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Word-Mappings of Level 2

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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 Sénizergues 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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Notes

  1. More accurately: that fulfills a condition dual to that of Definition 3.

  2. Here too, a formal definition would distinguish the system itself from the families of mappings that fulfill the system.

  3. The original formulation was slightly different since the constant series 1 of our statement was replaced by any rational bounded series.

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Acknowledgements

This work has been partly supported by the project ANR 2010 BLAN 0202 01 FREC. We thank the referees for their helpful remarks.

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Authors and Affiliations

  1. LIF, Université d’Aix-Marseille 1, Marseille, France

    Julien Ferté

  2. LaBRI, Université Bordeaux 1, Bordeaux, France

    Nathalie Marin & Géraud Sénizergues

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  1. Julien Ferté
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  2. Nathalie Marin
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  3. Géraud Sénizergues
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Correspondence to Géraud Sénizergues.

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Ferté, J., Marin, N. & Sénizergues, G. Word-Mappings of Level 2. Theory Comput Syst 54, 111–148 (2014). https://doi.org/10.1007/s00224-013-9489-5

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