Alternating Fixpoint Operator for Hybrid MKNF Knowledge Bases as an
  Approximator of AFT
  Approximation fixpoint theory (AFT) provides an algebraic framework for the
study of fixpoints of operators on bilattices and has found its applications in
characterizing semantics for various classes of logic programs and nonmonotonic
languages. In this paper, we show one more application of this kind: the
alternating fixpoint operator by Knorr et al. for the study of the well-founded
semantics for hybrid MKNF knowledge bases is in fact an approximator of AFT in
disguise, which, thanks to the power of abstraction of AFT, characterizes not
only the well-founded semantics but also two-valued as well as three-valued
semantics for hybrid MKNF knowledge bases. Furthermore, we show an improved
approximator for these knowledge bases, of which the least stable fixpoint is
information richer than the one formulated from Knorr et al.'s construction.
This leads to an improved computation for the well-founded semantics. This work
is built on an extension of AFT that supports consistent as well as
inconsistent pairs in the induced product bilattice, to deal with
inconsistencies that arise in the context of hybrid MKNF knowledge bases. This
part of the work can be considered generalizing the original AFT from symmetric
approximators to arbitrary approximators.
