How to Reason Credulously and Skeptically within a Single Extension

Consistency-based approaches in nonmonotonic reasoning may be expected to yield multiple sets of default conclusions for a given default theory. Reasoning about such extensions is carried out at the meta-level. In this paper, we show how such reasoning may be carried out at the object level for a large class of default theories. Essentially we show how one can translate a (normal) default theory∆, obtaining a second∆′, such that∆′ has a single extension that encodes every extension of ∆. Moreover, our translated theory is only a constant factor larger than the original (with the exception of unique names axioms). We prove that our translation behaves correctly. In the approach we can now encode the notion of extension from within the framework of standard default logic. Hence one can encode notions such as skeptical and credulous conclusions, and can reason about such conclusions within a single extension. This result has some theoretical interest, in that it shows how multiple extensions of normal default theories are encodable with manageable overhead in a single extension.