Abstract
We study the computational complexity of problems defined by formulas over fixed finite lattices. We consider evaluation, satisfiability, tautology, counting, and quantified formulas. It turns out that evaluation and tautology always can be decided in alternating logarithmic time. For satisfiability we obtain the following dichotomy result: If the lattice is distributive, satisfiability is in alternating logarithmic time. Otherwise, it is NP-complete. Counting is #-complete for every lattice with at least two elements. For quantified formulas over non-distributive lattices we obtain PSPACE-completeness, while the problem is in alternating logarithmic time, if the lattice is distributive.
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
Birkhoff, G.: Lattice Theory. In: Colloquium Publications, vol. XXV, American Mathematical Society, Providence (1967)
Barrington, D.M., McKenzie, P., Moore, C., Tesson, P., Therien, D.: Equation Satisfiability and Program Satisfiability for Finite Monoids. In: Proceedings of the 25th International Symposium on Mathematical Foundations of Computer Science, pp. 172–181 (2000)
Buss, S.R.: The Boolean formula value problem is in ALOGTIME. In: Proceedings of the 19th Symposium on Theory of Computing, pp. 123–131 (1987)
Cook, S.A.: The complexity of theorem-proving procedures. In: Proceedings of the Third ACM Symposium on Theory of Computing, pp. 151–158 (1971)
Garey, M.R., Johnson, D.S.: Computers and Intractability. W. H. Freeman and Company, New York (1979)
Goldmann, M., Russell, A.: The complexity of solving equations over finite groups. In: Proceedings of the 14th Annual IEEE Conference on Computational Complexity, pp. 80–86 (1999)
Papadimitriou, C.H.: Computational Complexity. Addison-Wesley, Reading (1994); [Ru81] Ruzzo, W.L.: On uniform circuit complexity. Journal of Computer System Sciences 22, 265–383 (1981)
Reith, S., Wagner, K.W.: The Complexity of Problems Defined by Subclasses of Boolean Functions, Technical Report 218, Inst. für Informatik, Univ. Würzburg (1999), Available via ftp from http://www.informatik.uniwuerzburg.de/reports/tr.html
Schwarz, B.: The Complexity of Satisfiability Problems over Finite Lattices, Technical Report 314, Institut für Informatik, Universität Würzburg (2004), Available via ftp from http://www.informatik.uni-wuerzburg.de/reports/tr.html
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
Schwarz, B. (2004). The Complexity of Satisfiability Problems over Finite Lattices. In: Diekert, V., Habib, M. (eds) STACS 2004. STACS 2004. Lecture Notes in Computer Science, vol 2996. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-24749-4_4
Download citation
DOI: https://doi.org/10.1007/978-3-540-24749-4_4
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-21236-2
Online ISBN: 978-3-540-24749-4
eBook Packages: Springer Book Archive