PERIODICITIES OF SOME ADDITIVE CELLULAR AUTOMATA

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...

Set of periods of additive cellular automata

It is shown that the set of periods of any additive cellular automata F, where the addition is done modulo a prime p, can be determined using some simple conditions on the coefficients in the linear expression of F. In particular, we establish that the set of periods has only four possibilities: {1,m} for some m where 1 m<p, N\{pm : m ∈ N}, N\{2pm : m ∈ N ∪ {0}} or the whole set N = {1, 2, 3, ....

Congruences, Automata and Periodicities

We show the deep connection between two apparently unrelated topics in automata theory and combinatorics on words: the equivalence problem of nite automata and the Fine and Wilf's periodicity theorem. In fact they both refer to the problem of checking whether two congruences agree on a nite set. This also allows us to introduce a notion of periodicity for labelled trees and to derive a periodic...

Additive Cellular Automata and Algebraic Series