I t e r a t e d T h e o r y Base C h a n g e : A C o m p u t a t i o n a l M o d e l
Mary-Anne Williams
Information Systems G r o u p
Department of Management
University of Newcastle, N S W 2308
Australia
Abstract
T h e A G M p a r a d i g m i s a f o r m a l approach t o ideal
a n d r a t i o n a l i n f o r m a t i o n change
F r o m a practical
p e r s p e c t i v e i t suffers f r o m t w o s h o r t c o m i n g s , the
first involves d i f f i c u l t i e s w i t h respect to t h e finite
representation
of i n f o r m a t i o n ,
and
the
second
involves t h e lack of s u p p o r t for the i t e r a t i o n of
change operators
In t h i s paper we show t h a t these
p r a c t i c a l p r o b l e m s can b e solved i n t h e o r e t i c a l l y
satisfying ways wholely w i t h i n the A G M paradigm
We
introduce
a
partial
entrenchment
ranking
w h i c h serves as a canonical r e p r e s e n t a t i o n for a the­
o r y base a n d a w e l l - r a n k e d episterruc e n t r e n c h m e n t ,
a n d we p r o v i d e a c o m p u t a t i o n a l m o d e l for a d j u s t ­
i n g p a r t i a l e n t r e n c h m e n t r a n k i n g s w h e n they receive
new i n f o r m a t i o n u s i n g a p r o c e d u r e based on t h e
p r i n c i p l e o f m i n i m a l change
T h e c o n n e c t i o n i between t h e s t a n d a r d A G M
t h e o r y change o p e r a t o r s a n d the t h e o r y base
change o p e r a t o r s developed herein suggest t h a t the
p r o p o s e d c o m p u t a t i o n a l m o d e l for i t e r a t e d t h e o r y
base change e x h i b i t s desirable b e h a v i o u r
1
Introduction
I n f o r m a t i o n systems can be used to represent a reasoning
agent's v i e w of t h e w o r l d Unless t h e agent has perfect
and complete i n f o r m a t i o n it w i l l require a mechanism to
s u p p o r t t h e m o d i f i c a t i o n o f i t s view a s m o r e i n f o r m a t i o n
a b o u t the w o r l d is a c q u i r e d M o r e o v e r , a c o m p u t a t i o n a l
m o d e l t o s u p p o r t the a c q u i s i t i o n o f new i n f o r m a t i o n
requires a f i n i t e r e p r e s e n t a t i o n of the i n f o r m a t i o n h e l d
b y t h e agent
T h i s representation must capture the
i n f o r m a t i o n c o n t e n t , t h e agent's c o m m i t t m e n t t o t h i s
i n f o r m a t i o n , a n d a n e n c o d i n g o f how the i n f o r m a t i o n
s h o u l d change u p o n t h e i n t r u s i o n o f new i n f o r m a t i o n
T h e A G M p a r a d i g m was o r i g i n a l l y developed b y
A l c h o u r r o n , G a r d e n f o r s a n d M a k i n s o n [1985] a n d has
b e c o m e one o f t h e s t a n d a r d f r a m e w o r k s for i n f o r m a t i o n
change
I t p r o v i d e s f o r m a l mechanisms for m o d e l i n g
the r a t i o n a l e v o l u t i o n o f a n i d e a l reasoner's view o f t h e
w o r l d I n p a r t i c u l a r , i t p r o v i d e s o p e r a t o r s for m o d e l i n g
the r e v i s i o n a n d c o n t r a c t i o n o f i n f o r m a t i o n W i t h i n t h e
A G M p a r a d i g m the f a m i l y o f revision operators, and
t h e f a m i l y o f c o n t r a c t i o n o p e r a t o r s are c i r c u m s c r i b e d b y
sets o f b y r a t i o n a l i t y p o s t u l a t e s T h e logical p r o p e r t i e s
o f a b o d y o f i n f o r m a t i o n are n o t s t r o n g enough t o
u n i q u e l y d e t e r m i n e a r e v i s i o n or c o n t r a c t i o n o p e r a t o r ,
therefore t h e p r i n c i p a l c o n s t r u c t i o n s for these o p e r a t o r s
rely o n some f o r m o f u n d e r l y i n g preference r e l a t i o n ,
such as a f a m i l y of selection f u n c t i o n s [ A l c h o u r r o n et at
1985], a s y s t e m of spheres [ G r o v e , 19881 a nice preorder
o n m o d e l s [ K a t s u n o a n d M e n d e l z o n 1992, Peppas a n d
W i l l i a m s , 1995], o r a n e p i s t e m i c e n t r e n c h m e n t o r d e r i n g
[ G a r d e n f o r s a n d M a k i n s o n , 1988]
F r o m a t h e o r e t i c a l p o i n t o f view t h e A G M p a r a d i g m
