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International Conference on Algebraic Informatics

CAI 2019: Algebraic Informatics pp 151–163Cite as

Commutative Regular Languages – Properties and State Complexity

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Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 11545))

Abstract

We consider the state complexity of intersection, union and the shuffle operation on commutative regular languages for arbitrary alphabets. Certain invariants will be introduced which generalize known notions from unary languages used for refined state complexity statements and existing notions for commutative languages used for the subclass of periodic languages. Our bound for shuffle is formulated in terms of these invariants and shown to be optimal, from this we derive the bound of \((2nm)^{|\varSigma |}\) for commutative languages of state complexities n and m respectively. This result is a considerable improvement over the general bound \(2^{mn-1} + 2^{(m-1)(n-1)}(2^{m-1} -1)(2^{n-1}-1)\).

We have no improvement for union and intersection for any alphabet, as was to be expected from the unary case. The general bounds are optimal here. Seeing commutative languages as generalizing unary languages is a guiding theme. For our results we take a closer look at a canonical automaton model for commutative languages.

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Notes

  1. 1.

    If not otherwise stated we assume that our alphabet has the form \(\varSigma = \{a_1, \ldots , a_k\}\) and k denotes the number of symbols.

  2. 2.

    These were introduced in [10] under the name of pure-group events.

  3. 3.

    It is not possible to give such an automaton for \(|\varSigma |\ge 1\), but allowing \(\varSigma =\emptyset \) the single-state automaton will do, or similar as \(\varSigma ^{*} = \{\varepsilon \}\) in this case.

  4. 4.

    Note that the notions of periodic and aperiodic languages appearing in this article are not meant to be related in dichotomous way.

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Acknowledgement

I thank my supervisor, Prof. Dr. Henning Fernau, for giving valuable feedback, discussions and research suggestions concerning the content of this article. I also thank the anonymous reviewers whose comments improved the presentation of this article. I also thank an anonymous reviewer of a previous version, whose feedback ultimately led to a new approach and stronger results.

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Authors and Affiliations

  1. Informatikwissenschaften, FB IV, Universität Trier, Universitätsring 15, 54296, Trier, Germany

    Stefan Hoffmann

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  1. Stefan Hoffmann
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Correspondence to Stefan Hoffmann .

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Editors and Affiliations

  1. University of Niš, Niš, Serbia

    Miroslav Ćirić

  2. University of Leipzig , Leipzig, Germany

    Manfred Droste

  3. Université Paris Denis Diderot and CNRS, Paris, France

    Jean-Éric Pin

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Hoffmann, S. (2019). Commutative Regular Languages – Properties and State Complexity. In: Ćirić, M., Droste, M., Pin, JÉ. (eds) Algebraic Informatics. CAI 2019. Lecture Notes in Computer Science(), vol 11545. Springer, Cham. https://doi.org/10.1007/978-3-030-21363-3_13

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  • DOI: https://doi.org/10.1007/978-3-030-21363-3_13

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-21362-6

  • Online ISBN: 978-3-030-21363-3

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