research-article

The Exact Complexity of the First-Order Logic Definability Problem

Authors:

Marcelo Arenas,

Gonzalo I. DiazAuthors Info & Claims

ACM Transactions on Database Systems (TODS), Volume 41, Issue 2

Article No.: 13, Pages 1 - 14

https://doi.org/10.1145/2886095

Published: 11 May 2016 Publication History

Abstract

We study the definability problem for first-order logic, denoted by FO-Def. The input of FO-Def is a relational database instance I and a relation R; the question to answer is whether there exists a first-order query Q (or, equivalently, a relational algebra expression Q) such that Q evaluated on I gives R as an answer.

Although the study of FO-Def dates back to 1978, when the decidability of this problem was shown, the exact complexity of FO-Def remains as a fundamental open problem. In this article, we provide a polynomial-time algorithm for solving FO-Def that uses calls to a graph-isomorphism subroutine (or oracle). As a consequence, the first-order definability problem is found to be complete for the class GI of all problems that are polynomial-time Turing reducible to the graph isomorphism problem, thus closing the open question about the exact complexity of this problem. The technique used is also applied to a generalized version of the problem that accepts a finite set of relation pairs, and whose exact complexity was also open; this version is also found to be GI-complete.

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Cited By

Cima GCroce FLenzerini M(2023)Separability and Its Approximations in Ontology-based Data ManagementSemantic Web10.3233/SW-233391(1-36)Online publication date: 8-Jun-2023
https://doi.org/10.3233/SW-233391
Jung JLutz CPulcini HWolter F(2022)Logical separability of labeled data examples under ontologiesArtificial Intelligence10.1016/j.artint.2022.103785313(103785)Online publication date: Dec-2022
https://doi.org/10.1016/j.artint.2022.103785
Barceló PBaumgartner ADalmau VKimelfeld B(2021)Regularizing conjunctive features for classificationJournal of Computer and System Sciences10.1016/j.jcss.2021.01.003119(97-124)Online publication date: Aug-2021
https://doi.org/10.1016/j.jcss.2021.01.003
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Index Terms

The Exact Complexity of the First-Order Logic Definability Problem
1. Information systems
  1. Data management systems
    1. Query languages
      1. Relational database query languages
2. Theory of computation
  1. Computational complexity and cryptography
    1. Problems, reductions and completeness

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Published In

cover image ACM Transactions on Database Systems

ACM Transactions on Database Systems Volume 41, Issue 2

Invited Paper from SIGMOD 2014 and Regular Papers

June 2016

271 pages

ISSN:0362-5915

EISSN:1557-4644

DOI:10.1145/2936309

Editor:
Christian S. Jensen
Aalborg University, Denmark

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Copyright © 2016 ACM.

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Association for Computing Machinery

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Publication History

Published: 11 May 2016

Accepted: 01 January 2016

Revised: 01 October 2015

Received: 01 May 2015

Published in TODS Volume 41, Issue 2

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Millennium Nucleus Center for Semantic Web Research
Becas Chile of CONICYT Chile

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Cited By

Cima GCroce FLenzerini M(2023)Separability and Its Approximations in Ontology-based Data ManagementSemantic Web10.3233/SW-233391(1-36)Online publication date: 8-Jun-2023
https://doi.org/10.3233/SW-233391
Jung JLutz CPulcini HWolter F(2022)Logical separability of labeled data examples under ontologiesArtificial Intelligence10.1016/j.artint.2022.103785313(103785)Online publication date: Dec-2022
https://doi.org/10.1016/j.artint.2022.103785
Barceló PBaumgartner ADalmau VKimelfeld B(2021)Regularizing conjunctive features for classificationJournal of Computer and System Sciences10.1016/j.jcss.2021.01.003119(97-124)Online publication date: Aug-2021
https://doi.org/10.1016/j.jcss.2021.01.003
Areces CCampercholi MPenazzi DVentura P(2020)The complexity of definability by open first-order formulasLogic Journal of the IGPL10.1093/jigpal/jzaa008Online publication date: 30-May-2020
https://doi.org/10.1093/jigpal/jzaa008
Campercholi MTellechea MVentura P(2020)Deciding Quantifier-free Definability in Finite Algebraic StructuresElectronic Notes in Theoretical Computer Science10.1016/j.entcs.2020.02.003348(23-41)Online publication date: Mar-2020
https://doi.org/10.1016/j.entcs.2020.02.003
Li MChan C(2020)Efficient Query Reverse Engineering Using Table FragmentsDatabase Systems for Advanced Applications10.1007/978-3-030-59419-0_25(406-422)Online publication date: 24-Sep-2020
https://dl.acm.org/doi/10.1007/978-3-030-59419-0_25
Jung JPapacchini FWolter FZakharyaschev MBouyer P(2019)Model comparison games for horn description logicsProceedings of the 34th Annual ACM/IEEE Symposium on Logic in Computer Science10.5555/3470152.3470156(1-14)Online publication date: 24-Jun-2019
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Barceló PBaumgartner ADalmau VKimelfeld BSuciu DSkritek SKoch C(2019)Regularizing Conjunctive Features for ClassificationProceedings of the 38th ACM SIGMOD-SIGACT-SIGAI Symposium on Principles of Database Systems10.1145/3294052.3319680(2-16)Online publication date: 25-Jun-2019
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Jung JPapacchini FWolter FZakharyaschev M(2019)Model Comparison Games for Horn Description Logics2019 34th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS)10.1109/LICS.2019.8785658(1-14)Online publication date: Jun-2019
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Surinx DVan den Bussche JVirtema J(2019)Fully Generic Queries: Open Problems and Some Partial AnswersModel and Data Engineering10.1007/978-3-030-32065-2_2(20-31)Online publication date: 28-Oct-2019
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