provides a v e r y elegant a n d s i m p l e m e c h a n i s m for
r a t i o n a l and i d e a l change
However, f r o m a practical
perspective its o p e r a t o r s are insufficient because t h e y
essentially take a preference r e l a t i o n t o g e t h e r w i t h
a sentence a n d p r o d u c e a t h e o r y
In o t h e r w o r d s ,
t h e u n d e r l y i n g preference r e l a t i o n i s l o s t
This
p r o p e r t y is a t t r a c t i v e in a t h e o r e t i c a l c o n t e x t because
i t allows the r e s u l t a n t t h e o r y t o a d o p t any preference
r e l a t i o n d e p e n d i n g o n t h e desired d y n a m i c b e h a v i o u r
In p r a c t i c e ,
however,
a policy for change w i l l be
necessary A s t r a i g h t f o r w a r d m e t h o d of s p e c i f y i n g such
a p o l i c y is to i m p o s e c o n s t r a i n t s on t h e u n d e r l y i n g
preference r e l a t i o n w h i c h w i l l i n t u r n d e t e r m i n e the
dynamic behaviour of the system
A s noted above
the A G M postulates do not uniquely determine revision
and c o n t r a c t i o n o p e r a t o r s , however G a r d e n f o r s a n d
M a k i n s o n [1986] showed t h a t a n e p i s t e m i c e n t r e n c h m e n t
o r d e r i n g does
A c c o r d i n g t o R o t t [1991a] w h a t s h o u l d
be at focus is n o t t h e o r y r e v i s i o n b u t e p i s t e m i c
entrenchment revision
T h e u n d e r l y i n g preference r e l a t i o n f u l l y characterizes
a i n f o r m a t i o n system's content, its c o m m i t t m e n t to the
i n f o r m a t i o n , a n d its desired d y n a m i c p r o p e r t i e s
We
refer to the process of c h a n g i n g an i n f o r m a t i o n s y s t e m ' s
u n d e r l y i n g preference r e l a t i o n as a transmutation
W i l l i a m s [1994a] showed t h a t a l l o w i n g the principle
of minimal change to c o m m a n d t h e p o l i c y for change
results t o t w o different
forms of transmutations,
conditionalization and adjustment
Conditionalization
was i n t r o d u c e d by S p o h n [1988] a n d is based on a
relative measure o f m i n i m a l change, o n the o t h e r h a n d
a d j u s t m e n t described by W i l l i a m s [1994a] is based on
an absolute measure of m i n i m a l change
Adjustments
are c o m p a r e d and c o n t r a s t e d w i t h c o n d i t i o n a l i z a t i o n i n
[ W i l l i a m s 1994a]
In our c u r r e n t theory base c o n t e x t we represent
the u n d e r l y i n g preference r e l a t i o n for a t h e o r y base
as a
partial
entrenchment ranking w h i c h serves as
a canonical r e p r e s e n t a t i o n of a w e l l - r a n k e d e p i s t e m i c
entrenchment
We show it can be used to s u p p o r t
t r a n s m u t a t i o n s of a t h e o r y base, a n d thus iterated theory
base change F u r t h e r m o r e , we p r o v i d e a c o m p u t a t i o n a l
model which modifies a partial entrenchment r a n k i n g
u s i n g a n a b s o l u t e measure o f m i n i m a l change
Any
t h e o r e m prover can be used to realize t h i s m o d e l
C o n s e q u e n t l y , we present a p r a c t i c a l s o l u t i o n to t h e
i t e r a t e d t h e o r y base change p r o b l e m , we n o t e however
t h a t a f u l l c o m p l e x i t y analysis of our p r o p o s e d p r o c e d u r e
is yet to be c o n d u c t e d
W e b r i e f l y o u t l i n e some t e c h n i c a l p r e l i m i n a r i e s a n d
t h e A G M p a r a d i g m i n section 2 , t h i s i s a s t a n d a r d
treatise where t w o e x t r a p o s t u l a t e s i n t r o d u c e d i n
[ W i l l i a m s 1994a] are presented, t h e reader f a m i l i a r w i t h
t h i s w o r k m a y s k i p t o t h e n e x t section
I n section 3 ,
we describe t h e r e p r e s e n t a t i o n of t h e o r y bases u s i n g
partial entrenchment rankings, and we demonstrate how
WILLIAMS
1541
they are related to epistemic entrenchment orderings
In section 4 we describe iterated theory base change
using transmutations of partial entrenchment rankings
In section 5 we describe a simple computational
model that implements a transmutation of a partial
