APPROXIMATIONS, STABLE OPERATORS, WELL-FOUNDED FIXPOINTS AND APPLICATIONS IN NONMONOTONIC REASONING
K. U. Leuven · 2000
Inthispaperwedevelopanalgebraicframeworkforstudyingsemantics of nonmonotonic logics. Our approach is formulated in the language oflattices, bilattices, operators and fixpoints. The goal is to describe fixpoints ofan operator 0 defined on a lattice. The key intuition is thatofan approximation,a pair (x, y) oflatticeelementswhich canbe viewed as an approximation to each lattice element z such that x ~ z ~ y. Thekeynotionisthatofan approximating operator,amonotone operatoronthebilatticeofapproximationswhosefixpointsapproximate the fixpoints ofthe operator O. The main contribution of the paper is an algebraic construction which assigns a certain operator, called the stableoperator, to every approximating operator on a bilattice of approximations. This construction leads to an abstract version ofthe well-founded semantics. In the paper we show that our theory offers a unified frameworkfor semanticstudiesoflogic programming, default logic and autoepistemic logic.