Bounds for the quantifier depth in finite-variable logics: Alternation
hierarchy
release_46kvqmghkbd4jbctpbvttbgthq
by
Christoph Berkholz, Andreas Krebs, Oleg Verbitsky
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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