entrenchment ranking
The transmutation employed
is an adjustment hence the proposed computational
model uses an absolute minimal measure of change
In section 6 we explore the connection between our
theory base change operators using an adjustment and
the standard A G M theory change operators, and we
argue that the established relationships demonstrate
that our proposed theory base change operators exhibit
desirable behaviour
In particular they maintain as
much of the theory base as would be retained in
the corresponding theory change, thus propagating as
much explicit information as possible Related work is
discussed in section 7, and a summary of our results is
given section 8
2 2
Epistemic Entrenchment
An epistemic entrenchment [Garden fors and Makinson,
1988] is an ordering or the sentences in C which attempts
to capture the relative importance of information in the
face of change In order to determine a unique revision
or contraction operation a theory is endowed w i t h an
epistemic entrenchment ordering, which can be used to
determine the information to be retracted, retained, and
acquired during the contraction of old information and
the acceptance of new information
1642
NON M0N0T0NIC REASONING
3
R e p r e s e n t i n g T h e o r y Bases
Identical canonical representations or epistemic en­
trenchment ordenngs were developed independently by
Rott [1991a] and Williams [1992, 1994c] Rott's specifi­
cation IS an E-Base and W i l l i a m s ' an ensconcement
An E-Base is a preference relation that provides a
canonical representation of an epistemic entrenchment
and a theory base
Williams showed that an E-Base
is capable of uniquely determining theory base change
operators, as well as theory change operators However,
the theory base operations provided do not take an
E-Base to an E-Base
In other words, the preference
relation on the theory base is not propagated during the
process of change, and hence iteration is not supported
Williams [1994a] extended the work of Spohn [1988],
based on observations in [Gardenfore, 1988], to arbitrary
iterated revision operators for theories, and referred
to the process of changing an epistemic entrenchment
ordering as a transmutation
In this section we demonstrate that a partial
entrenchment ranking which is a ranking of a theory
base can specify a well ranked (or finite) epistemic
entrenchment ordering on a theory
A partial entrenchment ranking formally defined
below, maps a set of sentences in C to ordinals
Intuitively, the higher the ordinal assigned to a sentence
the more firmly held it is Throughout the remainder of
the paper it will be understood that 0 is an ordinal
chosen to be sufficiently large for the purpose of the
discussion
WIIIIAMS
1543
The T referred to in the theorem above is the
explicit information set of the partial entrenchment
ranking that generates the epistemic entrenchment and
while obviously not unique, it must contain enough
sentences of 'the right stuff' in each natural partition
of the epistemic entrenchment ordering to enable its
regeneration In particular, it must satisfy the condition
in the theorem, at least one such T exists, namely T
itself
Stepwise constructions of epistemic entrenchment
orderings from partial entrenchment rankings, and vice
versa, can be derived from the those provided in [Rott,
1991b, Williams, 1994c]
4
I t e r a t e d C h a n g e f o r T h e o r y Bases
Recall from our previous discussion that from a practical
perspective the A G M paradigm does not provide a policy
to support the iteration of its change operations
In
this section we show how transmutations of partial
entrenchment rankings can be used to support iterated
theory base change, and we provide existence theorems
that demonstrate how transmutations are related to the
A G M theory revision and theory contraction operators
For revision and contraction operators the information input is a sentence We now define a transmutation
of partial entrenchment rankings where the information
input is a contingent sentence a and an ordinal i The
