Bounds on Non-surjective Cellular Automata

Abstract. Cellular automata (CA) are discrete, homogeneous dynamical systems. Non-surjective one-dimensional CA have nite words with no preimage (called orphans), pairs of dierent 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(n2 ), O(n3/2 ) and O(n) for the shortest orphan, diamond and unbalanced word, respectively.

• Type: Conference
• Year: 2009
• Where: MFCS
• Authors: Jarkko Kari, Pascal Vanier, Thomas Zeume
• Keywords: Homogeneous Dynamical Systems, MFCS 2009, Non-surjective One-dimensional Ca, Theoretical Computer Science, Unbalanced Word