We show that the 2-dimensional Weisfeiler-Leman algorithm stabilizes n-vertex graphs after at most @math iterations. This implies that if such graphs are distinguishable in 3-variable first order logic with counting, then they can also be distinguished in this logic by a formula of quantifier depth at most @math . For this we exploit a new refinement based on counting walks and argue that its iteration number differs from the classic Weisfeiler-Leman refinement by at most a logarithmic factor. We then prove matching linear upper and lower bounds on the number of iterations of the walk refinement. This is achieved with an algebraic approach by exploiting properties of semisimple matrix algebras. We also define a walk logic and a bijective walk pebble game that precisely correspond to the new walk refinement.
Regarding bounds, Berkholz and Nordstr " o m @cite_10 proved a lower bound on the number of iterations of the @math -dimensional WL algorithm for finite structures. Specifically, they show for sufficiently large @math the existence of @math -element relational structures distinguished by the @math -dimensional WL algorithm but for which @math iterations do not suffice. For a different logic, namely the @math -variable existential negation-free fragment of first-order logic, Berkholz also developed techniques to prove tight bounds @cite_14 . In contrast to these bounds, Fürer's lower bound @cite_4 of @math mentioned above is applicable to graphs and in fact also applies to all fixed dimensions @math .
@cite_14 @cite_10 @cite_4
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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] We consider the problem of finding a characterization for polynomial time computable queries on finite structures in terms of logical definability. It is well known that fixpoint logic provides such a characterization in the presence of a built-in linear order, but without linear order even very simple polynomial time queries involving counting are not expressible in fixpoint logic. Our approach to the problem is based on generalized quantifiers. A generalized quantifier isn-ary if it binds any number of formulas, but at mostnvariables in each formula. We prove that, for each natural numbern, there is a query on finite structures which is expressible in fixpoint logic, but not in the extension of first-order logic by any set ofn-ary quantifiers. It follows that the expressive power of fixpoint logic cannot be captured by adding finitely many quantifiers to first-order logic. Furthermore, we prove that, for each natural numbern, there is a polynomial time computable query which is not definable in any extension of fixpoint logic byn-ary quantifiers. In particular, this rules out the possibility of characterizing PTIME in terms of definability in fixpoint logic extended by a finite set of generalized quantifiers.