a survey of belief revision and updating rules in various uncertainty models

A Survey of Belief Revision and Updating Rules in Various Uncertainty Models*

Didier Dubois and Henri Prade lnsfitut de Recherche en lnformatique de Toulouse (I. R.I. T.), Universitb Paul Sabatier-CNRS, 7 18 route de Narbonne, 3 1062 Toulouse Cedex, France, Email: {dubois, prade}@irit fr

The paper proposes a parallel survey of revision and updating operations available in the probability theory and in the possibility theory frameworks. In these two formalisms the current state of knowledge is generally represented by a [0,1]-valued function whose domain is an exhaustive set of mutually exclusive possible states of the world. However, in possibility theory, the unit-interval can be viewed as a purely ordinal scale. Two general kinds of operations can be defined on this assignment function: conditioning, and imaging (or "projection"). The difference between these two operations is analogous to the one made between belief revision i3 la Gtirdenfors and updating i3 la Katsuno and Mendelzon in the logical framework. In the probabilistic framework these two operations are respectively Bayesian conditioning and Lewis' imaging. Counterparts to these operations are presented for the possibilistic framework including the case of conditioning upon uncertain observations, and justifications are given which parallel the ones existing for the probabilistic operations. More particularly, it is recalled that possibilistic conditioning satisfies all the postulates proposed by Alchourrh, Gilrdenfors and Makinson for belief revision (stated in possibilistic terms), and it is proved that possibilistic imaging satisfies all the postulates proposed by Katsuno and Mendelzon. The situation where our current knowledge is stated in terms of weighted logical propositions is discussed in connection to possibility theory. Revision in other more complex numerical formalisms. namely belief and plausibility functions, and upper and lower probabilities is also surveyed. Recent results on the revision of conditional knowledge bases are also reviewed. The frameworks of belief functions, upper and lower probabilities and conditional bases are more sophisticated than the previous ones because they enable to distinguish between factual evidence and generic knowledge in a cognitive state. This framework leads to two forms of belief revision respectively taking care of the revision of evidence and the revision of knowledge. 0 1994 John Wiley & Sons, Inc.

INTERNATIONAL JOURNAL OF INTELLIGENT SYSTEMS, VOL. 9,61-100 (1994) 0 1994 John Wiley & Sons, Inc. CCC 0884-8173/94/010061-40

* This paper is a revised and extended version of a paper entitled "Belief Revision and Updates in Numerical Formalisms -An overview. with new results for the possibilistic framework-'' published in the Proceedings of the 13th International Joint Conference on Artificial Intelligence (IJCAI'93), ChambBry. France, Aug.28-Sept.3. 1993, pp. 620-625.

62 DUBOIS AND PRADE

I. INTRODUCTION

In logical settings, a cognitive state, describing the current state of knowledge in a given case, is represented by means of sets of sentences. But sentences have a meaning, and should be related to the state of facts they describe. One usual way of doing it is to characterize the meaning of sentences in terms of possible worlds. The basic idea is that any statement, e.g. in natural language, refers to a set of worlds, one of which is the actual world where the statement is taken for granted. For instance "John is not married" corresponds to situations where "John is single, widower, etc." one of which is the actual situation. Each of these situations can be viewed as a possible state of the world or a possible world for short. A sentence like 7married(John) that supposedly describes the statement "John is not married" is considered as true in any possible world consistent with the statement. The reason for introducing models of cognitive states based on possible worlds is that most numerical settings use this kind of representation, rather than sets of sentences, despite recent advances in logics for uncertainty such as probabilistic or possibilistic logics, and conditional objects.

Possible world models can be described by means of a set R representing a set of possible situations called "worlds". These worlds w are supposed to be mutually exclusive and usually R is assumed to be exhaustive. Here, R will be a finite set. A cognitive state is a subset C of R that supposedly contains the actual world. In numerical models of uncertainty, all worlds in C are not always considered as equally likely, and each world can be attached a weight that expresses its degree of likelihood in some model of uncertainty. Both in probability theory and in possibility theory, to each world w is attached a degree d(w) E [OJ] which estimates the extent to which o may represent the real (state of the) world. These worlds can be put in correspondence with so-called "interpretations" or "models" used in logical formalisms. The assignment function d is such that d(w) = 0 means that we are completely certain that o cannot be the real (state of the) world. But the meaning of d(w) = 1 is completely different in probability theory where it means that w is the real world (complete knowledge), and in possibility theory where it only expresses that nothing prevents w from being the real world. More complex numerical models of cognitive states attach a weight to any subset S r; !2 of possible worlds, expressing the confidence in S as containing the actual world. In other words, a cognitive state is described by a set function m on R.

In the two simpler formalisms, the change of the current state of knowledge (called 'cognitive state') upon the arrival of a new information stating that the real world is in A G R, corresponds to a modification of the assignment function d into a new assignment d'. This change should obey some general principles which guarantee that i) d is of the same nature as d (preservation of the representation principles); ii) A, which denotes 'not A is excluded by d , i.e., V o e A, d'(w) = 0 (what is observed is hold as certain after the revision or the updating); iii) some informational distance between d and d is minimized (principle of minimal change). Counterparts to these principles are also at the basis of revision and updating in logical formalisms (Alchourrdn, Gikdenfors and Makinson', Gadenfors2, Katsuno and Mendelzon3).

Interestingly enough, the probabilistic framework offers at least two ways of modifying a probability distribution upon the arrival of a new and certain information: the Bayesian conditioning, and also what has been called 'imaging' by D. Lewis4 which consists in translating the weights originally on worlds outside A toward worlds which are their closest neighbours in A. It turns out, as it will be

A SURVEY OF BELIEF REVISION AND UPDATING RULES 63

shown in this paper, that the existence of these two modes, which can be also defined in the possibilistic framework, is analogous to the distinction between belief revision based on Alchourr6n, Gitrdenfors and Makinson's postulates2 and updating based on Katsuno and Mendelzon's postulates3.

This structured overview is organized in five main sections. The next section discusses general principles of cognitive change in the setting of numerical formalisms. Section I11 surveys basic results on conditioning and imaging in the probabilistic framework. Then Section IV introduces these two operations in the possibilistic framework and provides new results and justifications for them. It also relates the treatment of uncertain inputs in the possibilistic framework to the conditioning of Spohn's ordinal conditional functions5. Section V briefly considers belief and plausibility functions and upper and lower probabilities, and then emphasizes the existence of a third change operation, called 'focusing', in this framework. Section VI envisages some syntactical counterparts of the previously considered uncertainty models, especially possibilistic logic, and more briefly conditional knowledge bases.

11. GENERAL PRINCIPLES OF COGNITIVE CHANGE IN NUMERICAL SETTINGS

Numerical formalisms for representing cognitive states are altogether richer and poorer than purely symbolic formalisms. They are richer because shades are introduced between worlds which are considered as completely possible and worlds which are considered as impossible. As a consequence, the nomenclature of possible modes of belief change will be more complicated than in the symbolic case where only contractions, expansions and revisions occur. In the case of numerical representations, some forms of belief change will not correspond to any of these three modes, although extensions thereof do appear in numerical settings. Another point is the notion of input which is enriched too. Namely an input may be tainted with uncertainty, and belief change with uncertain inputs might differ in its basic principles from belief change with sure inputs. However, possible worlds model suffer from a lack of expressivity since two logically equivalent sets of sentences always correspond to the same set of possible worlds although they might not be equivalent from the point of view of revision. Hence the debate pervading the literature on logic-based approaches, between syntax-based and syntax-independent modes of belief change does not exist here. A measure of uncertainty may synthesize various pieces of information that were received in the past, but the memory of these pieces of information is lost, contrary to when a set of sentences is available, where the way information came in can be encoded in the language. In that sense possible worlds-based numerical models of cognitive states are poorer than syntax-based models. And principles of belief change can only be borrowed from the syntax- independent models.

The basic principle for belief change in a numerical setting is the principle of minimal change which is imbedded in the oldest belief change rule, i.e. Bayes rule of probability theory. The presence of numbers make it easier to define an informational distance between cognitive states, without demanding any metric-like (or topological) structure on the set of possible worlds. However cognitive change rules based on distance between possible worlds have been proposed as well, such as Lewis' imaging. Imaging seems to better capture world change rather than belief change.

64 DUBOIS AND PRADE

The minimal belief change principle in numerical settings can be expressed as follows. Given a set function d on R that describes an a priori cognitive state, and given an informational distance I(d,d'), the change of d upon learning that event A c R is true is defined by the set function dA on R such that dA(B) = 0 if B n A f 0 and I(d,dA) is minimal.

In contrast, the updating of a distribution d on R based on the imaging concept proceeds as follows. If o E A, then the weight d(w) is allocated to the world WA E

A closest to o in the sense of a metric notion equipping R, and dA(O') for 61' E A is defined by means of a suitable combination of d(w') and of {d(w), WA = o'). This method assumes that the cognitive state d can be entirely characterized by the distribution {d(w), 61 E a) , which is true in possibility theory (as well as with Spohn's ordinal conditional functions5), and in probability theory but not for other kinds of frameworks. Note that imaging requires that R be equipped with some sort of distance.

Usual belief change rules in numerical setting are based on a technical device called conditioning (for instance, conditional probability). There is, as a consequence, a natural approach to belief change outside the probabilistic setting, that consists in extending the concept of conditioning over to non-probabilistic settings, without bothering about informational distance. The change of the uncertainty measure g (which denotes the set function associated with the assignment or distribution function d), upon learning A, yields the conditional measure g(. I A). However it would be nice to reconcile conditioning and the principle of minimal change, as it is the case in probability theory. Moreover, it is not right away clear how to extrapolate conditioning to the case when the input information is uncertain. On the contrary, the use of an informational distance makes the problem of updating with uncertain information easy to specify as follows: given a cognitive state description d and an event A for which all is known is that g'(A) = a, the update g' of g is defined as minimizing I(d,d) with constraint g'(A) = a where d' is the distribution associated with g'. Note that this approach is difficult to be envisaged for imaging-like updating methods. Moreover the above principle may clearly increase the uncertainty of event A, since the input g'(A) = a is taken as a constraint (e.g., if A was previously considered as a sure fact by d, i.e., V 61 E A, d(o) = 0)).

