Domino games with an application to the complexity of boolean algebras with bounded quantifier alternations
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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.
KeywordsBoolean Algebra Turing Machine Winning Strategy Satisfiability Problem Quantifier Alternation
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