Belief Revision on Modal Accessibility Relations

Aaron Hunter

British Columbia Institute of Technology, Burnaby, Canada

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

 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

= {x | Rwx}. Accessibility re-

lations in KD45 have the property that R is universal

on the set R

, 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

each with a corresponding accessibility relation R

This gives the modal logic KD45

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

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 φ∧γ is satisﬁable, then φ∗ γ ≡ φ∧γ.

[R3] If γ is satisﬁable, then φ∗ γ is satisﬁable.

[R4] If φ

≡ φ

and γ

≡ γ

, then φ

∗ γ

≡ φ

∗ γ

[R5] (φ∗ γ) ∧ β implies φ∗ (γ ∧ β).

[R6] If (φ∗ γ)∧ β is satisﬁable, then φ∗ (γ∧β) implies

(φ∗ γ) ∧ β.

Let f be a function that maps every propositional

formula φ to a total pre-order 

over interpretations.

We say that f is a faithful assignment if and only if

1. If s

, s

|= φ, then s

2. If s

|= φ and s

6|= φ, then s

≺

3. If φ

≡ φ

, then 

=

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 

such that

s |= φ∗ γ ⇐⇒ s is a 

−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

= {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

by φ. Rather than writing

(φ, ψ), we will adopt the notation 

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 

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

satisfying these conditions:

1. w

= w and w

= v

2. Rw

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

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 |= 

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∗

γ = ψ

where ψ is a formula such that: M , w |= B(γ, ψ) if and

only if w |= φ.

We remark that ∗

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 ∗

is a well-deﬁned function on formulas.

We call the operator ∗

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

, there is exactly one w ∈ W such

that R

= α.

The following result says that complete structures

deﬁne AGM revision operators.

Proposition 3. Let M be a complete KD

AGM

struc-

ture. Then ∗

is an AGM revision operator.

The converse is also true.

Proposition 4. Let ∗ be an AGM revision operator.

Then ∗ = ∗

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.

REFERENCES

Alchourr´on, C., G ardenfors, P., and Makinson, D. (1985).

On the logic of theory change: Partial meet func-

tions for contraction and revision. Journal of Symbolic

Logic, 50(2):510–530.

Baltag, A. and Moss, L. (2004). Logic for epistemic pro-

grams. Synthese, 31(2):165–224.

Baltag, A., Moss, L., and Solecki, S. (1998). The logic of

public announcements, common knowledge and pri-

vate suspicions. In Proceedings of the 7th Conference

on Theoretical Aspects of Rationality and Knowledge

(TARK-98), pages 43–56.

Chellas, B. (1980). Modal Logic: An Introduction. Cam-

bridge University Press.

Herzig, A., Lang, J., and Marquis, P. (2004). Revision and

update in multi-agent belief structures. In Proceedings

of LOFT 6.

Katsuno, H. and Mendelzon, A. (1992). On the difference

between updating a knowledge base and revising it.

In G ardenfors, P., editor, Belief Revision, pages 183–

203. Cambridge University Press.

van Ditmarsch, H., van der Hoek, W., and Kooi, B. (2007).

Dynamic Epistemic Logic. Springer.

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