Ranking Functions for Belief Change

A Uniform Approach to Belief Revision and Belief Progression

Aaron Hunter

British Columbia Institute of Technology, Burnaby, Canada

Keywords:

Belief Revision, Reasoning about Action, Knowledge Representation.

Abstract:

In this paper, we explore the use of ranking functions in reasoning about belief change. It is well-known that

the semantics of belief revision can be deﬁned either through total pre-orders or through ranking functions

over states. While both approaches have similar expressive power with respect to single-shot belief revision,

we argue that ranking functions provide distinct advantages at both the theoretical level and the practical level,

particularly when actions are introduced. We demonstrate that belief revision induces a natural algebra over

ranking functions, which treats belief states and observations in the same manner. When we introduce belief

progression due to actions, we show that many natural domains can be easily represented with suitable ranking

functions. Our formal framework uses ranking functions to represent belief revision and belief progression in

a uniform manner; we demonstrate the power of our approach through formal results, as well as a series of

natural problems in commonsense reasoning.

1 INTRODUCTION

The study of belief revision is concerned with the

manner in which an agent’s beliefs change in response

to new information. Following the highly inﬂuen-

tial AGM model (Alchourr´on et al., 1985), many ap-

proaches to belief revision rely on some form of en-

trenchment ordering. The idea is simple: an a pri-

ori ordering over states is used to guarantee that an

agent always believes the formulas that are true in

the “most entrenched” states consistent with an ob-

servation. The literature is not always clear on ex-

actly what the ordering represents: in some cases, it

may represent the likelihood of each state, whereas

in other instances it may represent an agent’s strength

of belief. It is well known that AGM revision can

also be framed in terms of ranking functions (Spohn,

1988). In this paper, we illustrate that ranking func-

tions have signiﬁcant advantages in modelling en-

trenchment, particularly when agents are able to ex-

ecute state-changing actions. We present a uniform

approach to modeling belief revision as well as be-

lief progression, which is the change in belief that oc-

curs when an action is executed. We illustrate through

formal results and practical examples that there are

many situations where the choice between ranking

functions and entrenchment orderings is signiﬁcant.

1.1 Motivation

Belief revision is often described as the belief change

that occurs when an agent receives new information

about a static world. For example, an agent might be-

lieve that the lamp is off in a certain room behind a

closed door. If the door is opened to reveal the lamp

is on, then the agent must modify their beliefs to in-

corporate this fact. One way to model this form of

belief change is by assuming an underlying ordering

≺ over all possible states, where precedence is under-

stood to represent plausibility. The ≺-minimal states

would initially be ones in which the lamp was off. Af-

ter observing the light is on, the agent will believe the

actual state is among the least ≺-states that are con-

sistent with this observation.

We are interested in domains where plausibility

can not easily be captured by an ordering. For ex-

ample, there are cases where evidence is additive; the

agent might require two reports that the light is on be-

fore changing beliefs. In this case, an observation of

light might make certain states more plausible, with-

out actually changing the relative order of possible

states. Similiarly, there are cases were observations

are graded; light under the door could indicate the

lamp is on, or perhaps that the window is open. Fi-

nally there are cases where actions may have unlikely

effects; opening the door might accidentally turn the

412

Hunter A..

Ranking Functions for Belief Change - A Uniform Approach to Belief Revision and Belief Progression.

DOI: 10.5220/0004812704120419

In Proceedings of the 6th International Conference on Agents and Artiﬁcial Intelligence (ICAART-2014), pages 412-419

ISBN: 978-989-758-015-4

 2014 SCITEPRESS (Science and Technology Publications, Lda.)

light on. Each of these is difﬁcult to capture in a situ-

ation where uncertainty is represented by an ordering,

without any mechanism for comparing magnitudes of

unlikely effects. Our aim in this paper is to provide

a ﬂexible approach where uncertainty over states, ac-

tions, and observations is modeled by ranking func-

tions that can be compared and combined with basic

arithmetic.

