On the Decidability of Elementary Modal Logics
Article No.: 2, Pages 1 - 47
Abstract
We consider the satisfiability problem for modal logic over first-order definable classes of frames. We confirm the conjecture from Hemaspaandra and Schnoor [2008] that modal logic is decidable over classes definable by universal Horn formulae. We provide a full classification of Horn formulae with respect to the complexity of the corresponding satisfiability problem. It turns out, that except for the trivial case of inconsistent formulae, local satisfiability is either NP-complete or PSpace-complete, and global satisfiability is NP-complete, PSpace-complete, or ExpTime-complete. We also show that the finite satisfiability problem for modal logic over Horn definable classes of frames is decidable. On the negative side, we show undecidability of two related problems. First, we exhibit a simple universal three-variable formula defining the class of frames over which modal logic is undecidable. Second, we consider the satisfiability problem of bimodal logic over Horn definable classes of frames, and also present a formula leading to undecidability.
References
[1]
Vince Bárány, Balder ten Cate, and Luc Segoufin. 2011. Guarded negation. In Proceedings of the 38th International Colloquium on Automata, Languages and Programming (ICALP’11). Springer-Verlag, Berlin, Heidelberg, 356--367.
[2]
Robert Berger. 1966. The undecidability of the domino problem. Memmoirs of the American Mathematical Society 66.
[3]
Patrick Blackburn, Maarten de Rijke, and Yde Venema. 2001. Modal Logic. Cambridge Tracts in Theoretical Computer Science, Vol. 53. Cambridge University Press, Cambridge.
[4]
Erich Grädel. 1999. On the restraining power of guards. Journal of Symbolic Logic 64, 4 (1999), 1719--1742.
[5]
Erich Graedel, Martin Otto, and Eric Rosen. 1997. Two-variable logic with counting is decidable. In Proceedings of the 12th Annual IEEE Symposium on Logic in Computer Science (LICS’97). IEEE Computer Society, Washington, DC, 306--317.
[6]
Yuri Gurevich and Igor Koryakov. 1972. Remarks on Berger’s paper on the domino problem. Siberian Mathematical Journal 13 (1972), 319--321.
[7]
Edith Hemaspaandra. 1996. The price of universality. Notre Dame Journal of Formal Logic 37, 2 (1996), 174--203.
[8]
Edith Hemaspaandra and Henning Schnoor. 2008. On the complexity of elementary modal logics. In Proceedings of the 25th Annual Symposium on Theoretical Aspects of Computer Science (STACS’08). Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik, Dagstuhl, Germany, 349--360.
[9]
Edith Hemaspaandra and Henning Schnoor. 2011. A universally defined undecidable unimodal logic. In MFCS (LNCS), Vol. 6907. Springer, Berlin, 364--375.
[10]
Emanuel Kieroński, Jakub Michaliszyn, and Jan Otop. 2011. Modal logics definable by universal three-variable formulas. In FSTTCS (LIPIcs), Vol. 13. Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik, Dagstuhl, Germany, 264--275.
[11]
Emanuel Kieroński, Jakub Michaliszyn, Ian Pratt-Hartmann, and Lidia Tendera. 2012. Two-variable first-order logic with equivalence closure. In Proceedings of the 2012 27th Annual IEEE/ACM Symposium on Logic in Computer Science (LICS’12). IEEE Computer Society, Washington, DC, 431--440.
[12]
Emanuel Kieroński and Martin Otto. 2005. Small substructures and decidability issues for first-order logic with two variables. In Proceedings of the 20th Annual IEEE Symposium on Logic in Computer Science (LICS’05). IEEE Computer Society, Washington, DC, 448--457.
[13]
Richard E. Ladner. 1977. The computational complexity of provability in systems of modal propositional logic. Siam Journal on Computing 6 (1977), 467--480.
[14]
Jakub Michaliszyn and Emanuel Kieroński. 2012. Finite satisfiability of modal logic over horn-definable classes of frames. In Advances in Modal Logic. College Publications, London, UK, 464--482.
