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Modifying the Upper Bound on the Length of Minimal Synchronizing Word

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Book cover Fundamentals of Computation Theory (FCT 2011)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 6914))

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Abstract

A word w is called synchronizing (recurrent, reset, magic, directable) word of deterministic finite automaton (DFA) if w sends all states of the automaton to a unique state. In 1964 Jan Černy found a sequence of n-state complete DFA possessing a minimal synchronizing word of length (n − 1)2. He conjectured that it is an upper bound on the length of such words for complete DFA. Nevertheless, the best upper bound (n 3 − n)/6 was found almost 30 years ago.

We reduce the upper bound on the length of the minimal synchronizing word to n(7n 2 + 6n − 16)/48.

An implemented algorithm for finding synchronizing word with restricted upper bound is described. The work presents the distribution of all synchronizing automata of small size according to the length of an almost minimal synchronizing word.

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Authors and Affiliations

  1. Dep. of Math., Bar-Ilan University, 52900, Ramat Gan, Israel

    A. N. Trahtman

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  1. A. N. Trahtman
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Editors and Affiliations

  1. Department of Informatics, University of Oslo, Postboks 1080 Blindern, 0316, Oslo, Norway

    Olaf Owe  & Martin Steffen  & 

  2. Department of Informatics, University of Bergen, Postboks 7800, 5020, Bergen, Norway

    Jan Arne Telle

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Trahtman, A.N. (2011). Modifying the Upper Bound on the Length of Minimal Synchronizing Word. In: Owe, O., Steffen, M., Telle, J.A. (eds) Fundamentals of Computation Theory. FCT 2011. Lecture Notes in Computer Science, vol 6914. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-22953-4_15

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  • DOI: https://doi.org/10.1007/978-3-642-22953-4_15

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-22952-7

  • Online ISBN: 978-3-642-22953-4

  • eBook Packages: Computer ScienceComputer Science (R0)

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