Near-Optimal Lower Bounds on Quantifier Depth and Weisfeiler–Leman Refinement Steps

Abstract

We prove near-optimal trade-offs for quantifier depth versus number of variables in first-order logic by exhibiting pairs of n-element structures that can be distinguished by a k-variable first-order sentence but where every such sentence requires quantifier depth at least nΩ(k/ log k). Our trade-offs also apply to first-order counting logic, and by the known connection to the k-dimensional Weisfeiler–Leman algorithm imply near-optimal lower bounds on the number of refinement iterations. A key component in our proof is the hardness condensation technique recently introduced by [Razborov ’16] in the context of proof complexity. We apply this method to reduce the domain size of relational structures while maintaining the quantifier depth required to distinguish them.Categories and Subject Descriptors F.4.1 [Mathematical Logic]: Computational Logic, Model theory; F.2.3 [Tradeoffs between Complexity Measures]