Abstract
We consider properties of the solution set of a word equation with one unknown. We prove that the solution set of a word equation possessing infinite number of solutions is of the form (pq)* p where pq is primitive. Next, we prove that a word equation with at most four occurrences of the unknown possesses either infinitely many solutions or at most two solutions. We show that there are equations with at most four occurrences of the unknown possessing exactly two solutions. Finally, we prove that a word equation with at most 2k occurrences of the unknown possesses either infinitely many solutions or at most 8logk + O(1) solutions. Hence, if we consider a class \({\cal E}_k\) of equations with at most 2k occurrences of the unknown, then each equation in this class possesses either infinitely many solutions or a constant number of solutions. Our considerations allow to construct the first truly linear time algorithm for computing the solution set of an equation in a nontrivial class of equations.
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
Baader, F., Snyder, W.: Unification theory. In: Robinson, J.A., Voronkov, A. (eds.) Handbook of Automated Reasoning, vol. 1, pp. 447–533. Elsevier, Amsterdam (2001)
Bala, S.: Decision Problems on Regular Expressions. PhD thesis, Universytet Wroclawski, Wroclaw, Poland (2005)
Dabrowski, R., Plandowski, W.: On word equations in one variable. In: Diks, K., Rytter, W. (eds.) MFCS 2002. LNCS, vol. 2420, pp. 212–221. Springer, Heidelberg (2002)
Gleibman, A.: Knowledge representation via verbal description generalization: alternative programming in sampletalk language. In: Proceedings of Workshop on Inference for Textual Question Answering, Pittsburgh, Pennsylvania. AAAI 2005 - the Twentieth National Conference on Artificial Inteligence, pp. 59–68 (2005), http://www.sampletalk.com
Lothaire, M.: Combinatorics on Words. Encyclopedia of Mathematics and its Applications, vol. 17. Addison-Wesley, Reading (1983); Reprinted in the Cambridge Mathematical Library, Cambridge University Press, Cambridge (1997)
Lothaire, M.: Algebraic Combinatorics on Words. Encyclopedia of Mathematics and its Applications, vol. 90. Cambridge University Press, Cambridge (2002)
Eyono Obono, S., Goralcik, P., Maksimenko, M.N.: Efficient solving of the word equations in one variable. In: Privara, I., Ružička, P., Rovan, B. (eds.) MFCS 1994. LNCS, vol. 841, pp. 336–341. Springer, Heidelberg (1994)
Rajasekar, A.: Applications in constraint logic programming with strings. In: Borning, A. (ed.) PPCP 1994. LNCS, vol. 874, pp. 109–122. Springer, Heidelberg (1994)
Schaefer, M., Sedgwick, E., Stefankovic, D.: Recognizing string graphs is in NP. Journal of Computer and System Sciences 67(2), 365–380 (2003)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2009 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Laine, M., Plandowski, W. (2009). Word Equations with One Unknown. In: Diekert, V., Nowotka, D. (eds) Developments in Language Theory. DLT 2009. Lecture Notes in Computer Science, vol 5583. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-02737-6_28
Download citation
DOI: https://doi.org/10.1007/978-3-642-02737-6_28
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-02736-9
Online ISBN: 978-3-642-02737-6
eBook Packages: Computer ScienceComputer Science (R0)