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 p m and ∏ p m 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.
Address after March 88: Dipartamento di Informatica, Università di Pisa, Corso Italia 40, I-56100 Pisa
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References
R.Berger, The undecidability of the domino problem, Memoirs of the AMS 66, 1966
L. Berman, The complexity of logical theories, Theor. Comp. Sci. 11 (1980), 71–77
A.K. Chandra, D.C. Kozen & L. Stockmeyer, Alternation, J. ACM 28 (1981), 114–133
B. Chlebus, Domino-Tiling Games, J. Comp. System Sci. 32 (1986), 374–392
P. van Emde Boas, Dominoes are forever, Report 83-04, Department of Mathematics, University of Amsterdam 1983
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, 312–319
E.Grädel, The Complexity of Subclasses of Logical Theories, Dissertation Basel 1987
Y. Gurevich, The decision problem for standard classes, J. of Symbolic Logic 41 (1976), 460–464
D.Harel, Recurring dominoes: Making the highly undecidable highly understandable, in: Foundation of Computation Theory, Lecture Notes in Computer Science Nr. 158, Springer 1983, 177–194
A.S. Kahr, E.F. Moore and H. Wang, Entscheidungsproblem reduced to the ∀∃∀-case, Proc. Nat. Acad. Sci. USA 48 (1962), 365–377
D. Kozen, Complexity of Boolean algebras, Theor. Comp. Sci. 10 (1980), 221–247
H.R.Lewis, Unsolvable Classes of Quantificational Formulas, Reading 1979
H.R.Lewis, C.H.Papadimitriou, Elements of the Theory of Computation, Prentice-Hall 1981
E.D. Sontag, Real addition and the polynomial-time hierarchy, Inform. Process. Letters 20 (1985), 115–120
L. Stockmeyer, The polynomial-time hierarchy, Theor. Comp. Sci. 3 (1977), 1–22
A. Tarski, Arithmetical classes and types of Boolean algebras, Bull. AMS 55 (1949) 64, 1192
H. Wang, Proving theorems by pattern recognition II, The Bell System Technical Journal 40 (1961), 1–41
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© 1988 Springer-Verlag Berlin Heidelberg
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Grädel, E. (1988). Domino games with an application to the complexity of boolean algebras with bounded quantifier alternations. In: Cori, R., Wirsing, M. (eds) STACS 88. STACS 1988. Lecture Notes in Computer Science, vol 294. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0035836
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DOI: https://doi.org/10.1007/BFb0035836
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