Computational Complexity of Ehrenfeucht-Fraïssé Games on Finite Structures

  • Elena Pezzoli
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1584)


We show that deciding the winner of the r-moves Ehrenfeucht-Fraïssé game on two finite structures A and B, over any fixed signature Σ that contains at least one binary and one ternary relation, is PSPACE complete. We consider two natural modifications of the EF game, the one-sided r-moves EF game, where the spoiler can choose from the first structure A only, and therefore the duplicator wins only if B satisfies all the existential formulas of rank at most r that A satisfies; and the k-alternations r-moves EF game (for each fixed k), where the spoiler can choose from either structure, but he can switch structure at most k times, and therefore the duplicator wins iff A and B satisfy the same first order formulas of rank at most r and quantifier alternation at most k (defined in the paper). We show that deciding the winner in both the one-sided EF game and the k-alternations EF game is also PSPACE complete.


Truth Assignment Unary Relation Boolean Formula Isomorphism Problem Additional Vertex 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Ebbinghaus, H.-D., Flum, J.: Finite Model Theory. Perspectives in Mathematical Logic. Springer, Berlin (1995)zbMATHGoogle Scholar
  2. 2.
    Pezzoli, E.: On the computational complexity of type-two functionals and logical games on finite structures. Ph.D thesis. Stanford University, Stanford (June 1998)Google Scholar
  3. 3.
    Pezzoli, E.: Classical paramteric complexity of logical games (in preparation)Google Scholar
  4. 4.
    Grohe, M.: Equivalence in finite-variable logics is complete for polynomial time. In: Proceeding FOCS 1996 (1996)Google Scholar
  5. 5.
    Immerman, N., Kozen, D.: Definability with a bounded number of variables. Information and Computation 83, 121–139 (1989)zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Köbler, J., Schöning, U., Torán, J.: The Graph Isomorphism Problem. Birkhäuser (1993)Google Scholar
  7. 7.
    Stockmeyer, L.J., Meyer, A.R.: Word problems requiring exponential time. In: Fifth Annual ACM Symposium on theory of computing, Association for Computing Machinery, New york, pp. 1–9 (1973)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1999

Authors and Affiliations

  • Elena Pezzoli
    • 1
  1. 1.Boston CollegeChesnut HillUSA

Personalised recommendations