Postprint version. Published in Annals of Mathematics and Artificial Intelligence, Volume 5, Issue 3-4, January 1, 2009, pages 355-388.
Copyright © 2009 Springer.
The definitive version is available at http://dx.doi.org/10.1007/s10472-008-9104-7.
Two approaches to logic programming with probabilities emerged over time: Bayesian reasoning and probabilistic satisfiability (PSAT). The attractiveness of the former is in tying the logic programming research to the body of work on Bayes networks. The second approach ties, from the point of view of computation, reasoning about probabilities to linear programming, and allows for natural expression of imprecision in probabilities via the use of intervals. In this paper we construct precise semantics for one PSAT-based formalism for reasoning with interval probabilities: disjunctive probabilistic logic programs (dp-programs). It has two origins: (1) disjunctive logic programs, a powerful language for knowledge representation, first proposed by Minker in the early eighties  and (2) a logic programming language with interval probabilities originally considered by Ng and Subrahmanian [21,22]. We show that the probability ranges of atoms and formulas in dp-programs cannot be expressed as single intervals. We construct the precise description of the set of models for the class of dp-programs and study the computational complexity of this problem, as well as the problem of consistency of a dp-program. We also study the conditions under which our semantics coincides with the single-interval semantics originally proposed by Ng and Subrahmanian. Our work sheds light on the complexity of constructing reasoning formalisms for imprecise probabilities and suggests that interval probabilities alone are inadequate to support such reasoning.