Merging
The assumption implicit in the revision operator is that the new piece of information is always to be considered more reliable than the old knowledge base . This is formalized by the second of the AGM postulates: is always believed after revising with . More generally, one can consider the process of merging several pieces of information (rather than just two) that might or might not have the same reliability. Revision becomes the particular instance of this process when a less reliable piece of information is merged with a more reliable .
While the input to the revision process is a pair of formulae and, the input to merging is a multiset of formulae, etc. The use of multisets is necessary as two sources to the merging process might be identical.
When merging a number of knowledge bases with the same degree of plausibility, a distinction is made between arbitration and majority. This distinction depends on the assumption that is made about the information and how it has to be put together.
- arbitration
- the result of arbitrating two knowledge bases and entails ; this condition formalizes the assumption of maintaining as much as the old information as possible, as it is equivalent to imposing that every formula entailed by both knowledge bases is also entailed by the result of their arbitration; in a possible world view, the "real" world is assumed one of the worlds considered possible according to at least one of the two knowledge bases;
- majority
- the result of merging a knowledge base with other knowledge bases can be forced to entail by adding a sufficient number of other knowledge bases equivalent to ; this condition corresponds to a kind of vote-by-majority: a sufficiently large number of knowledge bases can always overcome the "opinion" of any other fixed set of knowledge bases.
The above is the original definition of arbitration. According to a newer definition, an arbitration operator is a merging operator that is insensitive to the number of equivalent knowledge bases to merge. This definition makes arbitration the exact opposite of majority.
Postulates for both arbitration and merging have been proposed. An example of an arbitration operator satisfying all postulates is the classical disjunction. An example of a majority operator satisfying all postulates is that selecting all models that have a minimal total Hamming distance to models of the knowledge bases to merge.
A merging operator can be expressed as a family of orderings over models, one for each possible multiset of knowledge bases to merge: the models of the result of merging a multiset of knowledge bases are the minimal models of the ordering associated to the multiset. A merging operator defined in this way satisfies the postulates for merging if and only if the family of orderings meets a given set of conditions. For the old definition of arbitration, the orderings are not on models but on pairs (or, in general, tuples) of models.
Read more about this topic: Belief Revision
Famous quotes containing the word merging:
“An administrator in a bureaucratic world is a man who can feel big by merging his non-entity in an abstraction. A real person in touch with real things inspires terror in him.”
—Marshall McLuhan (1911–1980)
“Adultery is the vice of equivocation.
It is not marriage but a mockery of it, a merging that mixes love and dread together like jackstraws. There is no understanding of contentment in adultery.... You belong to each other in what together you’ve made of a third identity that almost immediately cancels your own. There is a law in art that proves it. Two colors are proven complimentary only when forming that most desolate of all colors—neutral gray.”
—Alexander Theroux (b. 1940)
“You hear a lot of dialogue on the death of the American family. Families aren’t dying. They’re merging into big conglomerates.”
—Erma Bombeck (b. 1927)