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The Algebraic Theory of Parikh Automata

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Abstract

The Parikh automaton model equips a finite automaton with integer registers and imposes a semilinear constraint on the set of their final settings. Here the theories of typed monoids and of rational series are used to characterize the language classes that arise algebraically. Complexity bounds are derived, such as containment of the unambiguous Parikh automata languages in NC1. Affine Parikh automata, where each transition applies an affine transformation on the registers, are also considered. Relying on these characterizations, the landscape of relationships and closure properties of the classes at hand is completed, in particular over unary languages.

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Notes

  1. The restriction that N is finite is needed to preserve the property that the monoids at hand are finitely generated. It is possible to define a sensible product of two infinite typed monoids, see [25]. We will however only need this particular case.

  2. A full trio or cone is a class of languages closed under morphisms, inverse morphisms,and intersection with the regular languages.

  3. A wreath product with an infinite typed monoid on the right results in an uncountablemonoid, an undesirable property that was circumvented in [25]. We note that Theorem 4stays true with a definition of wreath product mimicking that of [25].

  4. A language L is bounded if L ⊆ w1∗w2∗⋯w k∗ for some words w 1,w 2,…,w k .

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Acknowledgements

We thank Michael Blondin and Charles Paperman for comments on early versions of this article. Part of this work was done during the Dagstuhl Seminar 15401 “Circuits, Logic and Games.”

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Authors and Affiliations

  1. Wilhelm-Schickard-Institut, Universität Tübingen, Sand 13, 72070, Tübingen, Germany

    Michaël Cadilhac & Andreas Krebs

  2. Département d’informatique et de recherche opérationnelle, Université de Montréal, CP 6128 succursale Centre-Ville, Montréal, QC, H3C 3J7, Canada

    Pierre McKenzie

Authors
  1. Michaël Cadilhac
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  2. Andreas Krebs
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  3. Pierre McKenzie
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Correspondence to Michaël Cadilhac.

Additional information

Extended version of the paper with the same title appearing in the proceedings of the 5th Conference on Algebraic Informatics (CAI’13), LNCS vol. 8080, pp. 60–73, Springer-Verlag (2013). The main changes in contents are as follows: Section 3 contains anew Chomsky-Schützenberger-like characterization of \(\protect \mathcal {L}_{\protect \text {CA}}\); Section 4 additionally shows that \(\protect \mathcal {L}_{\protect \text {DetAPA}} = \protect \mathcal {L}_{\protect \text {UnAPA}}\); Section 5 presents an expressiveness lemma that allows, in Section 6, to show separation of \(\protect \mathcal {L}_{\protect \text {DetAPA}}\) and \(\protect \mathcal {L}_{\protect \text {APA}}\), and new nonclosure results; moreover, most proofs were rewritten for added uniformity.

The first two authors are supported by the DFG Emmy Noether program (KR 4042/2). The third author is supported by the Natural Sciences and Engineering Research Council of Canada and by the “Chaire Digiteo, ENS Cachan -École Polytechnique.”

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Cadilhac, M., Krebs, A. & McKenzie, P. The Algebraic Theory of Parikh Automata. Theory Comput Syst 62, 1241–1268 (2018). https://doi.org/10.1007/s00224-017-9817-2

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