Bounds on non-surjective cellular automata

Abstract

Cellular automata (CA) are discrete, homogeneous dynamical systems. Non-surjective one-dimensional CA have finite words with no preimage (called orphans), pairs of different words starting and ending identically and having the same image (diamonds) and words with more/ fewer preimages than the average number (unbalanced words). Using a linear algebra approach, we obtain new upper bounds on the lengths of the shortest such objects. In the case of an n-state, non-surjective CA with neighborhood range 2 our bounds are of the orders O(n 2), O(n 3/2) and O(n) for the shortest orphan, diamond and unbalanced word, respectively. © 2009 Springer Berlin Heidelberg.