Belief Revision on Modal Accessibility Relations
Aaron Hunter
British Columbia Institute of Technology, Burnaby, Canada
K
eywords:
Belief Revision, Knowledge Representation and Reasoning.
Abstract:
In order to model the changing beliefs of an agent, one must actually address two distinct issues. First,
one must devise a model of static beliefs that accurately captures the appropriate notions of incompleteness
and uncertainty. Second, one must define appropriate operations to model the way beliefs are modified in
response to different events. Historically, the former is addressed through the use of modal logics and the
latter is addressed through belief change operators. However, these two formal approaches are not particularly
complementary; the normal representation of belief in a modal logic is not suitable for revision using standard
belief change operators. In this paper, we introduce a new modal logic that uses the accessibility relation to
encode epistemic entrenchment, and we demonstrate that this logic captures AGM revision. We consider the
suitability of our new representation of belief, and we discuss potential advantages to be exploited in future
work.
1 INTRODUCTION
The study of belief revision is concerned with the
manner in which an agent’s beliefschange in response
to new information. The formalization of dynamic be-
liefs has been studied extensively by modal logicians
and by AI researchers. In modal logic, there is an es-
tablished representation of beliefs in terms of acces-
sibility relations in Kripke structures(van Ditmarsch
et al., 2007). The standard approach to modeling
dyanmic beliefs is then through transformations on
structures that induce new beliefs. In this paper, we
argue that this representation of (static) beliefs is not
appropriate, if we are interested in capturing the well
known AGM approach to belief revision in a Kripke
structure. We propose an alternative use of accessi-
bility relations that encodes the relative plausibility of
different states, thereby allowing us to explictly cap-
ture AGM revision in a modal setting.
Broadly speaking, work based on modal logic has
tended to address dynamic beliefs through what could
be called a bottom-up approach. In this approach,
we start with an expressive representation of static
beliefs, and then work towards the development of
suitable change operations. Modal approaches to be-
lief change often focus on difficult problems related
to multi-agent belief revision, or the logic of public
announcements. In the AI community, research on
belief revision has taken a top-down approach: it is
common to represent the beliefs of an agent as a sim-
ple set of propositional formulas, but then the belief
change operators need to incorporate some notion of
plausibility or entrenchment.
The AGM approach to belief revision (Alchourr´on
et al., 1985) is a representative example of work in
the top-down approach, and there is no universally
accepted modal formulation of AGM revision. We
would like to formulate such a logic without intro-
ducing any new machinery in terms of ranking func-
tions or transformationson Kripke structures. In other
words, we would like to use the Kripke structure itself
to encode the entrenchment of beliefs. By doing so,
we are likely to lose the simple modal intuition of ac-
cessible alternative worlds. We argue, however, that
such a notion is not appropriate for AGM revision in
any event, as it allows too much explict information
about the structure of beliefs with respect to a change
in the world. The end result is a new model concep-
tion of belief that provides an exact characterization
of AGM revision. The utility of this model for prac-
tical application is left for future work, as this is pri-
marily an expository paper focused on introducing a
new modal approach to belief revision.
The main contribution of this position paper is
the introduction of a simple modal characterization
of AGM revision. This characterization has not been
specified previously because it is somewhat unnatural
from the perspective of modal logic. However, AGM
revision itself is somewhat unnatural from the per-
spective of modal logic; so this is not surprising. The
663
Hunter A..
Belief Revision on Modal Accessibility Relations.
DOI: 10.5220/0004917706630666
In Proceedings of the 6th International Conference on Agents and Artificial Intelligence (ICAART-2014), pages 663-666
ISBN: 978-989-758-015-4
Copyright
c
2014 SCITEPRESS (Science and Technology Publications, Lda.)
second contribution of this paper is a kind of com-
pleteness result, demonstrating that every suitable bi-
nary Kripke structure defines an AGM revision oper-
ator. However, we need to make substantive assump-
tions about the philosophical significance of “accessi-
bility. This work describes work in progress, return-
ing to the logical roots of AGM revision to solidfy the
foundations of the approach with respect to the formal
structures of modal logic.
