Almost everywhere elimination of probability quantifiers

Abstract

We obtain an almost everywhere quantifier elimination for (the noncritical fragment of) the logic with probability quantifiers, introduced by the first author in [10]. This logic has quantifiers like ∃^{≥ 3/4}y, which says that “for at least 3/4 of all y”. These results improve upon the 0-1 law for a fragment of this logic obtained by Knyazev [11]. Our improvements are:

• We deal with the quantifier ∃^{≥ r}y, where y is a tuple of variables.
• We remove the closedness restriction, which requires that the variables in y occur in all atomic subformulas of the quantifier scope.
• Instead of the unbiased measure where each model with universe n has the same probability, we work with any measure generated by independent atomic probabilities p_R for each predicate symbol R.
• We extend the results to parametric classes of finite models (for example, the classes of bipartite graphs, undirected graphs, and oriented graphs).
• We extend the results to a natural (noncritical) fragment of the infinitary logic with probability quantifiers.
• We allow each p_R, as well as each r in the probability quantifier (∃^{≥ r}y), to depend on the size of the universe.