1.2 Contributions

This paper makes several contributions to existing

work on belief change. First, we introduce a natural

algebra of belief revision that simpliﬁes the semantics

for the general case, while simultaneously subsuming

AGM revision. Second, we introduce a formal model

of belief progression due to actions that is Markovian,

but still allows belief revision to respect the action his-

tory by excluding certain states after actions are exe-

cuted. Finally, we demonstrate through a series of

commonsense reasoning examples that a great deal of

practical expressive power can be gained by allowing

plausibility functions to range not only over states, but

also over possible actions.

2 BACKGROUND

2.1 AGM Belief Revision

We focus on propositional belief revision, so we

assume an underlying propositional signature. One

of the most inﬂuential approaches to belief revision is

the AGM approach (Alchourr´on et al., 1985). In the

AGM approach, a belief set is a deductively closed

set of formulas. In this paper, we restrict attention to

ﬁnite propositional signatures, so a belief set can be

represented by a propositional formula φ. The new

information to be incorporated is also represented by

a single formula, say γ. An AGM revision operator is

a binary function ∗ that satisﬁes the AGM postulates.

The following reformulation of the postulates is due

to 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.

The class of AGM revision operators can be char-

acterized in terms of orderingson states, where a state

is just an interpretation of the underlying proposi-

tional signature. Let f be a function that maps every

propositional formula φ to a total pre-order 

over

states. 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 

=

The following characterization result indicates that

every AGM operator can be understood in terms of

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.

A similar characterization can be given using

Spohn’s ordinal conditional functions (Spohn, 1988),

which are functions mapping each state to an ordinal.

2.2 Belief Change Due to Actions

We assume set A of action symbols. The effects

of actions are described by a transition function f :

S× A → S. Hence, a transition function takes a state

and an action as arguments, then it returns a new state.

Informally, the output is the state that results from ex-

ecuting the given action in the given state. We are pri-

marily concerned with actions that have deterministic

effects, though we also allow non-deteministic effects

in some examples. Throughout this paper, we will

let the lower case letter a (possibly with subscripts)

range over actions. A propositional signature F to-

gether with a transition function f over S is called an

action signature.

In the literature, belief update refers to the belief

change that occurs when an agent receives informa-

tion about a change in the state of the world (Katsuno

and Mendelzon, 1991). The classic approach to belief

update deﬁnes the operation on propositional formu-

las, just as belief revision is deﬁned on propositional

formulas. This can lead to ambiguity because it is not

always clear when an agent should perform revision

and when they shoud perform update (Lang, 2007).

We avoid this ambiguity by modeling belief

change due to actions through the operation com-

monly called belief progression. Belief progression

operators take a belief state and an action as input,

and return a new belief state that is obtained by pro-

gressing each possible world in accordance with the

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413

effects of the action. We remark that belief change

in our formal approach is Markovian in that the new

beliefs are completely determined by the orginal be-

liefs as well as the event (action or observation) that

has occured. The basic model is not able to represent

a class Markov Decision Process because we do not

incorporate any notion of likelihood on action effects;

we address this by adding such a measure in one of

the examples in §4.

3 PLAUSIBILITY FUNCTIONS

3.1 An Algebra for Belief Revision

We formally deﬁne plausibility functions as follows.

Deﬁnition 1. Let X be a non-empty set. A plausibil-

ity function over X is a function r : X → N such that

r(x) = 0 for at least one x ∈ X.

If r is a plausibility function and r(x) ≤ r(y), then

we say that x is at least as plausible as y. Plausibil-

ity functions are similar to ordinal conditional func-

tions (Spohn, 1988), except that the domain can be

any arbitrary set and we restrict the range to the nat-

ural numbers. This deﬁnition is based on a similar

concept introduced in (Hunter and Delgrande, 2006).

When r is a plausibility function over states, we

can identify the minimal elements of r with the states

currently believed possible. Let

Bel(r) = {x | r(x) = 0}.

The degree of strength of a plausibility function r is

the least n such that n = r(v) for some v 6∈ Bel(r).

Hence, the degree of strength is a measure of how dif-

ﬁcult it would be for an agent to abandon the currently

believed set of states.

We use plausibility functions to represent initial

belief states, and also to represent new information

for revision. Hence, revision in this context is just a

binary operator on plausibility functions. Given any

pair of plausibility functions r

and r

, we can deﬁne

a new function r

such that (r

)(x) = r

(x)+

(x). Of course, the sum of two plausibility functions

need not be a plausbility function; but we can obtain

an equivalent plausibility function by normalizing.

