Abstract
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: the mappings which are computable by deterministic pushdown automata of level 2 the mappings which are solution of a system of catenative recurrence equations the mappings which are definable as a Lindenmayer system of type HDT0L. We illustrate the usefulness of this characterization by proving three statements about formal power series, rational sets of homomorphisms and equations in words.
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