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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...
Homomorphism Preservation Theorems
, 2008
"... The homomorphism preservation theorem (h.p.t.), a result in classical model theory, states that a firstorder formula is preserved under homomorphisms on all structures (finite and infinite) if and only if it is equivalent to an existentialpositive formula. Answering a longstanding question in fin ..."
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The homomorphism preservation theorem (h.p.t.), a result in classical model theory, states that a firstorder formula is preserved under homomorphisms on all structures (finite and infinite) if and only if it is equivalent to an existentialpositive formula. Answering a longstanding question in finite model theory, we prove that the h.p.t. remains valid when restricted to finite structures (unlike many other classical preservation theorems, including the ̷Lo´sTarski theorem and Lyndon’s positivity theorem). Applications of this result extend to constraint satisfaction problems and to database theory via a correspondence between existentialpositive formulas and unions of conjunctive queries. A further result of this article strengthens the classical h.p.t.: we show that a firstorder formula is preserved under homomorphisms on all structures if and only if it is equivalent to an existentialpositive formula of equal quantifierrank.
Syntax vs. Semantics on Finite Structures
 Structures in Logic and Computer Science. A Selection of Essays in Honor of A. Ehrenfeucht
, 1997
"... . Logic preservation theorems often have the form of a syntax /semantics correspondence. For example, the / LosTarski theorem asserts that a firstorder sentence is preserved by extensions if and only if it is equivalent to an existential sentence. Many of these correspondences break when one restr ..."
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Cited by 9 (1 self)
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. Logic preservation theorems often have the form of a syntax /semantics correspondence. For example, the / LosTarski theorem asserts that a firstorder sentence is preserved by extensions if and only if it is equivalent to an existential sentence. Many of these correspondences break when one restricts attention to finite models. In such a case, one may attempt to find a new semantical characterization of the old syntactical property or a new syntactical characterization of the old semantical property. The goal of this paper is to provoke such a study. 1 Introduction It is well known that famous theorems about firstorder logic fail in the case when only finite structures are allowed (see, for example, [?]). A more careful examination shows that it is wrong to lump all these failing theorems together. On one side we have theorems like completeness or compactness where the failure is really and truly hopeless. On the other side there are theorems like the / LosTarski theorem, which we...
On Preservation Theorems for TwoVariable Logic
 MATHEMATICAL LOGIC QUARTERLY
, 1999
"... We show that the existential preservation theorem fails for FO², twovariable firstorder logic. It is known that for all k 3, FO^k does not have an existential preservation theorem, so this settles the last open case, answering a question of Andr'eka, van Benthem, and N'emeti. In contrast, we pro ..."
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We show that the existential preservation theorem fails for FO², twovariable firstorder logic. It is known that for all k 3, FO^k does not have an existential preservation theorem, so this settles the last open case, answering a question of Andr'eka, van Benthem, and N'emeti. In contrast, we prove that the homomorphism preservation theorem holds for FO².
Homomorphisms and FirstOrder Logic
, 2007
"... We prove that the homomorphism preservation theorem (h.p.t.), a classical result of mathematical logic, holds when restricted to finite structures. That is, a firstorder formula is preserved under homomorphisms on finite structures if, and only if, it is equivalent in the finite to an existentialp ..."
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We prove that the homomorphism preservation theorem (h.p.t.), a classical result of mathematical logic, holds when restricted to finite structures. That is, a firstorder formula is preserved under homomorphisms on finite structures if, and only if, it is equivalent in the finite to an existentialpositive formula. This result, which contrasts with the known failure of other classical preservation theorems on finite structures, answers a longstanding question in finite model theory. The relevance of this result, however, extends beyond logic to areas of computer science, including constraint satisfaction problems and database theory; the database connection arises from a correspondence between existentialpositive formulas and unions of conjunctive queries (also known as selectprojectjoinunion queries). A second result of this article strengthens the classical h.p.t. by showing that a firstorder formula is preserved under homomorphisms on all structures if, and only if, it is equivalent to an existentialpositive formula of equal quantifierrank. Unlike traditional proofs of the classical h.p.t., the proof of this stronger “equirank ” theorem is compactnessfree and constructive. While these results are logical in nature, the technical development of the article takes place almost entirely within a combinatorial framework. The concept of treedepth, a graph parameter related to treewidth, plays an important role in our analysis (as a combinatorial counterpart to quantifierrank). We introduce new notions of nhomomorphism and ncore, which approximate the familiar concepts of homomorphism and core “up to treedepth n”. The key technical lemmas take a pair of nhomomorphically equivalent [finite] relational structures and construct corresponding [finite] coretracts which satisfy a certain backandforth property.
kuniversal finite graphs
 Logic and Random Structures
, 1997
"... Abstract. This paper investigates the class of kuniversal finite graphs, a local analog of the class of universal graphs, which arises naturally in the study of finite variable logics. The main results of the paper, which are due to Shelah, establish that the class of kuniversal graphs is not defi ..."
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Abstract. This paper investigates the class of kuniversal finite graphs, a local analog of the class of universal graphs, which arises naturally in the study of finite variable logics. The main results of the paper, which are due to Shelah, establish that the class of kuniversal graphs is not definable by an infinite disjunction of firstorder existential sentences with a finite number of variables and that there exist kuniversal graphs with no kextendible induced subgraphs. 1.
Finite Models and Finitely Many Variables
 Banach Center Publications
, 1999
"... We consider L  first order logic restricted to k variables, and interpreted in finite structures. The study of classes of finite structures axiomatisable with finitely many variables has assumed importance through connections with computational complexity. In particular, we investigate the relation ..."
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We consider L  first order logic restricted to k variables, and interpreted in finite structures. The study of classes of finite structures axiomatisable with finitely many variables has assumed importance through connections with computational complexity. In particular, we investigate the relationship between the size of a finite structure and the number of distinct types it realizes, with respect to L . Some open questions, formulated as finitary LöwenheimSkolem properties, are presented regarding this relationship. This is also investigated through finitary versions of an EhrenfeuchtMostowski property.
Extension preservation theorems on classes of acyclic finite structures
"... A class of structures satisfies the extension preservation theorem if, on this class, every first order sentence is preserved under extension iff it is equivalent to an existential sentence. We consider different acyclicity notions for hypergraphs (γ, β and αacyclicity and also acyclicity on hyperg ..."
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A class of structures satisfies the extension preservation theorem if, on this class, every first order sentence is preserved under extension iff it is equivalent to an existential sentence. We consider different acyclicity notions for hypergraphs (γ, β and αacyclicity and also acyclicity on hypergraph quotients) and estimate their influence on the validity of the extension preservation theorem on classes of finite structures. More precisely, we prove that γacyclic classes (with some closure properties) satisfy the extension preservation theorem, whereas βacyclic classes do not. We also extend the validity of the extension preservation theorem for a generalization of γacyclicity that we call γacyclic kquotient. To achieve this, we make a reduction from finite structures to their kquotients and we use combinatorial arguments on γacyclic hypergraphs. 1