Abstract
We present a simple PSPACE-algorithm for computing the most general solution of a system of quadratic trace equations with involution. We extend the known linear time algorithm for quadratic word equations with length constraints to cope with involutions. Finally, we show that the same linear-time result cannot be expected for trace equations. We obtain an NP-hardness result.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
V. Diekert, C. Gutiérrez, and C. Hagenah. The existential theory of equations with rational constraints in free groups is PSPACE-complete. In A. Ferreira and H. Reichel, editors, Proc. 18th Annual Symposium on Theoretical Aspects of Computer Science (STACS’01), Dresden (Germany), 2001, number 2010 in Lecture Notes in Computer Science, pages 170–182. Springer, 2001.
V. Diekert and A. Muscholl. Solvability of equations in free partially commutative groups is decidable. In F. Orejas, P. G. Spirakis, and J. van Leeuwen, editors, Proc. 28th International Colloquium on Automata, Languages and Programming (ICALP’01), number 2076 in Lecture Notes in Computer Science, pages 543–554, Berlin-Heidelberg-New York, 2001. Springer.
V. Diekert and G. Rozenberg, editors. The Book of Traces. World Scientific, Singapore, 1995.
Yu. Matiyasevich. A connection between systems of word and length equations and Hilbert’s Tenth Problem. Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI), 8:132–144, 1968. In Russian. English translation in: Sem. Math. V. A. Steklov, 8, 61-67, 1970.
A. Mazurkiewicz. Introduction to trace theory. In V. Diekert and G. Rozenberg, editors, The Book of Traces, chapter 1, pages 3–41. World Scientific, Singapore, 1995.
W. Plandowski. Satisfiability of word equations with constants is in PSPACE. In Proc. 40th Ann. Symp. on Foundations of Computer Science, FOCS’99, pages 495–500. IEEE Computer Society Press, 1999.
W. Plandowski and W. Rytter. Application of Lempel-Ziv encodings to the solution of word equations. In K. G. Larsen et al., editors, Proc. 25th International Colloquium on Automata, Languages and Programming (ICALP’98), Aalborg (Denmark), 1998, number 1443 in Lecture Notes in Computer Science, pages 731–742, Berlin-Heidelberg-New York, 1998. Springer.
J. M. Robson and V. Diekert. On quadratic word equations. In C. Meinel and S. Tison, editors, Proc. 16th Annual Symposium on Theoretical Aspects of Computer Science (STACS’99), Trier (Germany), 1999, number 1563 in Lecture Notes in Computer Science, pages 217–226. Springer-Verlag, 1999.
M. Schaefer, E. Sedgwick, and D. Štefankovič. Recognizing string graphs in NP. In Proceedings 34th Annual ACM Symposium on Theory of Computing, STOC’2002, pages 1–6, New York, 2002. ACM Press.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2003 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Diekert, V., Kufleitner, M. (2003). A Remark about Quadratic Trace Equations. In: Ito, M., Toyama, M. (eds) Developments in Language Theory. DLT 2002. Lecture Notes in Computer Science, vol 2450. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-45005-X_5
Download citation
DOI: https://doi.org/10.1007/3-540-45005-X_5
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-40431-6
Online ISBN: 978-3-540-45005-4
eBook Packages: Springer Book Archive