A Short Course in Belief Revision - PowerPoint Presentation

Slide1

A Short Course in Belief Revision

Pavlos Peppas

University of

Technology

Sydney

and

University of Patras

Slide2

John’s car is a BMW

BMWs are made in Germany

Germany is part of the EU

All cars made in EU take unleaded petrol

John’s car takes unleaded petrol

Rational

Belief

Revision

John’s car

takes leaded petrol

BMWs are made in Germany

Germany is part of the EU

All cars made in the EU take unleaded petrol

An Example

John’s car takes

leaded petrol

There is life on Mars.

Slide3

Some History

(

Carlos Alchourron and David Makinson

)

The children may watch TV only if they eat their dinner.

The children may eat their dinner only if they do their homework

Old Legal Code

?

New Legal Code

Amendment

:

On Fridays the children may watch TV without doing their homework .

Slide4

Ramsey Test:

The sentence "If A, then B" is true at belief state K,

iff

B is true at the state that results from the revision of K by A.

K

⊨

A > B iff (K*A) ⊨ B

A

K*A

K

*

¬A

B

Κ

A B A

 B

T T T

T F F

F T T

F F T

├

History II:

Semantics for

Counterfactuals

(

Peter

Gardernfors

)

If

Napoleon had

not invaded Russia

,

then

he would had conquered Europe

.

Slide5

The Birth of Belief Revision (1985)

The AGM Framework

f

or

Theory Change

C.

Alchourron, P.

Gardernfors, and D. Makinson, “On the logic of theory change: Partial meet functions for contraction and

revision”, Journal of Symbolic Logic, 1985

Slide6

Using Logic to Model Belief Revision

Rational Belief Revision

*

a

b

Κ

Κ*Α



a

b

(

ab) 

d(de)  f

a



b

a(ab)  d

(de)  f

d

Beliefs are modeled as sentences of propositional logic. Belief States are modeled as sets of sentences

closed under logical implication.

Slide7

Models for Belief Revision

Grove, 1988

Axiomatic

Models

AGM

Revision Functions

AGM

Contraction Functions

Preorders

on possible

worlds

Epistemic

Entrenchments

Constructive

Models

Levi / Harper Identity

Selection

Functions

Alchourron

, Gardenfors &

Makinson

, 1985, 1988

Slide8

The AGM Postulates for Belief Revision

*

A

Κ

Κ*Α

Principle of Minimal Change:

The new belief state differs

as little as possible

from the

old belief states, in light of the new information

(Κ*2) Α

 (K*A)

(K*7) K*(AB)  (K*A)+B

(K*8)

If

B  (K*A)

then (K*A)+B  K*(AB)

(K*1) Κ*Α is a theory

(K*5) If A is consistent then K*A is consistent

(K*6) If AB

then K*A = K*B

(K*3) K*A  K+A

(K*4)

If A  K then K+A  K*A

Slide9

The Plurality of AGM Revision Functions

*

A

Κ

Κ*Α

For a given belief state K and new information A, the AGM postulates (K*1) - (K*8)

do not specify

uniquely

the new belief state K*A.

Functions that satisfy the postulates

(Κ*1) - (Κ*8)

*

*

*

*

*

*

*

*

*

*

*

*

*

Slide10

Some Additional Conditions

*

A

Κ

Κ*Α

(K*M

) If

K

 H then K*A  H*A.

Theorem:

Condition (K*M) is inconsistent with (K*1) – (K*8).

(K*R)

If B

 K and B  K*A then B  K*A.

Theorem:

If * satisfies (K*1) – (K*8) and (K*R), then K*A is complete whenever

A  K.

