Spectrum of FO logic with quantifier depth 4 is finite

TL;DR

This paper investigates the properties of the spectrum of first-order logic with bounded quantifier depth, establishing that the spectrum becomes infinite starting at quantifier depth 5, which advances understanding of logical expressiveness in random graphs.

Contribution

It proves that the minimum quantifier depth for an infinite spectrum in FO logic is 5, providing a precise threshold for spectrum finiteness.

Findings

The 4-spectrum is finite.

The 5-spectrum is infinite.

Identifies the exact quantifier depth threshold for spectrum infiniteness.

Abstract

The $k$-spectrum is the set of all $\alpha>0$ such that $G(n,n^{-\alpha})$ does not obey the 0-1 law for FO sentences with quantifier depth at most $k$. In this paper, we prove that the minimum $k$ such that the $k$-spectrum is infinite equals 5.