Skeptical Inheritance: Computing the Intersection of Credulous Extensions

Ideally skeptical inheritance supports exactly those inferences true in every credulous extension of an inheritance hierarchy. We provide a formal definition of ideally skeptical inheritance. We show that two path-based approaches fail to capture ideally skeptical inheritance , and that there are inheritance hierarchies for which there are more always-true inferences than always-supported paths. We describe an ATMS-like scheme that computes ide ally skeptical inheritance and represents hierarchical dependencies using a limited form of Boolean satisfiability. Finally, we demonstrate a preemption (specificity) strategy for which ideally skeptical inheritance is polynomial time computable. 1 Introduction Inheritance, like other forms of defeasible reasoning, sanctions uncertain conclusions. In unambiguous contexts , these conclusions are only as reliable as the general rules from which they are derived: concluding that a particular bird can fly is reasonable only if birds generally fly. Ambiguity introduces another kind of uncertainty: if we have evidence that Charlie is a bird, and other evidence that he is not, our reasoning is still more uncertain because our assumptions are questionable. of these conclusions, however, we can reason further: if we assume that Nixon is a pacifist, we can conclude that, like a typical pacifist, he is anti-military. Credulous inheritance permits us to believe that Nixon is not a pacifist , or is a pacifist, and therefore is anti-military, but either way our beliefs must be internally consistent. While credulous inheritance offers many alternatives, skeptical inheritance yields a single, unambiguous set of conclusions for any inheritance hierarchy. Ideally skeptical inheritance supports exactly those conclusions true in every credulous extension. In the next section, we provide formal definitions of ambiguity, credulous extensions , and ideally skeptical inheritance. We then present two approximate approaches to skeptical inheritance, ambiguity blocking and ambiguity propagating inheritance. The failure of these path-based approaches to be both sound and complete for ideally skeptical inheritance indicates the importance of reasoning about conclusions, or inferences, rather than about their supporting paths, or arguments. In section 5, we demonstrate an ATMS-like labeling scheme for inheritance hierarchies, based on a limited form of Boolean satisfiability. This labeling gives a precise description of dependencies in the hierarchy, including an exact representation of ideally skeptical inheritance. Finally, in section 6, we present a definition of preemption, or specificity, and provide a polynomial-time algorithm for computing the results of specificity. Although the definition of specificity itself is neither skeptical nor credulous, the output of the …