Slide11

Models for Belief Revision

Axiomatic

Models

AGM

Revision Functions

AGM

Contraction Functions

Preorders

over

possible

worlds

EpistemicEntrenchmentsConstructive

Models

Selection

Functions

Slide12

A Nice Possible World

Australia

Greece

Germany

Slide13

Possible Worlds vs Sentences

All academics are rich

All academics are nice

All academics are rich

and

nice

Slide14

Belief Revision with Worlds

John’s car is a BMW

BMWs are made in Germany

Germany is part of the EU

All cars made in EU take unleaded petrol

John’s car

takes leaded petrol

Slide15

Plausibility Rankings

John’s car is a BMW

BMWs are made in Germany

Germany is part of the EU

All cars made in EU take unleaded petrol

≤

≤

≤

(S*) [K*A] = min([A], ≤)

John’s car

takes leaded petrol

Slide16

Representation Result

(S*)

Revision Functions

(

K*1) - (K*8)*

*

*

*

*Preorders on Possible Worlds













Slide17

Models for Belief Revision

Axiomatic

Models

AGM

Revision Functions

AGM

Contraction Functions

Preorders

o

n possible

worlds

EpistemicEntrenchments

Constructive

Models

Selection

Functions

Slide18

The AGM Postulates for Belief Contraction

-

A

Κ

Κ-Α

Principle of Minimal Change:

The new belief state differs

as little of possible

from the old belief state, in view of the sentence A that needs to be

removed.

(Κ-2)

K-A  K

(K-3)

If

A  K

then

K - A = K

(K-7) (K-A)(K-B)  K-(A  B)

(K-8)

If

A  K-(A  B)

then

K-(A  B)  K-A

(K-1) Κ-Α is a theory

(K-5)

If

A K

then

K  (K-A)+A

(K-6)

If

A≣B

then K-A = K-B

(K-4)

If

⊭

A

then

A  K-A

Slide19

Levi Identity

*

A

Κ

(Κ - (

A)) + A

K - (

A)

- (

A)

+ A

THEOREM :

(LI) K*A = (Κ - (

A)) + A

(K-1) - (K-8)

-

(K*1) - (K*8)

*

(LI)

-

-

-

-

*

*

*

*

Slide20

Harper Identity

-

A

Κ

(Κ * (

A))  K

K

*

(

A)

*

(

A)

 K

THEOREM :

(K-1) - (K-8)

-

(K*1) - (K*8)

*

(HI

)

-

-

-

-

*

*

*

*

(HI

)

K-A

= (Κ

*

(

A))



K

Slide21

Inter-definability

(K*1) - (K*8)

*

(K-1) - (K-8)

-

(LI)

*

*

*

*

-

-

-

-

(HI)

Slide22

Models for Belief Revision

Axiomatic

Models

AGM

Revision Functions

AGM

Contraction Functions

Preorders

o

n possible

worlds

EpistemicEntrenchments

Constructive

Models

Selection

Functions

Slide23

Epistemic Entrenchment

John

’

s car does not take unleaded petrol

Germany belongs to

Ε.

U

.

All BMW are made in Germany

All cars made in E.U. take unleaded petrolJohn’

s car is a BMW







John

’

s car does not take unleaded petrol

John

’

s car is not a BMWAll BMW are made in Germany

Germany belongs to E.U.All cars made in E.U. take unleaded petrol

Slide24

Epistemic Entrenchment

(EE1)

If

Α

 B and B  C, then A  C.(ΕΕ3)

Α  AB or B  AB.

(ΕΕ4)

If Κ

 L, then Α  K iff A  B, for all BL.

(ΕΕ2) If

Α

⊨ Β then Α  B.

(ΕΕ5) If Α  B for all A L,

then ⊨ B

(E-)

Κ







Α

Κ-Α

(Ε-)

Β

 (K-A)

iff

B∈K and

A

< A



B

or ⊨ A

(C-)

A



Β

iff

A

 K-(AB)

Slide25

Representation Result

(E-)

Contraction Functions

Axioms (K-1) - (K-8)

-

-

-

-

-

-

Epistemic Entrenchments

Axioms

(EE1) - (EE5)













Slide26

Models for Belief Revision

Axiomatic

Models

AGM

Revision Functions

AGM

Contraction Functions

Preorders

o

n possible

worlds

EpistemicEntrenchments

Constructive

Models

Selection

Functions

Slide27

(Semi-) Open Problems

Relevance-Sensitive Revision

Iterated-Revision

Revision over Weaker Logics

Implementations - Representational Cost

Slide28

Relevance-Sensitive Revision

Slide29

Relevance-Sensitive Belief Revision

*

A

K

K*Α

*

A

Κ

Κ*Α

An non-intuitive AGM revision function:

