Asymptotic Conditional Probabilities: The Unary Case

Adam J. Grove, Joseph Y. Halpern, Daphne Koller
1996 SIAM journal on computing (Print)  
Motivated by problems that arise in computing degrees of belief, we consider the problem of computing asymptotic conditional probabilities for rst-order sentences. Given rst-order sentences ' and , we consider the structures with domain f1; : : : ; Ng that satisfy , and compute the fraction of them in which ' 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 rst-order sentences, by considering
more » ... mptotic conditional probabilities. As shown by Liogon'ki 31] and Grove, Halpern, and Koller 22], 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 can use a nonunary predicate symbol. Liogon'ki 31] shows that if we condition on formulas involving unary predicate symbols only (but no equality or constant symbols), then the asymptotic conditional probability does exist and can be e ectively computed. This is the case even if we place no corresponding restrictions on '. We extend this result here to the case where involves equality and constants. We show that the complexity of computing the limit depends on various factors, such as the depth of quanti er nesting, or whether the vocabulary is nite or in nite. We completely characterize the complexity of the problem in the di erent cases, and show related results for the associated approximation problem. Key Words. asymptotic probability, 0-1 law, nite model theory, degree of belief, labeled structures, principle of indi erence, complexity. 1 of ' given , and then actually computing that probability, depends in part on the language and logic being considered. In decision theory, applications often demand the ability to express statistical knowledge (for instance, correlations between symptoms and diseases) as well as rst-order knowledge. Work in the eld of 0-1 laws (which, as discussed below, is closely related to our own) has examined some higher-order logics as well as rst-order logic. Nevertheless, the pure rst-order case is still di cult, and is important because it provides a foundation for all extensions. In this paper and in 22] we address the problem of computing conditional probabilities in the rst-order case. In a related paper 23], we consider the case of a rst-order logic augmented with statistical knowledge. The general problem of assigning probabilities to rst-order sentences has been well studied (cf. 15, 16]). In this paper, we investigate two speci c formalisms for computing probabilities, based on the same basic approach. Our approach is itself an instance of a much older idea, known as the the principle of insu cient reason 28] or the principle of indi erence 26]. This states that all possibilities should be given equal probability, and was once regarded as one of the most basic principles of probability theory. (See 24] for a discussion of the history of the principle.) We use this idea to assign equal degrees of belief to all basic \situations" consistent with the known facts. The two formalisms we consider di er only in how they interpret \situation". We discuss this in more detail below. In many applications, including the one of most interest to us, it makes sense to consider nite domains only. In fact, the case of most interest to us is the behavior of the formulas ' and over large nite domains. Similar questions also arise in the area of 0-1 laws. Our approach essentially generalizes the methods used in the work on 0-1 laws for rst-order logic to the case of conditional probabilities. (See Compton's overview 8] for an introduction to this work.) Assume, without loss of generality, that the domain is f1; : : :; Ng for some natural number N. As we said above, we consider two notions of \situation". In the random-worlds method, the possible situations are all the worlds, or rst-order models, with domain f1; : : :; Ng that satisfy the constraints . Based on the principle of indi erence, we assume that all worlds are equally likely. To assign a probability to ', we therefore simply compute the fraction of them in which the sentence ' is true. The random-worlds approach views each individual in f1; : : :; Ng as having a distinct name (even though the name may not correspond to any constant in the vocabulary). Thus, two worlds that are isomorphic with respect to the symbols in the vocabulary are still treated as distinct situations. In some cases, however, we may believe that all relevant distinctions are captured by our vocabulary, and that isomorphic worlds are not truly distinct. The randomstructures method attempts to capture this intuition by considering a situation to be a structure|an isomorphism class of worlds. This corresponds to assuming that individuals are distinguishable only if they di er with respect to properties de nable by the language. As before, we assign a probability to ' by computing the fraction of the structures that satisfy ' among those structures that satisfy . 1 Since we are computing probabilities over nite models, we have assumed that the domain is f1; : : :; Ng for some N. However, we often do not know the precise 1 The random-worlds method considers what has been called in the literature labeled structures, while the random-structures method considers unlabeled structures 8]. We choose to use our own terminology for the random-worlds and random-structures methods, rather than the terminology of labeled and unlabeled. This is partly because we feel it is more descriptive, and partly because there are other variants of the approach, that are useful for our intended application, and that do not t into the standard labeled/unlabeled structures dichotomy (see 2]). 2
doi:10.1137/s0097539793257034 fatcat:l2jjt36yxrfkxntwwmkli2ibcq