Improvement of a Result on Sequencing Elementary Cellular Automata Rules for Solving the Parity Problem

The understanding of how simple predefined computations can be achieved with cellular automata, either through individual rules, or with rules organised in spatial arrangements or in temporal sequences is a key conceptual underpinning in the general notion of emergent computation. In this context, the parity problem for cellular automata is considered here, through which the number of 1-bits in a binary string should be determined to be even or odd. Although it is known that no individual rule can solve this problem, a solution has been presented in the literature that is able to solve it, for any string length, by means of the sequential chaining of elementary cellular automata rules. Drawing on that solution, here we provide others, significantly simplified from the latter, our approach being based upon analyses of the behaviour of the rules used in the original work.