Complexity and Expressivity of Uniform One-Dimensional Fragment with Equality

  • Emanuel Kieroński
  • Antti Kuusisto
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8634)

Abstract

Uniform one-dimensional fragment \(UF_1^=\) is a formalism obtained from first-order logic by limiting quantification to applications of blocks of existential (universal) quantifiers such that at most one variable remains free in the quantified formula. The fragment is closed under Boolean operations, but additional restrictions (called uniformity conditions) apply to combinations of atomic formulas with two or more variables. \(UF_1^=\) can be seen as a canonical generalization of two-variable logic, defined in order to be able to deal with relations of arbitrary arities. \(UF_1^=\) was introduced recently, and it was shown that the satisfiability problem of the equality-free fragment UF1 of \(UF_1^=\) is decidable. In this article we establish that the satisfiability and finite satisfiability problems of \(UF_1^=\) are NEXPTIME-complete. We also show that the corresponding problems for the extension of \(UF_1^=\) with counting quantifiers are undecidable. In addition to decidability questions, we compare the expressivities of \(UF_1^=\) and two-variable logic with counting quantifiers FOC2. We show that while the logics are incomparable in general, \(UF_1^=\) is strictly contained in FOC2 when attention is restricted to vocabularies with the arity bound two.

Keywords

Two-variable logics complexity expressivity 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Andréka, H., van Benthem, J., Németi, I.: Modal languages and bounded fragments of predicate logic. Journal of Philosophical Logic 27(3), 217–274 (1998)CrossRefMATHMathSciNetGoogle Scholar
  2. 2.
    Bárány, V., ten Cate, B., Segoufin, L.: Guarded negation. In: Aceto, L., Henzinger, M., Sgall, J. (eds.) ICALP 2011, Part II. LNCS, vol. 6756, pp. 356–367. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  3. 3.
    Benaim, S., Benedikt, M., Charatonik, W., Kieroński, E., Lenhardt, R., Mazowiecki, F., Worrell, J.: Complexity of two-variable logic on finite trees. In: Fomin, F.V., Freivalds, R., Kwiatkowska, M., Peleg, D. (eds.) ICALP 2013, Part II. LNCS, vol. 7966, pp. 74–88. Springer, Heidelberg (2013)CrossRefGoogle Scholar
  4. 4.
    Charatonik, W., Witkowski, P.: Two-variable logic with counting and trees. In: LICS, pp. 73–82 (2013)Google Scholar
  5. 5.
    Ebbinghaus, H.-D., Flum, J.: Finite model theory. Perspectives in Mathematical Logic. Springer (1995)Google Scholar
  6. 6.
    Grädel, E., Kolaitis, P., Vardi, M.: On the decision problem for two-variable first-order logic. Bulletin of Symbolic Logic 3(1), 53–69 (1997)CrossRefMATHMathSciNetGoogle Scholar
  7. 7.
    Grädel, E., Otto, M., Rosen, E.: Two-variable logic with counting is decidable. In: LICS, pp. 306–317 (1997)Google Scholar
  8. 8.
    Hella, L., Kuusisto, A.: One-dimensional fragment of first-order logic. arXiv:1404.4004 (2014)Google Scholar
  9. 9.
    Henkin, L.: Logical systems containing only a finite number of symbols. Presses De l’Université De Montréal (1967)Google Scholar
  10. 10.
    Kieroński, E., Michaliszyn, J., Pratt-Hartmann, I., Tendera, L.: Two-variable first-order logic with equivalence closure. SIAM Journal of Computing 43(3) (2014)Google Scholar
  11. 11.
    Mortimer, M.: On languages with two variables. Mathematical Logic Quarterly 21(1), 135–140 (1975)CrossRefMATHMathSciNetGoogle Scholar
  12. 12.
    Pacholski, L., Szwast, W., Tendera, L.: Complexity of two-variable logic with counting. In: LICS, pp. 318–327. IEEE (1997)Google Scholar
  13. 13.
    Pratt-Hartmann, I.: Complexity of the two-variable fragment with counting quantifiers. Journal of Logic, Language and Information 14(3), 369–395 (2005)CrossRefMATHMathSciNetGoogle Scholar
  14. 14.
    Scott, D.: A decision method for validity of sentences in two variables. Journal Symbolic Logic 27, 477 (1962)Google Scholar
  15. 15.
    Szwast, W., Tendera, L.: FO2 with one transitive relation is decidable. In: STACS, pp. 317–328 (2013)Google Scholar

Copyright information

© Springer-Verlag GmbH Berlin Heidelberg 2014

Authors and Affiliations

  • Emanuel Kieroński
    • 1
  • Antti Kuusisto
    • 1
  1. 1.Institute of Computer ScienceUniversity of WrocławPoland

Personalised recommendations