Bounds for the quantifier depth in finite-variable logics: Alternation hierarchy release_46kvqmghkbd4jbctpbvttbgthq

by Christoph Berkholz, Andreas Krebs, Oleg Verbitsky

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2013  

Abstract

Given two structures G and H distinguishable in k (first-order logic with k variables), let A^k(G,H) denote the minimum alternation depth of a 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 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)>_k+1n-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 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 2 much more succinctly if the alternation number is increased just by one: while 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 k with i quantifier alternations, this can be done with quantifier depth n^2k-2.
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Date   2013-05-17
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arXiv  1212.2747v3
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