skip to main content
research-article

Bounds for the Quantifier Depth in Finite-Variable Logics: Alternation Hierarchy

Published: 21 April 2015 Publication History

Abstract

Given two structures G and H distinguishable in FOk (first-order logic with k variables), let Ak(G, H) denote the minimum alternation depth of a FOk formula distinguishing G from H. Let Ak(n) be the maximum value of Ak(G, H) over n-element structures. We prove the strictness of the quantifier alternation hierarchy of FO2 in a strong quantitative form, namely A2(n) > n/8 − 2, which is tight up to a constant factor. For each k ⩾ 2, it holds that Ak(n) > log k + 1n − 2 even over colored trees, which is also tight up to a constant factor if k ⩾ 3. For k ⩾ 3, the last lower bound holds also over uncolored trees, whereas the alternation hierarchy of FO2 collapses even over all uncolored graphs. We also show examples of colored graphs G and H on n vertices that can be distinguished in FO2 much more succinctly if the alternation number is increased just by one: Whereas in Σi it is possible to distinguish G from H with bounded quantifier depth, in Πi this requires quantifier depth Ω(n2). The quadratic lower bound is best possible here because, if G and H can be distinguished in FOk with i quantifier alternations, this can be done with quantifier depth n2k − 2 + 1 and the same number of alternations.

References

[1]
Christoph Berkholz, Andreas Krebs, and Oleg Verbitsky. 2013. Bounds for the quantifier depth in finite-variable logics: Alternation hierarchy. In Computer Science Logic 2013 (CSL 2013) (LIPIcs), Simona Ronchi Della Rocca (Ed.), Vol. 23. Leibniz-Zentrum für Informatik, Schloss Dagstuhl, 61--80.
[2]
Christoph Berkholz and Oleg Verbitsky. 2013. On the speed of constraint propagation and the time complexity of arc consistency testing. In MFCS’13, Proceedings (Lecture Notes in Computer Science), K. Chatterjee and J. Sgall (Eds.), Vol. 8087. Springer, Berlin, 159--170.
[3]
Jin-Yi Cai, Martin Fürer, and Neil Immerman. 1992. An optimal lower bound on the number of variables for graph identifications. Combinatorica 12, 4 (1992), 389--410.
[4]
Ashok K. Chandra and David Harel. 1982. Structure and complexity of relational queries. Journal of Computer Systems Science 25, 1 (1982), 99--128.
[5]
Rina Dechter and Judea Pearl. 1985. A Problem Simplification Approach that Generates Heuristics for Constraint-Satisfaction Problems. Technical Report. Cognitive Systems Laboratory, Computer Science Department, University of California, Los Angeles.
[6]
Heinz-Dieter Ebbinghaus and Jörg Flum. 1995. Finite Model Theory. Springer, Berlin.
[7]
Martin Grohe. 1999. Equivalence in finite-variable logics is complete for Polynomial Time. Combinatorica 19, 4 (1999), 507--532.
[8]
Neil Immerman. 1981. Number of quantifiers is better than number of tape cells. Journal of Computer System Science 22, 3 (1981), 384--406.
[9]
Neil Immerman and Eric Lander. 1990. Describing graphs: A first-order approach to graph canonization. In Complexity Theory Retrospective, Alan Selman (Ed.). Springer, Berlin, 59--81.
[10]
Øystein Ore. 1962. Theory of Graphs. Colloquium Publications, Vol. 38. American Mathematical Society (AMS), Providence, R.I.
[11]
Oleg Pikhurko, Joel Spencer, and Oleg Verbitsky. 2006. Succinct definitions in the first order theory of graphs. Annals of Pure Applied Logic 139, 1--3 (2006), 74--109.
[12]
Oleg Pikhurko and Oleg Verbitsky. 2011. Logical complexity of graphs: A survey. In Model Theoretic Methods in Finite Combinatorics, Martin Grohe and Janos Makowsky (Eds.). Contemporary Mathematics, Vol. 558. American Mathematical Society (AMS), Providence, RI, 129--179.
[13]
Ashok Samal and Tom Henderson. 1987. Parallel consistent labeling algorithms. International Journal of Parallel Programming 16, 5 (1987), 341--364.
[14]
Philipp Weis and Neil Immerman. 2009. Structure theorem and strict alternation hierarchy for FO2 on words. Logical Methods in Computer Science 5, 3 (2009), 1--23.

Cited By

View all
  • (2018)On the speed of constraint propagation and the time complexity of arc consistency testingJournal of Computer and System Sciences10.1016/j.jcss.2017.09.00391(104-114)Online publication date: Mar-2018

Index Terms

  1. Bounds for the Quantifier Depth in Finite-Variable Logics: Alternation Hierarchy

      Recommendations

      Comments

      Information & Contributors

      Information

      Published In

      cover image ACM Transactions on Computational Logic
      ACM Transactions on Computational Logic  Volume 16, Issue 3
      July 2015
      285 pages
      ISSN:1529-3785
      EISSN:1557-945X
      DOI:10.1145/2764956
      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 the author(s) 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].

      Publisher

      Association for Computing Machinery

      New York, NY, United States

      Publication History

      Published: 21 April 2015
      Accepted: 01 January 2015
      Revised: 01 July 2014
      Received: 01 February 2014
      Published in TOCL Volume 16, Issue 3

      Permissions

      Request permissions for this article.

      Check for updates

      Author Tags

      1. Alternation hierarchy
      2. finite-variable logic

      Qualifiers

      • Research-article
      • Research
      • Refereed

      Funding Sources

      • DFG

      Contributors

      Other Metrics

      Bibliometrics & Citations

      Bibliometrics

      Article Metrics

      • Downloads (Last 12 months)4
      • Downloads (Last 6 weeks)1
      Reflects downloads up to 26 Dec 2024

      Other Metrics

      Citations

      Cited By

      View all
      • (2018)On the speed of constraint propagation and the time complexity of arc consistency testingJournal of Computer and System Sciences10.1016/j.jcss.2017.09.00391(104-114)Online publication date: Mar-2018

      View Options

      Login options

      Full Access

      View options

      PDF

      View or Download as a PDF file.

      PDF

      eReader

      View online with eReader.

      eReader

      Media

      Figures

      Other

      Tables

      Share

      Share

      Share this Publication link

      Share on social media