Weak Nonmonotonic Probabilistic Logics

We present an approach where probabilistic logic is combined with default reasoning from conditional knowledge bases in Kraus et al.’s System P , Pearl’s System Z, and Lehmann’s lexicographic entailment. The resulting probabilistic generalizations of default reasoning from conditional knowledge bases allow for handling in a uniform framework strict logical knowledge, default logical knowledge, as well as purely probabilistic knowledge. Interestingly, probabilistic entailment in System P coincides with probabilistic entailment under g-coherence from imprecise probability assessments. We then analyze the semantic and nonmonotonic properties of the new formalisms. It turns out that they all are proper generalizations of their classical counterparts and have similar properties as them. In particular, they all satisfy the rationality postulates of System P and some Direct Inference property. Moreover, probabilistic entailment in System Z and probabilistic lexicographic entailment both satisfy the property of Rational Monotonicity and some Irrelevance property, while probabilistic entailment in System P does not. We also analyze the relationships between the new formalisms. Here, probabilistic entailment in System P is weaker than probabilistic entailment in System Z, which in turn is weaker than probabilistic lexicographic entailment. Moreover, they all are weaker than entailment in probabilistic logic where default sentences are interpreted as strict sentences. Under natural conditions, probabilistic entailment in System Z and lexicographic entailment even coincide with such entailment in probabilistic logic, while probabilistic entailment in System P does not. Finally, we also present algorithms for reasoning under probabilistic entailment in System Z and probabilistic lexicographic entailment, and we give a precise picture of its complexity. 1Dipartimento di Informatica e Sistemistica, Università di Roma “La Sapienza”, Via Salaria 113, I-00198 Rome, Italy; e-mail: [email protected]. Institut für Informationssysteme, Technische Universität Wien, Favoritenstraße 9-11, A-1040 Vienna, Austria; e-mail: [email protected]. Acknowledgements: This work was supported by the Marie Curie Individual Fellowship HPMF-CT-2001001286 of the European Union programme “Human Potential” (disclaimer: The author is solely responsible for information communicated and the European Commission is not responsible for any views or results expressed), by a Heisenberg Fellowship of the German Research Foundation, and by the project Z29-N04 of the Austrian Science Fund. I am thankful to Angelo Gilio and Lluis Godo for useful comments on an earlier version of this paper. Many thanks also to the reviewers of the KR-2004 abstract of this paper, whose constructive comments helped to improve this work. Copyright c © 2004 by the authors INFSYS RR 1843-02-02 I