![](bg4.png)
not be assured that every set of interpretations is de-
fined by a unique formula. However, if we restrict
attention to finite vocabularies, this is not a problem.
Proposition 2. If the underlying vocabulary is finite,
then ∗
M
is a well-defined function on formulas.
We call the operator ∗
M
a partial revision operator
because it does not define revision for every possible
initial belief set. Instead it only specifies the outcome
of belief revision for the collection of belief sets de-
fined by the structure M .
Definition 4. M = hW, R, vi is a complete structure
if, for each α ⊆ 2
P
, there is exactly one w ∈ W such
that R
w
= α.
The following result says that complete structures
define AGM revision operators.
Proposition 3. Let M be a complete KD
AGM
struc-
ture. Then ∗
M
is an AGM revision operator.
The converse is also true.
Proposition 4. Let ∗ be an AGM revision operator.
Then ∗ = ∗
M
for some complete KD
AGM
structure M .
Hence, KD
AMG
structures provide an equivalent
characterization of AGM revision. Syntactically, we
would define the modal logic KD
AGM
in terms of a set
of axioms. We leave this problem for future work.
4 DISCUSSION
There has been a large body of research on the re-
lation between belief revision and modal epistemic
logic. Of particular note is the work on dynamic epis-
temic logic, originating with (Baltag et al., 1998) and
(Baltag and Moss, 2004). However, most of the work
in this tradition is focused on providing some kind
of transformation on Kripke structures. The idea is
to represent the initial beliefs of agents in a Kripke
structure, and then provide a systematic way to define
a new Kripke structure that represents the beliefs after
some event occurs.
Relating work in dynamic epistemic logic to AGM
revision has proven to be a challenge. In this paper,
we propose that the reason this is a challenge is sim-
ply because the representation of belief in a Kripke
structure is not fundamentally equivalent to the repre-
sentation in AGM revision. In AGM revision, beliefs
are represented by a set of formulas with no additional
structure. While a Kripke structure associates a set
of formulas with a belief state, there is actually ad-
ditional structure in the form of a binary accessibility
relation that permits counterfactual reasoning. Hence,
we argue that a typical Kripke structure actually can
express some forms of reasoning about belief that can
not be represented in the AGM framework. At the
same time, the AGM model of belief revision implic-
itly requires an ordering over possible interpretations
in order to carry out revision. Such an ordering is not
immediately available in a the standard KD45 modal
approach. So the typical Kripke structure approach
does not capture all aspects of the AGM approach.
In this work, we suggest that Kripke structures can
in fact be used to reason about belief change in an
AGM-like setting; however, we need to take a differ-
ent perspective on accessibility. If we use the acces-
sibility relation to encode some notion of plausibil-
ity, then we have shown that a simple class of Kripke
structures can completely characterize the AGM ap-
proach to belief revision. However, in order for this to
be possible, we lose the Kripke-style intuition about
accessibility and “possible worlds.”
In future work, we intend to consider formal em-
beddings of this logic in standard doxastic logic.
More importantly, we also intend to explore the appli-
cation of this logic for reasoning about belief change
in multi-agent systems. One of the real disadvantages
of the AGM approach is that it provides no capability
for one agent to reason about the beliefs of another
agent. This has been studied extensively in the modal
approach, particularly with respect to public announc-
ments. It is our hope that this new modal formulation
of AGM can take advantage of higher level beliefs,
while maintaining an explicit notion of entrenchment.
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