M.: Ultimate approximations in nonmonotonic knowledge representation systems

We study xpoints of operators on lattices. To this end we introduce the notion of an approximation of an operator. We order ap-proximations by means of a precision order-ing. We show that each lattice operator O has a unique most precise or ultimate ap-proximation. We demonstrate that xpoints of this ultimate approximation provide useful insights into xpoints of the operator O. We apply our theory to logic program-ming and introduce the ultimate Kripke-Kleene, well-founded and stable semantics. We show that the ultimate Kripke-Kleene and well-founded semantics are more precise then their standard counterparts We argue that ultimate semantics for logic program-ming have attractive epistemological proper-ties and that, while in general they are com-putationally more complex than the standard semantics, for many classes of theories, their complexity is no worse.