K*A =

K+A, if

A 

K

Cn(A), otherwise

Slide30

Parikh’s Notion of Relevance

A = (

a

be

)  (a

b



e)

a

 c

d

¬a

gey

K

=

a



e

y

K

*Α

*

(P) If

K

=

Cn

(X,Y), L

X



L

Y

=



and A

 L

X

,

then

(

K

*A)

L

Y

=

K

L

Y.

Slide31

(SP) If Diff(

w

,r)  Diff(

w

,z) then r < z.



a



b

c

a



bc

ab



c



abc

a



b



c



ab



c



a



bc

abc

≤

≤

≤

≤

e.g.

Distance Between Worlds

(P) If

K

=

Cn

(X,Y), L

X



L

Y

=



and A

 L

X

,

then

(

K

*A)

L

Y

=

K

L

Y.

Diff(

w,r

) =

the set of variable that have different values in w and r.

e.g. , Diff(

abc

,

a



bc

) = {a, b}

Slide32

K

r = {

a, b,



c, d, g, e, f }

⇒

Diff(K, r) = {a, c, d}

(Q1) If Diff(

K

,r)  Diff(

K

,z) and Diff(

K,r

)∩

Diff

(

r

,

z

) =

∅

then r < z.

(Q2) If Diff(

K

,r) = Diff(

K

,z) and Diff(

K,r

)∩

Diff

(

r

,

z

) =

∅

then r ≈ z.



a



b



c

a



bc

ab



c



abc

a



b



c



ab



c



a



bc

abc

≤

≤

≤

≤

K =

Cn

(

a≣b

, c)

a

 c

d

¬a

g

e

b



f

Distance Between a Theory and a World

(SP) If Diff(

w

,r)  Diff(

w

,z) then r < z.

Slide33

Representation Result

Peppas,

Williams,

Chopra, and Foo (2015)

(S*)

Revision Functions

(K*1) – (K*8)

*

*

*

*

*

*

*

*

*

*

(P

)

Preorders in

Possible Worlds

















(

Q1) - (Q2)



*

*







Slide34

Strong (P)

(P) If

K

=

Cn

(X,Y), L

X

LY =

 and A  LX, then (K*A)L

Y = KLY.

A

K

K*A

*

X

Y

Y

A

H

H

*A

*

X

Z

Z

(

sP

) If

K

=

Cn

(X,Y), H =

Cn

(X,Z), L

X



L

Y

=

,

L

X



L

Z

=

,

and A

 L

X

,

then

(

K

*A)

L

X

=

(

H*A)

L

X

.

Slide35

Representation Result

Peppas,

Williams,

Chopra, and Foo (2015)

(S*)

Revision Functions

(K*1) – (K*8)

*

*

*

*

*

*

*

*

*

*

(P

)

Preorders in

Possible Worlds

















(Q1

)

– (Q2)



*

*

(

sP

)







(Q3)

Slide36

Future Work on Relevance

A

K

K*A

*

A

K

K*A

*

?

Slide37

Iterated Revision

Slide38

Iterative Belief Revision

≤

≤

≤

*

K

K*A

?

≤

B

A

Slide39

Iterative Belief Revision

≤

≤

≤

≤

≤

≤

≤

≤

≤

*

*

K

K*A

K*A*B

B

A

Slide40

Darwiche and Pearl’s Postulates for Iteration

≤

≤

≤

≤

≤

≤

≤

≤

≤

*

*

K

K*A

K*A*B

(IS1)

If

w,

r

 [

A]

then w

 r

iff

w ’ r.

(

IS2)

If

w,

r

 [



A]

then w

 r

iff

w ’ r.

(

IS3)

If

w

 [

A] and

r

 [



A]

then w

< r entails w <’ r.

(

IS4)

If

w

 [

A] and

r

 [



A]

then w

 r entails w ’ r.

(DP1)

If

B

⊨

A then K*A*B = K*B.

