Belief Revision - Contraction

Contraction

Contraction is the operation of removing a belief from a knowledge base ; the result of this operation is denoted by . The operators of revision and contractions are related by the Levi and Harper identities:

Eight postulates have been defined for contraction. Whenever a revision operator satisfies the eight postulates for revision, its corresponding contraction operator satisfies the eight postulates for contraction, and vice versa. If a contraction operator satisfies at least the first six postulates for contraction, translating it into a revision operator and then back into a contraction operator using the two identities above leads to the original contraction operator. The same holds starting from a revision operator.

One of the postulates for contraction has been longly discussed: the recovery postulate:

According to this postulate, the removal of a belief followed by the reintroduction of the same belief in the belief base should lead to the original belief base. There are some examples showing that such behavior is not always reasonable: in particular, the contraction by a general condition such as leads to the removal of more specific conditions such as from the belief base; it is then unclear why the reintroduction of should also lead to the reintroduction of the more specific condition . For example, if George was previously believed to have German citizenship, it was also believed to be European. Contracting this latter belief amounts to stop believing that George is European; therefore, that George has German citizenship is also retracted from the belief base. If George is later discovered to have Austrian citizenship, then the fact that he is European is also reintroduced. According to the recovery postulate, however, the belief that he also has German citizenship should also be reintroduced.

The correspondence between revision and contraction induced by the Levi and Harper identities is such that a contraction not satisfying the recovery postulate is translated into a revision satisfying all eight postulates, and that a revision satisfying all eight postulates is translated into a contraction satisfying all eight postulates, including recovery. As a result, if recovery is excluded from consideration, a number of contraction operators are translated into a single revision operator, which can be then translated back into exactly one contraction operator. This operator is the only one of the initial group of contraction operators that satisfies recovery; among this group, it is the operator that preserves as much information as possible.