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

TL;DR

This paper investigates the quantifier alternation hierarchy in finite-variable first-order logic, establishing tight bounds on the minimum alternation depth needed to distinguish structures, revealing hierarchy strictness and succinctness differences.

Contribution

It proves the strictness of the quantifier alternation hierarchy for $o 2$ and provides tight bounds for $A^k(n)$, advancing understanding of logical complexity in finite structures.

Findings

$A^2(n) rac{n}{8}-2$ lower bound for $o 2$ hierarchy

$A^k(n) > rac{ ext{log}_{k+1} n - 2}$ for $k  2$ over colored trees

Distinguishing graphs with increased alternation depth can require quadratic quantifier depth

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 $A^2(n)\ge n/8-2$, which is tight up to a constant factor. For each $k\ge2$, 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 $k\ge3$. For $k\ge 3$ 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 $\Omega(n^2)$. 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 $n^{2k-2}$.