Probabilities of first order sentences on sparse random relational structures: An application to definability on random CNF formulas

TL;DR

This paper extends the convergence law for sparse random structures to arbitrary relational languages and applies it to analyze the probability of properties in random CNF formulas, especially regarding satisfiability.

Contribution

It generalizes the convergence law to broader relational structures and applies this to understand definability and satisfiability in random CNF formulas.

Findings

First order properties have well-defined asymptotic probabilities in sparse random structures.

Limit probabilities are stable under parameter variations of relation densities.

Almost surely, certain first order properties implying unsatisfiability do not hold in large random k-CNF formulas.

Abstract

We extend the convergence law for sparse random graphs proven by Lynch to arbitrary relational languages. We consider a finite relational vocabulary $\sigma$ and a first order theory $T$ for $\sigma$ composed of symmetry and anti-reflexivity axioms. We define a binomial random model of finite $\sigma$-structures that satisfy $T$ and show that first order properties have well defined asymptotic probabilities when the expected number of tuples satisfying each relation in $\sigma$ is linear. It is also shown that these limit probabilities are well-behaved with respect to several parameters that represent the density of tuples in each relation $R$ in the vocabulary $\sigma$. An application of these results to the problem of random Boolean satisfiability is presented. We show that in a random $k$-CNF formula on $n$ variables, where each possible clause occurs with probability $\sim c/n^{k-1}$, independently any first order property of $k$-CNF formulas that implies unsatisfiability does almost surely not hold as $n$ tends to infinity.