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