Approximations, Stable Operators, Well-Founded Fixpoints and Applications in Nonmonotonic Reasoning
In this paper we develop an algebraic framework for studying semantics of nonmonotonic logics. Our approach is formulated in the language of lattices, bilattices, operators and fixpoints. The goal is to describe fixpoints of an operator O defined on a lattice. The key intuition is that of an approximation, a pair (x, y) of lattice elements which can be viewed as an approximation to each lattice element z such that x ≤ z ≤ y. The key notion is that of an approximating operator, a monotone operator on the bilattice of approximations whose fixpoints approximate the fixpoints of the operator O. The main contribution of the paper is an algebraic construction which assigns a certain operator, called the stable operator, to every approximating operator on a bilattice of approximations. This construction leads to an abstract version of the well–founded semantics. In the paper we show that our theory offers a unified framework for semantic studies of logic programming, default logic and autoepistemic logic.
KeywordsNonmonotonic logics operators on lattices fixpoints approximating operators well-founded fixpoint stable fixpoints
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