Probabilities of first order sentences on sparse random relational structures: An application to definability on random CNF formulas
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We extend the convergence law for sparse random graphs proven by Lynch to arbitrary relational languages. We consider a finite relational vocabulary σ and a first order theory T for σ composed of symmetry and anti-reflexivity axioms. We define a binomial random model of finite σ-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 σ 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 σ. 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 ∼ 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.
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