Asymptotic Conditional Probabilities: The Unary Case
Motivated by problems that arise in computing degrees of belief, we consider the problem of computing asymptotic conditional probabilities for first-order sentences. Given first-order sentences $\phi$ and $\theta$, we consider the structures with domain $\{1,\ldots, N\}$ that satisfy $\theta$, and 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 on 0-1 laws that considers the limiting probability of first-order sentences, by considering asymptotic conditional probabilities. As shown by Liogon'kii [Math. Notes Acad. USSR, 6 (1969), pp.\ 856--861] and by Grove, Halpern, and Koller [Res. Rep. RJ 9564, IBM Almaden Research Center, San Jose, CA, 1993], in the general case, asymptotic conditional probabilities do not always exist, and most questions relating to this issue are highly undecidable. These results, however, all depend on the assumption that $\theta$ can use a nonunary predicate symbol. Liogonkii\ [Math. Notes Acad. USSR, 6 (1969), pp.\ 856--861] shows that if we condition on formulas $\theta$ involving unary predicate symbols only (but no equality or constant symbols), then the asymptotic conditional probability does exist and can be effectively computed. This is the case even if we place no corresponding restrictions on $\phi$. We extend this result here to the case where $\theta$ involves equality and constants. We show that the complexity of computing the limit depends on various factors, such as the depth of quantifier nesting, or whether the vocabulary 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.