Flexible Approximators for Approximating Fixpoint Theory.

Approximation fixpoint theory AFT is an algebraic framework for the study of fixpoints of operators on bilattices, which has been applied to the study of the semantics for a number of nonmonotonic formalisms. A central notion of AFT is that of stable revision based on an underlying approximating operator called approximator, where the negative information used in fixpoint computation is by default. This raises a problem in systems that combine different formalisms, where both default negation and established negation may be present in reasoning. In this paper we extend AFT to allow more flexible approximators. The main idea is to formulate and propose ternary approximators, of which traditional binary approximators are a special case. The extra parameter allows separation of two kinds of negative information, by entailment and by default, respectively. The new approach is motivated by the need to integrate different knowledge representation and reasoning KRR systems, in particular to support combined reasoning by nonmonotonic rules with ontologies. However, this small change by allowing flexible approximators raises a mathematical question - whether the resulting AFT is a sound fixpoint theory. The main result of this paper is a proof that answers this question positively.