Deﬁnition 2. Let r

and r

be plausibility functions

over X, and let m be the minimum value of r

+ r

Then r

∗ r

is the function on X deﬁned as follows:

∗ r

(x) = r

(x) + r

(x) − m.

It should be clear that r

∗ r

is a plausibility func-

tion, because it attains a minimum value of 0. We use

the symbol ∗ for this operation, because it can be seen

as a generalization of AGM belief revision. To make

this explicit, we introduce a basic deﬁnition.

Deﬁnition 3. A plausibility function r is two-valued

iff the range of r is a set of size 2. If r is two-valued,

we write |r| = {s | r(s) = 0}.

A formula can be represented by a two-valued

plausibility function. We remark also that every plau-

sibility function deﬁnes a total pre-order. Hence,

AGM belief revision can be captured by taking a plau-

sibility function over states (the initial beliefs) and

adding a two-valuedplausibility function (the formula

for revision).

The class of plausibility functions is clearly closed

under ∗. We state some other basic properties.

Proposition 2. The operator ∗ is associative. i.e.

∗ r

) ∗ r

= r

∗ (r

∗ r

We remark that many approaches to iterated revi-

sion are not associative, so this result suggests that our

model of revision does not align directly with work in

this area. We accept this difference, as it has been ar-

gued that none of the existing appoaches to iterated

revision are completely satisfactory(Stalnaker, 2009).

In the following propositions, let r

be the plausi-

bility function such that r

(x) = 0 for all x. We refer

to r

as the identity function. As an initial belief state,

the identity function represents ignorance. As infor-

mation for revision, it represents a null observation.

Proposition 3. r

∗ r = r ∗ r

= r for any plausibility

function r.

Proposition 4. For any plausibility function r, there

is a plausibility function r

−1

such that r∗ r

−1

= r

−1

∗

r = r

In abstract algebra, any closed system with an op-

erator satisfying Propositions 2, 3 and 4 is called a

group. If the operator is also commutative, the sys-

tem is called an abelian group.

Proposition 5. The class of plausibility functions is

an abelian group under ∗.

The fact that ∗ deﬁnes an abelian group means that

we can exploit all of the known results about groups

to analyze the symmetries and structure of revision

under this deﬁnition.

3.2 Adding Actions

In this section, we assume an underlying set of ac-

tion symbols A as well as a transition function f that

describes the effects of the actions in A. A belief pro-

gression operator is a function that maps an initial be-

lief state to a new belief state, given that some action

has been executed. When actions are introduced, we

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414

need to account for the fact that certain states may not

be possible following the execution of a particular ac-

tion.

To address this issue, we introduce the notion of

an extended plausibility function.

Deﬁnition 4. An extended plausibility function over

X is a function r : X → N∪ {∞} such that r(x) = 0 for

at least one x ∈ X.

We deﬁne ∞ to be larger than every number in N.

Morever, we deﬁne addition as follows

p+ ∞ = ∞+ p = ∞ for any p ∈ N∪{∞}.

For any plausibility function r, let

imp(r) = {s | r(s) = ∞}.

Hence, imp(r) is the set of states that are “impossible”

according to the function r. On the other hand, we

will refer to the complement of imp(r) as the set of

“possible” states.

Proposition 6. If r

, r

are extended plausibility func-

tions, then imp(r

∗ r

) = imp(r

) ∪ imp(r

It follows that the set of possible states does not

change when we revise by a plausibility function r

with imp(r) =

0. So we can think of an extended plau-

sibility function as a plausibility function together

with a set I of impossible states that are assigned the

plausibility ∞.

We are now able to deﬁne belief progression with

respect to extended plausibility functions.

Deﬁnition 5. Let a ∈ A with deterministic transition

function f, and let r be an extended plausibility func-

tion over A. Then r · a is the extended plausibility

function:

r·a(s) =



min({r(s

′

) | f(a, s

′

) = s})

∞ otherwise

This deﬁnition says that the plausibility of each

state following an action is obtained by progressing

forward the effects of actions. Hence, for each state

s with plausibility k, we say that f(a, s) = k. This

just means that the plausibility of the state remains

the same before and after the execution of the action

a. However, since there may be more than one initial

state s with the same outcome, we take the minimum

possible value. We need to assign ∞ to some states is

because some states are not possible after executing a.