Another fruitful point of view on belief change with numerical representations of cognitive states is to start with GLdenfors' postulates in the logical setting, and extend them over to uncertainty settings. This kind of approach has been tried by G&denfors2 for the probabilistic representation of cognitive states, and by Dubois and Prade6 for the possibilistic representation. A crucial question is then to define what is an expansion, a revision and a contraction for a given uncertainty measure. Informally d+ is an expansion of d if it is more informative than d and does not contradict it; a contraction d- of d is the converse notion, i.e. d- is a contraction of d if d is an expansion of d-. A revision d* of d should disagree with d to some extent, i.e. there are subsets of possible worlds which d claim to be likely and d* unlikely, and other subsets of possible worlds for which the converse situation occurs. The main problem is then to give a precise meaning to expressions such as "d is more informative than d 'I, "d does not contradict d' 'I; "d disagrees with d' 'I.

To summarize the situation with the numerical approaches, the two views of change exist in the symbolic framework: conditioning and imaging. Conditioning, as a belief change rule is similar to GLdenfors'2 approach to expansion and revision since it works by suppressing possible worlds which are incompatible with the input


information. Imaging, on the contrary is the numerical extension of Katsuno and Mendelzon's3 updating method based on shifts to closest worlds. The next sections survey the existing results in the several available numerical settings for modeling revision and updating of uncertain cognitive states.

111. THE PROBABILISTIC FRAMEWORK

The setting of probability theory is the traditional one for representing uncertainty in Artificial Intelligence (e.g., Pearl7). This is the so-called Bayesian approach. In this setting, a cognitive state is represented by means of a single probability measure P on the set of possible worlds, i.e. a set-function P on R (here supposedly finite for simplicity) such that

V A, B c 52, A n B # 0 3 P(A u B) = P(A) + P(B).

A complete state of knowledge is represented by P( (00)) = 1 for some 00. An cognitive state is usually characterized by the probability distribution (p(o), o E SZ) from which P(A) = C, A p(o) can be computed. The interpretation of P(A) is either frequency-based @'(A) is the subjectively guessed frequency of the event "A contains the actual world"), or is defined throu h a betting-behavior procedure, using a scoring

to pay in a lottery game that gives you one dollar if A turns out to be true. rule (De Fine@, Savage9, Lindley' % ); P(A) is then the highest price one is willing

A. Probabilistic Conditioning

Upon learning that event A has occurred, i.e. we are certain that A is true, the a priori cognitive state P is revised into the conditional probability P(- I A) such that

where the second equality clearly shows that for any subset B of possible worlds, P(B) is changed into P(B I A). This operation is not defined if A is judged to be impossible by the a priori cognitive state. G&denfors2 tries to overcome this impossibility by means of Popper functions. However this mending of Bayes rule is a technical trick that has little intuitive appeal. In terms of a distribution, (1) writes

= O i f o E A.

This conditioning rule satisfies minimal requirements such as i) P(A I A) = 1, when P(A) > 0 (priority to the input information); ii) P(A) = 1 3 P(. I A) = P (an already known input information does not modify the cognitive state). Revision via a scaling factor as done in (1) is not so natural (except if one considers that P(B I A) should be a relative frequency) and needs some justification. Several types of justification exist.

66 DUBOIS AND PRADE

- Numeric&ustificatiQ: it is clear that Bayes rule is the only possibility if for B, C c A. P(B I A) / P(C I A) should be equal to P(B) / P(C), i.e. there is no relative change in the probability values of subsets of possible worlds. This condition obviously embodies a minimal change principle. Another numerical justification is given by Gadenfors2 who proves that conditioning is the only updating rule such that P(. 1 A u A ) is a convex combination of P(. I A) and P(. I A ) when A n A = 0. Indeed the following identity is then valid:

P(B I A v A ) = kP(B I A) +(1 -h)P(B I A ) (3)

where h = P(A) / P(A u A') = P(A) / (P(A) + P(A)). Giirdenfors' result assumes that P(A I A) = 1, and P(. I 12) = P. Thus conditioning on "A or A " can be expressed in terms of an average between the results of conditioning on A and conditioning on A', if A and A are mutually exclusive. Teller'' also gives a representation theorem whose central condition is that for B, C G A, P(B) = P(C) 3 P(B I A) = P(C I A).

- m c iustification: Cox12 and his followers (e.g. H ~ k e r m a n ' ~ ) have proved that Bayes rule, as well as probability theory itself can be justified through consistency between the Boolean structure of the subsets of R and three simple axioms for the measure of uncertainty g, and the conditioned one g(. I A), i.e.

i) g(B n A) = f(g(B 1 A), g(A)) ; ii) g(@ = s(g(A)), v A G 12 : iii) f is a continuous strictly monotonic function in both places; s is a

continuous, strictly decreasing function.

Then g should be a probability measure and f the product. However this is more a justification of conditional probability than of revision itself. See Heckerman13 for a Coxian-like justification of conditional probability as actually performing a revision. Namely Heckerman defines P(B I A) as a function of P(B) and a quantity U(B, A, Q) called an update function, such that U(B, A n A , Q) is a continuous. monotonic function of U(B, A, 12) and U(B, A', Q).

- Jnformation-theoretic dtifica-: the information content of a probabilistic representation of a cognitive state is the so-called Shannon entropy which, in the finite case, reads

S(P) = -& 12 P(w)LogP(w).

Basically, the less ambiguous is P, the smaller is S(P); particularly S(P) = 0 if P is a complete cognitive state and S(P) is maximal if and only if p(o) = p(o'), Vw, o' E 12, i.e., P is a uniform probability on 12, denoted P?. It has been extended over to an informational distance by Kulback and Leibler, i.e.

This quantity is such that S(P) = - I(P?,P) + Logn where 1121 = n and P7 is the maximally ambiguous cognitive state in 12. It has been proved that the probability measure P that minimizes I(P,P) under the constraint P(A) = 1 is the conditional probability P(. I A) (William~'~).


Note that the understanding of Bayesian conditioning as a revision process (as above) is challenged by many Bayesians who view conditioning as a mere change of reference class reflecting the available evidence, i.e. a "focusing process". This is also called hypothetical conditioning: all the posterior probabilities P(. I A) can be computed in advance, i.e. before the input is known. Then when A is known to be true, only the selection of a posterior takes place. However because of the existence of a single conditioning rule, there is no way to tell one form of belief change from the other. This distinction will appear in the more elaborate numerical settings, later on in this paper.

B. Probabilistic Imaging

Another path in the problem of updating probabilities is the one followed by Lewis4. Assume that the set R of possible worlds is such that for any world w E 0, and any set A E R, ~ W A such that OA is the closest world from w, that belongs to A. Then the principle of minimal change can be expressed as an advice to move probability weights as little as possible away from the worlds that become impossible upon learning that some event A E R has occurred. This updating rule can be formally expressed as

This rule is called 'imaging' because PA is the image of p on A obtained by moving the masses p(o) for o c A to WA E A, with the natural convention that OA = o if o E A. This rule actually comes from the study of conditional logics (Harper et al.15). It has been generalized by Giirdenfors2 to the case when the set of worlds in A closest to a given world o contains more than one element. If A(o) G A is the subset of closest worlds from o, p(o) can be shared among the various worlds o' E A(o) instead of being allocated to a unique world. Glirdenfors has proved that general imaging is the only updating rule that is homomorphic, i.e. such that

which expresses invariance of the updating under convex combination (PA is the measure associated with PA). Clearly, instead of sharing p(w) among w' E A(o), a less committed update is to allocate p(o) to A(o) itself (and none of its subsets). In that case the imaging process produces a basic probability assignment in the sense of Dempster's view of belief functions16. But this type of update is not consistent with Bayesian probabilities because the result of imaging is a family of probability distribution not a unique one. Note that imaging can turn impossible worlds into possible ones, i.e. one may have PA(@) > 0 while p(o) = 0 for some o, e.g., if WA is such that ~ ( o A ) = 0. As a consequence a sure fact B a priori, i.e. such that P(B) = 1 may become uncertain, i.e. PA(B) c 1. This is not the case with Bayesian conditionin!. In order to preserve this kind of monotonicity property, one idea (see Giirdenfors ) is to build PA as the image of P on A n S where S = (a I P(w) > 0) is the support of P. However this is no longer homomorphic. However as with the Bayesian rule, P(A) = 1 * PA = P; this is the updating version of the success postulate of Katsuno and Mendelzon. ( 5 ) is the probabilistic version of the

68 DUBOIS AND PRADE

disjunction postulate of these authors where disjunction is changed into convex combination. In fact all postulates of Katsuno and Mendelzon2 hold or have a natural counterpart for probabilistic cognitive states, except the postulate which expresses that the conjunction of B with the result of an updating by A entails the result of the updating by A and B.

C. Axioms of Probabilistic Change

The next problem to address is the status of Bayesian conditioning with respect to the Gkdenfors classification of belief change rules. Giirdenfors2 considers Bayesian conditioning as a generalized expansion. The main reason is that the support of P(. I A) is S n A if S is the support of P, and P(. I A) is defined only if P(A) > 0, i.e. if A is not a priori impossible. However it is not obvious to translate all the 6 postulates of expansion into the probabilistic setting. Denoting by P+A the result of the expansion of P by the input A. the obvious ones are

(P+1) (P'2) P'AfA) = 1 (P+q)

that are counterparts to expansion postulates. However, it is difficult to find a counterpart to axiom X 5; %+A (expressing that after expansion the belief set X, closed under deduction, includes the previous one) in the probabilistic setting when P(A) > 0, i.e. how to express that P+A is more informative than P in a more refined way than by comparing the supports of P and P+A. The Alchourr6n. Gkdenfors and Makinson's monotonicity postulate2 %, c X' 3 %.,+A c X'+A does not make sense since inclusion is not defined for probability measures. Similarly, the monotonicity of expansions with respect to the input, i.e. A E; B X+B 5; %.,+A makes no sense for probability measures due to the normalisation constraint Q

p(w) = 1; generally, there is no systematic inequality between P(B) and P(B I A) since P(B I A) can increase with respect to P(B) and P(B' I A) can decrease with respect to P(B') for another event B'. These points cast doubts on the interpretation of probabilistic conditioning in terms of expansion.

