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Homomorphism preservation theorems

Published: 06 August 2008 Publication History

Abstract

The homomorphism preservation theorem (h.p.t.), a result in classical model theory, states that a first-order formula is preserved under homomorphisms on all structures (finite and infinite) if and only if it is equivalent to an existential-positive formula. Answering a long-standing 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 Łoś--Tarski theorem and Lyndon's positivity theorem). Applications of this result extend to constraint satisfaction problems and to database theory via a correspondence between existential-positive formulas and unions of conjunctive queries. A further result of this article strengthens the classical h.p.t.: we show that a first-order formula is preserved under homomorphisms on all structures if and only if it is equivalent to an existential-positive formula of equal quantifier-rank.

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Jan De Beule

Model theory is the abstract study of mathematical structures that satisfy axioms stated using a first-order logic. A general question that is investigated in model theory is whether a property of a structure, expressed as logical sentences, is true in all structures that are related to the given structure, where this relation is expressed by an abstract homomorphism. Many important theorems in this field are only valid for infinite structures. Rossman shows in this paper that the homomorphism preservation theorem "remains valid when restricted to finite structures." The main result of the paper is stated as follows: A first-order sentence of quantifier-rank n is preserved under homomorphisms on finite structures if, and only if, it is equivalent in the finite to an existential-positive sentence of quantifier-rank __?__(n). This result answers an important open question in finite model theory. The paper contains a very detailed description of all necessary ingredients of and steps toward the proof of the main theorem. Related results in infinite model theory are clearly described. Possible extensions and related open problems are mentioned. Online Computing Reviews Service

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cover image Journal of the ACM
Journal of the ACM  Volume 55, Issue 3
July 2008
154 pages
ISSN:0004-5411
EISSN:1557-735X
DOI:10.1145/1379759
Issue’s Table of Contents
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. Copyrights for components of this work owned by others than ACM must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from [email protected]

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Association for Computing Machinery

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Publication History

Published: 06 August 2008
Accepted: 01 April 2008
Revised: 01 February 2008
Received: 01 May 2007
Published in JACM Volume 55, Issue 3

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Author Tags

  1. Finite model theory
  2. conjunctive queries
  3. first-order logic
  4. homomorphisms
  5. preservation theorems
  6. quantifier-rank
  7. tree-depth

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  • (2024)FORBIDDEN INDUCED SUBGRAPHS AND THE ŁOŚ–TARSKI THEOREMThe Journal of Symbolic Logic10.1017/jsl.2023.99(1-33)Online publication date: 4-Jan-2024
  • (2024)A logic-based framework for characterizing nexus of similarity within knowledge basesInformation Sciences: an International Journal10.1016/j.ins.2024.120331664:COnline publication date: 1-Apr-2024
  • (2024)Arboreal categories and equi-resource homomorphism preservation theoremsAnnals of Pure and Applied Logic10.1016/j.apal.2024.103423175:6(103423)Online publication date: Jun-2024
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  • (2022)Language-aware Indexing for Conjunctive Path Queries2022 IEEE 38th International Conference on Data Engineering (ICDE)10.1109/ICDE53745.2022.00054(661-673)Online publication date: May-2022
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