(DP2)

If

B

⊨



A then K*A*B = K*B.

(DP3)

If

A



K

*B then

A



K

*A*B.

B

A

(DP4)

If



A



K

*B then



A



K

*A*B.

(

IndR

)

If w

 [

A] and

r

 [

A] then w

 r entails w <’ r.

(

Ind) If A

 K*B then A  K

*A*B.

Jin

and

Thielscher

Slide41

Conflicts between Iteration and Relevance

Peppas

,

et. a

l. (2008) A

B

*

*

K

K*A

K*A*B

Theorem:

(

DP1

) – (DP4) and (P) are inconsistent.

Slide42

Revision

over Weaker Logics

Slide43

Belief Revision over Horn Theories

*

A

Κ

Κ*Α

Horn clause

: a

1

∧ . . . ∧a

n

⇒ a

Any Horn clause is a

Horn

sentence

. If

φ, ψ are Horn sentences then so is φ

∧ψ.A Horn theory K, is any set of Horn sentences closed under ⊨; i.e. if

φ is a Horn sentence and K⊨φ then

φ∈K.

Horn theoryHorn theory

Slide44

Problems with the “Naive” AGM Horn Revision

The correspondence between Horn revision functions and preorders over worlds breaks down:

There exists Horn revision functions that can not be constructed by any

preorder

over worlds.There exist preorders over worlds that induce non-Horn revision functions. Revision Functions(K*1) - (K*8)

*

*

*

**Preorders

on Possible Worlds











*

*





Slide45

The Solution to AGM Horn Revision

(

Delgrande

and Peppas, 2015)

(HC) If r ≈ r’, then r∩r’ <r.(Acyc) If (K*a1)+a0 ⊭⊥, . . . ,

(K*an

)+

an-1 ⊭⊥, and (K*a

0)+an ⊭⊥, then (K*a0)+an ⊭⊥

Delgrande, Peppas, and Woltran, “General Belief Revision”, 2017.

Revision Functions

(K*1) - (K*8)

***

Preorders over

Possible Worlds







*



*

*





*



Slide46

Implementations - Representational Cost

Slide47

Specifying a Revision Function

For a language with

n

propositional elements, there are 2

n worlds that need to be ordered. Even worse, there are 22n theories, and for each one we need to specify an ordering over worlds.

Revision Functions

*

*

**Preorders







*





Slide48

Solution I: Use an Off-the-Shelf Operator (Dalal

)

r

≤ r’ iff

for some w∈[K], |Diff(w,r)| ≤ |Diff(w’,r’)|, for all w’ ∈[K]. Pros: No information about the revision function needs to be specified.

Cons: Restricted range of applicability.



a



bc

a



bc

ab



c



abc

a



b



c



ab



c



a



bc

abc

≤

≤

≤

Slide49

Solution II: Parameterised Difference Operators

(Peppas and Williams, 2016)



a



b



c

a



bc

ab



c



abc

a



b



c



ab



c



a



bc

abc

≤

≤

≤

≤

≤



a



b



c

a



bc

ab



c



abc

a



b



c



ab



c



a



bc

abc

≤

≤

≤

Domain

Specific :

Info

c < a ≈ b



a



b



c

a



bc

ab



c



abc

a



b



c



ab



c



a



bc

abc

≤

≤

≤

Slide50

Properties of Parameterised Difference Operators

AGM Revision Functions

*

*

*

*

*

Preorders











*

*





PD preorders

PD revisions

A

single preorder over the

n

variables of the

language suffices to specify the preorders for all theories.

A natural generalization of

Dalal’s

operator.

At the same level in the polynomial hierarchy as

Dalal’s

operator (2

nd

level).

PD revisions satisfy Parikh’s relevance axiom (P).

A built-in solution to the iterated revision problem.

An axiomatic characterization of PD revisions.

Slide51

Conclusion

The AGM approach is a very elegant framework for studying Belief Revision.

More work is needed in (at least):

Relevance-Sensitive Revision

Iterated RevisionEasy to use Implementations

*

A

Κ

Κ*Α