Weisfeiler-lehman refinement requires at least a linear number of iterations

    Research output: Chapter in Book/Report/Conference proceedingConference contribution

    6 Citations (Scopus)

    Abstract

    Let Lk,m be the set of formulas of first order logic containing only variables from x1, x2, ... xk and having quantifier depth at most m. Let Ck,m be the extension of L k,m obtained by allowing counting quantifiers meaning that there are at least i distinct xj 's. It is shown that for constants h ≥ 1, there are pairs of graphs such that h-dimensional Weisfeiler-Lehman refinement (h-dim W-L) can distinguish the two graphs, but requires at least a linear number of iterations. Despite of this slow progress, 2h-dim W-L only requires O(n) iterations, and 3h-1-dim W-L only requires O(log n) iterations. In terms of logic, this means that there is a c > 0 and a class of non-isomorphic pairs (GhnHhn) of graphs with G hn and Hhn having O(n) vertices such that the same sentences of Lh+1cn and Ch+1cn hold (h + 1 variables, depth cn), even though Ghn and H hn can already be distinguished by a sentence of L k,m and thus Ckm for some k > h and m = O(log n).

    Original languageEnglish (US)
    Title of host publicationAutomata, Languages and Programming - 28th International Colloquium, ICALP 2001, Proceedings
    Pages322-333
    Number of pages12
    StatePublished - Dec 1 2001
    Event28th International Colloquium on Automata, Languages and Programming, ICALP 2001 - Crete, Greece
    Duration: Jul 8 2001Jul 12 2001

    Publication series

    NameLecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
    Volume2076 LNCS
    ISSN (Print)0302-9743
    ISSN (Electronic)1611-3349

    Other

    Other28th International Colloquium on Automata, Languages and Programming, ICALP 2001
    CountryGreece
    CityCrete
    Period7/8/017/12/01

    Fingerprint

    Refinement
    Quantifiers
    Iteration
    Graph in graph theory
    First-order Logic
    Counting
    Logic
    Distinct
    Class
    Meaning

    All Science Journal Classification (ASJC) codes

    • Theoretical Computer Science
    • Computer Science(all)

    Cite this

    Furer, M. (2001). Weisfeiler-lehman refinement requires at least a linear number of iterations. In Automata, Languages and Programming - 28th International Colloquium, ICALP 2001, Proceedings (pp. 322-333). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 2076 LNCS).
    Furer, Martin. / Weisfeiler-lehman refinement requires at least a linear number of iterations. Automata, Languages and Programming - 28th International Colloquium, ICALP 2001, Proceedings. 2001. pp. 322-333 (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)).
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    abstract = "Let Lk,m be the set of formulas of first order logic containing only variables from x1, x2, ... xk and having quantifier depth at most m. Let Ck,m be the extension of L k,m obtained by allowing counting quantifiers meaning that there are at least i distinct xj 's. It is shown that for constants h ≥ 1, there are pairs of graphs such that h-dimensional Weisfeiler-Lehman refinement (h-dim W-L) can distinguish the two graphs, but requires at least a linear number of iterations. Despite of this slow progress, 2h-dim W-L only requires O(n) iterations, and 3h-1-dim W-L only requires O(log n) iterations. In terms of logic, this means that there is a c > 0 and a class of non-isomorphic pairs (GhnHhn) of graphs with G hn and Hhn having O(n) vertices such that the same sentences of Lh+1cn and Ch+1cn hold (h + 1 variables, depth cn), even though Ghn and H hn can already be distinguished by a sentence of L k,m and thus Ckm for some k > h and m = O(log n).",
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    Furer, M 2001, Weisfeiler-lehman refinement requires at least a linear number of iterations. in Automata, Languages and Programming - 28th International Colloquium, ICALP 2001, Proceedings. Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), vol. 2076 LNCS, pp. 322-333, 28th International Colloquium on Automata, Languages and Programming, ICALP 2001, Crete, Greece, 7/8/01.

    Weisfeiler-lehman refinement requires at least a linear number of iterations. / Furer, Martin.

    Automata, Languages and Programming - 28th International Colloquium, ICALP 2001, Proceedings. 2001. p. 322-333 (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 2076 LNCS).

    Research output: Chapter in Book/Report/Conference proceedingConference contribution

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    Furer M. Weisfeiler-lehman refinement requires at least a linear number of iterations. In Automata, Languages and Programming - 28th International Colloquium, ICALP 2001, Proceedings. 2001. p. 322-333. (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)).