Additive cellular automata and algebraic series

Additive Cellular Automata and Algebraic Series

Theory of Additive Cellular Automata

This paper reports the complete characterization of additive cellular automata (ACA) that employ xor and xnor logic as the next state function. Compared to linear cellular automata (LCA) [3], which employs only xor logic in its next state function, an ACA display much wider varieties of state transition behavior and enhanced computing power. An analytical framework is developed to characterize ...

Decomposition of Additive Cellular Automata

Finite additive cellular automata with fixed and periodic boundary conditions are considered as endomorphisms over pattern spaces. A characterization of the nilpotent and regular parts of these endomorphisms is given in terms of their minimal polynomials. Generalized eigenspace decomposition is determined and relevant cyclic subspaces are described in terms of symmetries. As an application, the...

Additive Cellular Automata and Volume Growth

A class of dynamical systems associated to rings of S–integers in rational function fields is described. General results about these systems give a rather complete description of the well–known dynamics in one–dimensional additive cellular automata with prime alphabet, including simple formulæ for the topological entropy and the number of periodic configurations. For these systems the periodic ...

Cellular Automata as Algebraic Systems

Infinite cellular automata have been studied mostly using empirical and sta tistical techniques, with some combinatorial analysis. Here we show how concepts of universal algebra such as subdirect decomposition and chains of varieties can be applied to their study. Cellular automata with ultimately periodic behavior are shown to correspond to varieties of groupoids. Relat ionships between these ...