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

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.