The infinitary logic of sparse random graphs

Abstract

Let L/sub /spl infin//spl omega///sup /spl omega// be the infinitary language obtained from the first-order language of graphs by closure under conjunctions and disjunctions of arbitrary sets of formulas, provided only finitely many distinct variables occur among the formulas. Let p(n) be the edge probability of the random graph on n vertices. Previous articles have shown that when p(n) is constant or p(n)=n/sup -/spl alpha// and /spl alpha/>1, then every sentence in L/sub /spl infin//spl omega///sup /spl omega// has probability that converges as n gets large; however, when p(n)=n/sup -/spl alpha// and /spl alpha/<1 is rational, then there are first-order sentences whose probability does not converge. This article completes the picture for L/sub /spl infin//spl omega///sup /spl omega// and random graphs with edge probability of the form n/sup -/spl alpha//. It is shown that if /spl alpha/ is irrational, then every sentence in L/sub /spl infin//spl omega///sup /spl omega// has probability that converges to 0 or 1. It is also shown that if /spl alpha/=1, then there are sentences an the deterministic transitive closure logic (and therefore in L/sub /spl infin//spl omega///sup /spl omega//), whose probability does not converge.