2 BACKGROUND
2.1 Doxastic Logic
We assume the reader is familiar with modal logic,
as introduced in (Chellas, 1980). Briefly, modal logic
is an extension of propositional logic where φ is a
sentence whenever φ is a sentence. The semantics
of modal logic is defined in terms of Kripke struc-
tures. Let P be a propositional vocabulary. A Kripke
structure is a triple M = hW, R, vi where W is a set
of possible worlds, R is a binary relation on W, and
v is a valuation function that maps every world to an
interpretation of P. In modal logic, the entailment re-
lation |= is defined with respect to a pair consisting of
a Kripke structure and a possible world.
Different modal logics can be defined axiomat-
ically, or by restricting the accessibility relation R.
For our purposes, two modal logics will be impor-
tant. First, we will be interested in KD45, which is
the standard doxastic logic. The logic KD45 is char-
acterized by allowing only Kripke structures where R
has the following properties:
1. R is serial: For all w W there exists x W such
that Rwx.
2. R is transitive: For all w, x, y W, if Rwx and Rxy
then Rwy.
3. R is euclidean: For all w, x, y W, if Rwx and Rxz
then Ryz.
For any w W, let R
w
= {x | Rwx}. Accessibility re-
lations in KD45 have the property that R is universal
on the set R
w
, for each w. It is common to use the
symbol B for the modal operator in KD45, and inter-
pret M , w |= Bφ to mean “in world w of the structure
M , it is believed that φ is true. In order to model the
beliefs of n agents, we introduce n modal operators
B
i
each with a corresponding accessibility relation R
i
.
This gives the modal logic KD45
n
.
The second modal logic that will be important in
this paper is the modal logic KD, which is the logic
obtained by allowing only serial accessibility rela-
tions. Hence, every KD45 structure is a KD structure
but the converse is not true. The logic KD
n
is defined
in the obvious manner.
2.2 AGM Belief Revision
In the theory of belief revision, the focus is on the dy-
namics of belief as opposed to the representation of
belief. The basic scenario in belief revision involves
a single agent with some a priori beliefs along with
some piece of “new” information about the world.
The intuition is that the new information is more re-
liable than the old information, and it must therefore
be incorporated.
One of the most influential approaches to belief
revision is the AGM approach (Alchourr´on et al.,
1985). In this approach, the beliefs of an agent are
represent by a deductively closed set of formulas
called a belief set. If we assume a finite signature, we
can equivalently represent the beliefs of an agent by
a single propositional formula φ. An AGM revision
operator is a binary function that satisfies the AGM
postulates, which were reformulated as follows by
Katsuno and Mendelzon (Katsuno and Mendelzon,
1992).
[R1] φ γ implies γ.
[R2] If φγ is satisfiable, then φ γ φγ.
[R3] If γ is satisfiable, then φ γ is satisfiable.
[R4] If φ
1
φ
2
and γ
1
γ
2
, then φ
1
γ
1
φ
2
γ
2
.
[R5] (φ γ) β implies φ (γ β).
[R6] If (φ γ) β is satisfiable, then φ (γβ) implies
(φ γ) β.
Let f be a function that maps every propositional
formula φ to a total pre-order
φ
over interpretations.
We say that f is a faithful assignment if and only if
1. If s
1
, s
2
|= φ, then s
1
=
φ
s
2
.
2. If s
1
|= φ and s
2
6|= φ, then s
1
φ
s
2
,
3. If φ
1
φ
2
, then
φ
1
=
φ
2
.
Every AGM revision operator can be captured by
minimization over a faithful assignment.
Proposition 1. (Katsuno and Mendelzon, 1992) A re-
vision operator satisfies [R1]-[R6] just in case there
is a faithful assignment that maps each φ to an order-
ing
φ
such that
s |= φ γ s is a
φ
minimal model of γ.
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3 A MODAL LOGIC OF BELIEF
REVISION
3.1 Motivation
The main intuition underlying our approach is the
idea that an accessibility relation in a modal logic
can be used to define a total pre-order over worlds.
