Degrees of Belief, Random Worlds, and Maximum Entropy
DS '00 Proceedings of the Third International Conference on Discovery Science
From statistical knowledge bases to degrees of belief: an overview
Proceedings of the twenty-fifth ACM SIGMOD-SIGACT-SIGART symposium on Principles of database systems
Inference Processes for Quantified Predicate Knowledge
WoLLIC '08 Proceedings of the 15th international workshop on Logic, Language, Information and Computation
Random worlds and maximum entropy
Journal of Artificial Intelligence Research
IJCAI'05 Proceedings of the 19th international joint conference on Artificial intelligence
A note on the least informative model of a theory
CiE'10 Proceedings of the Programs, proofs, process and 6th international conference on Computability in Europe
Entailment in probability of thresholded generalizations
UAI'96 Proceedings of the Twelfth international conference on Uncertainty in artificial intelligence
There is a reason for everything (probably): on the application of maxent to induction
WCII'02 Proceedings of the 2002 international conference on Conditionals, Information, and Inference
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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.