Random graphs and the parity quantifier

Abstract

The classical zero-one law for first-order logic on random graphs says that for every first-order property φ in the theory of graphs and every p ∈ (0,1), the probability that the random graph G ( n , p ) satisfies φ approaches either 0 or 1 as n approaches infinity. It is well known that this law fails to hold for any formalism that can express the parity quantifier: for certain properties, the probability that G ( n , p ) satisfies the property need not converge, and for others the limit may be strictly between 0 and 1. In this work, we capture the limiting behavior of properties definable in first order logic augmented with the parity quantifier, FO[⌖], over G ( n , p ), thus eluding the above hurdles. Specifically, we establish the following “modular convergence law”. For every FO[⌖] sentence φ, there are two explicitly computable rational numbers a 0 , a 1 , such that for i ∈ {0,1}, as n approaches infinity, the probability that the random graph G (2 n + i , p ) satisfies φ approaches a i . Our results also extend appropriately to FO equipped with Mod q quantifiers for prime q . In the process of deriving this theorem, we explore a new question that may be of interest in its own right. Specifically, we study the joint distribution of the subgraph statistics modulo 2 of G ( n , p ): namely, the number of copies, mod 2, of a fixed number of graphs F 1 , …, F ℓ of bounded size in G ( n , p ). We first show that every FO[⌖] property φ is almost surely determined by subgraph statistics modulo 2 of the above type. Next, we show that the limiting joint distribution of the subgraph statistics modulo 2 depends only on n mod 2, and we determine this limiting distribution completely. Interestingly, both these steps are based on a common technique using multivariate polynomials over finite fields and, in particular, on a new generalization of the Gowers norm. The first step is analogous to the Razborov-Smolensky method for lower bounds for AC 0 with parity gates, yet stronger in certain ways. For instance, it allows us to obtain examples of simple graph properties that are exponentially uncorrelated with every FO[⌖] sentence, which is something that is not known for AC 0 [⌖].