Abstract
A word w is called synchronizing (recurrent, reset, directable) word of deterministic finite automaton (DFA) if w brings all states of the automaton to an unique state. Černy conjectured in 1964 that every n-state synchronizable automaton possesses a synchronizing word of length at most (n − 1)2. The problem is still open.
It will be proved that the minimal length of synchronizing word is not greater than (n − 1)2/2 for every n-state (n > 2) synchronizable DFA with transition monoid having only trivial subgroups (such automata are called aperiodic). This important class of DFA accepting precisely star-free languages was involved and studied by Schŭtzenberger. So for aperiodic automata as well as for automata accepting only star-free languages, the Černý conjecture holds true.
Some properties of an arbitrary synchronizable DFA and its transition semigroup were established.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Černy, J.: Poznamka k homogenym eksperimentom s konechnymi automatami. Math.-Fyz. Čas. 14, 208–215 (1964)
Dubuc, L.: Sur le automates circulaires et la conjecture de Černy, RAIRO Inform. Theor. Appl. 32(1-3), 21–34 (1998)
Frankl, P.: An extremal problem for two families of sets. Eur. J. Comb. 3, 125–127 (1982)
Kari, J.: Synchronizing finite automata on Eulerian digraphs. In: Sgall, J., Pultr, A., Kolman, P. (eds.) MFCS 2001. LNCS, vol. 2136, pp. 432–438. Springer, Heidelberg (2001)
Kljachko, A.A., Rystsov, I.K., Spivak, M.A.: An extremely combinatorial problem connected with the bound on the length of a recurrent word in an automata. Kybernetika 2, 16–25 (1987)
Lallement, G.: Semigroups and Combinatorial Applications. Wiley, Hoboken (1979)
Pin, J.E.: On two combinatorial problems arising from automata theory. Annals of Discrete Mathematics 17, 535–548 (1983)
Rystsov, I.K.: Almost optimal bound on recurrent word length for regular automata. Cybernetics and System An. 31(5), 669–674 (1995)
Rystsov, I.K.: Reset words for commutative and solvable automata. Theoret. Comput. Sci. 172, 273–279 (1997)
Salomaa, A.: Generation of constants and synchronization of finite automata. J. of Univers. Comput. Sci. 8(2), 332–347 (2002)
Schützenberger, M.P.: On finite monoids having only trivial subgroups. Inf. control 8, 190–194 (1965)
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)
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 2007 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Trahtman, A.N. (2007). Synchronization of Some DFA. In: Cai, JY., Cooper, S.B., Zhu, H. (eds) Theory and Applications of Models of Computation. TAMC 2007. Lecture Notes in Computer Science, vol 4484. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-72504-6_21
Download citation
DOI: https://doi.org/10.1007/978-3-540-72504-6_21
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-72503-9
Online ISBN: 978-3-540-72504-6
eBook Packages: Computer ScienceComputer Science (R0)