Cellular automata on irregular tessellations

Abstract

To this day, most articles on cellular automata (CA) employ regular tessellations of ℝ n , notwithstanding this kind of tessellation suffers from a few serious drawbacks limiting the appraisal of regular CA as full-fledged modelling tools. In this article, we propose an extension to the classical CA paradigm, by allowing irregular tessellations of ℝ n , which, in essence, encloses the regular tessellations as a special case. After a short review on the major shortcomings of CA on regular tessellations, we establish a definition for CA on irregular tessellations of ℝ n that is followed by a profound elaboration of its constituents. In addition, we propose a graph representation that is applicable to an important family of irregular CA, and, in accordance with the framework developed for CA on regular tessellations, we present an enumeration scheme for the k-state, θ-sum irregular (outer-)totalistic CA. Illustrative examples conclude this article.