Formal Languages and Finite Cellular Automata

A bst ract. A one-dimensional cellular automaton rule with specified boundary conditions can be considered as acting simultaneously on all finite lattices, which gives a mapping between formal languages. Regular lang uages are always mapped to regular langu ages, contex t-free to context-free, context-sensitive to context-sensitive, and recursive sets to recursive sets . In particular, the finite time sets on finite lattices are regular languages. The limit set on finite lattices (t he periodic set) is shown to be neither a regular nor an unambiguous context-free language for certain additive rules with chaotic behavior, and for rules that can simulate one of these additive rules t hrough a finit e blocking transformation. The relation between cellular automata on finite and infinite lattices is discussed .