Failure of 0-1 law for sparse random graph in strong logics

TL;DR

This paper investigates the breakdown of the 0-1 law in sparse random graphs when considering stronger logical systems beyond first-order logic, such as infinitary logics and least fixpoint logic.

Contribution

It demonstrates the failure of the 0-1 law for these advanced logics in the context of sparse random graphs with edge probability $1/n^eta$.

Findings

0-1 law holds for first-order logic in these graphs

Failure of 0-1 law occurs for stronger logics like $ ext{L}_{ ext{infty},k}$ and LFP

Results highlight limitations of classical probabilistic zero-one laws in complex logical frameworks.

Abstract

Let $\alpha\in(0,1)_\mathbb{R}$ be irrational and $G_n = G_{{n, 1/n}^\alpha}$ be the random graph with edge probability $1/n^\alpha$; we know that it satisfies the 0-1 law for first order logic. We deal with the failure of the 0-1 law for stronger logics: $\mathbb{L}_{ \infty, k}, k$ large enough and the LFP, least fix point logic.