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Most Complex Deterministic Union-Free Regular Languages

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Descriptional Complexity of Formal Systems (DCFS 2018)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 10952))

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Abstract

A regular language L is union-free if it can be represented by a regular expression without the union operation. A union-free language is deterministic if it can be accepted by a deterministic one-cycle-free-path finite automaton; this is an automaton which has one final state and exactly one cycle-free path from any state to the final state. Jirásková and Masopust proved that the state complexities of the basic operations reversal, star, product, and boolean operations in deterministic union-free languages are exactly the same as those in the class of all regular languages. To prove that the bounds are met they used five types of automata, involving eight types of transformations of the set of states of the automata. We show that for each \(n \geqslant 3\) there exists one ternary witness of state complexity n that meets the bound for reversal and product. Moreover, the restrictions of this witness to binary alphabets meet the bounds for star and boolean operations. We also show that the tight upper bounds on the state complexity of binary operations that take arguments over different alphabets are the same as those for arbitrary regular languages. Furthermore, we prove that the maximal syntactic semigroup of a union-free language has \(n^n\) elements, as in the case of regular languages, and that the maximal state complexities of atoms of union-free languages are the same as those for regular languages. Finally, we prove that there exists a most complex union-free language that meets the bounds for all these complexity measures. Altogether this proves that the complexity measures above cannot distinguish union-free languages from regular languages.

This work was supported by the Natural Sciences and Engineering Research Council of Canada grant No. OGP0000871.

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Notes

  1. 1.

    Terminology changed to that of the present paper.

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Authors and Affiliations

  1. David R. Cheriton School of Computer Science, University of Waterloo, Waterloo, ON, N2L 3G1, Canada

    Janusz A. Brzozowski

  2. Department of Pure Mathematics, University of Waterloo, Waterloo, ON, N2L 3G1, Canada

    Sylvie Davies

Authors
  1. Janusz A. Brzozowski
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  2. Sylvie Davies
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Correspondence to Sylvie Davies .

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Editors and Affiliations

  1. Department of Mathematics and Computing Science, Saint Mary’s University, Halifax, Nova Scotia, Canada

    Stavros Konstantinidis

  2. Dipartimento di Informatica e Comunicazi, Universita degli Studi di Milano, Milan, Italy

    Giovanni Pighizzini

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Brzozowski, J.A., Davies, S. (2018). Most Complex Deterministic Union-Free Regular Languages. In: Konstantinidis, S., Pighizzini, G. (eds) Descriptional Complexity of Formal Systems. DCFS 2018. Lecture Notes in Computer Science(), vol 10952. Springer, Cham. https://doi.org/10.1007/978-3-319-94631-3_4

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  • DOI: https://doi.org/10.1007/978-3-319-94631-3_4

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  • Print ISBN: 978-3-319-94630-6

  • Online ISBN: 978-3-319-94631-3

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