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Modal Languages And Bounded Fragments Of Predicate Logic
, 1996
"... Model Theory. These are nonempty families I of partial isomorphisms between models M and N , closed under taking restrictions to smaller domains, and satisfying the usual BackandForth properties for extension with objects on either side  restricted to apply only to partial isomorphisms of size ..."
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Cited by 210 (12 self)
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Model Theory. These are nonempty families I of partial isomorphisms between models M and N , closed under taking restrictions to smaller domains, and satisfying the usual BackandForth properties for extension with objects on either side  restricted to apply only to partial isomorphisms of size at most k . 'Invariance for kpartial isomorphism' means having the same truth value at tuples of objects in any two models that are connected by a partial isomorphism in such a set. The precise sense of this is spelt out in the following proof. 21 Proof (Outline.) kvariable formulas are preserved under partial isomorphism, by a simple induction. More precisely, one proves, for any assignment A and any partial isomorphism IÎI which is defined on the Avalues for all variables x 1 , ..., x k , that M, A = f iff N , IoA = f . The crucial step in the induction is the quantifier case. Quantified variables are irrelevant to the assignment, so that the relevant partial isomorphism can be res...
Definability with Bounded Number of Bound Variables
 Information and Computation
, 1989
"... A theory satisfies the kvariable property if every firstorder formula is equivalent to a formula with at most k bound variables (possibly reused). Gabbay has shown that a model of temporal logic satisfies the kvariable property for some k if and only if there exists a finite basis for the tempora ..."
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Cited by 76 (5 self)
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A theory satisfies the kvariable property if every firstorder formula is equivalent to a formula with at most k bound variables (possibly reused). Gabbay has shown that a model of temporal logic satisfies the kvariable property for some k if and only if there exists a finite basis for the temporal connectives over that model. We give a modeltheoretic method for establishing the kvariable property, involving a restricted EhrenfeuchtFraisse game in which each player has only k pebbles. We use the method to unify and simplify results in the literature for linear orders. We also establish new kvariable properties for various theories of boundeddegree trees, and in each case obtain tight upper and lower bounds on k. This gives the first finite basis theorems for branchingtime models of temporal logic. 1 Introduction A firstorder theory \Sigma satisfies the kvariable property if every firstorder formula is equivalent under \Sigma to a formula with at most k bound variables (pos...
On the Decision Problem for TwoVariable FirstOrder Logic
, 1997
"... We identify the computational complexity of the satisfiability problem for FO², the fragment of firstorder logic consisting of all relational firstorder sentences with at most two distinct variables. Although this fragment was shown to be decidable a long time ago, the computational complexity ..."
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Cited by 48 (1 self)
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We identify the computational complexity of the satisfiability problem for FO², the fragment of firstorder logic consisting of all relational firstorder sentences with at most two distinct variables. Although this fragment was shown to be decidable a long time ago, the computational complexity of its decision problem has not been pinpointed so far. In 1975 Mortimer proved that FO² has the finitemodel property, which means that if an FO²sentence is satisfiable, then it has a finite model. Moreover, Mortimer showed that every satisfiable FO²sentence has a model whose size is at most doubly exponential in the size of the sentence. In this paper, we improve Mortimer's bound by one exponential and show that every satisfiable FO²sentence has a model whose size is at most exponential in the size of the sentence. As a consequence, we establish that the satisfiability problem for FO² is NEXPTIMEcomplete.
Back and Forth Between Modal Logic and Classical Logic
, 1994
"... Model Theory. That is, we have a nonempty family I of partial isomorphisms between two models M and N, which is closed under taking restrictions to smaller domains, and where the standard BackandForth properties are now restricted to apply only to partial isomorphisms of size at most k. Proof. ..."
