6.8 KiB
| title | chunk | source | category | tags | date_saved | instance |
|---|---|---|---|---|---|---|
| Belief revision | 6/7 | https://en.wikipedia.org/wiki/Belief_revision | reference | science, encyclopedia | 2026-05-05T14:44:59.576343+00:00 | kb-cron |
Spohn rejected revision this non-numerical proposal has been first considered by Spohn, who rejected it based on the fact that revisions can change some orderings in such a way the original ordering cannot be restored with a sequence of other revisions; this operator change a preference ordering in view of new information
P
{\displaystyle P}
by making all models of
P
{\displaystyle P}
being preferred over all other models; the original preference ordering is maintained when comparing two models that are both models of
P
{\displaystyle P}
or both non-models of
P
{\displaystyle P}
; Natural revision while revising a preference ordering by a formula
P
{\displaystyle P}
, all minimal models (according to the preference ordering) of
P
{\displaystyle P}
are made more preferred by all other ones; the original ordering of models is preserved when comparing two models that are not minimal models of
P
{\displaystyle P}
; this operator changes the ordering among models minimally while preserving the property that the models of the knowledge base after revising by
P
{\displaystyle P}
are the minimal models of
P
{\displaystyle P}
according to the preference ordering; Transmutations Williams provided the first generalization of belief revision iteration using transmutations. She illustrated transmutations using two forms of revision, conditionalization and adjustment, which work on numerical preference orderings; revision requires not only a formula but also a number or ranking of an existing belief indicating its degree of plausibility; while the preference ordering is still inverted (the lower a model, the most plausible it is) the degree of plausibility of a revising formula is direct (the higher the degree, the most believed the formula is); Ranked revision a ranked model, which is an assignment of non-negative integers to models, has to be specified at the beginning; this rank is similar to a preference ordering, but is not changed by revision; what is changed by a sequence of revisions are a current set of models (representing the current knowledge base) and a number called the rank of the sequence; since this number can only monotonically non-decrease, some sequences of revision lead to situations in which every further revision is performed as a full meet revision.
== Merging == The assumption implicit in the revision operator is that the new piece of information
P
{\displaystyle P}
is always to be considered more reliable than the old knowledge base
K
{\displaystyle K}
. This is formalized by the second of the AGM postulates:
P
{\displaystyle P}
is always believed after revising
K
{\displaystyle K}
with
P
{\displaystyle P}
. 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
K
{\displaystyle K}
is merged with a more reliable
P
{\displaystyle P}
. While the input to the revision process is a pair of formulae
K
{\displaystyle K}
and
P
{\displaystyle P}
, the input to merging is a multiset of formulae
K
{\displaystyle K}
,
T
{\displaystyle T}
, 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
K
{\displaystyle K}
and
T
{\displaystyle T}
entails
K
∨
T
{\displaystyle K\vee T}
; 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
K
{\displaystyle K}
with other knowledge bases can be forced to entail
K
{\displaystyle K}
by adding a sufficient number of other knowledge bases equivalent to
K
{\displaystyle K}
; 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.