research-article

Complexity of Two-Variable Logic on Finite Trees

Authors:

Michael Benedikt,

Witold Charatonik,

Emanuel Kieroński,

Rastislav Lenhardt,

Filip Mazowiecki,

James WorrellAuthors Info & Claims

ACM Transactions on Computational Logic (TOCL), Volume 17, Issue 4

Article No.: 32, Pages 1 - 38

https://doi.org/10.1145/2996796

Published: 11 November 2016 Publication History

Abstract

Verification of properties expressed in the two-variable fragment of first-order logic FO² has been investigated in a number of contexts. The satisfiability problem for FO² over arbitrary structures is known to be NEXPTIME-complete, with satisfiable formulas having exponential-sized models. Over words, where FO² is known to have the same expressiveness as unary temporal logic, satisfiability is again NEXPTIME-complete. Over finite labelled ordered trees, FO² has the same expressiveness as navigational XPath, a popular query language for XML documents. Prior work on XPath and FO² gives a 2EXPTIME bound for satisfiability of FO² over trees. This work contains a comprehensive analysis of the complexity of FO² on trees, and on the size and depth of models. We show that different techniques are required depending on the vocabulary used, whether the trees are ranked or unranked, and the encoding of labels on trees. We also look at a natural restriction of FO², its guarded version, GF². Our results depend on an analysis of types in models of FO² formulas, including techniques for controlling the number of distinct subtrees, the depth, and the size of a witness to satisfiability for FO² sentences over finite trees.

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Cited By

Kieroński E(2023)A Uniform One-Dimensional Fragment with Alternation of QuantifiersElectronic Proceedings in Theoretical Computer Science10.4204/EPTCS.390.1390(1-15)Online publication date: 30-Sep-2023
https://doi.org/10.4204/EPTCS.390.1
van Bremen TKuželka O(2023)Lifted inference with tree axiomsArtificial Intelligence10.1016/j.artint.2023.103997324(103997)Online publication date: Nov-2023
https://doi.org/10.1016/j.artint.2023.103997
Kieroński EKuusisto A(2022)One-dimensional fragment over words and treesJournal of Logic and Computation10.1093/logcom/exac00232:5(902-941)Online publication date: 19-Feb-2022
https://doi.org/10.1093/logcom/exac002
Show More Cited By

Index Terms

Complexity of Two-Variable Logic on Finite Trees
1. Theory of computation
  1. Logic
  2. Models of computation
    1. Computability

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cover image ACM Transactions on Computational Logic

ACM Transactions on Computational Logic Volume 17, Issue 4

November 2016

292 pages

ISSN:1529-3785

EISSN:1557-945X

DOI:10.1145/2996393

Editor:
Orna Kupferman
School of Computer Science and Engineering, Hebrew University, Israel

Issue’s Table of Contents

Copyright © 2016 ACM.

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Association for Computing Machinery

New York, NY, United States

Publication History

Published: 11 November 2016

Accepted: 01 September 2016

Revised: 01 July 2016

Received: 01 October 2014

Published in TOCL Volume 17, Issue 4

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Engineering and Physical Science Research Council projects “Enforcement of Constraints on XML Streams”
Polish National Science Centre
DBOnto: Bridging Ontologies and Databases

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Cited By

Kieroński E(2023)A Uniform One-Dimensional Fragment with Alternation of QuantifiersElectronic Proceedings in Theoretical Computer Science10.4204/EPTCS.390.1390(1-15)Online publication date: 30-Sep-2023
https://doi.org/10.4204/EPTCS.390.1
van Bremen TKuželka O(2023)Lifted inference with tree axiomsArtificial Intelligence10.1016/j.artint.2023.103997324(103997)Online publication date: Nov-2023
https://doi.org/10.1016/j.artint.2023.103997
Kieroński EKuusisto A(2022)One-dimensional fragment over words and treesJournal of Logic and Computation10.1093/logcom/exac00232:5(902-941)Online publication date: 19-Feb-2022
https://doi.org/10.1093/logcom/exac002
Baelde DLick ASchmitz SSuciu DSkritek SKoch C(2019)Decidable XPath Fragments in the Real WorldProceedings of the 38th ACM SIGMOD-SIGACT-SIGAI Symposium on Principles of Database Systems10.1145/3294052.3319685(285-302)Online publication date: 25-Jun-2019
https://dl.acm.org/doi/10.1145/3294052.3319685

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