On failure of 0-1 laws

TL;DR

This paper investigates the failure of 0-1 laws in certain stronger logical frameworks for specific random graphs, extending known results beyond first-order logic.

Contribution

It demonstrates the failure of 0-1 laws for stronger logics like 0_{,k} and inductive logic in the context of random graphs with edge probability 1/n^lpha.

Findings

0-1 law holds for first-order logic on G_{n,1/n^lpha}

Failure of 0-1 law occurs for 0_{,k} and inductive logic

Results extend understanding of logical properties of random graphs

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 inductive logic.