For example, in a deterministic world, all states where

the door is open will be impossible after an agent per-

forms a closedoor action. Therefore, r · a is the nat-

ural shifting of r by the effects of s for states in the

range of a; for states that are not possible following a,

the plausibility is deﬁned to be ∞.

In many applications, it is important to know if

an extended plausibility function is consistent with a

sequence of actions. We let

a denote a sequence of

actions of indeterminate length, and we let r· a denote

the sequential progression of r by each element of

Deﬁnition 6. An extended plausibility function r is

consistent with the action sequence a just in case

imp(r) = imp(r

a).

Hence, r is consistent with a if the execution of

a from a state of total ignorance leads to the same

collection of impossible states. Since we are primar-

ily interested in extended plausibility functions that

result from actions, we simply say that an extended

plausibility function is consistent if it is consistent

with some action sequence.

Proposition 7. If the extended plausibility function r

is consistent, then r · a and r ∗ r

′

are also consistent,

for any action a any plausibility function r

′

Hence, if every state is initially possible, then we

need only be concerned with consistent functions.

The following result says that the outcome of a se-

quence of actions is consistent provided all states

were initially possible.

Proposition 8. Let r

, r

be plausibility functions and

let

a be a sequence of actions. The extended plausi-

bility function r

· a∗ r

is consistent with

A set of so-called “interaction properties” have

been proposed to ensure that observations following

actions are incorporated in a sensible manner (Hunter

and Delgrande, 2011). The main postulate can be re-

formulated in our notation as the following:

P5. Bel(r· a ∗ r

′

) ⊆ Bel(r

· a)

The postulate P5 asserts that, regardless of the ini-

tial belief state and the observation, the ﬁnal belief

state must be a possible outcome of the action a. It

is straightforward to show that Proposition 8 entails

than interaction property P5.

3.3 Plausibility Functions Over Actions

A plausibility function over action symbols can be

used to represent an agent’s uncertainty about the ac-

tions that have been executed. We let A range over

plausibility functions over actions; so we think of A

as a partially observed action. We extend the use of ·

in the following deﬁnition.

Deﬁnition 7. Let r be an extended plausibility func-

tion and let A be a plausibility function over A. Then

r· A is the extended plausibility function such that:

r· A(s) = min({r(s

′

) + A(a) | f(a, s

′

) = s}).

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415

Informally, the deﬁnition just says that the most

likely ﬁnal states are the states that result starting from

the most likely initial states and carrying out the most

likely actions.

Proposition 9. Let A be an extended plausibility func-

tion over actions, and suppose that Bel(A) = {a} and

A(a

′

) = ∞ for a

′

6= a. Then for any extended plausi-

bility function r, it follows that r· A = r · a.

This proposition deals with the case where A es-

sentially picks out a single action a. In this case, we

get the same result we would get if we simply used

progression by the action a.

The following proposition gives some indication

of the role played by ∞ when we have uncertainty over

actions.

Proposition 10. If r is a plausibility function (i.e.

with domain N), then r · A(s) = ∞ just in case one of

the following holds:

1. A is inconsistent, or

2. If f(a, s

′

) = s, then A(a) = ∞.

Hence, if A is consistent, then the only states ex-

cluded by r·A are those that are not possible outcomes

of any action that is possible according to A.

Note that the model proposed here is only appro-

priate for action domains where the effects of an ac-

tion can not fail. We are using plausibility functions

to rank the likelihood that an action has occurred; if

an agent believes an action has occurred, then the ef-

fects of that action must hold. We will see in the ex-

amples, however, that it is possible to include non-

deterministic and failed actions by adding some addi-

tional ranking functions for effects.

4 REPRESENTING NATURAL

ACTION DOMAINS

In this section, we demonstrate how sequences of

plausibility functions can be used to represent natu-

ral action domains. In terms of notation, we use INIT

to represent the initial plausibility function. We use A

and O (possibly with subscripts) to represent plausi-

bility functions over actions and states, respectively.