interpretation [Gardenfore 1986] of this is that the sen­
tence a is the information to be accepted, and i is the
degree of firmness w i t h which it is to be incorporated
into the transmuted partial entrenchment ranking
5
A Computational Model
In this section we describe a particular type of
transmutation called an adjustment which uses a policy
for change based on an absolute minima/ measure In
particular, it involves the absolute minimal change of
a partial entrenchment ranking that is required to
incorporate the desired new information
A semantic
account of adjustments can be found in [Williams,
1994 a]
We present a computational model for adjustments
which can be used as the basis for a computer-based
implementation
Any theorem prover can be used
to realize it
The model itself is stated in its most
perspicuous form, and can be optimised in several
obvious ways, see [Williams, 1993] for details
For
the purpose of our model we focus on finite epistemic
rankings, and the following theorem describes the degree
of acceptance of sentences w i t h respect to finite rankings
Theorem 6
Let a be a nontaulologtcal sentence If
B€B is finite, then degree(B, a)
1044
N0N M0N0T0NIC REASONING
Baaed on Theorem 6 the following function uaes a
■imple procedural algorithm to determine the degree of
a sentence w i t h respect to a finite partial entrenchment
ranking
The function takes two input parameters,
namely a finite partial entrenchment ranking, B, and
a nontautological sentence, a, it calculates and returns
the degree of acceptance of a in D
The algorithm
requires the support of your favourite theorem prover
to implement the logical implication relation, I-
This algorithm can compute the generated epistemic
entrenchment ordering of sentences based on the
information encoded in a partial entrenchment ranking
A transmutation which is suitable for theory base
change is an adjustment defined in the theorem below
When D is finite then its definition constitutes a
computational model which, loosely speaking, involves
successive calls to the function degree for each sentence
in the domain of D
6
Connections with Theory Change
The theorems in this section establish the explicit
relationship between theory base change operators based
on adjustments and theory change operators based on
the generated epistemic entrenchment ordering
The following theorem illustrates the interrelation­
ships between theory base revision and theory base con­
traction based on adjustments In particular, Theorem
8 ( J ) IS analogous to the Harper Identity and it captures
the dependence of theory base contraction on the in­
formation content of the theory base, that is, the ex­
plicit information set, exp(B) Similarly Theorem 8(11)
is analogous to the Levi identity, and 8(111) says that a
Levi Identity also exists at the deeper preference relation
level In particular, precisely the same partial entrench­
ment ranking is obtained when a partial entrenchment
ranking is adjusted to accept new information a with
firmness 0 < 1 < O, as when we adjust the partial en­
trenchment ranking to remove a, and then adjust the
resultant partial entrenchment ranking to accept new in-
Theorem 9(1) captures the dependence of a theory
base contraction on the content of the theory base
that is, exp(B) In particular, a sentence is retained
in the theory base contraction if and only if it is
member of the theory base and it would be retained
in the corresponding theory contraction
Theoren
10(1) establishes a similar result for revision
This
substantiates our claim that adjustments retain as 'much
as possible' of the original theory base
WILLAMS
1645
Consequently, a partial entrenchment ranking can
be used to model belief change for a limited reasoner
In particular, partial entrenchment rankings can be
used to represent the reasoner's explicit information, its
commitment to that information, and an encapsulation
of the desired dynamic behaviour In fact, depending
on the nature of the theorem prover adopted for
an implementation some resource bounds could also
be introduced [Williams, 1993]
Theorems 9(1) and