This intuition differs from the standard intuition of
the logic KD45, where the accessibility relation un-
ambiguously associates a set of believed worlds with
each possible world. We suggest that this is not what
is actually required for AGM revision. In AGM re-
vision, the beliefs of an agent correspond to a set
of believed interpretations (as opposed to believed
worlds). The main advantage of KD45 is that it per-
mits agents to reason counterfactually in a consistent
manner. However, this kind of reasoning is neither
possible nor desirable in an AGM setting. The only
structure on worlds in AGM revision is a pre-order
where the believed worlds are all minimal. However,
each believed world leads to a different set of plausi-
ble alternatives. Agents can not reason counterfactu-
ally among the interpretations initially believed possi-
ble. As such, we suggest that KD45 does not provide
a natural logical foundation for AGM belief revision.
In the remainder of this section, we would like to
consider whether we need any restrictions on R in or-
der to define belief sets in terms of the set R
w
= {x |
Rwx}. First, we suggest that R does need to be serial.
If R is not serial, then it is possible that no worlds are
accessible. In this case, the corresponding belief set is
inconsistent. This is not permitted in AGM revision.
The rationale for both transitivity and euclideanness
depends on allowing an agent to perform some kind
of counterfactual reasoning. This is not possible in
AGM revision: a belief set is just a set of formulas.
Our approach differs from existing work, such as
(Herzig et al., 2004), where the emphasis is on trans-
formations on Kripke structures. A modal logic for
AGM revision need only specify the belief set that re-
sults from a revision; it need not specify any structure
on possible worlds following the revision. Indeed,
this is a major weakness of the AGM approach which
has lead to a great deal of work on iterated revision.
3.2 A Binary Modal Approach
We define a modal logic that has a single, serial ac-
cessibility relation (for each agent). As such, we view
the logic as a variation on the modal logic KD. How-
ever, instead of the standard unary modal operator, we
define a binary modal operator (φ, ψ). Informally
M , w |= (φ, ψ) will be interpreted to mean that ψ is
believed after revising R
w
by φ. Rather than writing
(φ, ψ), we will adopt the notation
φ
ψ in order to
maintain a close connection with standard KD syntax.
We define the modal logic KD
AGM
for a fixed
propositional signature P. Formulas are defined as
follows:
1. p is a formula for p P
2. If φ is a formula, then ¬φ is a formula.
3. If φ and ψ are formulas, then φ ψ and
φ
ψ are
both formulas.
For a Kripke structure M and a world w, the entail-
ment relation |= is defined in the usual way for atomic
formulas, negations and conjunctions. To define the
semantics of modal formulas, we need some defini-
tions.
Definition 1. For any w and v, define d(w, v) to be the
minimal number n such that there exists a sequence
w
1
, . . . , w
n
satisfying these conditions:
1. w
1
= w and w
n
= v
2. Rw
i
w
i+1
for 1 i n.
If no such path exists, then we say d(w, v) = .
We say that d(w, v) is the distance between w and
v.
Definition 2. For w W and a formula ψ, define
D(w, φ) to be the set of all v W satisfying:
1. M , v |= φ
2. d(w, v) is minimal among worlds satisfying 1.
Therefore D(w, φ) denotes the set of worlds satis-
fying φ that are minimally distant from w.
For modal formulas, we define
M , w |=
φ
ψ M , v |= ψ for all v D(w, φ).
Note that this definition is a straightforward extension
of the normal definition of ψ, except that the quan-
tification is now relativized to the set D(w, φ). We
write ψ as a shorthand notation for ψ. This “in-
duced” unary modality is clearly a KD modal opera-
tor.
3.3 Relationship with AGM Revision
We associate a revision operator with every Kripke
structure, according to the following definition.
Definition 3. For any structure M , define φ
M
γ = ψ
where ψ is a formula such that: M , w |= B(γ, ψ) if and
only if w |= φ.
We remark that
M
is not well-defined if the vo-
cabulary in infinite; in such a case, there may not be a
single formula ψ satisfying the definition. The prob-
lem is that, if the vocabulary is infinite, then we can
BeliefRevisiononModalAccessibilityRelations
665
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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