66 research outputs found

    The Church Problem for Countable Ordinals

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    A fundamental theorem of Buchi and Landweber shows that the Church synthesis problem is computable. Buchi and Landweber reduced the Church Problem to problems about &#969;-games and used the determinacy of such games as one of the main tools to show its computability. We consider a natural generalization of the Church problem to countable ordinals and investigate games of arbitrary countable length. We prove that determinacy and decidability parts of the Bu}chi and Landweber theorem hold for all countable ordinals and that its full extension holds for all ordinals < \omega\^\omega

    The Church Synthesis Problem with Parameters

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    For a two-variable formula &psi;(X,Y) of Monadic Logic of Order (MLO) the Church Synthesis Problem concerns the existence and construction of an operator Y=F(X) such that &psi;(X,F(X)) is universally valid over Nat. B\"{u}chi and Landweber proved that the Church synthesis problem is decidable; moreover, they showed that if there is an operator F that solves the Church Synthesis Problem, then it can also be solved by an operator defined by a finite state automaton or equivalently by an MLO formula. We investigate a parameterized version of the Church synthesis problem. In this version &psi; might contain as a parameter a unary predicate P. We show that the Church synthesis problem for P is computable if and only if the monadic theory of is decidable. We prove that the B\"{u}chi-Landweber theorem can be extended only to ultimately periodic parameters. However, the MLO-definability part of the B\"{u}chi-Landweber theorem holds for the parameterized version of the Church synthesis problem

    The Church Synthesis Problem with Metric

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    Church\u27s Problem asks for the construction of a procedure which, given a logical specification S(I,O) between input strings I and output strings O, determines whether there exists an operator F that implements the specification in the sense that S(I,F(I)) holds for all inputs I. Buechi and Landweber gave a procedure to solve Church\u27s problem for MSO specifications and operators computable by finite-state automata. We consider extensions of Church\u27s problem in two orthogonal directions: (i) we address the problem in a more general logical setting, where not only the specifications but also the solutions are presented in a logical system; (ii) we consider not only the canonical discrete time domain of the natural numbers, but also the continuous domain of reals. We show that for every fixed bounded length interval of the reals, Church\u27s problem is decidable when specifications and implementations are described in the monadic second-order logics over the reals with order and the +1 function
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