Bounds for the Quantifier Depth in Finite-Variable Logics

Abstract

Given two structures G and H distinguishable in FO k (first-order logic with k variables), let A k ( G , H ) denote the minimum alternation depth of a FO k formula distinguishing G from H . Let A k ( n ) be the maximum value of A k ( G , H ) over n -element structures. We prove the strictness of the quantifier alternation hierarchy of FO 2 in a strong quantitative form, namely A 2 ( n ) > n /8 − 2, which is tight up to a constant factor. For each k ⩾ 2, it holds that A k ( n ) > log k + 1 n − 2 even over colored trees, which is also tight up to a constant factor if k ⩾ 3. For k ⩾ 3, the last lower bound holds also over uncolored trees, whereas the alternation hierarchy of FO 2 collapses even over all uncolored graphs. We also show examples of colored graphs G and H on n vertices that can be distinguished in FO 2 much more succinctly if the alternation number is increased just by one: Whereas in Σ i it is possible to distinguish G from H with bounded quantifier depth, in Π i this requires quantifier depth Ω( n 2 ). The quadratic lower bound is best possible here because, if G and H can be distinguished in FO k with i quantifier alternations, this can be done with quantifier depth n 2 k − 2 + 1 and the same number of alternations.