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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.
Homomorphism Closed vs. Existential Positive
 In Proc. of the 18th IEEE Symp. on Logic in Computer Science
, 2003
"... Preservations theorems, which establish connection between syntactic and semantic properties of formulas, are a major topic of investigation in model theory. In the context of finitemodel theory, most, but not all, preservation theorems are known to fail. It is not known, however, whether the L/os ..."
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Preservations theorems, which establish connection between syntactic and semantic properties of formulas, are a major topic of investigation in model theory. In the context of finitemodel theory, most, but not all, preservation theorems are known to fail. It is not known, however, whether the L/osTarskiLyndon Theorem, which asserts that a 1storder sentence is preserved under homomorphisms iff it is equivalent to an existential positive sentence, holds with respect to finite structures. Resolving this is an important open question in finitemodel theory.
Reasoning about Consistency in Model Merging
"... Models undergo a variety of transformations throughout development. One of the key transformations is merge, used when developers need to combine a set of models with respect to the overlaps between them. A major question about model transformations in general, and merge in particular, is what consi ..."
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Models undergo a variety of transformations throughout development. One of the key transformations is merge, used when developers need to combine a set of models with respect to the overlaps between them. A major question about model transformations in general, and merge in particular, is what consistency properties are preserved across the transformations and what consistency properties may need to be rechecked (and if necessary, reestablished) over the result. In previous work [18], we developed a technique based on categorytheoretic colimits for merging sets of interrelated models. The use of category theory leads to the preservation of the algebraic structure of the source models in the merge; however, this does not directly provide a characterization of the (in)consistency properties that carry over from the source models to the result, because consistency properties are predominantly expressed as logical formulas. Hence, an investigation of the connections between the “algebraic ” and “logical ” properties of model merging became necessary. In this paper, we undertake such an investigation and use techniques from finite model theory [9] to show that the use of colimits indeed leads to the preservation of certain logical properties. Our results have implications beyond the merge framework in [18] and are potentially useful for the broad range of techniques in the graph transformation and algebraic specification literature that use colimits as the basis for model manipulations.
Tarski’s influence on computer science
"... The following is the text of an invited lecture for the LICS 2005 meeting held in ..."
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The following is the text of an invited lecture for the LICS 2005 meeting held in
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
SOME CONNECTIONS BETWEEN FINITE AND INFINITE MODEL THEORY
"... Most of the work in model theory has, so far, considered infinite structures and the methods and results that have been worked out in this context can usually not be transferred to the study of finite structures in any obvious way; in addition, some basic results from infinite model theory fail with ..."
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Most of the work in model theory has, so far, considered infinite structures and the methods and results that have been worked out in this context can usually not be transferred to the study of finite structures in any obvious way; in addition, some basic results from infinite model theory fail within the context of finite models. The theory about finite structures has
On Preservation under Homomorphisms and . . .
, 2006
"... Unions of conjunctive queries, also known as selectprojectjoinunion queries, are the most frequently asked queries in relational database systems. These queries are definable by existential positive firstorder formulas and are preserved under homomorphisms. A classical result of mathematical log ..."
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Unions of conjunctive queries, also known as selectprojectjoinunion queries, are the most frequently asked queries in relational database systems. These queries are definable by existential positive firstorder formulas and are preserved under homomorphisms. A classical result of mathematical logic asserts that the existential positive formulas are the only firstorder formulas (up to logical equivalence) that are preserved under homomorphisms on all structures, finite and infinite. After resisting resolution for a long time, it was eventually shown that, unlike other classical preservation theorems, the homomorphismpreservation theorem holds for the class of all finite structures. In this paper, we show that the homomorphismpreservation theorem holds also for several restricted classes of finite structures of interest in graph theory and database theory. Specifically, we show that this result holds for all classes of finite structures of bounded degree, all classes of finite structures of bounded treewidth, and, more generally, all classes of finite structures whose cores exclude at least one minor.