Skip to main content
SpringerLink
Log in

On the Probability of Being Synchronizable

  • Conference paper
Algorithms and Discrete Applied Mathematics (CALDAM 2016)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 9602))

Included in the following conference series:

Abstract

We prove that a random automaton with n states and any fixed non-singleton alphabet is synchronizing with high probability. Moreover, we also prove that the convergence rate is exactly \(1-\varTheta (\frac{1}{n})\) as conjectured by Cameron [4] for the most interesting binary alphabet case.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Notes

  1. 1.

    Here and below by independence of two objects \(O_1(\mathcal {A})\) and \(O_2(\mathcal {A})\), we mean the independence of the events \(O_1(\mathcal {A}) = O_1\) and \(O_1(\mathcal {A}) = O_2\) for each instances \(O_1, O_2\) from the corresponding probability spaces.

  2. 2.

    The reason why we consider 1-branches instead of trees is that the state r would not be completely defined by the unique highest tree of a.

References

  1. Araújo, J., Bentz, W., Cameron, P.: Groups synchronizing a transformation of non-uniform kernel. Theor. Comp. Sci. 498, 1–9 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  2. Berlinkov, M.: On the probability of being synchronizable (2013). arXiv:1304.5774

  3. Berlinkov, M.: Highest trees of random mappings (2015). arXiv:1504.04532

  4. Cameron, P.J.: Dixon’s theorem and random synchronization. arXiv:1108.3958

  5. Carayol, A., Nicaud, C.: Distribution of the number of accessible states in a random deterministic automaton. In: Leibniz International Proceedings in Informatics (LIPIcs), STACS 2012, vol. 14, pp. 194–205 (2012)

    Google Scholar 

  6. Černý, J.: Poznámka k homogénnym eksperimentom s konečnými automatami. Matematicko-fyzikalny Časopis Slovensk. Akad. Vied 14(3), 208–216 (1964) (in Slovak)

    Google Scholar 

  7. Kari, J.: Synchronization and stability of finite automata. J. Univers. Comp. Sci. 2, 270–277 (2002)

    MathSciNet  MATH  Google Scholar 

  8. Nicaud, C.: Fast synchronization of random automata (2014). arXiv:1404.6962

  9. Skvortsov, E., Zaks, Y.: Synchronizing random automata. Discrete Math. Theor. Comput. Sci. 12(4), 95–108 (2010)

    MathSciNet  MATH  Google Scholar 

  10. Skvortsov, E., Zaks, Y.: Synchronizing random automata on a 4-letter alphabet. J. Math. Sci. 192, 303–306 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  11. Timashov, A.: Asymptotic expansions for the distribution of the number of components in random mappings and partitions. Discrete Math. Appl. 21(3), 291–301 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  12. Trahtman, A.: The road coloring problem. Israel J. Math. 172(1), 51–60 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  13. Volkov, M.V.: Synchronizing automata and the Černý conjecture. In: Martín-Vide, C., Otto, F., Fernau, H. (eds.) LATA 2008. LNCS, vol. 5196, pp. 11–27. Springer, Heidelberg (2008)

    Chapter  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institute of Mathematics and Computer Science, Ural Federal University, 620000, Ekaterinburg, Russia

    Mikhail V. Berlinkov

Authors
  1. Mikhail V. Berlinkov
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Mikhail V. Berlinkov .

Editor information

Editors and Affiliations

  1. Indian Institute of Science, Bangalore, India

    Sathish Govindarajan

  2. Carleton University, Ottawa, ON, Canada

    Anil Maheshwari

Rights and permissions

Reprints and permissions

Copyright information

© 2016 Springer International Publishing Switzerland

About this paper

Cite this paper

Berlinkov, M.V. (2016). On the Probability of Being Synchronizable. In: Govindarajan, S., Maheshwari, A. (eds) Algorithms and Discrete Applied Mathematics. CALDAM 2016. Lecture Notes in Computer Science(), vol 9602. Springer, Cham. https://doi.org/10.1007/978-3-319-29221-2_7

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-29221-2_7

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-29220-5

  • Online ISBN: 978-3-319-29221-2

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation