The Representation Role for Basic Operations Embodied in Cellular Automata: A Suitability Example for Addition in Redundant Numeral Systems vs Conventional Ones

Abstract

Cellular Automata (CA) are both a parallel computational paradigm and an archetype for modelling complex systems, that evolve on the basis of local interactions. CA can embody different numeral representations and perform related basic arithmetical operations. However, conventional numeral representations are thought as intrinsically sequential in such operations, which implies that CA parallelism is underexploited when CA evolution mimics the sequentiality of calculation, while some redundant numeral representations could exalt the CA parallelism in a space/time trade-off, where the time complexity of some operations is constant on input length. The problem then arises when the result of an operation must be utilized in the conventional representation since, usually, the migration toward an advantageous redundant numeric representation is costless, but the inverse one implies necessarily a cost that cancels the benefits in terms of computation time. This paper explores the properties of the conventional binary positional representation embodied in a CA together with the addition operation and the corresponding ones of a redundant binary positional representation, the rules and time cost for the passage from conventional numeral system to redundant one and vice versa. The results permit to individuate the CA computation context, when redundancy could be exploited advantageously. It regards cases where a longest sequence of additions (or operations based on addition, e.g., fast Fourier transforms) has to be performed in well-defined short times as for the automatic control of mobile devices.