A Max-Plus Model of Asynchronous Cellular Automata
Ebrahim L. Patel
Mathematical Institute, University of Oxford, Woodstock Road
Oxford OX2 6GG, UK
David Broomhead
School of Mathematics, University of Manchester, Oxford Road
Manchester M13 9PL, UK
Abstract
This paper presents a new framework for asynchrony. This has its origins in our attempts to better harness the internal decision-making process of cellular automata (CA). Thus, we show that a max-plus algebraic model of asynchrony arises naturally from the CA requirement that a cell receive the state of each neighbor before updating. The significant result is the existence of a bijective mapping between the asynchronous system and the synchronous system classically used to update CA. Consequently, although the CA outputs look qualitatively different, when surveyed on "contours" of real time, the asynchronous CA replicates the synchronous CA. Moreover, this type of asynchrony is simple—it is characterized by the underlying network structure of the cells, and long-term behavior is deterministic and periodic due to the linearity of max-plus algebra. The findings lead us to proffer max-plus algebra as: (i) a more accurate and efficient underlying timing mechanism for models of patterns seen in nature; and (ii) a foundation for promising extensions and applications.