Based on conditioning, taken as an expansion function, GLdenfors2 introduces notions of contractions and revisions of probability functions, in such a way as to remain as parallel as possible to the properties of these notions in the setting of classical logic. Basically, contraction is viewed as a reverse conditioning so that conditioning a contraction PA of P leads to recover P if P(A) = 1 ( P A ( . I A) = P). Postulates for revision that apply if P(A) = 0 are also proposed; they are very close to Popper's postulates of conditional probability that encompasses the case when P(A) = 0. These postulates, together with those for contraction enable Giirdenfors to recover Levi's identity (expressing that revision by the input A is equivalent to a contraction by A followed by an expansion) as, P*A = FA(. I A) where P*A is the revision of P and A is the complement of A, However the construction of explicit probability revision functions looks rather tedious, somewhat artificial, and not appealing in practice.

P+A is a probability measure

i fP(A) = 1 then P+A = P


D. Example

Let us illustrate the difference between conditioning and imaging on an example (due to Morreau). A box contains either an apple (a) or a banana (b). Let 01 .02 , 03,014 denote the states where a A b is true, a A Tb is me, la A b is true, -a A l b is true respectively. Our cognitive state is represented by ~ ( 0 1 ) = p(w4) = 0, p(02) > 0, p(03) > 0. Assume for instance that p(w2) = 0.7, p(03) = 0.3, i.e. an apple is more probably present than a banana in the box. Upon the occurrence of A = (03.04) (no apple) Bayes rule yields p(03 1 A) = 1; i.e. there is a banana in the box (Gadenfors' revision leads to the same result in the logical setting). Let us now apply Lewis' imaging. Proximity on R can be defined by means of Winslett's relation (see Lta S ~ m b t ' ~ for instance). The closest "neighbour" of 0 2 (E A) in A is 0 4 (both agrees that b is false). Then moving p(02) to 02A = 0 4 (and 03A = 0 3 since a3 is in A) gives the update

PA(04) = 0.7 ; PA(03) = 0.3

i.e., the most probable situation is that the box is empty. This is in agreement with a reasoning by case: either the box was containing an apple or a banana; if the apple (if any) has been taken out of the box, either the box is now empty or there is still the banana. This is also in agreement with Katsuno and Mendelzon3's approach to updating, In Bayes rule, A is understood as "there is no apple" (static world), while with imaging, A rather means "there is no longer any apple" (world change).

E. Uncertain Inputs

The Bayesian setting has been extended to the case of uncertain inputs. The simplest uncertain input corresponds to an event A G R along with a probability a that this event did happen. The updated probability measure P(B I (A,a)) can be computed using Jeffrey's rule18 as

P(B I (A,cx)) = aP(B I A) + (1 - a)P(B I A) (9

where P(B I A) and P(B I A) are obtained by regular conditioning. The initial cognitive state must be such that P(A) > 0, P(A) > 0. This formula has been extended to the case where the input is under the form of a partition ( A 1, A2, . . . , An] of R, and the probability attached to each Ai is ai. The result of the revision is then

with Cai = 1. Again, the condition P(Ai) > 0 must be satisfied. In a strict Bayesian view of (6) and (7), a i is interpreted as a conditional probability P(Ai I E) where E denotes the (sure) event underlying the uncertain information. That is: an event E has occurred, the result of which is that P(A;) = ail i = 1.n. Then (7) assumes that P(B I Ai) = P(B I Ai n E), i.e. that for all Ai, E is independent of B in the context Ai (e.g., Pearl7). (6) and (7) have been justified by Wil l iam~'~ on the basis of the

70 DUBOIS AM) PRADE

informational distance I(P,P) under the constraints P'(Ai) = ai. Formula (7) can be also justified at the formal level by the fact that the only way of combining the conditional probabilities P(B I Ai) in an eventwise manner (i.e. using the same combination law for all events B) is to use a linear weighted combination such as (7) (Lehrer and Wagner19). Note that the uncertain input is viewed as a constraint that forces the probability measure to bear certain values on a partition of R. Jeffrey's rule ensures that probabilities do not change in relative value for possible worlds within each partition element Ai, In that view, the ai's cannot be interpreted as the reliability of input Ai as being the true one; the uncertain input (Ai,ai), i = I ,n is rather lo be interpreted as a correction of the prior probability. However, (7) could as well be viewed as an expected value of the conditional probability P(B I A) when A is a random event whose realizations belong to the partition (A1, . . ., An). A is then a genuine unreliable observation, ai being the probability that Ai is the true input. It may look strange that the two views coincide in their implementation. Besides, the requirement that (A1, .. ., An) forms a partition does not look compulsory when the uncertain input is viewed as an uncertain (random) observation. It is compulsory only when the uncertain input is a constraint in order to ensure that the result obeys the constraint P(Ai I ((Ai,ai), i = 1,n)) = ai (since C ~ = I , ~ a i = 1).

Lastly, it is obvious that, pushing (7) to the limit, by assuming R = ( w 1 , . . . , an) and Ai = (q). i.e. choosing the finest partition in R, then letting a i = P( (o i ) ) for a probability measure P', Jeffrey's rule (7) comes down to a simple substitution of P by P . In other words, the new piece of evidence totally destroys the cognitive state. It emphasizes the dissymmetry of the belief change process and emphasizes the role of the uncertain input as a constraint; priority is given to the new information, and levels of uncertainty are part of this information. Conversely, when the partition is reduced to a single element (a), the input information is vacuous and the revision through Jeffrey's rule leaves the prior probability untouched. This fact points out that a uniform probability on R is not equivalent to the absence of known probability on R, as some Bayesian views tend to suggest. Jeffrey's rule is a good tool for telling one from the other.

There is a whole field of investigation called "probability kinematics" which generalizes the type of problem addressed by Jeffrey's rule, and that use informational distances that sometimes differ from Kulback-Leibler relative information index. Other types of constraints than those dealt with by Jeffrey's rule can also be envisaged. More on this topic can be found in Van Fraassen20, Domotor21.22.

IV. THE POSSIBILISTIC FRAMEWORK

Let us turn now to the case when a cognitive state is modelled by a possibility distribution TC, that is, a mapping from R to a totally ordered set V containing a greatest element (denoted 1) and a least element (denoted 0), e.g. V = [0,1]. However any finite, or infinite and bounded, chain will do as well. A consistent cognitive state K is such that TC(W) = 1 for some w, i.e., at least one of the worlds is considered as completely possible in a. If x(w) > x(w'), then w is a more plausible state of the world than w'. Let K and x' be two possibility distributions on R describing cognitive states. When K I K', TC is said to be more specific than zt' (Yager23), i.e., the cognitive state described by K is more complete, contains more information than

A SURVEY OFBELIEF REVISION AND UPDATING RULES 71

the cognitive state described by x'. Especially, if there is a world og E R such that x(o0) = 1, and A(W) = 0 if o f 00, x corresponds to a complete cognitive state. Conversely the vacuous cognitive state is expressed by the least specific possibility distribution on R, i.e., xT(o) = 1, 'do. It corresponds to the state of total ignorance. Lastly, the absurd cognitive state, where nothing is possible is "1 such that " ~ ( 0 ) = 0, 'd w E R. In the following, by cognitive state we mean either a consistent cognitive state or the absurd one.

The three basic forms of belief dynamics described by G&denfors2, namely expansion, contraction and revision can easily be depicted in the possibilistic framework. The result of an expansion, that stems from receiving new information consistent with a previously available cognitive state described by A, is another possibility distribution x' that is more specific than x . Note that a', by definition, is also such that x'(w) = 1. Hence if we let C(x) = { o I x(o) = 1) be the core of x (i.e. the set of most plausible worlds in a given cognitive state), we have C(d) # 0 and C(x') G C(x). A contraction, i.e. the result of forgetting some piece of information among those that form a cognitive state, will be expressed by going from x to a less specific possibility distribution x' 2 x. The term revision will be interpreted as any other belief change which is neither a contraction nor an expansion. Namely, it is when from x , we reach x' where neither x 2 A' nor A' 5 x hold. More specifically we shall encounter a special kind of revision that will be termed "strict", such that C(x) n C(x') = 0. This type of revision may be met when changing a cognitive state upon receiving a piece of information under the form of a proposition taken as absolutely true, but which is not completely plausible a priori.

However expansions, contractions and strict revisions do not exhaust the set of possible belief changes in the possibilistic setting; more refined changes may take place such as slightly increasing the possibility of one world while decreasing the possibility of another world without altering C(x) nor the support S(x) = (a I x ( o ) > 0) containing the worlds considered as somewhat possible.

Similarly to the probabilistic case, a possibility distribution generates a set function n called a possibility measure (Zadeh24) defined by (for simplicity V = [0,1])

and satisfying n(A u B) = max(n(A),n(B)) as a basic axiom. n(A) evaluates to what extent the subset A of possible worlds is consistent with the cognitive state x. n(A) = 0 indicates that A is impossible. n(A) = 1 only means that A is totally consistent with x , and it may happen that n(A) = n(A) = 1 where A is the complement of A, in which case, it expresses ignorance about A (and A). In case of total ignorance modelled by x = q, we have V A # 0, n(A) = 1. The degree of certainty of A is measured by means of the necessity function N(A) = 1 - n(A), whose characteristic axiom is N(A n B) = min(N(A),N(B)). A is considered as a sure (hence not defeasible) fact in a cognitive state A whenever N(A) = 1. If S(x) is the support (o I x(w) > 0) of x , then N(A) = 1 e S(x) E A, while N(A) > 0 w C(x) c A (in a finite setting) means that A is an accepted belief (that could be defeated by future inputs if N(A) < 1). The axiom of necessity measures indicates that if A and B are accepted beliefs then A n B should be accepted too.

There are two main differences between possibilistic and probabilistic representations of cognitive states: the possibilistic setting is ordinal and not additive. Moreover the possibilistic setting captures total ignorance under the form of the

72 DUBOIS AND PRADE

vacuous cognitive state, which a single probability distribution cannot model, because the probability attached to a proposition always accounts for the number of possible worlds in which the proposition is true; hence it is never true that for any non-tautological and non-contradictory propositions A and B, P(A) = P(B) holds.

