9.2 KiB
| title | chunk | source | category | tags | date_saved | instance |
|---|---|---|---|---|---|---|
| Belief revision | 2/7 | https://en.wikipedia.org/wiki/Belief_revision | reference | science, encyclopedia | 2026-05-05T14:44:59.576343+00:00 | kb-cron |
== The AGM postulates == The AGM postulates (named after their proponents Alchourrón, Gärdenfors, and Makinson) are properties that an operator that performs revision should satisfy in order for that operator to be considered rational. The considered setting is that of revision, that is, different pieces of information referring to the same situation. Three operations are considered: expansion (addition of a belief without a consistency check), revision (addition of a belief while maintaining consistency), and contraction (removal of a belief). The first six postulates are called "the basic AGM postulates". In the settings considered by Alchourrón, Gärdenfors, and Makinson, the current set of beliefs is represented by a deductively closed set of logical formulae
K
{\displaystyle K}
called belief set, the new piece of information is a logical formula
P
{\displaystyle P}
, and revision is performed by a binary operator
∗
{\displaystyle *}
that takes as its operands the current beliefs and the new information and produces as a result a belief set representing the result of the revision. The
+
{\displaystyle +}
operator denoted expansion:
K
+
P
{\displaystyle K+P}
is the deductive closure of
K
∪
{
P
}
{\displaystyle K\cup \{P\}}
. The AGM postulates for revision are:
Closure:
K
∗
P
{\displaystyle K*P}
is a belief set (i.e., a deductively closed set of formulae); Success:
P
∈
K
∗
P
{\displaystyle P\in K*P}
Inclusion:
K
∗
P
⊆
K
+
P
{\displaystyle K*P\subseteq K+P}
Vacuity:
If
(
¬
P
)
∉
K
,
then
K
∗
P
=
K
+
P
{\displaystyle {\text{If }}(\neg P)\not \in K,{\text{ then }}K*P=K+P}
Consistency:
K
∗
P
{\displaystyle K*P}
is inconsistent only if
P
{\displaystyle P}
is inconsistent Extensionality:
If
P
and
Q
are logically equivalent, then
K
∗
P
=
K
∗
Q
{\displaystyle {\text{If }}P{\text{ and }}Q{\text{ are logically equivalent, then }}K*P=K*Q}
(see logical equivalence) Superexpansion:
K
∗
(
P
∧
Q
)
⊆
(
K
∗
P
)
+
Q
{\displaystyle K*(P\wedge Q)\subseteq (K*P)+Q}
Subexpansion:
If
(
¬
Q
)
∉
K
∗
P
then
(
K
∗
P
)
+
Q
⊆
K
∗
(
P
∧
Q
)
{\displaystyle {\text{If }}(\neg Q)\not \in K*P{\text{ then }}(K*P)+Q\subseteq K*(P\wedge Q)}
A revision operator that satisfies all eight postulates is the full meet revision, in which
K
∗
P
{\displaystyle K*P}
is equal to
K
+
P
{\displaystyle K+P}
if consistent, and to the deductive closure of
P
{\displaystyle P}
otherwise. While satisfying all AGM postulates, this revision operator has been considered to be too conservative, in that no information from the old knowledge base is maintained if the revising formula is inconsistent with it.
== Conditions equivalent to the AGM postulates == The AGM postulates are equivalent to several different conditions on the revision operator; in particular, they are equivalent to the revision operator being definable in terms of structures known as selection functions, epistemic entrenchments, systems of spheres, and preference relations. The latter are reflexive, transitive, and total relations over the set of models. Each revision operator
∗
{\displaystyle *}
satisfying the AGM postulates is associated to a set of preference relations
≤
K
{\displaystyle \leq _{K}}
, one for each possible belief set
K
{\displaystyle K}
, such that the models of
K
{\displaystyle K}
are exactly the minimal of all models according to
≤
K
{\displaystyle \leq _{K}}
. The revision operator and its associated family of orderings are related by the fact that
K
∗
P
{\displaystyle K*P}
is the set of formulae whose set of models contains all the minimal models of
P
{\displaystyle P}
according to
≤
K
{\displaystyle \leq _{K}}
. This condition is equivalent to the set of models of
K
∗
P
{\displaystyle K*P}
being exactly the set of the minimal models of
P
{\displaystyle P}
according to the ordering
≤
K
{\displaystyle \leq _{K}}
. A preference ordering
≤
K
{\displaystyle \leq _{K}}
represents an order of implausibility among all situations, including those that are conceivable but yet currently considered false. The minimal models according to such an ordering are exactly the models of the knowledge base, which are the models that are currently considered the most likely. All other models are greater than these ones and are indeed considered less plausible. In general,
I
<
K
J
{\displaystyle I<_{K}J}
indicates that the situation represented by the model
I
{\displaystyle I}
is believed to be more plausible than the situation represented by
J
{\displaystyle J}
. As a result, revising by a formula having
I
{\displaystyle I}
and
J
{\displaystyle J}
as models should select only
I
{\displaystyle I}
to be a model of the revised knowledge base, as this model represent the most likely scenario among those supported by
P
{\displaystyle P}
.
== Contraction == Contraction is the operation of removing a belief
P
{\displaystyle P}
from a knowledge base
K
{\displaystyle K}
; the result of this operation is denoted by
K
−
P
{\displaystyle K-P}
. The operators of revision and contractions are related by the Levi and Harper identities:
K
∗
P
=
(
K
−
¬
P
)
+
P
{\displaystyle K*P=(K-\neg P)+P}
K
−
P
=
K
∩
(
K
∗
¬
P
)
{\displaystyle K-P=K\cap (K*\neg P)}
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:
K
=
(
K
−
P
)
+
P
{\displaystyle K=(K-P)+P}