10(1) clearly demonstrate that theory base adjustments
contain as many of the explicit beliefs as would be
retained in the corresponding theory change operations
In particular, we have established that each explicit
belief retained in a theory change is also retained via
theory base adjustments
Theorems 9(11) and 10(n) show that theory change
operators can be formulated in terms of theory base
change using adjustments, and they provide explicit
constructions for the Theorems 4 and 5, respectively
For revision the adjustments of equivalent partial
entrenchment rankings result in equivalent implicit
information sets
Furthermore, it turns out that the
second parts of Theorems 9 and 10 can be generalised
to equivalent partial entrenchment rankings, however
we provided the current readings to for the sake of
simplicity
7
Related W o r k
Gardenfore and Rott [1992] provide a comprehensive
analysis of various prominent approaches [Hansson 1989,
Fuhrmann 1991, Nebel, 1991] to theory base change, and
the interested reader should consult their work
Nayak [1993] and Boutilier [1993] explore iterated
theOTy change, and their approaches are closely related
to transmutations because they focus on changes at the
preference relation level, and as a consequence if the
underlying preference relation is well-ranked then both
of their procedures can be expressed as transmutations
Boutiher's natural revision being a special case of
adjustment, and Nayak's method being a special case
of conditionalization For example, if we use Theorem
3 to capture the relationship between an epistemic
entrenchment ordering and a partial entrenchment
ranking, then Boutilier's approach can be characterized
as an ( a , l)-adjustrnent
Boutilier's new information is always accepted mini­
mally firmly [Spohn 1988, p 114], and Nayak's new in­
formation is always accepted maximally firmly [Spohn
1988, p l l 4 ] In other words, they both suffer from prob­
lems described by Spohn, since their representation is
essentially a simple conditional function, and their in­
formation input a sentence alone Spohn claims that
both of these extreme schemes are undesirable because
we don't always want to accept new information with
the same degree of firmness
In defense of Nayak and Boutilier we should point
out that the degree of firmness for new information may
not always be available, in which case we could adopt
default firmnesses for newly acquired information when
1546
NON MONOTONIC REASONING
t h e q u a l i t y o f t h e evidence i s u n k n o w n
I f the u n d e r l y i n g preference r e l a t i o n i s w e l l - r a n k e d
t h e n t r a n s m u t a t i o n s c h e m a * can c a p t u r e any conceivable
change w i t h i n t h e A G M p a r a d i g m
For t h e p u r p o s e o f
an implementation we w i l l almost certainly be dealing
w i t h a w e l l - r a n k e d , p r o b a b l y f i n i t e , preference r e l a t i o n
AN alternative approach to the p r o b l e m of iterated
r e v i s i o n i s a d o p t e d b y D a r w i c h e a n d P e a r l [1994], a n d
F r e u n d a n d L e h m a n n [1994]
I n p a r t i c u l a r , t h e y have
s t u d i e d the i t e r a t e d r e v i s i o n p r o b l e m a t t h e a x i o m a t i c
level, a n d t h e y suggest n e w m e t a - p o s t u l a t e s for i t e r a t e d
r e v i s i o n r a t h e r t h a n c o n s t r a i n t s o n the m o d i f i c a t i o n o f
preference r e l a t i o n s
W i l l i a m s et al
[1995b] c a p t u r e S p o h n ' s n o t i o n of
reason for [1983] u s i n g a d j u s t m e n t s of e n t r e n c h m e n t
rankings, and the c o m p u t a t i o n a l model provided in
section 5 can be used to i m p l e m e n t S p o h m a n reasons
F i n a l l y , w e h i g h l i g h t some c o n n e c t i o n s w i t h n o n m o n o t o n i c reasoning G a r d e n f o r s a n d M a k i n s o n [1994] use an
comparative expectation ordering to construct a non­
m o n o t o n i c inference r e l a t i o n W i l l i a m s [1995a] defines a
partial expectation ranking to be is a f u n c t i o n B f r o m