Several possible interpretive frameworks exist for possibility theory. Zadeh24 explains how to generate possibility distributions from vague propositions such as "John is tall", that describes a cognitive state. The possibility distribution a is then equated to the membership function ptall of the fuzzy extension of "tall". Purely qualitative distributions can be derived as natural semantics of layered propositional knowledge bases (see Dubois, Lang and Prade25) as will be seen in Section VI. Numerical possibility distributions also model approximations of generalized probabilistic representations of cognitive states, as the ones studied in Section V. Namely x(o) can be understood as an upper bound of an ill-known probability p(w) (see Dubois and Prade26). Lastly, degrees of possibility can be related to orders of magnitude of infinitesimal probabilities (Dubois and Prade27); see Subsection E.

A. Possibilistic Expansion and Conditioning

The expansion "+A of a non-absurd cognitive state a upon learning the sure fact A is defined as:

if 3 o E A, X(O) = 1 then X+A(W) = x(w) if w E A = 0 otherwise

else '&A(O) = 0, vm.

In case the new information A is such that n(A) = I , i.e. A is consistent with the cognitive state A, a is expanded into R' given by

where p~ is the characteristic function of the subset A. Otherwise min(pA,x) < 1 and K+A given by (9) is no longer the description of a consistent cognitive state. In that case expansion results in the absurd belief state by convention (SC+A = x i ) . Note that this definition, which is in the spirit of Giirdenfors' theory can be questioned in the last case. Indeed when the new information contradicts the old cognitive state, it might look as reasonable to reach a vacuous cognitive state ICT rather than to reach an absurd cognitive state, since conflicting information may result in the awareness of ignorance (see Yager2* for this type of conflict resolution), However, the above definition is natural in the scope of successive expansions that result in more and more specific possibility distributions (X+A I a), and can never decrease specificity.

This definition of expansions of possibility distributions verifies the 6 Gkdenfors' postulates of ex ansion. that we express in terms of possibility distributions (Dubois and Prade t ). (n+l) K+A is a possibilily distribution describing a cognitive state (stability)


( F 2 ) N+A(A) = I (priority to the new information) ( p 3 ) n + ~ I a (improvement of knowledge) (F4) i f N ( A ) = 1 then K+A = a(invariance i f the input is already known) (P5) a _< d * Z+A _< K'+A (monotonicity) (n+6) for any a and A , &A is the least specific possibility distribution that

satisfies NI+i ) - (P5)

Hence we obtain for expansions in the possibilistic setting the same result as Gkdenfors. It is easy to check that if n is a non-fuzzy possibility distribution, i.e., corresponds to the set of models of a belief set % (as in equation (9)), then (n+l)- (n+d specialize in exactly Giirdenfors' expansion axioms.

Revision in possibility theory is performed by means of a conditioning device similar to the probabilistic one, obeying an equation of the form

VB, n(A n B) = n ( B I A) * n(A). (10)

Possible choices for * are min ( H i ~ d a l ~ ~ ) and the product; see Dubois and Prade30 for justifications. In case of * = min, equation (10) already appears in Shackle's3' book under a different guise, namely, using the so-called degree of surprise s(A) = N(A) instead of n(A). However since this equation may have more than one solution in terms of n ( B I A), Dubois and Rade32 have proposed to select as the most reasonable solution the least specific solution to (lo), (i.e. the solution with the greatest possibility degrees in agreement with the constraint (10)). It yields, when n(A) > 0,

n (B I A) = 1 i fn(A n B) = n(A) > 0 = n (A n B) otherwise.

In particular n ( B I A) = 0 if A n B = 0. The conditional necessity function is defined by N(B I A) = 1 - n(B I A), by duality. The possibility distribution underlying the conditional possibility measure n(. I A) is defined by

x ( o I A) = 1 if x(w) = n(A), w E A

=Oifwcz A. = x(w) if x(w) c ll(A), w E A (1 1)

To see it, one just has to let B = ( 0 ) in (10) and to choose maximal values for n( (0 ) I A). Note that if n(A) = 0 then A(. I A) is still a solution to (10) and is equal to PA. In this case, PA is simply substituted to n. This includes the case when A = 0, which results in n(. I A) = x ~ . This definition of conditional possibility, using * = min, is natural in a purely qualitative setting. When * = product, the corresponding expression is

74 DUBOIS AND PRADE

provided that n(A) # 0. This is Dempster rule of conditioning. specialized to possibility measures, i.e. consonant plausibility measures of Shafer33. The corresponding revised possibility distribution is

= 0 otherwise.

This rule is much closer to Bayesian conditioning than the ordinal rule (1 1) which is purely based on comparing numbers; (12) requires more of the structure of the unit interval (a product operation). In both cases the set function itself remains ordering- based. (12) makes sense if Q is a continuous universe, if R is continuous and revision is required to preserve continuity.

x( . I A) = x (no revision for an information already known with certainty). Counterparts of the Gadenfors' postulates2 for revision hold as well with the two definitions. It can be shown (Dubois and Prade6) that if K*A denotes a possibility distribution obtained by revising x with input A, it makes sense to let "*A = x(. I A). Let us first translate the axioms of belief revision into the possibilistic setting

(Pi) for any subset A _c 0, @A represents a cognitive state (stability) (P2) N*A(A) = I (priority to the new information) (F3) R*A 2 X+A (revising does not give more specific results than expanding) (P4) f n(A) = 1 then n * ~ 5 "+A (if A is not rejected by n, revision reduces to

expansion) (Ps) ~ Z C A = IC_L if and only i f A = 8

(P7) fl~d 2 ("*A)+B

Note that we have N(A) = 1

(P6) A = B * l r * ~ = l r * g

if IItB I AJ = 1 then @AM 5 ~ I ~ C A ) + B

Note that (n*3) and (n*4) are particular cases of (n*7) and (n*8) respectively, letting A = $2, since x*n = R. In fact, (n*7) and (n*8) correspond to the fact that if B is consistent with the revised cognitive state accepting A, then x*A,.-,B = min(x*A,pB). R*A = x(. I A) as defined by (7) is not the unique possibility distribution that satisfies (n* 1)-(n*8). However (1 1) embodies a principle of minimal change. If x and x' define two real-valued possibility distributions on a finite universe Q then the Hamming distance between x and x' is defined by H(x,x') = EmE Q Ix(w) - x'(w)I. Then we have the following result (Dubois and Prade6): x(. I A) is the possibility distribution the closest to n that complies with counterparts of Giirdenfors' postulates, as long as there is a single world WA where IC(WA) = n(A). H(x; x(. I A)) is thus minimal under the constraint N(A I A) = 1.

It is worth noticing that the revision rule x(- I A) satisfies the counterpart to the property (3) of probabilistic conditioning with respect to the disjunction. It can be checked (see Appendix A) that

n(B I A u A') = max(min(a, n ( B I A)), min(a', n ( B I A')) (14)

A SURVEY OF B E E F REVISION AND UPDATING RULES 75

with a = n(A') 4 n(A) and a' = n(A) -+ n(A'); 3 is the multiple-valued implication a + b = 1 if a I b and b otherwise. The function from [0,1]* to [O,l] defined by Ma,al(x,y) = max(min(a,x), min(a',y)), where max(a,a') = 1 is the possibilistic counterpart of the weighted arithmetic mean (or convex mixture) in probability theory. Condition max(a, a') = 1 is indeed verified in (14). The behavior of the product based definitions of n ( B I A) with respect to a disjunction of input is similar to the min-based definition: identity (14) remains true provided that we define a = n(A) + n(A) and a' = n(A) + n(A) by means of the implication a + b =

mid 1, :) and the min is changed into product (see Appendix A). While (13) looks as a more natural counterpart to probabilistic conditioning,

N(B I A) stemming from (11) is more closely akin to the concept of "would counterfactual" following Lewis, and denoted A o+ B, which is intended to mean "if it were the case that A, then it would be the case that B". Lewis proposes to consider A D+ B as true in world w if and only if some accessible world in A n B is closer to w than any world in A n B, if there are worlds in A. Let us interpret "closer to world w" as "preferred" in the sense of possibility degrees. And let us notice that we do have when n(A) > 0 (as pointed out in Dubois and P ~ i d e ~ ~ ) ,

where B is the complement of B and N(. I A) is the necessity measure based on JC(. I A). The latter inequality means that there is a world in B n A which is more possible than any world in B n A, and this works if n(A) > 0. Hence N(B I A) > 0 agrees with the truth of A 04 B. The counterpart of Lewis' "might conditional" A O+ B is of course I7(B I A) in the sense of (10) with * = min. However there is a difference in scope between Lewis' construct and possibilistic conditioning: in the former, the conditional A o-+ B is evaluated with respect to a precisely known world (the actual world) while the latter is evaluated with respect to an incomplete cognitive state.

Let us point out the proximity between the possibilistic approach and Spohn'ss well-ordered partitions (WOP). It is easy to check that given a WOP, Eo, El, E2, . . . , En which partitions R, and assuming V w E Ei, X(O) = a i with a0 = impossible and an = completely possible, our approach to revision is to consider, as a result, a WOP on A, namely (Ei n A, i = 0,n) where we must delete empty terms. Spohn objected to this kind of definition because in the revised state all worlds in A are completely disbelieved, and because the revision is not reversible. However possibilistic belief changes are commutative. Moreover the input information A is taken as absolute truth in this section, and reversibility makes no sense in such a context. The above results suggest that Spohn's criticisms also apply to the Alchourr6n, Gadenfors and Makinson's theory itself.

76 DUBOIS AND PRADE

B. Possibilistic Contraction

Contraction of a possibility distribution with respect to A c Q corresponds to forgetting that A is true if A was known to be true. In other words, the result IC-A of the contraction must lead to a possibility measure n - ~ such that ~ - A ( A ) = ~ - A ( A ) = 1, i.e. complete ignorance about A. Intuitively if n(A) = n(A) = 1 already, then we should have IC-A = IC. Besides if n(A) = 1 > n(A) then we should have IC-A(O) = 1 for some w in A, and especiaIIy €or those w such that n(A) = IC(W). It leads to the following proposal

IC-A(W) = 1 if ~ ( w ) = n(A), E A = ~ ( w ) otherwise. (16)

Again let us translate Giirdenfors' postulates for contraction into the possibilistic

for any subset A E a, KA represents a cognitive state r~ 2 R (z-A is not more informative than R) i f N ( A ) = 0 then K A = K (N(A) = 0 means that A is not sure under cognitive state R) N-A(A) = 0 unless A = i fN(A) > 0 then K 2 min(pA,rA) (retracting A followed by an expansion on A shoufd be coherent with the original cognitive state) A = B m a x ( T A , r B ) 2 T A ~ (retracting A n B leads to a cognitive state that is at least as informative as retracting A or retracting B ) i fN-AnB(A) = 0 then ~A,- ,B 2 f f ~ (when retracting A n B , i f A is no longer certain then we do not lose more information by retracting A directly).

