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Journal ArticleDOI

The Church Synthesis Problem with Parameters

14 Nov 2007-Logical Methods in Computer Science (Centre pour la Communication Scientifique Directe (CCSD))-Vol. 3, Iss: 4
TL;DR: For a two-variable formula ψ(X,Y) of Monadic Logic of Order (MLO), the authors showed that the Church synthesis problem for P is computable if and only if the monadic theory of fixme is decidable.
Abstract: For a two-variable formula ψ(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 ψ(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 ψ 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.

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Citations
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Book ChapterDOI
02 Sep 2009
TL;DR: For such a predicate P where (N, +1, P) occurs in the k-th level of the hierarchy, an effective determinacy result is provided and it is shown that winning strategies can be implemented by deterministic level-k pushdown automata.
Abstract: Given a set P of natural numbers, we consider infinite games where the winning condition is a regular ω-language parametrized by P. In this context, an ω-word, representing a play, has letters consisting of three components: The first is a bit indicating membership of the current position in P, and the other two components are the letters contributed by the two players. Extending recent work of Rabinovich we study here predicates P where the structure (N, +1, P) belongs to the pushdown hierarchy (or "Caucal hierarchy"). For such a predicate P where (N, +1, P) occurs in the k-th level of the hierarchy, we provide an effective determinacy result and show that winning strategies can be implemented by deterministic level-k pushdown automata.

30 citations

Proceedings ArticleDOI
11 Jul 2010
TL;DR: The main result of this paper is that, over bounded time domains, language emptiness for alternating timed automata is decidable (but nonelementary).
Abstract: Alternating timed automata are a powerful extension of classical Alur-Dill timed automata that are closed under all Boolean operations. They have played a key role, among others, in providing verification algorithms for prominent specification formalisms such as Metric Temporal Logic. Unfortunately, when interpreted over an infinite dense time domain (such as the reals), alternating time dautomata have an undecidable language emptiness problem. The main result of this paper is that, over bounded time domains, language emptiness for alternating timed automata is decidable (but nonelementary). The proof involves showing decidability of a class of parametric McNaughton games that are played over timed words and that have winning conditions expressed in the monadic logic of order augmented with the distance-one relation. As a corollary, we establish the decidability of the time-bounded model-checking problem for Alur-Dill timed automata against specifications expressed as alternating timed automata.

25 citations

Journal ArticleDOI
TL;DR: The synthesis problem for Halpern and Shoham's modal logic of intervals extended with an equivalence relation over time points, abbreviated HSeq, is introduced and it is proved that the fragment ABBbareq is decidable (non-primitive recursive hard), while the fragment AAbarBBbar turns out to be undecidable.
Abstract: We introduce the synthesis problem for Halpern and Shoham's modal logic of intervals extended with an equivalence relation over time points, abbreviated HSeq. In analogy to the case of monadic second-order logic of one successor, the considered synthesis problem receives as input an HSeq formula phi and a finite set Sigma of propositional variables and temporal requests, and it establishes whether or not, for all possible evaluations of elements in Sigma in every interval structure, there exists an evaluation of the remaining propositional variables and temporal requests such that the resulting structure is a model for phi. We focus our attention on decidability of the synthesis problem for some meaningful fragments of HSeq, whose modalities are drawn from the set A (meets), Abar (met by), B (begins), Bbar (begun by), interpreted over finite linear orders and natural numbers. We prove that the fragment ABBbareq is decidable (non-primitive recursive hard), while the fragment AAbarBBbar turns out to be undecidable. In addition, we show that even the synthesis problem for ABBbar becomes undecidable if we replace finite linear orders by natural numbers.

5 citations

Proceedings ArticleDOI
01 Jan 2011
TL;DR: It is shown that for every fixed bounded length interval of the reals, Church's problem is decidable when specifications and implementations are described in the monadic second-order logics over the real with order and the +1 function.
Abstract: Church's 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's problem for MSO specifications and operators computable by finite-state automata. We consider extensions of Church's 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's problem is decidable when specifications and implementations are described in the monadic second-order logics over the reals with order and the +1 function.

4 citations

Journal ArticleDOI
24 Aug 2014
TL;DR: In this paper, the authors introduce the synthesis problem for Halpern and Shoham's modal logic of intervals extended with an equivalence relation over time points, abbreviated HSeq.
Abstract: We introduce the synthesis problem for Halpern and Shoham's modal logic of intervals extended with an equivalence relation over time points, abbreviated HSeq. In analogy to the case of monadic second-order logic of one successor, the considered synthesis problem receives as input an HSeq formula phi and a finite set Sigma of propositional variables and temporal requests, and it establishes whether or not, for all possible evaluations of elements in Sigma in every interval structure, there exists an evaluation of the remaining propositional variables and temporal requests such that the resulting structure is a model for phi. We focus our attention on decidability of the synthesis problem for some meaningful fragments of HSeq, whose modalities are drawn from the set A (meets), Abar (met by), B (begins), Bbar (begun by), interpreted over finite linear orders and natural numbers. We prove that the fragment ABBbareq is decidable (non-primitive recursive hard), while the fragment AAbarBBbar turns out to be undecidable. In addition, we show that even the synthesis problem for ABBbar becomes undecidable if we replace finite linear orders by natural numbers.

4 citations