a subset of sentences in C i n t o O such t h a t ( P E R I )
and ( P E R 2 ) are satisfied
W e also n o t e t h a t Reseller's
plausibility indexes [1976] are essentially p a r t i a l e x p e c t a ­
tion rankings Given a partial expectation ranking B, if
we define a n o n m o n o t o n i c inference r e l a t i o n h, as af—/3
if a n d o n l y if e x p ( B * ( α , i ) ) (- β where 0 < 1, t h e n (-v
is consistency preserving a n d rational, a n d t h e associ­
a t e d m o d e l s t r u c t u r e i s nice [ G a r d e n f o r s a n d M a k i n s o n ,
1994]
W i l l i a m s [1995a] shows t h a t a d j u s t m e n t s can b e
a p p l i e d t o p a r t i a l e x p e c t a t i o n r a n k i n g s thus p r o v i d i n g
a
mechanism
for
changing
nonmonotonic
inference
relations in an absolute a n d m i n i m a l way
B o u t i h e r [1993], P e a r l [1994], F r e u n d a n d L e h m a n n
[1994] also address t h e idea o f c h a n g i n g d e f a u l t
information, arguing that default information is usually
q u i t e s t a b l e , for e x a m p l e a l t h o u g h o u r i n f o r m a t i o n a b o u t
t h e f l i g h t coefficient of a p a r t i c u l a r b i r d m a y change
d r a m a t i c a l l y , the d e f a u l t t h a t typically birds fly w i l l
i n v a r i a b l y r e m a i n u n c h a n g e d , therefore changes s h o u l d
m a i n t a i n a s m u c h d e f a u l t i n f o r m a t i o n a s possible
An
a d j u s t m e n t n o t o n l y concords w i t h t h i s p e r s p e c t i v e , b u t
i t preserves t h e p r o p e r t i e s o f consistency p e r s e r v a t i o n
a n d r a t i o n a l i t y o f t h e inference r e l a t i o n
8
Discussion
W e have s h o w n t h a t the A G M p a r a d i g m can b e
e x t e n d e d t o solve t w o o u t s t a n d i n g p r a c t i c a l p r o b l e m s
t h a t arise i n t h e d e v e l o p m e n t o f a c o m p u t a t i o n a l m o d e l
for b e l i e f r e v i s i o n
T h i s e x t e n s i o n focuses on a finite
representation of an e p i s t e m i c e n t r e n c h m e n t o r d e r i n g ,
a n d the d e t e r m i n a t i o n of a policy for change based on
the p r i n c i p l e o f m i n i m a l change
W e established t h a t p a r t i a l e n t r e n c h m e n t r a n k i n g s
can be used to c o n s t r u c t t h e o r y base change o p e r a t o r s ,
t h e o r y change o p e r a t o r s , as w e l l as n o n m o n o t o n i c i n f e r ­
ence r e l a t i o n s M o r e o v e r , we p r o v i d e d r e p r e s e n t a t i o n re­
sults for t h e c o n s t r u c t i o n o f w e l l - b e h a v e d a n d v e r y w e l l behaved t h e o r y change o p e r a t o r s based o n p a r t i a l en­
trenchment rankings
W e used p a r t i a l e n t r e n c h m e n t r a n k i n g s t o represent
a w e l l - r a n k e d e p i s t e m i c e n t r e n c h m e n t o r d e r i n g , a n d we
p r o v i d e d a computational model for m o d i f y i n g p a r t i a l
e n t r e n c h m e n t r a n k i n g s based on an absolute measure
o f m i n i m a / change w h i c h d e a l t w i t h t h e r e m o v a l o f o l d
i n f o r m a t i o n a n d t h e acquiescence o f new i n f o r m a t i o n
W e established t h a t t h e o r y base r e v i s i o n a n d t h e o r y
base contraction operators based on adjustments are
related via Levi and Harper Identities at both the
information content and the preference relation levels
Finally, we demonstrated that theory base operators
based on an adjustment maintain as much explicit
information as is retained by the corresponding theory
change operator
Acknowledgements
The author gratefully acknowledges, Peter Garden fore
for suggesting how Spohn'B work might be used in the
development of a computational model, David Makinson
for a myriad of insightful suggestions concerning
theory base change
Many thanks are also due to
Craig Boutilier, Didier D U B O I S , Norman Foo, Georg
Gottlob, Abhaya Nayak, Bernhard Nebel, Erik Olsson,
Maurice Pagnucco, Judea Pearl, Pavlos Pep pas, Wlodek
Rabinowicz and Hans R o t t for their helpful comments
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