Word-Mappings of Level 2

摘要

A sequence of natural numbers is said to have level k, for some natural integer k, if it can be computed by a deterministic pushdown automaton of level k (Fratani and Senizergues in Ann Pure Appl. Log. 141:363-411,2006). We show here that the sequences of level 2 are exactly the rational formal power series over one undeterminate. More generally, we study mappings from words to words and show that the following classes coincide: 1. the mappings which are computable by deterministic pushdown automata of level 2 2. the mappings which are solution of a system of catenative recurrence equations 3. the mappings which are definable as a Lindenmayer system of type HDTOL. We illustrate the usefulness of this characterization by proving three statements about formal power series, rational sets of homomorphisms and equations in words.