Homomorphism preservation theorems

Rossman, Benjamin

doi:10.1145/1379759.1379763

Cited by 103 publications

(69 citation statements)

References 35 publications

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“…This long open problem was solved by Benjamin Rossman [20], with very sophisticated finitary saturation methods. These also fall in the category of constructive finitary counterparts of classical saturation arguments that we are interested in.…”

Section: Two Examplesmentioning

confidence: 99%

“…The obvious differences have often been stressed: classical expressive completeness results for fragments of first-order logic are typically compactness-based; alternative, more constructive and combinatorial arguments are required in those cases where expressive completeness results can be established in finite model theory. Meanwhile the growing number of qualified expressive completeness results in finite model theory [19,15,16,17,20,10,3,4,18] calls for a re-assessment of the earlier primarily negative view that focused on "failures in the finite" in comparison to the well-known classical preservation theorems. Also the major methodological differences may have hidden some interesting commonality that does prevail more often than had first been appreciated.…”

Section: Introductionmentioning

confidence: 99%

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Expressive completeness through logically tractable models

Otto¹

2013

Annals of Pure and Applied Logic

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How can we prove that some fragment of a given logic has the power to define precisely all structural properties that satisfy some characteristic semantic preservation condition? This issue is a fundamental one for classical model theory and applications in non-classical settings alike. While methods differ greatly, and while the classical methods can usually not be matched for instance in the setting of finite model theory, this note surveys some interesting commonality revolving around the use and availability of tractable representatives in the relevant model classes. The construction of models in which simple invariants like partial types based on some weak fragment control all the relevant structural properties, may be seen at the heart of such questions. We highlight some constructions involving degrees of acyclicity and saturation that can be achieved in finite model constructions, and discuss their uses towards expressive completeness w.r.t. bisimulation based equivalences in hypergraphs and relational structures. The emphasis is on the combinatorial challenges in such more constructive approaches that work in non-classical settings and especially in finite model theory. One new result concerns expressive completeness w.r.t. guarded negation bisimulation, a back-and-forth equivalence involving local homomorphisms.

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Section: Two Examplesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Expressive completeness through logically tractable models

Otto¹

2013

Annals of Pure and Applied Logic

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“…As a matter of fact, we give two different proofs of this result. The first proof entails combining the main technical result in Rossman's proof of the preservationunder-homomorphisms theorem in the finite [Ros08] with a result from the full, unpublished version of [ABFL04] to the effect that the chase transformation for GLAV mappings is local in a sense of first-order equivalence. This proof yields an N that is a stack of exponentials in n, because this type of blow-up already occurs in Rossman's proof [Ros08], and no smaller bounds are presently known.…”

Section: Introductionmentioning

confidence: 99%

“…The first proof entails combining the main technical result in Rossman's proof of the preservationunder-homomorphisms theorem in the finite [Ros08] with a result from the full, unpublished version of [ABFL04] to the effect that the chase transformation for GLAV mappings is local in a sense of first-order equivalence. This proof yields an N that is a stack of exponentials in n, because this type of blow-up already occurs in Rossman's proof [Ros08], and no smaller bounds are presently known. We therefore give a different and direct proof of the CQlocality of the chase procedure for a GLAV mapping that also yields an N that is bounded by a polynomial in the size of n. In fact, the degree of the polynomial is equal to the maximum arity of the relation symbols of the target schema.…”

Section: Introductionmentioning

confidence: 99%

Local transformations and conjunctive-query equivalence

Fagin

Kolaitis

2012

Proceedings of the 31st ACM SIGMOD-SIGACT-SIGAI Symposium on Principles of Database Systems

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Over the past several decades, the study of conjunctive queries has occupied a central place in the theory and practice of database systems. In recent years, conjunctive queries have played a prominent role in the design and use of schema mappings for data integration and data exchange tasks. In this paper, we investigate several different aspects of conjunctive-query equivalence in the context of schema mappings and data exchange.In the first part of the paper, we introduce and study a notion of a local transformation between database instances that is based on conjunctive-query equivalence. We show that the chase procedure for GLAV mappings (that is, schema mappings specified by sourceto-target tuple-generating dependencies) is a local transformation with respect to conjunctive-query equivalence. This means that the chase procedure preserves bounded conjunctive-query equivalence, that is, if two source instances are indistinguishable using conjunctive queries of a sufficiently large size, then the target instances obtained by chasing these two source instances are also indistinguishable using conjunctive queries of a given size. Moreover, we obtain polynomial bounds on the level of indistinguishability between source instances needed to guarantee indistinguishability between the target instances produced by the chase. The locality of the chase extends to schema mappings specified by a secondorder tuple-generating dependency (SO tgd), but does not hold for schema mappings whose specification includes target constraints.In the second part of the paper, we take a closer look at the composition of two GLAV mappings. In particular, we break GLAV mappings into a small number of well-studied classes (including LAV and GAV), and complete the picture as to when the composition of schema mappings from these various classes can be guaranteed to be a GLAV mapping, and when they can be guaranteed to be conjunctive-query equivalent to a GLAV mapping.We also show that the following problem is decidable: given a schema mapping specified by an SO tgd and a GLAV mapping, are they conjunctive-query equivalent? In contrast, the following problem is known to be undecidable: given a schema mapping specified by an SO tgd and a GLAV mapping, are they logically equivalent? Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. To copy otherwise, to republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee.

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“…[EF95,Lib04] and the survey of the uses of the latter can be found in [Mak04]. For other failures of classical theorems of First Order Logic when restricted to the finite case, such as the compactness theorem and various preservation theorems, see Y. Gurevich [Gur88] and B. Rossman [Ros08].…”

Section: Model Theorymentioning

confidence: 99%

From Hilbert’s Program to a Logic Toolbox

Makowsky

Logic for Programming, Artificial Intelligence, and Reasoning

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Abstract. In this paper I discuss what, according to my long experience, every computer scientists should know from logic. We concentrate on issues of modeling, interpretability and levels of abstraction. We discuss what the minimal toolbox of logic tools should look like for a computer scientist who is involved in designing and analyzing reliable systems. We shall conclude that many classical topics dear to logicians are less important than usually presented, and that less known ideas from logic may be more useful for the working computer scientist.For Witek Marek, first mentor, then colleague and true friend, on the occasion of his 65th birthday.

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Homomorphism preservation theorems

Cited by 103 publications

References 35 publications

Expressive completeness through logically tractable models

Expressive completeness through logically tractable models

Local transformations and conjunctive-query equivalence

From Hilbert’s Program to a Logic Toolbox

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