Simulating Logspace-Recursion with Logarithmic Quantifier Depth

Bergerem, Steffen van; Grohe, Martin; Kiefer, Sandra; Luca, Oeljeklaus,

doi:10.1109/lics56636.2023.10175818

2023 38th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS) 2023

DOI: 10.1109/lics56636.2023.10175818

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Simulating Logspace-Recursion with Logarithmic Quantifier Depth

Steffen van Bergerem

Martin Grohe

Sandra Kiefer

et al.

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2024

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The Iteration Number of the Weisfeiler-Leman Algorithm

Grohe,

Lichter,

Neuen

2024

ACM Trans. Comput. Logic

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We prove new upper and lower bounds on the number of iterations the \(k\) -dimensional Weisfeiler-Leman algorithm ( \(k\) -WL) requires until stabilization. For \(k\geq 3\) , we show that \(k\) -WL stabilizes after at most \(O(kn^{k-1}\log n)\) iterations (where \(n\) denotes the number of vertices of the input structures), obtaining the first improvement over the trivial upper bound of \(n^{k}-1\) and extending a previous upper bound of \(O(n\log n)\) for \(k=2\) [Lichter et al., LICS 2019]. We complement our upper bounds by constructing \(k\) -ary relational structures on which \(k\) -WL requires at least \(n^{\Omega(k)}\) iterations to stabilize. This improves over a previous lower bound of \(n^{\Omega(k/\log k)}\) [Berkholz, Nordström, LICS 2016]. We also investigate tradeoffs between the dimension and the iteration number of WL, and show that \(d\) -WL, where \(d=\lceil\frac{3(k+1)}{2}\rceil\) , can simulate the \(k\) -WL algorithm using only \(O(k^{2}\cdot n^{\lfloor k/2\rfloor+1}\log n)\) many iterations, but still requires at least \(n^{\Omega(k)}\) iterations for any \(d\) (that is sufficiently smaller than \(n\) ). The number of iterations required by \(k\) -WL to distinguish two structures corresponds to the quantifier rank of a sentence distinguishing them in the \((k+1)\) -variable fragment \(\mathsf{C}_{k+1}\) of first-order logic with counting quantifiers. Hence, our results also imply new upper and lower bounds on the quantifier rank required in the logic \(\mathsf{C}_{k+1}\) , as well as tradeoffs between variable number and quantifier rank.

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The Iteration Number of the Weisfeiler-Leman Algorithm

Grohe,

Lichter,

Neuen

2024

ACM Trans. Comput. Logic

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Simulating Logspace-Recursion with Logarithmic Quantifier Depth

Cited by 1 publication

References 33 publications

The Iteration Number of the Weisfeiler-Leman Algorithm

The Iteration Number of the Weisfeiler-Leman Algorithm

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