We refer to O as an observation, as it is a ranking

function on states that provides new information. To

be clear, athough INIT, A and O are all plausibility

functions, it may be the case that INIT · A ∗ O is an

extended plausibility function.

We now introduce a sequence of examples. In

each case, we assume that the actions and observa-

tions are (simple) plausiblity functions, and the ﬁnal

belief state is an extended plausbility function where

some states are excluded. The ﬁnal plausiblity values

can be obtained by minimizing the sum of all plausi-

bility values over actions and states, restricting atten-

tion to sequences of actions that are actually possible

in the underlying transition system. The agent there-

fore maintains a consistent representation of the plau-

sibility of a world together with the effects of actions.

Example (Additive Evidence). Bob believes that he

turned the lamp off in his ofﬁce, but he is not com-

pletely certain. As he is leaving the building, he talks

ﬁrst to Alice and then to Eve. If only Alice tells him

his lamp is still on, then he will believe that she is

mistaken. Similarly, if only Eve tells him his lamp

is still on, then he will believe that she is mistaken.

However, if both Alice and Eve tell Bob that his lamp

is still on, then he will believe that it is in fact still on.

The action signature contains, among others, a

propositional variable LampOn and an action sym-

bol TurnLampOf f. The underlying transition system

deﬁnes the effects of turning the lamp off in the obvi-

ous manner. Let ON denote the set of states in which

LampOn is true. The following plausibility functions

describe this action domain.

1. INIT(s) = 0 if s ∈ ON, INIT = 10 otherwise

2. A

(a) = 0 if a = TurnLampOf f, A

(a) = 3 oth-

erwise

3. O

(s) = 0 if s ∈ ON, O

(s) = 2 otherwise

4. A

(a) = 0 if a = null, A

(a) = 10 otherwise

5. O

(s) = 0 if s ∈ ON, O

(s) = 2 otherwise

It is easy to verify that, under this representation, two

observations of ON are required to make Bob believe

that he did not turn the lamp off.

Example (Graded Evidence). Bob receives a gift that

he estimates to be worth $7. He is curious about the

price, so he tries to glance quickly at the receipt with-

out anyone noticing. He believes that the receipt says

the price is $3. This is far too low, so Bob concludes

that he must have mis-read the receipt. Since a “3”

looks very similar to an “8”, he concludes that the

price on the receipt must have been $8.

To represent this example, it is useful to assume

that the set of actions includes a distinguished action

symbol null that is just the identity function over the

set S of states. Deﬁne the plausibility function A

such

that A

(null) = 0 and A

(a) = 10 for every non-null

action a, because Bob believes that no actions have

occurred. We assume that there are propositional vari-

ables Cost1, Cost2, . . . , Cost9 interpreted to represent

the cost of the gift. We deﬁne a plausibility function

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416

INIT representing Bob’s initial beliefs.

INIT(w) =







0 if w = {Cost7}

1 if w = {Cost6} or w = {Cost8}

3 otherwise

Note that Bob initially believes that the cost is $7, but

it is comparativelyplausible that this cost is one dollar

more or less. Finally, we deﬁne a plausibility function

representing the observation of the receipt.

(w) =







0 if w = {Cost3}

1 if w = {Cost8}

3 otherwise

Bob believes that the observed digit was most likely

a “3”, with the most plausible alternative being the

visually similar digit “8”.

Given these plausibility functions, the most plau-

sible conclusion is that the actual price is $8; this is

the result obtained through minimization. In order

to draw this conclusion, Bob needs graded evidence

about states of the world and he needs to be able to

weight this information against his initial beliefs.

The preceding examples illustrate that there are

commonsense reasoning problems in which an agent

needs to consider aggregate plausibilities over a se-

quence of actions and observations. Plausibility func-

tions are well-suited for reasoning about such prob-

lems. Total pre-orders over states, on the other hand,

are not. In the case of graded evidence, the important

point is that we need to be able to distinguish between

initial plausibilities and some measure of similarity

between observations. This is easy to represent us-

ing ranking functions; it is more difﬁcult to represent

using orderings, as we need to introduce some mech-

anism for combining the “levels” of an ordering.

4.1 Non-deterministic and Failed

Actions

In this section, we consider actions with non-

deterministic effects, including actions that may fail.

