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13
On decidability of monadic logic of order over the naturals extended by monadic predicates
, 2007
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Church Synthesis Problem with Parameters
"... Abstract. The following problem is known as the Church Synthesis problem: Input: an MLO formula ψ(X, Y). Task: Check whether there is an operator Y = F (X) such that ..."
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Abstract. The following problem is known as the Church Synthesis problem: Input: an MLO formula ψ(X, Y). Task: Check whether there is an operator Y = F (X) such that
Automatabased presentations of infinite structures
, 2009
"... The model theory of finite structures is intimately connected to various fields ..."
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The model theory of finite structures is intimately connected to various fields
Model Transformations in Decidability Proofs for Monadic Theories
"... Abstract. We survey two basic techniques for showing that the monadic secondorder theory of a structure is decidable. In the first approach, one deals with finite fragments of the theory (given for example by the restriction to formulas of a certain quantifier rank) and – depending on the fragment ..."
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Abstract. We survey two basic techniques for showing that the monadic secondorder theory of a structure is decidable. In the first approach, one deals with finite fragments of the theory (given for example by the restriction to formulas of a certain quantifier rank) and – depending on the fragment – reduces the model under consideration to a simpler one. In the second approach, one applies a global transformation of models while preserving decidability of the theory. We suggest a combination of these two methods. 1
Path Logics with Synchronization
"... Abstract. Over trees and partial orders, chain logic and path logic are systems of monadic secondorder logic in which secondorder quantification is applied to paths and to chains (i.e., subsets of paths), respectively; accordingly we speak of the path theory and the chain theory of a structure. We ..."
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Abstract. Over trees and partial orders, chain logic and path logic are systems of monadic secondorder logic in which secondorder quantification is applied to paths and to chains (i.e., subsets of paths), respectively; accordingly we speak of the path theory and the chain theory of a structure. We present some known and some new results on decidability of the path theory and chain theory of structures that are enhanced by features of synchronization between paths. We start with the infinite twodimensional grid for which the finitepath theory is shown to be undecidable. Then we consider the infinite binary tree expanded by the binary ”equal level predicate ” E. We recall the (known) decidability of the chain theory of a regular tree with the predicate E and observe that this does not extend to algebraic trees. Finally, we study refined models in which the time axis (represented by the sequence of tree levels) or the tree levels themselves are supplied with additional structure. 1
Parametrized Regular Infinite Games and HigherOrder Pushdown Strategies
"... Abstract. Given a set P of natural numbers, we consider infinite games where the winning condition is a regular ωlanguage parametrized by P. In this context, an ωword, representing a play, has letters consisting of three components: The first is a bit indicating membership of the current position ..."
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Abstract. Given a set P of natural numbers, we consider infinite games where the winning condition is a regular ωlanguage parametrized by P. In this context, an ωword, 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 kth level of the hierarchy, we provide an effective determinacy result and show that winning strategies can be implemented by deterministic levelk pushdown automata. 1
Infinite Games and Uniformization
"... Abstract. The problem of solvability of infinite games is closely connected with the classical question of uniformization of relations by functions of a given class. We work out this connection and discuss recent results on infinite games that are motivated by the uniformization problem. The fundame ..."
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Abstract. The problem of solvability of infinite games is closely connected with the classical question of uniformization of relations by functions of a given class. We work out this connection and discuss recent results on infinite games that are motivated by the uniformization problem. The fundamental problem in the effective theory of infinite games was posed by Church in 1957 (“Church’s Problem”; see [2,3]). It refers to twoplayer games in the sense of Gale and Stewart [6] in which the two players 1 and 2 build up two sequences α = a0a1..., respectively β = b0b1..., where ai,bi belong to a finite alphabet Σ. Player 1 picks a0, then Player 2 picks b0, then Player 1 picks a1, and so forth in alternation. A play is an ωword over Σ × Σ of the form () () () a0 a1 a2...; we also write α β. A game is specified by a relation b0 b1 b2 R ⊆ Σω × Σω, or equivalently by the ωlanguage LR = {α⌢β  (α,β) ∈ R}. Player 2 wins the play α⌢β if (α,β) ∈ R.
Decidable expansions of labelled linear orderings
, 2010
"... Let M =(A, <, P)where(A, <) is a linear ordering and P denotes a finite sequence of monadic predicates on A. We show that if A contains an interval of order type ω or −ω, and the monadic secondorder theory of M is decidable, then there exists a nontrivial expansion M ′ of M by a monadic pred ..."
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Let M =(A, <, P)where(A, <) is a linear ordering and P denotes a finite sequence of monadic predicates on A. We show that if A contains an interval of order type ω or −ω, and the monadic secondorder theory of M is decidable, then there exists a nontrivial expansion M ′ of M by a monadic predicate such that the monadic secondorder theory of M ′ is still decidable.
A Hierarchy of Automatic Words having a Decidable MSO Theory
, 2006
"... We investigate automatic presentations of infinite words. Starting points of our study are the works of Rigo and Maes, and Carton and Thomas concerning the lexicographic presentation, respectively the decidability of the MSO theory of morphic words. Refining their techniques we observe that the lexi ..."
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We investigate automatic presentations of infinite words. Starting points of our study are the works of Rigo and Maes, and Carton and Thomas concerning the lexicographic presentation, respectively the decidability of the MSO theory of morphic words. Refining their techniques we observe that the lexicographic presentation of a (morphic) word is canonical in a certain sense. We then go on to generalize our techniques to a hierarchy of classes of infinite words enjoying the above mentioned properties. We introduce klexicographic presentations, and morphisms of level k stacks and show that these are intertranslatable, thus giving rise to the same classes of klexicographic or level k morphic words. We prove that these presentations are also canonical, which implies decidability of the MSO theory of every klexicographic word as well as closure of these classes under restricted MSO interpretations, e.g. closure under deterministic sequential mappings. The classes of klexicographic words are shown to form an infinite hierarchy. 1
Decidable Extensions of Church’s Problem
"... Abstract. For a twovariable formula B(X,Y) of Monadic Logic of Order (MLO) the Church Synthesis Problem concerns the existence and construction of a finitestate operator Y=F(X) such that B(X,F(X)) is universally valid over Nat. Büchi and Landweber (1969) proved that the Church synthesis problem is ..."
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Abstract. For a twovariable formula B(X,Y) of Monadic Logic of Order (MLO) the Church Synthesis Problem concerns the existence and construction of a finitestate operator Y=F(X) such that B(X,F(X)) is universally valid over Nat. Büchi and Landweber (1969) proved that the Church synthesis problem is decidable. We investigate a parameterized version of the Church synthesis problem. In this extended version a formula B and a finitestate operator F might contain as a parameter a unary predicate P. A large class of predicates P is exhibited such that the Church problem with the parameter P is decidable. Our proofs use Composition Method and game theoretical techniques. 1