(A is forgotten ifpossible)

CA = f f ~

By construction, when defined by (16), IC-A again corresponds to the idea of minimally changing IC so as to forget A, when there is a unique w E A such that 1 > n( A) = x(w). When there are several elements in ( w e A, ~ ( w ) = n( A)), minimal change contractions correspond to letting I~A(W) = 1 for any selection of such world, and IC-A corresponds to considering the envelope of the minimal change solutions. If II( A) = 0, what is obtained is the fullmeet contraction (GBdenfors*).

As in the classical case, Levi and Harper's identities remain valid, namely

R(* I A) = (IC-A)>+A ; IC-A = max(lc,x(* I A)).

An alternative contraction rule to (16) is


= ~ ( w ) otherwise

that is the companion to the Bayesian-like possibilistic revision rule. Again Levi's and Harper's identity hold between the two Bayesian-like rules.

C. Possibilistic Imaging

It is easy to envisage the possibilistic counterpart to Lewis' imaging since this type of belief change is based on mapping each possible world to the closest one that accomodates the input information. As in Section 111, define for any w E R, and non- empty set A E; f2 the closest world WA E A to a. Then the image of a cognitive state K in A is such that

K'A(o') = maxOl,wA x(o) if w' E A

= O i f o ' e A.

If there is more than one world OA closest to o, then the weight x(w) can be allocated to each of the closest worlds forming the set A(w), and the above updating rule becomes

A'A(o') = maxu A(O) if O' E A W'E A( w)

= 0 if w' e A. (18)

It is easy to check that the above updating rule defined by (17) satisfies all postulates of Katsuno and Mendelson's updates3 namely (see Appendix B):

Defining Vw, A(w) precisely as (w' I ~(w') = n(A) = max(x(o"), a" E A)), which does not depend on w, then K'A = K(. I A), i.e., we recover the revision based on conditioning. Clearly in this setting, we see that Katsuno and Mendelzon's approach subsumes the Alchourrdn-Giirdenfors-Makinson's (AGM) framework.

In the setting of possibility theory, the differences between Katsuno and Mendelson framework and the AGM approach are patent. While the AGM approach tries to use the input information so as to reduce incompleteness, the update view tries to carry the incompleteness of the cognitive state over to a new cognitive state

78 DUBOIS AND PRADE

that agrees with the input information, assuming that the shift of the "real world" is minimal.

If we define RA as the relation that to each w assigns its closest neighbours in A, the above update formula is nothing but Z a d e h ' ~ ~ ~ extension principle that characterizes the fuzzy image of the fuzzy set whose membership function is R, i.e. if x = p~ then R'A = p~~~ F with ~ R ~ ( o , o ' ) = ~ A ( ~ ) ( w ' ) and ~ R ~ ( w , w ' ) = 0 if w' e A. This suggests the extension of (18) when A(o) is a fuzzy subset of A, by

R'A(w') = maxW x(w) * p ~ ( ~ ) ( w ' ) if w' E A = O i f o ' e A

where * = min or product. An example of this kind of updating is the following. A man is known to have a very high income. Then you learn that his income is upper- bounded by an amount which is only weakly compatible with your previous information, then your update corresponds to values which are as close as possible to those representing the idea of very high income but less than the upper bound.

It can be easily checked that the updating defined by (17) is invariant under weighted max-combination, i.e.

[maxi (hi * Ri)]'A = maxi hi * (R~)'A with maxi 4 = 1.

Clearly U8 is a particular case of this invariance under max-weighted combination. The possibilistic framework would enable us to deal with the apple and banana

example in a way very similar to the probabilistic solution, although in a more qualitative way. The example is too elementary to exhibit significant differences between the two approaches. Indeed the support of the distribution has only one element in common with the subset representing the input information and one element outside, which is insufficient to exhibit the effects of the different ways distributions are normalized in possibility and in probability theory.

D. Uncertain Inputs

Belief change can be extended to the case of uncertain inputs of the form N(A) = a. The main question as in the probabilistic case is how to interpret such an uncertain input. The two already met interpretations make sense:

i) N(A) = a is taken as a constraint that the new cognitive state must satisfy; it means that if R' is obtained by revising x with information (A,a), the resulting necessity measure N must be such that "(A) = a;

ii) N(A) = a is interpreted as an extra piece of information that may be useful or not useful to refine the current cognitive state; in that case a is viewed as a degree of reliability or priority of information A.

Interpretation i) is in the spirit of Jeffrey's rule. Clearly N(A) = 1 will lead to an expansion of R into K + A or a revision R ( . I A), while N(A) = 0 will force a contraction. In contrast, ii) corresponds to either a revision or an expansion but is never a contraction, since if a is too low, the input information will be discarded.

A SURVEY OF BELIEF REVISION AMI UPDATING RULES 79

The input information is not modelled in the same way whether it is a constraint or an additional information. In the first case, N(A) = a is interpreted as n(A) = 1 and n(A) = 1 - a, and the belief change rule is of the form

x(w I (A,a)) = n(o I A) if w E A = (1 - a) * x(w I A) if o E A (19)

where * = min or product according to whether x ( o I A) is the ordinal or Bayesian- like revised possibility distribution. Note that when a = 1, ~ ( o I (A,a)) = R(W I A), but when a = 0, we obtain a possibility distribution less specific than x such that the associated necessity of A is zero.

', In the second case where N(A) = a is viewed as an additional information

represented by a fuzzy set F with membership function p~

~ F ( w ) = 1 if o E A = 1 - a otherwise.

Letting FA = (o I ~ F ( w ) 2 A ) , each Fk is viewed as the (non-fuzzy) regular input information underlying F, with plausibility k and the revised cognitive state x(. I F) is defined by analogy with Jeffrey's rule as

where the convex mixing is changed into the weighted maximum and * is min or product again. In our particular case, it gives, for a > 0

x(ofF) = x ( w l A ) i f w E A = x ( o ) * (1 - a) if o E A.

Note that n(w I FJ I p~ = max(pA, 1 - a); moreover x ( o IF) = n(w) if a = 0 since then F = 0, i.e., the operation is never a contraction. This behavior is very different from the case when N(A) = a is taken as a constraint.

The first revision rule (19) under uncertain inputs can be extended to a set of constraints n(Ai) = hi, i = 1 ,n, where (Ai, i = 1 ,n) forms a partition of R, and it gives

(21) IC(O I ((Ai,hi))) = hi * R(O 1 Ai), b' w E Ai

where * = minimum or product whether x(w I Ai) is ordinal or Bayesian-like. In the limit case when Ai = ( o i l , b'i, the input is equivalent to a fuzzy input F with VF(O~) = hi. And the above belief change rule reduces to a simple substitution of x by p ~ , just as for Jeffrey's rule in the probabilistic setting.

To conclude, while the belief change rule (20) is formally analogous to Jeffrey's rule, its behavior is very much akin to a revision h la Giirdenfors. On the contrary the other rule (19)-(21) is very close to the spirit of Jeffrey's rule, and has been proposed in another setting by Spohn5 who uses the integers as a scale rather than [0,11 with the convention that 0 corresponds to the minimum impossibility (i.e. the maximal possibility), see Dubois and P ~ - a d e ~ ~ .

80 DUBOIS AND PRADE

E. Spohn's Model of Cognitive States

An ordinal conditional function (OCF for short) is a function K from a complete field of propositions into the class of ordinals. Here, for simplicity we consider a function from a finite Boolean algebra % to the set o'i of natural integers n. This Boolean algebra consists of a family of subsets of a universe R induced by a finite partition (Al, . . ,, Am) of R. By definition an OCF verifies the following properties ( S p ~ h n ~ , ~ ~ ) :

i) Vi, Vw, w' E Ai, K(W) = K(w')

ii) 3 Ai G R, K(A~) = 0 iii) V A E R, K(A) = min(K(w) I w E A},

It is easy to see that the set function N, defined by NK(A) = 1 - 2-K(A) (any k > 1 could be used instead of 2 as well) is a necessity measure, with values in a subset of the unit interval. Moreover because K(A) E Pi, N,(A) c 1, V A f R. The set { ~ ( w ) I w E R) is the counterpart of a possibility distribution II on R. Namely, let for instance nK(A) = 1 - NK(&) = 2-K(A), it is easy to check that x,(w) is equal to 2-"(O), where xK is the possibility distribution associated with nK. K(W) can be viewed as a degree of impossibility of w, and K(A) = 0 means A is completely possible. Since K(O) E fA, ~ ~ ( 0 ) > 0 for all w's, i.e., nothing is considered as fully impossible in Spohn's approach.

As pointed out by S h e n ~ y ~ ~ , the function K is a kind of measure of disbelief. Spohns has introduced a notion of measure of belief P taking its values on Z and built from K:

P(A) = -K(A) if K(A) > 0 = K(A) if K(A) = 0.

It is simple to see that the function p: Z + [-1,+1] defined by

P(P(A)) = 2P(A) - 1 if P(A) 5 o = 1 - 2-P(A) if P(A) 2 o

is such that p(P(A)) = nK(A) - &(A) = CF(A), the certainty factor of MYCIN (Buchanan and Sh~rt l i f fe~~) as reinterpreted in the framework of possibility theory. It turns out that this set-function is not easy to handle beyond binary universes (Dubois and Prade?.