Let f be a non-deterministic transition function, so

f(a, s) is a set of states that represents the possible

outcomes when action a is executed in state s. Given

such a transition function along with a plausibility

function over actions, it is not possible to give a clear

categorical procedure for choosing the effects of each

action in the most plausible world histories. This

problem can be solved by following (Boutilier, 1995),

and attaching a plausibility value to the possible ef-

fects of each action.

Deﬁnition 8. An effect ranking function is a function

δ that maps every action-state pair (a, s) to a plausi-

bility function over f(a, s).

Informally, an effect ranking function gives the

likelihood of each possible effect for each action. A

non-deterministic plausibility function is a pair hr, δi

where r is a plausibility function over actions and δ is

an effect ranking function.

Example (Unlikely Action Effects). Consider an

action domain involving a single propositional vari-

able LampOn indicating whether or not a certain lamp

is turned on. There are two action symbols Press

and ThrowPaper respectively representing the acts of

pressing on the light switch, or throwing a ball of pa-

per at the light switch. Informally, throwing a ball

of paper at the light switch is not likely to turn on

the lamp. But suppose that an agent has reason to be-

lieve that a piece of paper was thrown at the lamp and,

moreover, the lamp has been turned on. Deﬁne A

that ThrowPaper is the most likely action at time 1.

λ Press ThrowPaper

10 1 0

Next deﬁne INIT and O

so that initially the light is

off, and then the light is on.

LightOn

INIT 10 0

0 10

Finally, we deﬁne an effect ranking function δ cap-

turing the fact that pressing is more likely to turn the

light on. This ranking function says nothing about

which action has actually occurred.

{LightOn}

δ(Press, LightOn) 0 10

δ(Press,

0) 1 2

δ(ThrowPaper, LightOn) 0 10

δ(ThrowPaper,

0) 2 1

Introducing effect ranking functions makes the

distinction between action occurrences and action ef-

fects explicit, which in turn gives a straightforward

treatment of failed actions.

5 DISCUSSION

5.1 Prioritizing Plausibility Functions

In formalizing commonsense examples, we had a se-

quence of plausibility functions over states and ac-

tions. We executed a sequence of operations itera-

tively by minimization over plausibility values at each

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417

step. It is not clear if this is always appropriate, partic-

ularly if we know in advance that there will be several

plausibility functions to combine.

This issue has been discussed for belief revision

in (Delgrande et al., 2006), where it is suggested that

sequences of observations should should ﬁrst be com-

bined through some form of prioritization, and then

the combination should be used as input for revision.

In our context, this could be accomplished by using a

large scalar multiple to prioritize certain ranking func-

tions, followed by summation to merge all informa-

tion together. It may also be interesting to consider

non-summation based aggregates; the important point

is that using a different aggregate does not introduce

a fundamental change to our framework.

We remark, however, that there is a distinction that

is lost here. In particular,there is a difference between

an action that fails to occur and an action that occurs,

but fails to produce an expected effect. Consider an

agent that tries to drop a glass on the ground to break

it. One possible outcome is that the agent executes

the drop action but it fails to occur; perhaps the glass

sticks to the agent’s hand. Alternatively, the glass

could be successfully dropped without breaking. In

our framework, both of these events are represented

by a dropping action with the null effect. In some

cases, this might not be appropriate.

5.2 Action Formalisms

The issues addressed in this paper have been ad-

dressed in related action formalisms. We have al-

ready mentioned related work that has been done in

the context of transition systems (Hunter and Del-

grande, 2006). Similar work has also been done in the

Situation Calculus (SitCalc). The SitCalc is an action

formalism based on ﬁrst-order logic, summarized in

(Levesque et al., 1998). While the original formalism

does not incorporate any epistemic notions, knowl-

edge and belief have been added in extended versions.

The most relevant work for comparison with our ap-

proach is the framework for iterated belief change in

(Shapiro et al., 2011). In order to reason about belief

change, a ranked set of possible initial situations is in-

troduced and this set is reﬁned over time as an agent

performs sensing actions.

On the surface, our work is distinguished from the

work in the SitCalc in that we use a less expressive

representation of action effects. By using transition

systems, we hope to focus entirely on the role played

by the relevant ranking functions in belief change.

