Asymptotic Conditional Probabilities: The Non-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 φ and θ, we consider the structures with domain {1, . . . , N} that satisfy θ, and compute the fraction of them in which φ 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’kĭı [Lio69], if there is a non-unary predicate symbol in the vocabulary, asymptotic conditional probabilities do not always exist. We extend this result to show that asymptotic conditional probabilities do not always exist for any reasonable notion of limit. Liogon’kĭı also showed that the problem of deciding whether the limit exists is undecidable. We analyze the complexity of three problems with respect to this limit: deciding whether it is well defined, whether it exists, and whether it lies in some nontrivial interval. Matching upper and lower bounds are given for all three problems, showing them to be highly undecidable. ∗Some of this research was done while all three authors were at IBM Almaden Research Center. During this work, the first author was at Stanford University, and was supported by an IBM Graduate Fellowship. This research was sponsored in part by the Air Force Office of Scientific Research (AFSC), under Contract F4962091-C-0080. The United States Government is authorized to reproduce and distribute reprints for governmental purposes. Some of this research appeared in preliminary form in a paper entitled “Asymptotic conditional probabilities for first-order logic”, which appears in Proceedings 24th ACM Symp. on Theory of Computing, 1992, pages 294–305. This paper appears in the Journal of Symbolic Logic.