Spohn also introduces conditioning concepts, especially:

- the A-part of K such that V w E A, K(O I A) = K(O) - K(A)

-the (A,n)-conditionalization of K, say ~ ( w I (A,n)) defined by K(O I (A,n)) = (0 I A) if w E A

= n + K ( w I A ) i f ~ € A


It is interesting to translate this notion into the possibilistic setting. Definitions (22) and (23) respectively become

= 0 otherwise

with a = 2-". (24) is the Bayesian-like revision rule of possibility theory while (25) is one of the belief change rules of possibility theory, the one (19) where the uncertain input is taken as a constraint. Rule (23) has been extended over to an ordinal conditional function X defined on the partition {Al, . . ., An) and that acts as uncertain evidence:

K(O 1 h) = h(Ai) + K(O I Ai), V o E Ai, i = 1,n (26)

This rule can be exactly mapped to the possibilistic belief change rule (21) where * =

S p ~ h n ~ ~ has in fact related his model to probability theory, and not to possibility theory. Namely K(W) is the exponent of an infinitesimal probability so that ~ ( w ) = n 2 1 means P( ( 0 ) ) = E-" where E is infinitesimally small. Then (22) exactly corresponds to the conditional probability P(o I A), and (26) is the infinitesimal version of Jeffrey's rule. No wonder if the possibilistic belief change rule (21) bears a strong analogy to Jeffrey's rule in its behavior. This view of Spohn's calculus has been used in an extensive way by Goldsmitz and in their logic of defaults called System Z+.

product and a i = 2- UAi)

V. EVIDENCE THEORY AND UPPER AND LOWER PROBABILITIES

Let us now consider Shafer's evidence theory. The set of possible worlds is called frame of discernment. In this framework the available knowledge is represented in terms of a basic probability assignment m, which is a set function from the set of subsets 2Q to [0,1] with the constraints m(0) = 0 and ZA m(A) = 1. The subsets A G Q such that m(A) > 0 are called focal elements. Note that there is no constraint on the structure of the set 5 of focal elements (here supposed to be finite and which does not make a partition in general). Each focal element Ai represents the most accurate description, with certainty m(Ai). of the available evidence pertaining to the location of the actual world in R. In other words, m(Ai) is the probability that Ai is the current cognitive state regarding the location of the actual world. The subsets Ai are not maximally specific due to some imperfection in the observations that lead to incompleteness of the available information. Hence, Ai is not necessarily a singleton.

82 DUBOIS AND PRADE

A plausibility function PI as well as a belief function Bel, attached to each event (or each proposition of interest) can be bijectively associated with m (Shafer33) and are defined by

= CA:AnB#@ m(A) (27) Bel(B) = 1 - PI@) = &+A,-B - m(A). (28)

In Dempster-Shafer-like approaches, a cognitive state is thus represented by a family of sets of possible worlds (each set representing a classical logic-like cognitive state), one of which is the current cognitive state, and the basic probability assignment expresses what are the most likely candidates.

On the contrary, in the setting of upper and lower probabilities, a cognitive state is represented by a set of probability distributions, one of which is the accurate representation of the cognitive state. Let ? be such a set of probability distributions on R. Then each subset of possible worlds A, representing an event, can be evaluated

P*(B) = suppE 5' P(B) ; P*(B) = infpE 5' P(B) = 1 - P*(B) bY

where P* and P* are called upper and lower probabilities. Note that the knowledge of P* (resp.: P*) for all subsets of d generally do not allow to recover the set .P exactly, namely 5' c (P, VB, P(B) I P*(B)) = (P, VB, P(B) 2 P*(B)), i.e., some precision is lost by upper and lower probabilities. Moreover upper and lower probabilities are more general than plausibility and belief functions viewed as probability bounds. For instance belief functions are super-additive of order n for all n E N while P* is superadditive only at order 1, generally. The most interesting class of upper and lower probabilities are when lower (resp.: upper) bounds are superadditive (resp.: subadditive) of order 2, i.e.:

P+(A u B) + P*(A n B) 2 P*(A) + P*(B)

and the converse inequality for P*. For this class, the knowledge of the probability bounds is equivalent to the set of probability distributions that induce them. See Huber"l2 for instance.

A. Revision, Updating and Focusing

In terms of plausibility functions, revision is expressed by Dempster rule of conditioning

PI@ IA) = B, : Bel(B I A) = 1 - Pl(B I A) P W )

This rule of conditioning can be justified on the basis of Cox's axiom that defines a conditional function associated to any uncertainty measure g defined on d as in Section 2 (Dubois and Prade30). Cox's axiom justifies Dempster's conditioning rule as well as the geometric rule of conditioning (Suppes and Z a n ~ t t i ~ ~ )


Belg(B I A) = B, ; Plg(B I A) = l-Belg(B I A) Bel(A)

In terms of basic probability assignments, P1(. I A) defined by (29) is obtained by transfemng all masses m(B) over to A n B, followed by a normaliation step, while Belg(. I A) is obtained by letting mg(B I A) = m(B) if B c A and 0 otherwise, followed by normalization, i.e. a more drastic way of conditioning. Dempster's rule of conditioning looks more attractive from the point of view of revision since Pl(B I A) is undefined only if PI(A) = 0 (i.e. A is impossible) while Belg(B I A) is undefined as soon as Bel(A) = 0 (i.e. A is unknown). This unability to revise with a vacuous prior is counterintuitive, with the geometric rule.

Dempster rule of conditioning is a mixture of AGM-type expansion (when m(B) carries over to A n B if A becomes true) and Bayesian updating. On the contrary, the geometric rule has nothing to do with an expansion on Q and is more in the spirit of imaging since Belg(. I A) = Be1 if and only if 'd Ai E F , Ai r A, which is an extension of postulate K&2 of Katsuno and Mendelzon3. Dempster rule of conditioning subsumes conditional possibility based on product and coincides with (12).

A direct extension of AGM revision and Katsuno and Mendelzon's updating could be envisaged in the belief function setting. Namely one might revise (or update) each focal set, as being a candidate initial cognitive state and then carry each weight m(Ai) from Ai over to the resulting set of possible worlds. Especially, in the case of updating one can thus generalize Lewis imaging which corresponds to when focal sets are singletons.

Another approach to conditioning has been proposed by De Campos et al.44 and Fagin and ~ d p e r n ~ ~ under the form

Pl(A n B) PI(A n B) + Bel(A n B)

P*(B I A) =

Bel(A n B) Bel(A n B) + Pl(A n B)

P*(B I A) =

These definitions can be justified by interpreting belief and plausibility functions as lower and upper probabilities, since it has been proved that

P*(B I A) = sup{P(B I A) I P E P(Be1)) P*(B I A) = inf (P(B I A) I P E P(Bel))

(33) (34)

where p(Be1) = (P I Bel(B) I P(B) 5 Pl(B), VB). These conditional functions are actually u r and lower conditional probabilities and have been considered by Dempster G i m s e l f and R u ~ p i n i ~ ~ . Fagin and H a l ~ e r n ~ ~ and J a f f r a ~ ~ ~ have proved that P*(. I A) is still a belief function.

Although very satisfying from a probabilistic point of view, this third definition leads to a rather uninformative conditioning process since P*(. I A) 2 P1(- I A) 2 Bell. I A) 2 P*(. I A) as proved by, e.g., Kybure8. Especially. complete ignorance is

84 DUBOIS AND PRADE

obtained on A (P*(B I A) = 1, P*(B I A) = 0, V B c A) as soon as Bel(A n B) = 0 and Bel(A n B) = 0, i.e., as soon as the conditioning set B intersects each focal element without including any one of them. It has been shown elsewere that (33)-(34) is not a rule for revision but a "focusing rule", by which one only changes the reference class (Dubois and Prade49, De Campos et al?4), withoutforcing P(A) = 0. Especially (33)-(34) does not modify the constraints specified by the belief functions. In contrast, Dempster rule of conditioning in the setting of upper and lower probability comes down to add the constraint P(A) = 1 to the set P(Bel), in the case when Pl(A) = 1, i.e., A is viewed as a new piece of information to be integrated in the current knowledge, and leads to a revision, and not only a change of reference class. When Pl(A) # 1, the constraint P(A) = 1 is incompatible with P(Be1) and a Bayesian revision takes place. Namely, Dempster rule (29) proves to make sense in the setting of upper and lower probabilities, provided that the lower probabilities be order 2-superadditive at least. In that situation Dempster rule comes down to apply (33-34) with the additional constraint P(A) = Pl(A), i.e. only the most likely probabilities (P such that P(A) is maximal) are selected. This interpretation of Dempster rule was found by Gilboa and Schmeidlerso. The geometric rule can be interpreted likewise, applying (33-34) with the additional constraint P(A) = P*(A). It is then a minimum likelihood revision. A more refined proposal can be found in (Moral and De CamposS1) where the distributions which do not maximize P(A) are also somewhat taken into account.

The difference between revision and focusing seems natural if we envisage a cognitive state as encoding generic knowledge. In that perspective input information can be either factual or generic, and the processing of this input differs in each caqe. A factual input points out the current particular situation as belonging to a certain class of situations, and focusing addresses the question answering problem "what can be said about the current situation". On the contrary, a generic input corresponds to a modification of the cognitive state that will affect its future behavior in all future cases the generic cognitive state will meet. Note that a purely Bayesian setting handles focusing and revision with the same tool: Bayes rule. Clearly focusing, as distinct from revision does not exist either in the propositional setting nor in the framework of qualitative possibility theory (min-based conditioning).

N.B.: From the point of view of evidence theory, belief functions are supposed to reflect a degree of certainty that uses a convention differing from probability functions (Bel(A) = 1 means certainty, Bel(A) = 0 means uncertainty) and that is not viewed as a lower probability (although from a mathematical point of view it is so). This point, i.e., that any set function can be used to represent certainty (up to further foundational issues) without referring to an unreachable probability function has often been overlooked by belief function opponents. Belief functions can be used as a model for evaluating certainty (this view is advocated by S m e t ~ ~ ~ ~ ~ ~ ) or as a model for capturing imprecision in probability (this view is that of Fagin and H a l ~ e r n ~ ~ , among others). Adopting the first point of view, Dempster rule of conditioning can be justified from a set of intuitive axioms ( S m e t ~ ~ ~ ) different from Cox's and that never uses any set of probabilities underlying the mathematical model of the belief functions. Similarly (28) and (29) have been justified by De Campos et al.44 as upper and lower bounds on belief functions.


B. Uncertain Inputs

Let us turn to revision in evidence theory under uncertain inputs. Here there are two generalizations of Bayes rule: Dempster rule of combination which is symmetric and the asymmetric extension of Jeffrey’s rule.

