Abstract
We present a few classes of synchronizing automata exhibiting certain extremal properties with regard to synchronization. The first is a series of automata with subsets whose shortest extending words are of length \(\varTheta (n^2)\), where n is the number of states of the automaton. This disproves a conjecture that every subset in a strongly connected synchronizing automaton is cn-extendable, for some constant c, and in particular, shows that the cubic upper bound on the length of the shortest reset words cannot be improved generally by means of the extension method. A detailed analysis shows that the automata in the series have subsets that require words as long as \(n^2/4+O(n)\) in order to be extended by at least one element.
We also discuss possible relaxations of the conjecture, and propose the image-extension conjecture, which would lead to a quadratic upper bound on the length of the shortest reset words. In this regard we present another class of automata, which turn out to be counterexamples to a key claim in a recent attempt to improve the Pin-Frankl bound for reset words.
Finally, we present two new series of slowly irreducibly synchronizing automata over a ternary alphabet, whose lengths of the shortest reset words are \(n^2-3n+3\) and \(n^2-3n+2\), respectively. These are the first examples of such series of automata for alphabets of size larger than two.
Andrzej Kisielewicz—Supported in part by Polish MNiSZW grant IP 2012 052272.
Marek Szykuła—Supported in part by Polish NCN grant DEC-2013/09/N/ST6/01194.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
First Russian-Finnish Symposium on Discrete Mathematics (RuFiDiM 2011).
References
Ananichev, D., Gusev, V., Volkov, M.: Slowly synchronizing automata and digraphs. In: Hliněný, P., Kučera, A. (eds.) MFCS 2010. LNCS, vol. 6281, pp. 55–65. Springer, Heidelberg (2010)
Ananichev, D.S., Volkov, M.V., Gusev, V.V.: Primitive digraphs with large exponents and slowly synchronizing automata. J. Math. Sci. 192(3), 263–278 (2013)
Ananichev, D.S., Volkov, M.V., Zaks, Y.I.: Synchronizing automata with a letter of deficiency 2. In: Ibarra, O.H., Dang, Z. (eds.) DLT 2006. LNCS, vol. 4036, pp. 433–442. Springer, Heidelberg (2006)
Béal, M.P., Berlinkov, M.V., Perrin, D.: A quadratic upper bound on the size of a synchronizing word in one-cluster automata. Int. J. Foundations Comput. Sci. 22(2), 277–288 (2011)
Berlinkov, M., Szykuła, M.: Algebraic synchronization criterion and computing reset words. In: Italiano, G.F., et al (eds.) MFCS 2015. Lecture Notes in Computer Science, vol. 9234, pp. 103–115 (2015)
Berlinkov, M.V.: On a conjecture by Carpi and D’Alessandro. Int. J. Foundations Comput. Sci. 22(7), 1565–1576 (2011)
Berlinkov, M.V.: Synchronizing quasi-eulerian and quasi-one-cluster automata. Int. J. Foundations Comput. Sci. 24(6), 729–745 (2013)
Černý, J.: Poznámka k homogénnym eksperimentom s konečnými automatami. Matematicko-fyzikálny Čas. Slovenskej Akad. Vied 14(3), 208–216 (1964). in Slovak
Dubuc, L.: Sur les automates circulaires et la conjecture de C̆erný. Informatique théorique et Appl. 32, 21–34 (1998). in French
Gonze, F., Jungers, R.M., Trahtman, A.N.: A note on a recent attempt to improve the Pin-Frankl bound. Discrete Math. Theoret. Comput. Sci. 17(1), 307–308 (2015)
Gusev, V.V., Pribavkina, E.V.: Reset thresholds of automata with two cycle lengths. In: Holzer, M., Kutrib, M. (eds.) CIAA 2014. LNCS, vol. 8587, pp. 200–210. Springer, Heidelberg (2014)
Kari, J.: Synchronizing finite automata on Eulerian digraphs. Theoret. Comput. Sci. 295(1–3), 223–232 (2003)
Kari, J., Volkov, M.V.: Černý’s conjecture and the road coloring problem. In: Handbook of Automata. European Science Foundation (2013, to appear)
Kisielewicz, A., Kowalski, J., Szykuła, M.: Computing the shortest reset words of synchronizing automata. J. Comb. Optim. 29(1), 88–124 (2015)
Roman, A.: A note on Černý conjecture for automata over 3-letter alphabet. J. Automata Lang. Comb. 13(2), 141–143 (2008)
Rystsov, I.K.: Quasioptimal bound for the length of reset words for regular automata. Acta Cybernetica 12(2), 145–152 (1995)
Steinberg, B.: The averaging trick and the Černý conjecture. Int. J. Foundations Comput. Sci. 22(7), 1697–1706 (2011)
Steinberg, B.: The Černý conjecture for one-cluster automata with prime length cycle. Theoret. Comput. Sci. 412(39), 5487–5491 (2011)
Trahtman, A.N.: An efficient algorithm finds noticeable trends and examples concerning the Černy conjecture. In: Královič, R., Urzyczyn, P. (eds.) MFCS 2006. LNCS, vol. 4162, pp. 789–800. Springer, Heidelberg (2006)
Trahtman, A.N.: Modifying the upper bound on the length of minimal synchronizing word. In: Owe, O., Steffen, M., Telle, J.A. (eds.) FCT 2011. LNCS, vol. 6914, pp. 173–180. Springer, Heidelberg (2011)
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)
Acknowledgment
We are grateful to Jakub Kowalski and anonymous referees for careful proofreading.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2015 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Kisielewicz, A., Szykuła, M. (2015). Synchronizing Automata with Extremal Properties. In: Italiano, G., Pighizzini, G., Sannella, D. (eds) Mathematical Foundations of Computer Science 2015. MFCS 2015. Lecture Notes in Computer Science(), vol 9234. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-48057-1_26
Download citation
DOI: https://doi.org/10.1007/978-3-662-48057-1_26
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-662-48056-4
Online ISBN: 978-3-662-48057-1
eBook Packages: Computer ScienceComputer Science (R0)