Imprecise probabilistic horn clause logic

Approaches for extending logic to deal with uncertainty immanent to many real-world problems are often on the one side purely qualitative, such as modal logics, or on the other side quantitative, such as probabilistic logics. Research on combinations of qualitative and quantitative extensions to logic which put qualitative constraints on probability distributions, has mainly remained theoretical until now. In this paper, we propose a practically useful logic, which supports qualitative as well as quantitative uncertainty and can be extended with modalities with varying level of quantitative precision. This language has a solid semantic foundation based on imprecise probability theory. While in general imprecise probabilistic inference is much harder than the precise case, this is the first expressive imprecise probabilistic formalism for which probabilistic inference is shown to be as hard as corresponding precise probabilistic problems. A second contribution of this paper is an inference algorithm for this language based on the translation to a weighted model counting (WMC) problem, an approach also taken by state-of-the-art probabilistic inference methods for precise problems.