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

Authors Christoph Berkholz, Andreas Krebs, Oleg Verbitsky

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Christoph Berkholz
Andreas Krebs
Oleg Verbitsky

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Christoph Berkholz, Andreas Krebs, and Oleg Verbitsky. Bounds for the quantifier depth in finite-variable logics: Alternation hierarchy. In Computer Science Logic 2013 (CSL 2013). Leibniz International Proceedings in Informatics (LIPIcs), Volume 23, pp. 61-80, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2013)


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, while 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: while in Sigma_i it is possible to distinguish G from H with bounded quantifier depth, in Pi_i this requires quantifier depth Omega(n2). 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^(2k-2).
  • Alternation hierarchy
  • finite-variable logic


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