First order complexity of finite random structures
Danila Demin, Maksim Zhukovskii
TL;DR
The paper develops a unified framework for the first-order (FO) logical complexity of random structures by placing FO truth probabilities in the Banach space $\ell^{\infty}/c_0$ and introducing the FO complexity $\mathrm{FOC}(D_n)=\mathcal{D}/c_0$. It introduces stochastic FO reductions as a preorder that preserves asymptotic FO behaviour, enabling transfer of FO limit laws across diverse models, including graphs, hypergraphs, and parametric sentences, in both dense and sparse regimes. The authors prove a rich hierarchy of FO complexities, showing that $G(n,p)$ and conditioned models $G(n\mid\varphi)$ can realize a spectrum from the FO0-1 law to dense, infinite-dimensional, and non-totally-bounded complexities, and they establish non-recursivity for deciding 0-1 laws in the conditioned setting. They extend the transfer framework to $H$-hypergraphs around connectivity thresholds, deriving sharp results $c_0^H = \frac{|H|}{d!}c_0^{S_d}$ and density of limit sets $L_c^H$ for $c\ge c_0^H$, thereby generalizing prior LMN findings. A central part of the work constructs two FO sentences $\varphi_1$ and $\varphi_2$ to realize a totally bounded but infinite-dimensional FO complexity, using EF games and locality to demonstrate the existence of a sentence with a highly intricate limiting profile. Together, these contributions offer a versatile toolkit for analyzing logical limit laws in random combinatorial structures and for mapping complexity landscapes across models.
Abstract
For a sequence of random structures with $n$-element domains over a relational signature, we define its first order (FO) complexity as a certain subset in the Banach space $\ell^{\infty}/c_0$. The well-known FO zero-one law and FO convergence law correspond to FO complexities equal to $\{0,1\}$ and a subset of $\mathbb{R}$, respectively. We present a hierarchy of FO complexity classes, introduce a stochastic FO reduction that allows to transfer complexity results between different random structures, and deduce using this tool several new logical limit laws for binomial random structures. Finally, we introduce a conditional distribution on graphs, subject to a FO sentence $\varphi$, that generalises certain well-known random graph models, show instances of this distribution for every complexity class, and prove that the set of all $\varphi$ validating 0--1 law is not recursively enumerable.