Abstract
We study the computational complexity of counting the number of solutions to systems of equations over a fixed finite semigroup. We show that if the semigroup is a group, the problem is tractable if the group is Abelian and #P-complete otherwise. If the semigroup is a monoid (that is not a group) the problem is #P-complete. In the case of semigroups where all elements have divisors we show that the problem is tractable if the semigroup is a direct product of an Abelian group and a rectangular band, and #P-complete otherwise. The class of semigroups where all elements have divisors contains most of the interesting semigroups e.g. regular semigroups. These results are proved by the use of powerful techniques from universal algebra.
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
Bulatov, A.: A dichotomy theorem for constraints on a three-element set. In: Proc. 43rd IEEE Symposium on Foundations of Computer Science, FOCS 2002, pp. 649– 658 (2002)
Bulatov, A., Dalmau, V.: Towards a dichotomy theorem for the counting constraint satisfaction problem. In: Proc. 44th IEEE Symposium on Foundations of Computer Science, FOCS 2003 (2003)
Goldmann, M., Russel, A.: The complexity of solving equations over finite groups. Inform. and Comput. 178(1), 253–262 (2002)
Howie, J.M.: Fundamentals of Semigroup Theory. Claredon Press, Oxford (1995)
Jeavons, P., Cohen, D., Gyssens, M.: Closure properties of constraints. Journal of the ACM 44, 527–548 (1997)
Moore, C., Tesson, P., Thérien, D.: Satisfiability of systems of equations over finite monoids. In: Sgall, J., Pultr, A., Kolman, P. (eds.) MFCS 2001. LNCS, vol. 2136, pp. 537–547. Springer, Heidelberg (2001)
Tesson, P.: Computational Complexity Questions Related to Finite Monoids and Semigroups. PhD thesis, School of Computer Science, McGill University, Montreal (2003)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2004 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Nordh, G., Jonsson, P. (2004). The Complexity of Counting Solutions to Systems of Equations over Finite Semigroups. In: Chwa, KY., Munro, J.I.J. (eds) Computing and Combinatorics. COCOON 2004. Lecture Notes in Computer Science, vol 3106. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-27798-9_40
Download citation
DOI: https://doi.org/10.1007/978-3-540-27798-9_40
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-22856-1
Online ISBN: 978-3-540-27798-9
eBook Packages: Springer Book Archive