean derivatives, Taylor and MacLaurin expansions, and applies these to improve cellular automata simulation algorithms.

Findings

RSE provides a more compact Boolean function representation.

Algorithms with fewer terms are identified for cellular automata.

Applications to multi-site coding techniques improve simulation efficiency.

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 analized, and expressions with fewer terms are generally found.