Nonmonotonic reasoning from conditional knowledge bases with system W

Abstract

For nonmonotonic reasoning in the context of a knowledge base mathcal {R} containing conditionals of the form If A then usually B, system P provides generally accepted axioms. Inference solely based on system P, however, is inherently skeptical because it coincides with reasoning that takes all ranking models of mathcal {R} into account. System Z uses only the unique minimal ranking model of mathcal {R}, and c-inference, realized via a complex constraint satisfaction problem, takes all c-representations of mathcal {R} into account. C-representations constitute the subset of all ranking models of mathcal {R} that are obtained by assigning non-negative integer impacts to each conditional in mathcal {R} and summing up, for every world, the impacts of all conditionals falsified by that world. While system Z and c-inference license in general different sets of desirable entailments, the first major objective of this article is to present system W. System W fully captures and strictly extends both system Z and c-inference. Moreover, system W can be represented by a single strict partial order on the worlds over the signature of mathcal {R}. We show that system W exhibits further inference properties worthwhile for nonmonotonic reasoning, like satisfying the axioms of system P, respecting conditional indifference, and avoiding the drowning problem. The other main goal of this article is to provide results on our investigations, underlying the development of system W, of upper and lower bounds that can be used to restrict the set of c-representations that have to be taken into account for realizing c-inference. We show that the upper bound of n − 1 is sufficient for capturing c-inference with respect to mathcal {R} having n conditionals if there is at least one world verifying all conditionals in mathcal {R}. In contrast to the previous conjecture that the number of conditionals in mathcal {R} is always sufficient, we prove that there are knowledge bases requiring an upper bound of 2n− 1, implying that there is no polynomial upper bound of the impacts assigned to the conditionals in mathcal {R} for fully capturing c-inference.