Abstract
Separating words with automata is a longstanding open problem in combinatorics on words. In this paper we present a related algebraic problem. What is the minimal length of a nontrivial identical relation in the symmetric group S n ?
Our main contribution is an upper bound \(2^{O(\sqrt n\log n)}\) on the length of the shortest nontrivial identical relation in S n . We also give lower bounds for words of a special types. These bounds can be applied to the problem of separating words by reversible automata. In this way we obtain an another proof of the Robson’s square root bound.
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
Bach, E., Shallit, J.: Algorithmic number theory. In: Efficient algorithms, vol. 1. MIT Press, Cambridge (1996)
Currie, J., Petersen, H., Robson, J.M., Shallit, J.: Separating words with small grammars. J. Automata, Languages, and Combinatoric 4, 101–110 (1999)
Krasikov, I., Roditty, Y.: On a reconstruction problem for sequences. J. Comb. Theory, Ser. A. 77(2), 344–348 (1997)
Leontév, V.K., Smetanin Yu, G.: Problems of information on the set of words. Journal of Mathematical Sciences 108(1), 49–70 (2002)
Leontév, V.K., Smetanin Yu, G.: Recognition Model with Representation of Information in the Form of Long Sequences. Pattern Recognition and Image Analysis: Advances in Mathematical Theory and Applications 12(3), 250–263 (2002)
Lyndon, R.C., Shupp, P.E.: Combinatorial group theory. Springer, Heidelberg (2001)
Robson, J.M.: Separating strings with small automata. Information Processing Letters 30, 209–214 (1989)
Robson, J.M.: Separating words with machines and groups. Informatique théorique et Applications 30, 81–86 (1996)
Smetanin Yu, G.: Refined logarithmic bound for the length of fragments ensuring the unambiguous reconstruction of unknown words. Pattern Recognition and Image Analysis: Advances in Mathematical Theory and Applications 11(1), 100–102 (2001)
Vinogradov, I.M.: Elements of Number Theory. Dover Publications, Mineola (2003)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2010 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Gimadeev, R.A., Vyalyi, M.N. (2010). Identical Relations in Symmetric Groups and Separating Words with Reversible Automata. In: Ablayev, F., Mayr, E.W. (eds) Computer Science – Theory and Applications. CSR 2010. Lecture Notes in Computer Science, vol 6072. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-13182-0_14
Download citation
DOI: https://doi.org/10.1007/978-3-642-13182-0_14
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-13181-3
Online ISBN: 978-3-642-13182-0
eBook Packages: Computer ScienceComputer Science (R0)