The representation of belief change in the SitCalc

does not use rankings to measure an agents percep-

tion of the action that has been executed, nor does it

attempt to merge multiple forms of uncertainty due

to graded evidence and epistemic entrenchment. Of

course this is not a limitation of the approach: the Sit-

Calc is a very expressive formalism that can be used

to capture such notions. For instance, recent work has

provided a treatment of beliefs about failed actions

in the SitCalc (Delgrande and Levesque, 2012). Our

view is that it can be simpler to iron out the main is-

sues with respect to believe change in a simple AGM-

like framework ﬁrst, before migrating the solutions to

a sophisticated formal framework such as the SitCalc.

5.3 Dynamic Epistemic Logic

Dynamic Epistemic Logic (DEL) is a broad term that

generally refers to formal models of changing knowl-

edge and belief following in the tradition of (Baltag

et al., 1998). For a complete discussion of work in

this area, we refer the reader to the extensiveintroduc-

tion in (van Ditmarsch et al., 2007). Broadly, work on

DEL is distinguished from the work presented here in

that DEL is based on the use of Kripke structures to

model knowledge and belief.

Recent work in DEL has incorporated key notions

of plausibility from the AGM tradition, as well as no-

tions of graded belief. For example, (Lorini, 2011)

provides an interesting example of graded belief in

DEL. This work is actually quite similar in spirit to

ours, and is fueled by the same kind of commonsense

reasoning examples. By using simple ranking functi-

nos over sets, our hope is to highlight the signiﬁcant

aspects of belief change that need to be modeled be-

fore committing to the representation of belief that is

embodied by a Kripke structure.

5.4 Reasoning with Ordinals

The plausibility functions used in this paper are re-

ally a variation of Spohn’s ordinal conditional func-

tions(OCFs). In this paper, we have taken the posi-

tion that a single inﬁnite value denoted by ∞ can be

useful for representing impossiblity. Adding this no-

tion of impossibility makes our model largely equiva-

lent to work in possibilistic logic, where quantitative

measures of likelihood are combined with a “neces-

sity measure” of 0 (Dubois and Prade, 2004).

While our focus in this paper has been on reason-

ing with a single inﬁnite value, we propose that there

are situations where we actually want greater expres-

sive power for discussing impossible states. For ex-

ample, we may want to reason about such states hypo-

thetically; in these situations, it can actually be useful

to allow plausibility values to range over all ordinals.

We suggest, for example, that it may be useful

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for an agent to reason counterfactually in worlds that

haveplausibility starting from the ordinal ω. Consider

a statement of the form “The present king of France

is bald.” We can reasonably expect an agent to revise

their beliefs about the present king of France, even if

they do not believe in his existence. For example, one

might be told “All french monarchs just had a hair

transplant.” In this case, it could be counterfactually

concluded that the present king of France is no longer

bald. This kind of reasoning can be modelled by iden-

tifying each limit ordinal (such as ω) with a hypothet-

ical world conﬁguration. In this manner, ordinals of

the form ω + i can then be used as plausibility val-

ues to represent events of varying degrees of implau-

sibility within these hypothetical worlds. The order-

ing on limit ordinals indicates which hypetheticals are

the most outlandish, but we are able to perform revi-

sion across each in a uniform manner. We leave this

application of plausibility functions for future work.

6 CONCLUSIONS

In this paper, we have discussed the use of plausiblity

functions for reasoning about belief change, with a

particular focus on action domains. We have demon-

strated that Spohn-style ranking functions can be used

to deﬁne an algebra of belief change, which can then

be extended to reason about belief progression due

to actions. We then used the same kind of rank-

ing functions to represent uncertainty over the actions

that have been executed. Through commonsense ex-

amples we demonstrated that ranking functions are a

ﬂexible tool that can capture many different kinds of

uncertainty. At a formal level, it has long been known

that AGM revision can be deﬁned in terms of rank-

ing functions or total pre-orders; but there is a sense

in which pre-orders are simpler, and they have tended

to be more popular in the literature. Our examples

give many situations that are easier to represent with

ranking functions, because the gap between different

levels of plausibility is important. In many cases, our

uniform treatment of belief states and observations

simpliﬁes the algebra of belief change.

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