Simulating Logspace-Recursion with Logarithmic Quantifier Depth
Abstract
The fixed-point logic LREC= was developed by Grohe et al. (CSL 2011) in the quest for a logic to capture all problems decidable in logarithmic space. It extends FO+C, first-order logic with counting, by an operator that formalises a limited form of recursion. We show that for every LREC=-definable property on relational structures, there is a constant k such that the k-variable fragment of first-order logic with counting quantifiers expresses the property via formulae of logarithmic quantifier depth. This yields that any pair of graphs separable by the property can be distinguished with the k-dimensional Weisfeiler-Leman algorithm in a logarithmic number of iterations. In particular, it implies that a constant dimension of the algorithm identifies every interval graph and every chordal claw-free graph in logarithmically many iterations, since every such graph admits LREC=-definable canonisation.
Keywords
Cite
@article{arxiv.2304.12948,
title = {Simulating Logspace-Recursion with Logarithmic Quantifier Depth},
author = {Steffen van Bergerem and Martin Grohe and Sandra Kiefer and Luca Oeljeklaus},
journal= {arXiv preprint arXiv:2304.12948},
year = {2023}
}