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Domino games with an application to the complexity of boolean algebras with bounded quantifier alternations

  • Erich Grädel
Contributed Papers Complexity
Part of the Lecture Notes in Computer Science book series (LNCS, volume 294)

Abstract

We consider domino games which describe computations of alternating Turing machines in the same way as dominoes (tiling systems) encode computations of deterministic and nondeterministic Turing machines. The domino games are two person games in the course of which the players build up domino-tilings of a square of prescribed size. Acceptance of an alternating Turing machine corresponds to a winning strategy for one player — the number of moves in the game is the number of alternations of the Turing machine.

For complexity classes ATIME(T(n), A(n)) we find complete sets of domino games. In particular we present domino games which are complete in the classes Σ m p and ∏ m p of the polynomial time hierarchy. This corresponds to the approach of van Emde Boas and Lewis/Papadimitriou who showed that the theory of NP-completeness may also be founded on a finite domino problem instead of the satisfiability problem for propositional formulas.

Finally domino games are used as a tool to prove that the subclasses with bounded quantifier alternations in the theory of Boolean algebras have essentially the same complexity as the whole theory, in contrast to other decidable first order theories where a restriction of quantifier alternation leads to an exponential decrease of complexity.

Keywords

Boolean Algebra Turing Machine Winning Strategy Satisfiability Problem Quantifier Alternation 
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.

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References

  1. [1]
    R.Berger, The undecidability of the domino problem, Memoirs of the AMS 66, 1966Google Scholar
  2. [2]
    L. Berman, The complexity of logical theories, Theor. Comp. Sci. 11 (1980), 71–77Google Scholar
  3. [3]
    A.K. Chandra, D.C. Kozen & L. Stockmeyer, Alternation, J. ACM 28 (1981), 114–133Google Scholar
  4. [4]
    B. Chlebus, Domino-Tiling Games, J. Comp. System Sci. 32 (1986), 374–392Google Scholar
  5. [5]
    P. van Emde Boas, Dominoes are forever, Report 83-04, Department of Mathematics, University of Amsterdam 1983Google Scholar
  6. [6]
    M.Fürer, The computational complexity of the unconstrained limited domino problem (with implications for logical decision problems), in: Logic and Machines. Decision problems and complexity, Lecture Notes in Computer Science Nr.171, Springer 1984, 312–319Google Scholar
  7. [7]
    E.Grädel, The Complexity of Subclasses of Logical Theories, Dissertation Basel 1987Google Scholar
  8. [8]
    Y. Gurevich, The decision problem for standard classes, J. of Symbolic Logic 41 (1976), 460–464Google Scholar
  9. [9]
    D.Harel, Recurring dominoes: Making the highly undecidable highly understandable, in: Foundation of Computation Theory, Lecture Notes in Computer Science Nr. 158, Springer 1983, 177–194Google Scholar
  10. [10]
    A.S. Kahr, E.F. Moore and H. Wang, Entscheidungsproblem reduced to the ∀∃∀-case, Proc. Nat. Acad. Sci. USA 48 (1962), 365–377Google Scholar
  11. [11]
    D. Kozen, Complexity of Boolean algebras, Theor. Comp. Sci. 10 (1980), 221–247Google Scholar
  12. [12]
    H.R.Lewis, Unsolvable Classes of Quantificational Formulas, Reading 1979Google Scholar
  13. [13]
    H.R.Lewis, C.H.Papadimitriou, Elements of the Theory of Computation, Prentice-Hall 1981Google Scholar
  14. [14]
    E.D. Sontag, Real addition and the polynomial-time hierarchy, Inform. Process. Letters 20 (1985), 115–120Google Scholar
  15. [15]
    L. Stockmeyer, The polynomial-time hierarchy, Theor. Comp. Sci. 3 (1977), 1–22Google Scholar
  16. [16]
    A. Tarski, Arithmetical classes and types of Boolean algebras, Bull. AMS 55 (1949) 64, 1192Google Scholar
  17. [17]
    H. Wang, Proving theorems by pattern recognition II, The Bell System Technical Journal 40 (1961), 1–41Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1988

Authors and Affiliations

  • Erich Grädel
    • 1
  1. 1.Mathematisches Institut der Universität BaselBasel

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