  Forgetting is an important concept in knowledge representation and automated
reasoning with widespread applications across a number of disciplines. A
standard forgetting operator, characterized in [Lin and Reiter'94] in terms of
model-theoretic semantics and primarily focusing on the propositional case,
opened up a new research subarea. In this paper, a new operator called weak
forgetting, dual to standard forgetting, is introduced and both together are
shown to offer a new more uniform perspective on forgetting operators in
general. Both the weak and standard forgetting operators are characterized in
terms of entailment and inference, rather than a model theoretic semantics.
This naturally leads to a useful algorithmic perspective based on quantifier
elimination and the use of Ackermman's Lemma and its fixpoint generalization.
The strong formal relationship between standard forgetting and strongest
necessary conditions and weak forgetting and weakest sufficient conditions is
also characterized quite naturally through the entailment-based, inferential
perspective used. The framework used to characterize the dual forgetting
operators is also generalized to the first-order case and includes useful
algorithms for computing first-order forgetting operators in special cases.
Practical examples are also included to show the importance of both weak and
standard forgetting in modeling and representation.
