research-article

Upper Bounds on the Quantifier Depth for Graph Differentiation in First Order Logic

Authors:

Pascal SchweitzerAuthors Info & Claims

LICS '16: Proceedings of the 31st Annual ACM/IEEE Symposium on Logic in Computer Science

Pages 287 - 296

https://doi.org/10.1145/2933575.2933595

Published: 05 July 2016 Publication History

Abstract

We show that on graphs with n vertices the 2-dimensional Weisfeiler-Leman algorithm requires at most O(n2 / log(n)) iterations to reach stabilization. This in particular shows that the previously best, trivial upper bound of O(n2) is asymptotically not tight. In the logic setting this translates to the statement that if two graphs of size n can be distinguished by a formula in first order logic with counting with 3 variables (i.e., in C3) then they can also be distinguished by a C3-formula that has quantifier depth at most O(n2 / log(n)).

To prove the result we define a game between two players that enables us to decouple the causal dependencies between the processes happening simultaneously over several iterations of the algorithm. This allows us to treat large color classes and small color classes separately. As part of our proof we show that for graphs with bounded color class size, the number of iterations until stabilization is at most linear in the number of vertices. This also yields a corresponding statement in first order logic with counting.

Similar results can be obtained for the respective logic without counting quantifiers, i.e., for the logic L3.

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Cited By

Böker JLevie RHuang NVillar SMorris COh ANaumann TGloberson ASaenko KHardt MLevine S(2023)Fine-grained expressivity of graph neural networksProceedings of the 37th International Conference on Neural Information Processing Systems10.5555/3666122.3668144(46658-46700)Online publication date: 10-Dec-2023
https://dl.acm.org/doi/10.5555/3666122.3668144
Berkholz CNordström J(2023)Near-optimal Lower Bounds on Quantifier Depth and Weisfeiler–Leman Refinement StepsJournal of the ACM10.1145/319525770:5(1-32)Online publication date: 11-Oct-2023
https://dl.acm.org/doi/10.1145/3195257
Grohe MLichter MNeuen D(2023)The Iteration Number of the Weisfeiler-Leman Algorithm2023 38th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS)10.1109/LICS56636.2023.10175741(1-13)Online publication date: 26-Jun-2023
https://doi.org/10.1109/LICS56636.2023.10175741
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Upper Bounds on the Quantifier Depth for Graph Differentiation in First Order Logic
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cover image ACM Conferences

LICS '16: Proceedings of the 31st Annual ACM/IEEE Symposium on Logic in Computer Science

July 2016

901 pages

ISBN:9781450343916

DOI:10.1145/2933575

Conference Chair:
Eric Koskinen,
General Chair:
Martin Grohe,
Program Chair:
Natarajan Shankar

Copyright © 2016 ACM.

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EACSL: European Association for Computer Science Logic
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Published: 05 July 2016

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LICS '16: 31st Annual ACM/IEEE Symposium on Logic in Computer Science

July 5 - 8, 2016

NY, New York, USA

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Cited By

Böker JLevie RHuang NVillar SMorris COh ANaumann TGloberson ASaenko KHardt MLevine S(2023)Fine-grained expressivity of graph neural networksProceedings of the 37th International Conference on Neural Information Processing Systems10.5555/3666122.3668144(46658-46700)Online publication date: 10-Dec-2023
https://dl.acm.org/doi/10.5555/3666122.3668144
Berkholz CNordström J(2023)Near-optimal Lower Bounds on Quantifier Depth and Weisfeiler–Leman Refinement StepsJournal of the ACM10.1145/319525770:5(1-32)Online publication date: 11-Oct-2023
https://dl.acm.org/doi/10.1145/3195257
Grohe MLichter MNeuen D(2023)The Iteration Number of the Weisfeiler-Leman Algorithm2023 38th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS)10.1109/LICS56636.2023.10175741(1-13)Online publication date: 26-Jun-2023
https://doi.org/10.1109/LICS56636.2023.10175741
Grohe MLichter MNeuen DSchweitzer P(2023)Compressing CFI Graphs and Lower Bounds for the Weisfeiler-Leman Refinements2023 IEEE 64th Annual Symposium on Foundations of Computer Science (FOCS)10.1109/FOCS57990.2023.00052(798-809)Online publication date: 6-Nov-2023
https://doi.org/10.1109/FOCS57990.2023.00052
Morris CRattan GMutzel PLarochelle HRanzato MHadsell RBalcan MLin H(2020)Weisfeiler and leman go sparseProceedings of the 34th International Conference on Neural Information Processing Systems10.5555/3495724.3497555(21824-21840)Online publication date: 6-Dec-2020
https://dl.acm.org/doi/10.5555/3495724.3497555
Kiefer S(2020)The Weisfeiler-Leman algorithmACM SIGLOG News10.1145/3436980.34369827:3(5-27)Online publication date: 16-Nov-2020
https://dl.acm.org/doi/10.1145/3436980.3436982
Kiefer SPonomarenko ISchweitzer P(2019)The Weisfeiler--Leman Dimension of Planar Graphs Is at Most 3Journal of the ACM10.1145/333300366:6(1-31)Online publication date: 27-Nov-2019
https://dl.acm.org/doi/10.1145/3333003
Lichter MPonomarenko ISchweitzer P(2019)Walk refinement, walk logic, and the iteration number of the Weisfeiler-Leman algorithm2019 34th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS)10.1109/LICS.2019.8785694(1-13)Online publication date: Jun-2019
https://doi.org/10.1109/LICS.2019.8785694
Kiefer SPonomarenko ISchweitzer PAceto LIngólfsdóttir A(2017)The weisfeiler-leman dimension of planar graphs is at most 3Proceedings of the 32nd Annual ACM/IEEE Symposium on Logic in Computer Science10.5555/3329995.3330042(1-12)Online publication date: 20-Jun-2017
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Kiefer SPonomarenko ISchweitzer P(2017)The Weisfeiler-Leman dimension of planar graphs is at most 32017 32nd Annual ACM/IEEE Symposium on Logic in Computer Science (LICS)10.1109/LICS.2017.8005107(1-12)Online publication date: Jun-2017
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