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Spectrum of FO logic with quantifier depth 4 is finite

Yury Yarovikov, Maksim Zhukovskii

TL;DR

The paper resolves a long-standing question about the spectral behavior of first-order properties on sparse random graphs by proving that the minimum quantifier depth $k$ for which the $k$-spectrum is infinite is $5$, making the $4$-spectrum finite. The authors develop a framework based on Ehrenfeucht–Fraïssé games, density analysis, and a carefully constructed finite family $oldsymbol{\mathcal{G}}$ of ‘bad’ neighbourhoods to control Spoiler's moves and to guarantee Duplicator's win for $oldsymbol{\alpha}$ near $oldsymbol{3/5}$. They define precise extensions (including $oldsymbol{\alpha}$-safe/rigid/neutral) and generic extensions, and they prove three key lemmas that underpin a 4-round strategy in the EHR game, culminating in the main theorem. The result sharpens the understanding of FO expressiveness in random graphs and informs the thresholds and 0-1 laws for fragments of FO logic on $G(n,n^{-oldsymbol{\alpha}})$, with implications for finite-model theory and descriptive complexity.

Abstract

The $k$-spectrum is the set of all $α>0$ such that $G(n,n^{-α})$ 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.

Spectrum of FO logic with quantifier depth 4 is finite

TL;DR

The paper resolves a long-standing question about the spectral behavior of first-order properties on sparse random graphs by proving that the minimum quantifier depth for which the -spectrum is infinite is , making the -spectrum finite. The authors develop a framework based on Ehrenfeucht–Fraïssé games, density analysis, and a carefully constructed finite family of ‘bad’ neighbourhoods to control Spoiler's moves and to guarantee Duplicator's win for near . They define precise extensions (including -safe/rigid/neutral) and generic extensions, and they prove three key lemmas that underpin a 4-round strategy in the EHR game, culminating in the main theorem. The result sharpens the understanding of FO expressiveness in random graphs and informs the thresholds and 0-1 laws for fragments of FO logic on , with implications for finite-model theory and descriptive complexity.

Abstract

The -spectrum is the set of all such that does not obey the 0-1 law for FO sentences with quantifier depth at most . In this paper, we prove that the minimum such that the -spectrum is infinite equals 5.
Paper Structure (13 sections, 12 theorems, 30 equations, 2 figures)

This paper contains 13 sections, 12 theorems, 30 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Known definitions and statements
  3. Ehrenfeucht game
  4. Subgraphs and extensions
  5. New statements and constructions
  6. Building the set $\mathcal{G}$
  7. Occurrence of graphs of $\mathcal{G}$ in an arbitrary graph
  8. Three main lemmas
  9. Lemma 1
  10. Lemmas 2 and 3
  11. Proof of the main theorem
  12. Conclusion
  13. Acknowledgements

Key Result

Theorem 1

Let $\alpha>0$, $p = n^{-\alpha}$. If $\alpha$ is irrational, then $G(n,p)$ obeys FO 0-1 law. If $\alpha\leq 1$ is rational, then $G(n,p)$ does not obey FO 0-1 law. If $\alpha>1$, then $G(n,p)$ does not obey the law if and only if $\alpha=1+\frac{1}{m}$ for some positive integer $m$.

Figures (2)

  • Figure 1: A generic extension
  • Figure 2: Pairs $(K_1, T_1)$ and $(K_2, T_2)$

Theorems & Definitions (48)

  • Definition
  • Theorem 1: S. Shelah, J. Spencer, 1988, spencershelah
  • Definition
  • Definition
  • Theorem 2: M. E. Zhukovskii, 2012, zhukovskii
  • Definition
  • Theorem 3: A. D. Matushkin, M. E. Zhukovskii, 2017, matushkin
  • Theorem 4: Y. N. Yarovikov, 2021, yu-rovikov
  • Theorem 5
  • Theorem 6
  • ...and 38 more