Power Defaults 1

We show how two general notions of domain theory, the logic induced from the Scott topology on a domain, together with the Smyth powerdomain, interact with the theory of default reasoning introduced by Reiter 19]. We deene extensions, which are xed points of a certain non-monotonic operator on a Scott domain induced by a normal default set over that domain. We specialize this deenition to the Smyth powerdomain, which consists of all nonempty compact sets in the Scott topology, ordered by reverse subset inclusion. Our main result is a general \splitting theorem" showing how any extension E of a union of compact sets U and V can be written as a union E = E U E V , where E U and E V are either empty or extensions of U and V respectively. This result holds in arbitrary Scott domains, provided the set of \power defaults" is obtained from \default constraints", a natural way of introducing defaults over the Smyth powerdomain. We apply our methodology in rst-order systems. As in 20], we equip standard rst-order logic with a Kleene 3-valued partial model semantics. Then, using our methodology, we add power defaults to this semantic system. We show how, in this application, the Extension Splitting Theorem sanctions default reasoning by cases. We point out, though, that our methodology applies much more generally than the rst-order case. It is a valid method for default reasoning in any setting appropriately modeled by Scott domains.