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TR16-135 | 31st August 2016 04:20
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#### 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 n^?(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 minimal quantifier depth to distinguish them in finite variable logics.