Iterated GSM Mappings: A Collapsing Hierarchy

With motivations from various areas (Lindenmayer systems, iterated reading of literary works, self-generated in nite sequences, \computing by carving" as suggested in the DNA computing area), in several places mechanisms based on iterated (non-deterministic) nite state sequential transducers were considered. It is known that such mechanisms can characterize the family of recursively enumerable languages. We continue here the study of such devices, investigating the hierarchy on the number of states. We nd that this hierarchy collapses: four states are enough in order to characterize the recursively enumerable languages, three states lead to non-recursive languages and cover the ET0L languages, while two states can cover the E0L (hence also context-free) languages. The case of deterministic transducers remains open.