Transitivity and Equivalence in Decidable Fragments of First-Order Logic: A Survey

  • Ian Pratt-HartmannEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11600)


In this talk, I survey recent work on extensions of various well-known decidable fragments of first-order logic, in which certain distinguished predicates are required to denote transitive relations or equivalence relations. I explain the origins of this work in modal logic, and outline the current state-of-the-art.


First-order logic Transitivity Equivalence Complexity 


  1. 1.
    Andréka, H., van Benthem, J., Németi, I.: Modal languages and bounded fragments of predicate logic. J. Philos. Logic 27, 217–274 (1998)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Church, A.: A note on the Entscheidungsproblem. J. Symb. Log. 1(1), 40–41 (1936)CrossRefGoogle Scholar
  3. 3.
    Gödel, K.: Zum Entscheidungsproblem des logischen Funktionenkalküls. Monatshefte für Mathematik und Physik 40, 433–443 (1933)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Goldfarb, W.: The unsolvability of the Gödel class with identity. J. Symbolic Logic 49, 1237–1252 (1984)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Grädel, E.: On the restraining power of guards. J. Symbolic Logic 64, 1719–1742 (1999)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Grädel, E., Kolaitis, P., Vardi, M.: On the decision problem for two-variable first-order logic. Bull. Symbolic Logic 3(1), 53–69 (1997)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Grädel, E., Otto, M., Rosen, E.: Two-variable logic with counting is decidable. In: Proceedings of the 12th IEEE Symposium on Logic in Computer Science, pp. 306–317. IEEE Online Publications (1997)Google Scholar
  8. 8.
    Hilbert, D., Ackermann, W.: Grundzüge der theoretischen Logik. Springer, Heidelberg (1928)zbMATHGoogle Scholar
  9. 9.
    Hilbert, D., Ackermann, W.: Grundzüge der Theoretischen Logik. DGW, vol. 27. Springer, Heidelberg (1938). Scholar
  10. 10.
    Kazakov, Y.: A polynomial translation from the two-variable guarded fragment with number restrictions to the guarded fragment. In: Alferes, J.J., Leite, J. (eds.) JELIA 2004. LNCS (LNAI), vol. 3229, pp. 372–384. Springer, Heidelberg (2004). Scholar
  11. 11.
    Kazakov, Y., Pratt-Hartmann, I.: A note on the complexity of the satisfiability problem for graded modal logic. In: 24th IEEE Symposium on Logic in Computer Science, pp. 407–416. IEEE Online Publications (2009)Google Scholar
  12. 12.
    Ladner, R.: The computational complexity of provability in systems of modal propositional logic. SIAM J. Comput. 6, 467–480 (1980)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Mortimer, M.: On languages with two variables. Zeitschrift für Mathematische Logik und Grundlagen der Mathematik 21, 135–140 (1975)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Pacholski, L., Szwast, W., Tendera, L.: Complexity of two-variable logic with counting. In: Proceedings of the 12th IEEE Symposium on Logic in Computer Science, pp. 318–327. IEEE Online Publications (1997)Google Scholar
  15. 15.
    Pacholski, L., Szwast, W., Tendera, L.: Complexity results for first-order two-variable logic with counting. SIAM J. Comput. 29(4), 1083–1117 (1999)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Pratt-Hartmann, I.: Complexity of the two-variable fragment with counting quantifiers. J. Logic Lang. Inform. 14, 369–395 (2005)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Pratt-Hartmann, I.: Complexity of the guarded two-variable fragment with counting quantifiers. J. Logic Comput. 17, 133–155 (2007)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Scott, D.: A decision method for validity of sentences in two variables. J. Symbolic Logic 27, 477 (1962)Google Scholar
  19. 19.
    Tobies, S.: PSPACE reasoning for graded modal logics. J. Logic Comput. 11(1), 85–106 (2001)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Trakhtenbrot, B.: The impossibility of an algorithm for the decision problem for finite models. Dokl. Akad. Nauk SSSR 70, 596–572 (1950). English translation in: AMS Trans. Ser. 2, vol. 23, 1–6 (1963)Google Scholar
  21. 21.
    Turing, A.M.: On computable numbers, with an application to the Entscheidungsproblem. Proc. London Math. Soc. 42(2), 230–265 (1936)MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.School of Computer ScienceUniversity of ManchesterManchesterUK
  2. 2.Wydział Matematyki, Informatyki i MechanikiUniwersytet WarszawskiWarsawPoland

Personalised recommendations