ASYMPTOTIC CONDITIONAL PROBABILITIES - THE UNARY CASE

Motivated by problems that arise in computing degrees of belief, we co nsider the problem of computing asymptotic conditional probabilities f or first-order sentences. Given first-order sentences phi and theta, w e consider the structures with domain (1,...,N) that satisfy theta, an d compute the fraction of them in which phi is true. We then consider what happens to this fraction as N gets large. This extends the work o n 0-1 laws that considers the limiting Probability of first-order sent ences, by considering asymptotic conditional probabilities. As shown b y Liogon'kii [Math. Notes Acad. USSR, 6 (1969), pp. 856-861] and by Gr ove, Halpern, and Koller [Res. Rep. RJ 9564, IBM Almaden Research Cent er, San Jose, CA, 1993], in the general case, asymptotic conditional p robabilities do not always exist, and most questions relating to this issue are highly undecidable. These results, however, all depend on th e assumption that theta can use a nonunary predicate symbol. Liogon'ki i [Math. Notes Acad. USSR, 6 (1969), pp. 856-861] shows that if we con dition on formulas theta involving unary predicate symbols only (but n o equality or constant symbols), then the asymptotic conditional proba bility does exist and can be effectively computed. This is the case ev en if we place no corresponding restrictions on phi. We extend this re sult here to the case where theta involves equality and constants. We show that the complexity of computing the limit depends on various fac tors, such as the depth of quantifier nesting, or whether the vocabula ry is finite or infinite. We completely characterize the complexity of the problem in the different cases, and show related results for the associated approximation problem.