Dempster rule of combination can be defined as a normalized intersection of two independent random sets (F1,ml) and (F2,m2)

This rule has been justified by S m e t ~ ~ ~ from axiomatic arguments. When the random set (F2.m2) associated with m2 reduces to the ordinary subset A, i.e. m2(A) = 1 and V A’ f A, m2(A’) = 0, it can be easily checked that (35) extends Dempster revision (29) to the case of an uncertain observation represented by (s2,m2), but in a symmetrical manner. This is unfortunate from a revision point of view. Indeed Dempster rule of combination embodies the combination of evidence from parallel sources that play the same role, while the idea of revision is basically dissymetrical: new information does not play the same role as a priori information.

A non-symmetrical extension of (29) in case of uncertain observation, in the spirit of Jeffrey’s rule, is provided by the formula

I (92@2)) = ZAEQ m2(A) * PI 1 (B I A) (36)

where PIl(B I A) = P1l(A ’). The expression (36) can be interpreted in the

following way: the subset A is the accurate description of what is observed with probability m2(A) and (36) is nothing but the expected plausibility of B given the uncertain observation. Formula (36) is discussed by Ishihashi and Tanaka54 among different alternatives to Dempster’s rule. The two combination rules (35) and (36) coincide when the normalization factor of Dempster rule is 1. This is what happens in the AGM theory when revision coincides with expansion upon receiving an input that is compatible with the cognitive state. Note indeed that expansion is symmetric.

It is important to point out that conditioning is meaningful only when observation does not completely contradict a priori knowledge. This is the case for Bayes rule where P(B I A) is defined only if P(A) > 0, or for Dempster rule of conditioning where Pl(B I A) is defined only if Pl(A) > 0. This is still the case for Jeffrey’s rule where in (7) we should have P(Ai) > 0 as soon as a i > 0, as well as for its extended version (36) which is defined only if VA, m2(A) > 0,3C, ml(C) > 0 and A n C # 0 (i.e., Pl1(A) > 0). Note that Dempster’s rule of combination is less demanding since it is still defined when 3A, m2(A) > 0 and P11(A) = 0 (provided that it is not true for all A): the latter condition may seem a bit disturbing since the new information claims as somewhat probable something which was held as certainly false according to previous information.

We now examine the difference in behaviour between Dempster’s rule of combination and of the extended Jeffrey’s rule on a small example.

PI1 (A)

86 DUBOIS AND PRADE

Example: Let $1 = (A1,BI) with ml(A1) = a and ml(B1) = 1 - a, $2 = (A2,B2} with m2(A2) = p and m2(B2) = 1 - p. Let us assume that A1 n A2 = 0 ; A1 n B1 #

0 ; A1 n B 2 # 0 ; B1 n B 2 # 0 ; A 2 n B 1 # 0 ; A 2 n B 2 f 0 .

Dempster's rule yields m = m l @ m2 with

while the extended Jeffrey's rule gives

1 - a - 1 - a

m(B1 n A2 I A;?) =-- 1 if a# 1 ; m(A1 n B2 I B2) = a; m(B1 n B 2 I B2) = 1 -a

and finally

As it can be seen on this example, the basic probability assignments obtained by Dempster's rule and the extended Jeffrey's rule have exactly the same focal elements but their weights are different. Moreover the extended Jeffrey's rule gives a non- symmetrical result as expected. When ct = 1 this latter rule does not apply since then F 1 = (A1) and one of the focal elements of 5 2 , namely, A2 is such that PIl(A2) = 0 due to A1 n A2 = 0. In this case Dempster's rule gives m(A1 n B2) = 1 whatever the value of p (provided that p f l), i.e., a conclusion which is not pervaded with uncertainty in spite of the fact we may have a strong conflict between and 9 2 if p is close to 1 (then 1 - ap is close to 0). Also in the example, the extended Jeffrey's rule looks more robust partly because it does not apply when the behaviour of Dempster's rule is particularly questionable, and also because the normalization is performed in a global way in Dempster's rule, while in the other case it takes place at the level of each focal elements of the body of evidence corresponding to the uncertain observation u n which we conditionalize.

particular case of Dempster rule of combination applied to two belief functions Be1 1 and Bel2, where Bell, viewed as a cognitive state is a probability measure and Be12 is such that its sets of focal elements are unions of sets belonging to a partition (El. E2, ..., Ei) of R. More specifically. given any probability distributions p and p1 such that p results from some uncertain input acting on p1 via Jeffrey's rule, there exists a belief function Be12 such that p is the result of Dempster rule of combination applied to pi and Bel2. On such a basis Shafer claims that the opposition between the dissymmetry of Jeffrey's rule and the symmetry of Dempster combination rule is due to a <<superficial understanding of the relation between [them],,. This claim is debatable. Indeed given any two probability distributions p and p1 on R such that p(w) > 0, Vm, there always exists a partition (A1, ..., An) of R and probability weights a l , ..., CXn such that p is computed from Jeffrey's rule and input ( (Aiq ) ,

Shafer5 p" has pointed out that the probabilistic Jeffrey's rule can be viewed as a


i = l,n}. In other words, Jeffrey's rule can capture any probabilistic belief change. It turns out that in the same conditions this is also true for Dempster rule of combination. This does not mean that they address the same kind of problems.

Wagner56 has justified a special case of the extension (36) of Jeffrey's rule when the belief function Bell is just a probability. This justification is based on the interpretation of belief changes Bel2(A) as lower probability bounds. First notice that if Pll(B I A) = P1(B I A) is a standard conditional probability, then so is the resulting belief function, say P. Wagner requests that P 2 Bell (so that the result is consistent with the input evidence), and a conservation of the conditional probabilities P1(B I Ai) for all Ai E 5 2 on the product space U x Q underlying the belief function Be12 (following Dempster's16 view whereby a belief function is induced by a single probability measure and a multivalued mapping from U to 0). Wagner56 also criticizes the general form of (36) as not respecting any counterpart to the property of conservation of conditional probabilities that Jeffrey's rule obeys. However, rule (36) should not be understood as revising a cognitive state by a forced uncertain input, but as the average belief change resulting from an unreliable input, m2(A) being the probability that the actual input is A. The extension of Jeffrey's rule with forced uncertain inputs to belief functions has been devised by SmetsS7, with the restriction that the input belief function Be12 is defined on a partition of R, say A l , ..., An. Besides WagneS6 has extended his generalized Jeffrey's rule to the case where the uncertain input can be modelled by a lower probability that is superadditive of order 2.

VI. REVISION IN LOGICS OF UNCERTAINTY

In this section, we envisage the syntactical counterpart of some of the uncertainty models that appeared in the previous sections. The aim is to lay bare how revision tools developed at the semantic level can be expressed at the level of a knowledge base expressed in a logic that captures uncertainty.

The links between cognitive states viewed as sets of sentences, and possible worlds is easily described as follows. Suppose that sentences are expressed in a given propositional language, so that the sentences, up to logical equivalence, form a Boolean algebra [B. V p E [B, let [p] be the subset of worlds in which p is true. We shall adopt usual definitions such as [p A ql = [PI n [ql, [-PI = [PI. [TI = R, [I1 = 0 (the empty set), etc. A cognitive state can then be viewed as a subset A of possible worlds such that A = [ X I where X is a set of sentences in [B and [ X I is the set of possible worlds in which all sentences in X are simultaneously true. [ X3 is viewed by G2rdenfors2 as "the largest set of possible worlds that is compatible with the individual's convictions" (as modelled by X). If % = (PI, p2, ..., pn}, it is obvious that [ X I = [PI] n [p21 n... n [pnl.

Note that the Boolean algebra [B defines a partition of R whose elements correspond to atoms of IB. Let A be the set of atoms of the Boolean algebra IB, i.e., A = ( P E [ B , p # I , $ q z I , q # p , a n d q t p ] . T h e s e t ( [ p ] , p ~ A ] formsa partition of S Z , and it is the finest one induced by [B. Namely o, o' E [p] for p E A cannot be distinguished using the language that generated [B. The set A is usually called the set of interpretations of X.

-

88 DUBOIS AND PRADE

The uncertainty framework where the connection between semantic and syntactic forms of belief change can be most easily laid bare is possibility theory. This is the topic of Section V1.A. Unsurprizingly, the most straightforward approach to possibilistic belief change is in full accordance with the Alchoun6n, Gkdenfors and Makinson's theory. More recently the concept of belief revision has been extended to conditional logics that are closely related to Spohn's theory of ordinal conditional functions. hence again to possibility theory. The main asset of the latter framework is that the distinction between focusing and revising, that generalized probabilistic models point out, can be laid bare in a purely symbolic setting.

A. Revision in Possibilistic Logic

Possibilistic logic (Dubois, Lang and Prade25-5S) is a logic of incomplete evidence. Its syntax consists of sentences in the first order calculus to which are attached lower bounds on the degree of necessity or of possibility of these sentences (Dubois and Prade59*60). Here we consider only the fragment of possibilistic logic with propositional sentences to which lower bounds of degrees of necessity are attached. If p E 3 is a propositional sentence, (p a) is a short notation for N([p]) 2 a. When a goes from 0 to 1. p goes from uncertainty to certainty, (p 1) standing for the sure assertion of p. In order to express that p is unknown, one must assert (p 0) and (-p 0). However, only (p a) such that a > 0 is explicitly used in the language. Reasoning in possibilistic logic is done by means of an extension of the resolution principle to weighted clauses:

(c a); (c' P) I- (Res(c,c') min(a,P))

where c and c' are propositional clauses, and Res(c,c') is their resolvent. For instance, (-p v q a); (p v r P) k (q v r min(a,p)). This inference rule presupposes that when in a possibilistic formula (p a) p is not in a clausal form, it can be turned into a set ((ci a), i=l,n) of weighted clauses such that p is equivalent to c1 A c2 ... A cn. This is justified by the semantics of propositional logic, and by the fact that N(p) 2 a is equivalent to N(ci) 2 a, i=l,n (from now on, we write N(p) instead of N([p]) for short). A possibilistic belief base is a set % = ((pi ai), i = 1,111.

Note that any belief base (i.e. set of propositional sentences) equipped with a complete partial ordering can be mapped to a possibilistic belief base. As already pointed out the unit interval could be changed into any bounded, totally ordered set; the possibility/necessity duality is then expressed by reversing the ordering.

