Skip to main content
SpringerLink
Log in

Complexity of formula classes in first order logic with functions

  • Conference paper
  • First Online:
Fundamentals of Computation Theory (FCT 1989)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 380))

Included in the following conference series:

Abstract

We consider the complexity of deciding satisfiability of formulas in full first order logic (including function symbols and equality) which obey restrictions on their quantifier prefix and their relation and function symbols (prefix vocabulary classes). This extends results of H. Lewis and M.Fürer on the complexity of the classical solvable cases of the decision problem.

We obtain complexity results for the maximal solvable cases and some of their subcases. In particular we give a complete classification of the prefix vocabulary classes in P and in NP.

Address after fall 1989: Mathematisches Institut der Universität Basel, Rheinsprung 21, CH-4051 Basel, Switzerland; e-mail: graedel@urz.unibas.ch. This work was supported by the Swiss National Science Foundation.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. W. Ackermann, Über die Erfüllbarkeit gewisser Zählausdrücke, Math. Annalen 100 (1928), 638–649.

    Article  Google Scholar 

  2. P. Bernays and M. Schönfinkel, Zum Entscheidungsproblem der mathematischen Logik, Math. Annalen 99 (1928), 342–372.

    Article  Google Scholar 

  3. E. Börger, Decision problems in predicate logic, in: “Logic Colloquium 82”, Elsevier (North Holland) 1984, 263–301.

    Google Scholar 

  4. K. Compton and C. W. Henson, A uniform method for proving lower bounds on the computational complexity of logical theories, to appear in Annals of Pure and Applied Logic.

    Google Scholar 

  5. B. Dreben and W. Goldfarb, The decision problem, Addison-Wesley, Reading (MA) 1979.

    Google Scholar 

  6. M. Fürer, Alternation and the Ackermann case of the decision problem, in: “Logic and Algorithmic”, Monographie Nr. 30 de L'Enseignement Mathématique, Genève 1982, 161–186.

    Google Scholar 

  7. K. Gödel, Ein Spezialfall des Entscheidungsproblems der theoretischen Logik, Ergebnisse eines mathematischen Kolloquiums 2 (1932), 27–28.

    Google Scholar 

  8. W. Goldfarb, The unsolvability of the Gödel class with identity, J. Symbolic Logic 49 (1984), 1237–1252.

    Google Scholar 

  9. E. Grädel, Satisfiability of formulas with one ∀ is decidable in exponential time, submitted for publication.

    Google Scholar 

  10. Y. Gurevich, Ob effektivnom rasponznavanii vipolnimosti formul UIP, Algebra i Logika 5 (1966), 25–55 (in Russian).

    Google Scholar 

  11. Y. Gurevich, The decision problem for the logic of predicates and operations, Algebra and Logic 8 (1969), 294–308.

    Google Scholar 

  12. Y. Gurevich, Formuly s odnim ∀ (Formulas with one ∀), in: “Izbrannye voprosy algebry i logiki”, (Selected Questions in Algebra and Logic; in memory of A. Malćev), Nauka, Novosibirsk 1973, 97–110 (in Russian).

    Google Scholar 

  13. Y. Gurevich, The decision problem for standard classes, J. Symbolic Logic 41 (1976), 460–464.

    Google Scholar 

  14. A. Kahr, E. Moore and H. Wang, Entscheidungsproblem reduced to the ∀∃∀ case, Proc. Nat. Acad. Sci. U.S.A. 48 (1962), 365–377.

    Google Scholar 

  15. P. Kolaitis and M. Vardi, 0–1 Laws and Decision Problems for Fragments of Second-Order Logic, Proceedings of 3rd IEEE Symposium on Logic in Computer Science 1988, 2–11.

    Google Scholar 

  16. H. Lewis, Unsolvable Classes of Quantificational Formulas, Addison Wesley, Reading (MA) 1979.

    Google Scholar 

  17. H. Lewis, Complexity Results for Classes of Quantificational Formulas, J. of Computer and System Sciences 21 (1980), 317–353.

    Article  Google Scholar 

  18. L. Löwenheim, Über Möglichkeiten im Relativkalkül, Math. Annalen 76 (1915), 447–470.

    Article  Google Scholar 

  19. A. Meyer, The inherent computational complexity of theories of ordered sets, in: “Proc. 1974 Int. Cong. of Mathematicians”, Vancouver (1974), 477–482.

    Google Scholar 

  20. S. Shelah, Decidability of a portion of the predicate calculus, Israel J. Math 28 (1977), 32–44.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Dipartimento di Informatica, Università di Pisa, Corso Italia 40, 56100, Pisa

    Erich Grädel

Authors
  1. Erich Grädel
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

J. Csirik J. Demetrovics F. Gécseg

Rights and permissions

Reprints and permissions

Copyright information

© 1989 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Grädel, E. (1989). Complexity of formula classes in first order logic with functions. In: Csirik, J., Demetrovics, J., Gécseg, F. (eds) Fundamentals of Computation Theory. FCT 1989. Lecture Notes in Computer Science, vol 380. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-51498-8_21

Download citation

  • DOI: https://doi.org/10.1007/3-540-51498-8_21

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-51498-5

  • Online ISBN: 978-3-540-48180-5

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation