Iterated Theory Base Change: A Computational Model Mary ... - IJCAI

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

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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

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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

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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)

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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

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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

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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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