Modal Logic and the Two-Variable Fragment

  • Carsten Lutz
  • Ulrike Sattler
  • Frank Wolter
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2142)


We introduce a modal language L which is obtained from standard modal logic by adding the difference operator and modal operators interpreted by boolean combinations and the converse of accessibility relations. It is proved that L has the same expressive power as the two-variable fragment FO 2 of first-order logic but speaks less succinctly about relational structures: if the number of relations is bounded, then L- satisfiability is ExpTime-complete but FO 2 satisfiability is NE xp Time-complete. We indicate that the relation between L and FO 2 provides a general framework for comparing modal and temporal languages with first-order languages.


Modal Logic Temporal Logic Modal Parameter Expressive Power Accessibility Relation 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    H. Andréka, I. Németi, and J. van Benthem. Modal languages and bounded fragments of predicate logic. Journal of Philosophical Logic, 27:217–274, 1998.zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    P. Blackburn, M. de Rijke, and Y. Venema. Modal logic. In print.Google Scholar
  3. 3.
    E. Börger, E. Grädel, and Yu. Gurevich. The Classical Decision Problem. Perspectives in Mathematical Logic. Springer, 1997.Google Scholar
  4. 4.
    A. Borgida. On the relative expressivness of description logics and predicate logics. Artificial Intelligence, 82(1-2):353–367, 1996.CrossRefMathSciNetGoogle Scholar
  5. 5.
    J.P. Burgess. Basic tense logic. In D.M. Gabbay and F. Guenthner, editors, Handbook of Philosophical Logic, volume 2, pages 89–133. Reidel, Dordrecht, 1984.Google Scholar
  6. 6.
    Maarten de Rijke. The modal logic of inequality. The Journal of Symbolic Logic, 57(2):566–584, June 1992.Google Scholar
  7. 7.
    K. Etessami, M. Vardi, and T. Wilke. First-order logic with two variables and unary temporal logic. In Proceedings of 12th. IEEE Symp. Logic in Computer Science, pages 228–235, 1997.Google Scholar
  8. 8.
    M. Fürer. The computational complexity of the unconstrained limited domino problem (with implications for logical decision problems). In Logic and Machines: Decision problems and complexity, pages 312–319. Springer, 1984.Google Scholar
  9. 9.
    D. Gabbay, I. Hodkinson, and M. Reynolds. Temporal Logic: Mathematical Foundations and Computational Aspects, Volume 1. Oxford University Press, 1994.Google Scholar
  10. 10.
    D.M. Gabbay. Expressive functional completeness in tense logic. In U. Mönnich, editor, Aspects of Philosophical Logic, pages 91–117. Reidel, Dordrecht, 1981.Google Scholar
  11. 11.
    G. Gargov and S. Passy. A note on boolean modal logic. In D. Skordev, editor, Mathematical Logic and Applications, pages 253–263, New York, 1987. Plenum Press.Google Scholar
  12. 12.
    R.I. Goldblatt. Logics of Time and Computation. Number 7 in CSLI Lecture Notes, Stanford. CSLI, 1987.zbMATHGoogle Scholar
  13. 13.
    E. Grädel. Why are modal logics so robustly decidable? Bulletin of the European Association for Theoretical Computer Science, 68:90–103, 1999.zbMATHGoogle Scholar
  14. 14.
    E. Grädel, P. Kolaitis, and M. Vardi. On the Decision Problem for Two-Variable First-Order Logic. Bulletin of Symbolic Logic, 3:53–69, 1997.zbMATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    E. Grädel and M. Otto. On Logics with Two Variables. Theoretical Computer Science, 224:73–113, 1999.zbMATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    J. Halpern and Y. Shoham. A propositional modal logic of time intervals. Journal of the ACM, 38:935–962, 1991.zbMATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    I. L. Humberstone. Inaccessible worlds. Notre Dame Journal of Formal Logic, 24(3):346–352, 1983.zbMATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    H. Kamp. Tense Logic and the Theory of Linear Order. Ph. D. Thesis, University of California, Los Angeles, 1968.Google Scholar
  19. 19.
    R.E. Ladner. The computational complexity of provability in systems of modal logic. SIAM Journal on Computing, 6:467–480, 1977.zbMATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    C. Lutz and U. Sattler. The complexity of reasoning with boolean modal logics. In Frank Wolter, Heinrich Wansing, Maarten de Rijke, and Michael Zakharyaschev, editors, Advances in Modal Logics Volume 3. CSLI Publications, Stanford, 2001.Google Scholar
  21. 21.
    C. Lutz, U. Sattler, and F. Wolter. Modal logic and the two-variable fragment. LTCS-Report 01-04, LuFG Theoretical Computer Science, RWTH Aachen, Germany, 2001. See Scholar
  22. 22.
    M. Mortimer. On languages with two variables. Zeitschrift für Mathematische Logik und Grundlagen der Mathematik, 21:135–140, 1975.zbMATHMathSciNetGoogle Scholar
  23. 23.
    Anil Nerode and Richard A. Shore. Logic for Applications. Springer Verlag, New York, 1997.zbMATHGoogle Scholar
  24. 24.
    D. Scott. A decision method for validity of sentences in two variables. Journal of Symbolic Logic, 27(377), 1962.Google Scholar
  25. 25.
    A. Sistla and E. Clarke. The complexity of propositional linear temporal logics. Journal of the Association for Computing Machinery, 32:733–749, 1985.zbMATHMathSciNetGoogle Scholar
  26. 26.
    E. Spaan. Complexity of Modal Logics. PhD thesis, Department of Mathematics and Computer Science, University of Amsterdam, 1993.Google Scholar
  27. 27.
    J. van Benthem. Modal Logic and Classical Logic. Bibliopolis, Napoli, 1983.Google Scholar
  28. 28.
    J. van Benthem. Correspondence theory. In D.M. Gabbay and F. Guenthner, editors, Handbook of Philosophical Logic, volume 2, pages 167–247. Reidel, Dordrecht, 1984.Google Scholar
  29. 29.
    M. Vardi. Why is modal logic so robustly decidable? In Descriptive Complexity and Finite Models, pages 149–184. AMS, 1997.Google Scholar
  30. 30.
    F. Wolter. Tense logics without tense operators. Mathematical Logic Quarterly, 42:145–171, 1996.zbMATHCrossRefMathSciNetGoogle Scholar
  31. 31.
    F. Wolter. Completeness and decidability of tense logics closely related to logics containing K4. Journal of Symbolic Logic, 62:131–158, 1997.zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2001

Authors and Affiliations

  • Carsten Lutz
    • 1
  • Ulrike Sattler
    • 1
  • Frank Wolter
    • 2
  1. 1.LuFG Theoretical Computer ScienceRWTH AachenAachenGermany
  2. 2.Institut für InformatikUniversität LeipzigLeipzigGermany

Personalised recommendations