Inductive Inference from Weakly Consistent Belief Bases.

We consider nonmonotonic inferences from belief bases that contain conditionals enforcing some of the possible worlds to be infeasible and thus completely implausible. In contrast to belief bases satisfying the strong notion of consistency requiring every world to be at least somewhat plausible, we call such belief bases weakly consistent. First, we review the treatment of weakly consistent belief bases by the seminal approaches of p-entailment, which coincides with system P, and of system Z, which coincides with rational closure. Then we focus on c-inference, an inductive inference operator that has been shown to exhibit many desirable properties put forward for nonmonotonic reasoning. It is based on c-representations, which are a special kind of ranking model ordering worlds according to their plausibility. While c-representation is defined for strongly consistent belief bases only, in this article, we extend the notions of c-representation and of c-inference to cover also weakly consistent belief bases. We adapt a constraint satisfaction problem (CSP) characterizing c-representations to capture extended c-representations, and we show how this extended CSP can be used to characterize extended c-inference, providing a basis for its implementation. We show various properties of extended c-inference and in particular, we prove that also the extended notion of c-inference fully satisfies syntax splitting. Furthermore, we extend and evaluate credulous and weakly skeptical c-inference to weakly consistent belief bases and provide characterizations for them as CSPs.