A Bounds for the quantifier depth in finite-variable logics: Alternation hierarchy

Given two structures G and H distinguishable in FOk (first-order logic with k variables), let Ak(G,H) denote the minimum alternation depth of a FOk formula distinguishing G from H. Let Ak(n) be the maximum value of Ak(G,H) over n-element structures. We prove the strictness of the quantifier alternation hierarchy of FO2 in a strong quantitative form, namely A2(n) > n/8−2, which is tight up to a constant factor. For each k ≥ 2, it holds that Ak(n) > logk+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, while the alternation hierarchy of FO2 collapses even over all uncolored graphs. We also show examples of colored graphs G and H on n vertices that can be distinguished in FO2 much more succinctly if the alternation number is increased just by one: while in Σi it is possible to distinguish G from H with bounded quantifier depth, in Πi this requires quantifier depth Ω(n2). The quadratic lower bound is best possible here because, if G and H can be distinguished in FOk with i quantifier alternations, this can be done with quantifier depth n2k−2 + 1 and the same number of alternations.