Existential Fixed-Point Logic as a Fragment of Second-Order Logic

  • Andreas BlassEmail author
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9300)


The standard translation of existential fixed-point formulas into second-order logic produces strict universal formulas, that is, formulas consisting of universal quantifiers on relations (not functions) followed by an existential first-order formula. This form implies many of the pleasant properties of existential fixed-point logic, but not all. In particular, strict universal sentences can express some co-NP-complete properties of structures, whereas properties expressible by existential fixed-point formulas are always in P. We therefore investigate what additional syntactic properties, beyond strict universality, are enjoyed by the second-order translations of existential fixed-point formulas. In particular, do such syntactic properties account for polynomial-time model-checking?


Atomic Formula Conjunctive Normal Form Parse Tree Predicate Symbol Horn Clause 
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.



Because of the last-minute discovery of the conjunction problem, this paper was submitted after the official deadline, leaving less than the normal time for refereeing. Nevertheless, the referee provided a very useful report. I thank him or her for the report, in particular for informing me about the existence and relevance of [4].


  1. 1.
    Barwise, J.: Admissible Sets and Structures: An Approach to Definability Theory. Perspectives in Mathematical Logic. Springer-Verlag, Berlin (1975)CrossRefzbMATHGoogle Scholar
  2. 2.
    Blass, A.: Existential fixed-point logic, universal quantifiers, and topoi. In: Blass, A., Dershowitz, N., Reisig, W. (eds.) Fields of Logic and Computation. LNCS, vol. 6300, pp. 108–134. Springer, Heidelberg (2010) CrossRefGoogle Scholar
  3. 3.
    Blass, A., Gurevich, Y.: Existential fixed-point logic. In: Börger, E. (ed.) Computation Theory and Logic. LNCS, vol. 270, pp. 20–36. Springer, Heidelberg (1987)CrossRefGoogle Scholar
  4. 4.
    Grädel, E.: Capturing complexity classes by fragments of second-order logic. Theoret. Computer Sci. 101, 35–57 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Moschovakis, Y.N.: Elementary Induction on Abstract Structures. Studies in Logic and the Foundations of Mathematics. North-Holland, New York (1974) zbMATHGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  1. 1.Mathematics DepartmentUniversity of MichiganAnn ArborUSA

Personalised recommendations