Article

The complexity of generalized satisfiability for linear temporal logic

Authors:

Michael Bauland,

Thomas Schneider,

Henning Schnoor,

Heribert VollmerAuthors Info & Claims

FOSSACS'07: Proceedings of the 10th international conference on Foundations of software science and computational structures

Pages 48 - 62

Published: 24 March 2007 Publication History

Abstract

In a seminal paper from 1985, Sistla and Clarke showed that satisfiability for Linear Temporal Logic (LTL) is either NP-complete or PSPACE-complete, depending on the set of temporal operators used. If, in contrast, the set of propositional operators is restricted, the complexity may decrease. This paper undertakes a systematic study of satisfiability for LTL formulae over restricted sets of propositional and temporal operators. Since every propositional operator corresponds to a Boolean function, there exist infinitely many propositional operators. In order to systematically cover all possible sets of them, we use Post's lattice. With its help, we determine the computational complexity of LTL satisfiability for all combinations of temporal operators and all but two classes of propositional functions. Each of these infinitely many problems is shown to be either PSPACE-complete, NP-complete, or in P.

References

[1]

E. Böhler, N. Creignou, S. Reith, and H. Vollmer. Playing with Boolean blocks, part I: Post's lattice with applications to complexity theory. SIGACT News, 34(4):38-52, 2003.

Digital Library

[2]

M. Bauland, E. Hemaspaandra, H. Schnoor, and I. Schnoor. Generalized modal satisfiability. In B. Durand and W. Thomas, editors, STACS, volume 3884 of Lecture Notes in Computer Science, pages 500-511. Springer, 2006.

Digital Library

[3]

M. Bauland, H. Schnoor, I. Schnoor, T. Schneider, and H. Vollmer. The complexity of generalized satisfiability for linear temporal logic. Technical Report TR06-153, Electronic Colloquium on Computational Complexity, 2006.

[4]

S. A. Cook. The complexity of theorem proving procedures. In Proceedings 3rd Symposium on Theory of Computing, pages 151-158. ACM Press, 1971.

Digital Library

[5]

V. Dalmau. Computational Complexity of Problems over Generalized Formulas . PhD thesis, Department de Llenguatges i Sistemes Informàtica, Universitat Politécnica de Catalunya, 2000.

[6]

H. Lewis. Satisfiability problems for propositional calculi. Mathematical Systems Theory, 13:45-53, 1979.

[7]

G. Nordh. A trichotomy in the complexity of propositional circumscription. In Proceedings of the 11th International Conference on Logic for Programming , volume 3452 of Lecture Notes in Computer Science, pages 257-269. Springer Verlag, 2005.

Digital Library

[8]

N. Pippenger. Theories of Computability. Cambridge University Press, Cambridge, 1997.

Digital Library

[9]

A. Pnueli. The temporal logic of programs. In FOCS, pages 46-57. IEEE, 1977.

Digital Library

[10]

E. Post. The two-valued iterative systems of mathematical logic. Annals of Mathematical Studies, 5:1-122, 1941.

[11]

S. Reith. Generalized Satisfiability Problems. PhD thesis, Fachbereich Mathematik und Informatik, Universität Würzburg, 2001.

[12]

S. Reith and H. Vollmer. Optimal satisfiability for propositional calculi and constraint satisfaction problems. Information and Computation, 186(1):1- 19, 2003.

Digital Library

[13]

S. Reith and K. W. Wagner. The complexity of problems defined by Boolean circuits. In Proceedings International Conference Mathematical Foundation of Informatics, (MFI99); World Science Publishing, 2005.

[14]

A. Sistla and E. Clarke. The complexity of propositional linear temporal logics. Journal of the ACM, 32(3):733-749, 1985.

Digital Library

[15]

H. Schnoor. The complexity of the Boolean formula value problem. Technical report, Theoretical Computer Science, University of Hannover, 2005.

[16]

L. Stockmeyer. The polynomial-time hierarchy. Theoretical Computer Science , 3:1-22, 1977.

Cited By

Bauland MMundhenk MSchneider TSchnoor HSchnoor IVollmer H(2011)The tractability of model checking for LTLACM Transactions on Computational Logic (TOCL)10.1145/1877714.187771912:2(1-28)Online publication date: 27-Jan-2011
https://dl.acm.org/doi/10.1145/1877714.1877719
Bauland MMundhenk MSchneider TSchnoor HSchnoor IVollmer H(2009)The Tractability of Model-checking for LTLElectronic Notes in Theoretical Computer Science (ENTCS)10.1016/j.entcs.2009.02.041231(277-292)Online publication date: 1-Mar-2009
https://dl.acm.org/doi/10.1016/j.entcs.2009.02.041

The complexity of generalized satisfiability for linear temporal logic
1. Theory of computation
  1. Computational complexity and cryptography
  2. Logic

Recommendations

On the Translation of Automata to Linear Temporal Logic
Foundations of Software Science and Computation Structures
Abstract
While the complexity of translating future linear temporal logic (LTL) into automata on infinite words is well-understood, the size increase involved in turning automata back to LTL is not. In particular, there is no known elementary bound on the ...
Efficient Normalization of Linear Temporal Logic
In the mid 1980s, Lichtenstein, Pnueli, and Zuck proved a classical theorem stating that every formula of Past LTL (the extension of Linear Temporal Logic (LTL) with past operators) is equivalent to a formula of the form \(\bigwedge _{i=1}^n {\mathbf {G}}...
Parametrised Complexity of Satisfiability in Temporal Logic

We apply the concept of formula treewidth and pathwidth to computation tree logic, linear temporal logic, and the full branching time logic. Several representations of formulas as graphlike structures are discussed, and corresponding notions of ...

Comments

Information & Contributors

Information

Published In

cover image Guide Proceedings

FOSSACS'07: Proceedings of the 10th international conference on Foundations of software science and computational structures

March 2007

378 pages

Editor:
Helmut Seidl
TU München,Institut für Informatik, München, Germany

Sponsors

Fundacao para a Ciencia e Tecnologia
TAP Air Portugal
Enabler
CISCO

Publisher

Springer-Verlag

Berlin, Heidelberg

Publication History

Published: 24 March 2007

Author Tags

Qualifiers

Article

Contributors

Other Metrics

View Article Metrics

Bibliometrics & Citations

Bibliometrics

Article Metrics

2
Total Citations
View Citations
0
Total Downloads

Downloads (Last 12 months)0
Downloads (Last 6 weeks)0

Reflects downloads up to 06 Jan 2025

Other Metrics

View Author Metrics

Citations

Cited By

Bauland MMundhenk MSchneider TSchnoor HSchnoor IVollmer H(2011)The tractability of model checking for LTLACM Transactions on Computational Logic (TOCL)10.1145/1877714.187771912:2(1-28)Online publication date: 27-Jan-2011
https://dl.acm.org/doi/10.1145/1877714.1877719
Bauland MMundhenk MSchneider TSchnoor HSchnoor IVollmer H(2009)The Tractability of Model-checking for LTLElectronic Notes in Theoretical Computer Science (ENTCS)10.1016/j.entcs.2009.02.041231(277-292)Online publication date: 1-Mar-2009
https://dl.acm.org/doi/10.1016/j.entcs.2009.02.041

View Options

View options

Media

Figures

Other

Tables

View Table of Contents