Hostname: page-component-cd9895bd7-q99xh Total loading time: 0 Render date: 2024-12-23T17:58:56.422Z Has data issue: false hasContentIssue false
  • English
  • Français

On the Restraining Power of Guards

Published online by Cambridge University Press:  12 March 2014

Erich Grädel
Erich Grädel*
Affiliation:
Mathematische Grundlagen Der InformAtik, Rwth Aachen, D-52056 Aachen, Germany, E-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

Guarded fragments of first-order logic were recently introduced by Andréka, van Benthem and Németi; they consist of relational first-order formulae whose quantifiers are appropriately relativized by atoms. These fragments are interesting because they extend in a natural way many propositional modal logics, because they have useful model-theoretic properties and especially because they are decidable classes that avoid the usual syntactic restrictions (on the arity of relation symbols, the quantifier pattern or the number of variables) of almost all other known decidable fragments of first-order logic.

Here, we investigate the computational complexity of these fragments. We prove that the satisfiability problems for the guarded fragment (GF) and the loosely guarded fragment (LGF) of first-order logic are complete for deterministic double exponential time. For the subfragments that have only a bounded number of variables or only relation symbols of bounded arity, satisfiability is Exptime-complete. We further establish a tree model property for both the guarded fragment and the loosely guarded fragment, and give a proof of the finite model property of the guarded fragment.

It is also shown that some natural, modest extensions of the guarded fragments are undecidable.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 64 , Issue 4 , December 1999 , pp. 1719 - 1742
Copyright
Copyright © Association for Symbolic Logic 1999

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1]Andréka, H., Hodkinson, I., and Németi, I., Finite algebras of relations are representable on finite sets, this Journal, vol. 64 (1999), pp. 243–267.Google Scholar
[2]Andréka, H., Van Benthem, J., and Németi, I., Modal languages and bounded fragments of predicate logic, Journal of Philosophical Logic, vol. 27 (1998), pp. 217–274.CrossRefGoogle Scholar
[3]Balcázar, J., Díaz, J., and Gabarró, J., Structural complexity II, Springer, 1990.CrossRefGoogle Scholar
[4]van Benthem, J., Modal logic an classical logic, Bibliopolis, Napoli, 1983.Google Scholar
[5]van Benthem, J., Exploring logical dynamics, CSLI-Publications, Stanford, 1996.Google Scholar
[6]van Benthem, J., Dynamic bits and pieces, ILLC Research Report, 1997.Google Scholar
[7]Berger, R., The undecidability of the domino problem, Memoirs of the American Mathemtical Society, no. 66, 1966.CrossRefGoogle Scholar
[8]Börger, E., Grädel, E., and Gurevich, Y., The classical decision problem, Springer, 1997.CrossRefGoogle Scholar
[9]Donnini, F., Lenzerini, M., Nardi, D., and Schaerf, A., Reasoning in description logics, Principles of knowledge representation (Brewka, G., editor), Studies in Logic, Language and Information, CSLI Publications, 1996, pp. 193–238.Google Scholar
[10]Grädel, E., Dominoes and the complexity of subclasses of logical theories, Annals of Pure and Applied Logic, vol. 43 (1989), pp. 1–30.CrossRefGoogle Scholar
[11]Grädel, E., Kolaitis, P., and Vardi, M., On the decision problem for two-variable first-order logic, The Bulletin of Symbolic Logic, vol. 3 (1997), pp. 53–69.CrossRefGoogle Scholar
[12]Grädel, E. and Otto, M., On logics with two variables, Theoretical Computer Science, vol. 224 (1999), pp. 73–113.CrossRefGoogle Scholar
[13]Grädel, E., Otto, M., and Rosen, E., Two-variable logic with counting is decidable, Proceedings of 12th IEEE Symposium on Logic in Computer Science LICS '97, Warsaw, 1997.Google Scholar
[14]Grädel, E., Otto, M., and Rosen, E., Undecidability results on two-variable logics, Archive of Mathematical Logic, vol. 38 (1999), pp. 313–354.Google Scholar
[15]Grädel, E. and Rosen, E., On preservation theorems for two-variable logic, Mathematical Logic Quarterly, vol. 45 (1999), pp. 315–325.CrossRefGoogle Scholar
[16]Grohe, M. and Mariño, J., Definability and descriptive complexity on databases of bounded treewidth, submitted for publication.Google Scholar
[17]Gurevich, Y. and Koryakov, I., Remarks on Berger's paper on the domino problem, Siberian Mathematics Journal, vol. 13 (1972), pp. 319–321.CrossRefGoogle Scholar
[18]Harel, D., Recurring dominoes: Making the highly undecidable highly understandable, Annals of Discrete Mathematics, vol. 24 (1985), pp. 51–72.Google Scholar
[19]Harel, D., Effective transformations on infinite trees, with applications to high undecidability, dominoes and fairness, Journal of the ACM, vol. 33 (1986), pp. 224–248.CrossRefGoogle Scholar
[20]Herwig, B., Extending partial isomorphisms on finite structures, Combinatorica, vol. 15 (1995), pp. 365–371.CrossRefGoogle Scholar
[21]Herwig, B., Extending partial isomorphisms for the small index property of many ω-categorical structures, Israel Journal of Mathematics (submitted).Google Scholar
[22]Hrushowski, E., Extending partial isomorphisms of graphs, Combinatorica, vol. 12 (1992), pp. 411–416.CrossRefGoogle Scholar
[23]Kahr, A., Moore, E., and Wang, H., Entscheidungsproblem reduced to the ∀∃∀ case, Proceedings of the National Academy of Sciences U.S.A., vol. 48 (1962), pp. 365–377.CrossRefGoogle Scholar
[24]Ladner, R., The computational complexity of provability in systems of modal logic, SIAM Journal of Computing, vol. 6 (1977), pp. 467–480.CrossRefGoogle Scholar
[25]Mortimer, M., On languages with two variables, Zeitschrift. für mathematische Logik und Grundlagen der Mathematik, vol. 21 (1975), pp. 135–140.CrossRefGoogle Scholar
[26]Reed, B., Tree width and tangles: A new connectivity measure and some applications, Surveys in combinatorics (Bailey, R.A., editor), Cambridge University Press, 1997, pp. 87–162.Google Scholar
[27]Trakhtenbrot, B., On recursive separability, Doklady Akademii Nauk SSSR, vol. 88 (1953), pp. 953–955, In Russian.Google Scholar
[28]Vardi, M., Why is modal logic so robustly decidable, DIMACS Series in Discrete Mathematics and Theoretical Computer Science, vol. 31, American Mathematical Society, 1997.Google Scholar