[15]
Jakub Michaliszyn and Jan Otop. 2012. Decidable elementary modal logics. In Proceedings of the 27th Annual IEEE/ACM Symposium on Logic in Computer Science (LICS’12). IEEE Computer Society, Washington, DC, 491--500.
[16]
Michael Mortimer. 1975. On languages with two variables. Mathematical Logic Quarterly 21, 1 (1975), 135--140.
[17]
Martin Otto. 2001. Two variable first-order logic over ordered domains. Journal of Symbolic Logic 66, 2 (2001), 685--702.
[18]
Leszek Pacholski, Wieslaw Szwast, and Lidia Tendera. 1997. Complexity of two-variable logic with counting. In LICS. IEEE Computer Society, Washington, DC, 318--327.
[19]
Ian Pratt-Hartmann. 2005. Complexity of the two-variable fragment with counting quantifiers. Journal of Logic, Language and Information 14, 3 (2005), 369--395.
[20]
Henrik Sahlqvist. 1975. Completeness and correspondence in the first and second order semantics for modal logic. Studies in Logic and the Foundations of Mathematics 82 (1975), 110--143.
[21]
Walter J. Savitch. 1970. Relationships between nondeterministic and deterministic tape complexities. Journal of Computer and System Sciences 4, 2 (April 1970), 177--192.
[22]
Wieslaw Szwast and Lidia Tendera. 2013. FO2 with one transitive relation is decidable. In Proceedings of the 30th International Symposium on Theoretical Aspects of Computer Science (STACS’13). Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik, Dagstuhl, Germany, 317--328.
[23]
Peter van Emde Boas. 1997. The convenience of tilings. In Complexity, Logic, and Recursion Theory. Marcel Dekker Inc, New York, 331--363.
Index Terms
- On the Decidability of Elementary Modal Logics
Recommendations
Decidable Elementary Modal Logics
LICS '12: Proceedings of the 2012 27th Annual IEEE/ACM Symposium on Logic in Computer ScienceIn this paper, the modal logic over classes of structures definable by universal first-order Horn formulas is studied. We show that the satisfiability problems for that logics are decidable, confirming the conjecture from [E. Hemaspaandra and H. Schnoor,...
Decidability of Quasi-Dense Modal Logics
LICS '24: Proceedings of the 39th Annual ACM/IEEE Symposium on Logic in Computer ScienceThe decidability of axiomatic extensions of the modal logic K with modal reduction principles, i.e. axioms of the form ⋄kp → ⋄np, has remained a long-standing open problem. In this paper, we make significant progress toward solving this problem and show ...
Labelled Sequent Calculi for Inquisitive Modal Logics
Logic, Language, Information, and ComputationAbstractWe present cut-free labelled sequent calculi for various systems of inquisitive modal logic, including inquisitive epistemic logic and inquisitive doxastic logic. Inquisitive modal logic extends the framework of standard modal logic with a ...
Comments
Information & Contributors
Information
Published In
December 2015
258 pages
Copyright © 2015 ACM.
Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than the author(s) must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from [email protected].
Publisher
Association for Computing Machinery
New York, NY, United States
Publication History
Published: 23 September 2015
Accepted: 01 July 2015
Revised: 01 June 2015
Received: 01 July 2013
Published in TOCL Volume 17, Issue 1
Check for updates
Author Tags
Qualifiers
- Research-article
- Research
- Refereed
Funding Sources
- Austrian Science Fund (FWF)NFN
- Polish National Science Center
- European Research Council (ERC)
Contributors
Other Metrics
Bibliometrics & Citations
Bibliometrics
Article Metrics
- 0Total Citations
- 249Total Downloads
- Downloads (Last 12 months)7
- Downloads (Last 6 weeks)0
Reflects downloads up to 23 Feb 2025
Other Metrics
Citations
View Options
Login options
Check if you have access through your login credentials or your institution to get full access on this article.
Sign in