DEFAULTS IN DOMAIN THEORY

This paper uses ideas from artificial intelligence to show how default notions can be defined over Scott domains. We combine these ideas wit h ideas arising in domain theory to shed some light on the properties of nonmonotonicity in a general model-theoretic setting. We consider i n particular a notion of default nonmonotonic entailment between prime open sets in the Scott topology of a domain. We investigate in what w ays this notion obeys the so-called laws of cautious monotony and caut ious cut, proposed by Gabbay, Kraus, Lehmann, and Magidor. Our notion of nonmonotonic entailment does not necessarily satisfy cautious monot ony, but does satisfy cautious cut. In fact, we show that any reasonab le notion of nonmonotonic entailment on prime opens over a Scott domai n, satisfying in particular the law of cautious cut, can be concretely represented using our notion of default entailment. We also give a va riety of sufficient conditions for defaults to induce cumulative entai lments, those satisfying cautious monotony. In particular, we show tha t defaults with unique extensions are a representation of cumulative n onmonotonic entailment. Furthermore, a simple characterization is give n for those default sets which determine unique extensions in coherent domains. Finally, a characterization is given for Scott domains in wh ich default entailment must be cumulative. This is the class of daisy domains; it is shown to be cartesian closed, a purely domain-theoretic result.