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Syntactic Complexity of Bifix-Free Languages

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Implementation and Application of Automata (CIAA 2017)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 10329))

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Abstract

We study the properties of syntactic monoids of bifix-free regular languages. In particular, we solve an open problem concerning syntactic complexity: We prove that the cardinality of the syntactic semigroup of a bifix-free language with state complexity n is at most \((n-1)^{n-3}+(n-2)^{n-3}+(n-3)2^{n-3}\) for \(n \geqslant 6\). The main proof uses a large construction with the method of injective function. Since this bound is known to be reachable, and the values for \(n \leqslant 5\) are known, this completely settles the problem. We also prove that \((n-2)^{n-3} + (n-3)2^{n-3} - 1\) is the minimal size of the alphabet required to meet the bound for \(n \geqslant 6\). Finally, we show that the largest transition semigroups of minimal DFAs which recognize bifix-free languages are unique up to renaming the states.

M. Szykuła—Supported in part by the National Science Centre, Poland under project number 2014/15/B/ST6/00615.

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Authors and Affiliations

  1. Institute of Computer Science, University of Wrocław, Joliot-Curie 15, 50-383, Wrocław, Poland

    Marek Szykuła

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

    John Wittnebel

Authors
  1. Marek Szykuła
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  2. John Wittnebel
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Correspondence to Marek Szykuła .

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Editors and Affiliations

  1. LIGM (UMR 8049), CNRS, Université Paris-Est, Marne-la-Vallée Cedex 2, France

    Arnaud Carayol

  2. LIGM (UMR 8049), Université Paris-Est, Marne-la-Vallée Cedex 2, France

    Cyril Nicaud

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Szykuła, M., Wittnebel, J. (2017). Syntactic Complexity of Bifix-Free Languages. In: Carayol, A., Nicaud, C. (eds) Implementation and Application of Automata. CIAA 2017. Lecture Notes in Computer Science(), vol 10329. Springer, Cham. https://doi.org/10.1007/978-3-319-60134-2_17

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  • DOI: https://doi.org/10.1007/978-3-319-60134-2_17

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  • Online ISBN: 978-3-319-60134-2

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