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Cited by 30 (3 self)
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Model Theory. That is, we have a nonempty family I of partial isomorphisms between two models M and N, which is closed under taking restrictions to smaller domains, and where the standard BackandForth properties are now restricted to apply only to partial isomorphisms of size at most k. Proof. (A complete argument is in [16].) An outline is reproduced here, for convenience. First, kvariable formulas are preserved under partial isomorphism, by a simple induction. More precisely, one proves, for any assignment A and any partial isomorphism I 2 I which is defined on the Avalues for all variables x 1 ; : : : ; x k , that M;A j= OE iff N; I ffi A j= OE: The crucial step in the induction is the quantifier case. Quantified variables are irrelevant to the assignment, so that the relevant partial isomorphism can be restricted to size at most k \Gamma 1, whence a matching choice for the witness can be made on the opposite side. This proves "only if". Next, "if" has a proof analogous to...
On the Relationship between Description Logic and Predicate Logic Queries
 In Proceedings of CIKM94
, 1994
"... Description Languages (DLs) are descendants of the klone [15] knowledge representation system, and form the basis of several objectcentered knowledge base management systems developed in recent years, including ones in industrial use. Originally used for conceptual modeling (to define views), DLs ..."
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Cited by 18 (2 self)
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Description Languages (DLs) are descendants of the klone [15] knowledge representation system, and form the basis of several objectcentered knowledge base management systems developed in recent years, including ones in industrial use. Originally used for conceptual modeling (to define views), DLs are seeing increased use as query languages for retrieving information. This paper, aimed at a general audience that includes database researchers, considers the relationship between the expressive power of DLs and that of query languages based on Predicate Calculus. We show that all descriptions built using constructors currently considered in the literature can be expressed as formulae of the First Order Predicate Calculus with at most three variable symbols, though we have to allow numeric quantifiers and infinitary disjunction in order to handle a couple of special constructors. Conversely, we show that all firstorder queries (formulae with one free variable) built up from unary and bin...
Local Variations on a Loose Theme: Modal Logic and Decidability
"... This chapter is about decidability and complexity issues in modal logic; more specifically, we confine ourselves to satisfiability (and the complementary validity) problems. The satisfiability problem is the following: for a fixed class of ..."
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Cited by 8 (1 self)
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This chapter is about decidability and complexity issues in modal logic; more specifically, we confine ourselves to satisfiability (and the complementary validity) problems. The satisfiability problem is the following: for a fixed class of
NonRepresentable Algebras of Relations
, 1997
"... this dissertation. More precisely, we are referring to what is called the orthodox version of these logics in these works ..."
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Cited by 4 (0 self)
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this dissertation. More precisely, we are referring to what is called the orthodox version of these logics in these works
EPIMORPHISMS IN CYLINDRIC ALGEBRAS AND DEFINABILITY IN FINITE VARIABLE LOGIC
, 2008
"... Abstract. The main result gives a sufficient condition for a class K of finite dimensional cylindric algebras to have the property that not every epimorphism in K is surjective. In particular, not all epimorphisms are surjective in the classes CAn of ndimensional cylindric algebras and the class of ..."
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Abstract. The main result gives a sufficient condition for a class K of finite dimensional cylindric algebras to have the property that not every epimorphism in K is surjective. In particular, not all epimorphisms are surjective in the classes CAn of ndimensional cylindric algebras and the class of representable algebras in CAn for finite n> 1, solving Problem 10 of [28] for finite n. By a result of Németi, this shows that the Bethdefinability property fails for the finitevariable fragments of first order logic as long as the number n of variables available is> 1 and we allow models of size ≥ n + 2, but holds if we allow only models of size ≤ n + 1. We also use our results in the present paper to prove that several results in the literature concerning injective algebras and definability of polyadic operations in CAn are best possible. We raise several open problems. §0. INTRODUCTION AND THE MAIN RESULTS In algebra, the properties of epimorphisms (in the categorial sense) being surjective and the amalgamation property in a class of algebras are well investigated, see e.g. [1] and [37]. In algebraic logic these properties turn out