The set of possible worlds in which a possibilistic logic sentence (p a) is true is a fuzzy set [p a1 on a defined by

where [p] is the set of possible worlds where p is true: prP is the least specific possibility distribution K such that N([p]) = infoe [ p ~ 1 - x(w) 2 a. The fuzzy set of worlds which satisfy a possibilistic belief base = ((pi ai), i=l,m] is defined in terms of a possibility distribution x such that


V o E R, x(w) = mini=l,m m a ~ ( p [ ~ ~ ] ( w ) , 1 - ai) (37)

which extends [%I = [PI] n [p2] n... n [pn] from a set of sentences to a set of weighted sentences. x is the least specific possibility distribution such that V (p a) E

5%, N(p) 2 a, where N is computed with IC. Semantic entailment is defined in terms of specificity ordering (x 5 x'). Namely, 5% l= (p a) if and only if x I max(plp], 1 - a). This notion of semantic entailment is exactly the one of Zadeh61.

Consistent possibilistic belief bases are such that x(o) = 1 for some w E R. Consistency of 5% is equivalent to the consistency of the classical knowledge base 5%* obtained by removing the weights (see Dubois and Prades9). When maxWE ~ ( w ) = T c 1, 5% is said to be partially inconsistent, T being the degree of consistency of 5%. When 7 = 0, 5% is completely inconsistent. At the syntactic level the consistency of 5% can be checked by means of repeated uses of the resolution principle until the empty clause is attained, with some positive weight. This is denoted 5% t- (I a). By definition, 3% t- (p a) means that X u ((7p 1)) t (I a) (refutation method). The degree of inconsistency inc(5%) is then defined by max (a I 5% t (I a)). Possibilistic logic is sound and complete with respect to refutation based on resolution (Dubois et aL5*). Namely, it can be checked that

inc(K) = 1 - = 1 - maXWE Q x(w) 5% t- (p a) if and only if X F (p a).

When 3% is consistent, we can define an ordered belief set as cons(X) = ((p a), X + (p a)). When 5% is partially inconsistent, i.e., inc(%) > 0, non-trivial deductions can still be made from 5%, namely all (p a) such that % t- (p a), and for which a > inc(X). Indeed (p a) is then the consequence of a consistent subpart of 5%. Non-trivial inference of p from 5% is denoted 5% Epref p. When 5% is partially (but not totally) inconsistent, the associated consistent ordered belief set is con%edX) = I(P a), 5% tpref P,

In that case maxmER n(o) = 1 - inc(5%) < 1 and Vp, min(N(p),N(7p)) = inc(%). Let E be a possibility distribution on R defined by

t (P a)).

E(o) = x(w) if x(w) c 1 - inc(X) = 1 otherwise

then it is easy to verify that the necessity measures N and R based on x (defined by (37)) and E respectively are related by the following relation

N(p) > N(7p) * R(p) = N(p) > N ( - I ~ ) = inc( 5%) > R ( - I ~ ) = 0.

In fact, we have that

Note that the above framework can be equivalently expressed in terms of Spohnian functions. Instead of using weights in the unit interval, we can use integers from 0 to

90 DUBOIS AND PRADE

n where n is the number of layers in the ordered belief base. An obvious understanding of p being in layer i is that K(lp) 2 i, which can be made equivalent to N(p) 2 a, where i = -Log2(l- a). The most entrenched sentences are then in layer n. The minimally specific possibility distribution IC that is induced by an ordered belief base is then changed into a ranking of the possible words, namely, letting )(pi) the layer number of pi

K(O) = 0 if w satisfies all pi's in X - - maxi:e7pi )(pi) otherwise.

This is called "minimal ranking function" by and it corresponds to the minimally specific possibility distribution x.

Belief change in possibilistic logic can then be envisaged as the syntactic counterpart of change for possibility distributions. Let % be a consistent set of possibilistic formulae. Expansion of YC by p consists of forming % u ((p 1)), provided that inc(X u ((p 1))) = 0. Clearly, the possibility distribution x' that restricts the fuzzy set of worlds that satisfy X u ((p I ) ) is x' = min(x, p[ ) -

Let us consider the case when tpC is consistent, but X ' = % u ((p 1)) is not, and let a = inc(X u ((p 1))) > 0. The following identity is easy to prove (Dubois and P ~ a d e ~ ~ ) :

PI - X+[PI.

X u ((p 1)) F (q p) with p > a if and only ifN(q I p) > 0

where N(q I p) is the necessity measure induced from x(. I [p]), i.e., the possibility distribution expressing the content of X , revised with respect to the set of models of p. Indeed let x' be the possibility distribution on R induced by Y€', then

x'= min(sc, p ) 0 c maxw,=a x'(w) = 1 - a c 1

[PI

and the possibility distribution %' induced from the consistent part of 2%' made of sentences whose weight is higher than a, is defined as

E'(o) = R(W) if w E [p] and d(w) c 1 - a = l i f w ~ [p ]andx(w)= l -a = R'(w) = 0 otherwise.

Hcnce ii' = x(. I [p]), the result of revising x by [p]. The corresponding revision is rather drastic since all sentences (pi ai) with weights ai 501 are thrown away, and replaced by (p 1). Note that when n(p) > 0, N(q I p) > 0 is equivalent to N(7p v q) > N(lp v -q), i.e., in terms of epistemic entrenchment (G&denfors2), lp v q is more entrenched than -.p v 4. and corresponds to a characteristic condition for having q in the (ordered) belief set obtained by revising cons(%) with respect to p. in Gardenfors2. However this revision is easily implemented in the possibilistic belief base itself, without making the underlying ordered belief set explicit. This result goes against the often encountered claim that representing epistemic entrenchment


orderings explicitly would be intractable. The above revision method would differ from applying Spohn's conditioning to the minimal ranking function since the latter corresponds to Dempster rule of conditioning. See Goldsmidt and Pearl63 for a syntactic treatment of Spohn's conditioning.

A more parsimonious revision scheme for possibilistic belief bases 3% receiving an input p is to consider all subsets of 3% that fail to infer (-p a) for a > 0. If .% is such a subset, then the result of the revision could be % u ((p,l)). We may take advantage of the ordering in % to make the selection. Namely we may restrict ourselves to .% such that V (q a) t X, % u ((4 a), (p 1)) t- (I a), i.e., (q a) is involved in the contradiction. This proposal, made independently in Dubois et al.64 corresponds to selecting a preferred subbase in the sense of B r e ~ k a ~ ~ . This selection process leads to a unique solution if 2% is totally ordered. A further refinement can be made using a lexicographic ordering of the weights of the sentences not in 3, as proposed in Dubois et al.64. The revision processes, that also relate to Nebel% syntax-based revision schemes, are systematically studied in Benferhat et al.67.

This kind of revision process cannot be expressed at the semantic level, where all sentences in the knowledge base 3% u ((p 1)) have been combined into a possibility distribution on R, to be revised using the conditional possibility approach. Especially if (q p) E % and j3 c inc(3% u ((p 1))) then

min(x:, cL[p]) 4 max(P[q]. 1 - PI (38)

i.e.. all happens as if (q p) had never been in 3%. The sentence-ordering based revision rule, explained above breaks the minimal inconsistent subsets of 3% u ((p 1)) in a parsimonious way, enabling pieces of evidence like (q p) to be spared when they are not involved in the inconsistency of 3% u ((p 1)).

Example: Consider the elementary knowledge base 2% = [(-p a), (q p)) with pca.

Then x(w) = min(max(1- ~ [ ~ l ( w ) . 1 - a), m a ~ ( ~ [ ~ l ( w ) , 1 - p)> = l i f U k ~ p h q = 1 - a i f u k p =I-pifw~=:ph*.

Revising by input p at the semantic level leads to consider

Hence x(w I [p]) = 1 if w t= p and 0 otherwise. Hence 3%*p = ((p I)) . Acting at the syntactic level, the preferred sub-base of (@ l), (-p a), (q p)) that contains p is ((p l), (q p)) . Note that although equation (38) holds we no longer have n(w I [p]) S min(~[~](w), 1 - p), i.e., adding the low certainty formulas consistent with X * p leads to a non-trivial expansion of n(. I [pl). It points out the already mentioned weakness of the semantic views of revision, which is particularly true with numerical approaches: the representation of the cognitive state is lumped, i.e., the pieces of

92 DUBOIS AND PRADE

belief are no longer available and the semantic revision process cannot account for the structure of the cognitive state that is made explicit in the ordered belief base.

B. Towards Belief Change in Conditional Knowledge Bases

More recently, the concept of conditional knowledge base has emerged ( L e h ~ n a n n ~ ~ t ~ ~ , Kraus, Lehmann and Magidor70, Conditional knowledge bases have the potential to considerably broaden the notion of belief change at the syntactic level. A conditional knowledge base contains syntactic objects that correspond to the idea of generic rules with exceptions, of the form p -+ q (where the arrow is not material implication) where p and q are propositional sentences. One possible understanding of p + q is "if all is known is p, plausibly infer 9". A rule can be understood as the non-monotonic deduction of q from p (Lehmanr@), the statement that the conditional probability P(q I p) is infinitesimally close to 1

or that the conditional necessity N(q I p) is strictly positive (Benferhat et al?l). More simply it can be viewed as a conditional sentence that is true if both p and q are true, false if q is false when p is true and is considered inapplicable otherwise (De Fined). The diversity of these understandings goes along with the unity of the underlying deductive structures, whose properties laid bare by Lehmann and colleagues are in full agreement with S h ~ h a m ~ ~ ' s preferential entailment for non-monotonic reasoning.

The main asset of the conditional belief base framework is that it enables factual evidence to be explicitly told apart from generic knowledge. Namely, a cognitive state is then syntactically described by a pair (E,A) where E is a set of propositional sentences describing factual evidence (e.g. the results of medical tests for a patient) and A is set of conditional sentences describing generic rules (e.g. the medical knowledge). From @,A) a fact q can be plausibly deduced if from A it is possible to derive (using inference rules that govern conditional sentences in A) a generic rule of the form E + q (where E is assimilated to the conjunction of sentences in E). A may also contain strict rules without exception (e.g. Goldszmidt and Pea1-1~~). Factual revision consists in changing E into E' = E u (p) upon receiving a new piece of evidence p, supposedly consistent with E. It is clear that the new inferred facts q' from (E',A) may differ from or even contradict the ones inferred from (E,A). More specifically several results (Goldszmidt and Pearl73, Benferhat et als7I) indicate that inference from @,A) can be achieved through the encoding of A as an ordered belief base X ( A ) such that @,A) infers q if and only if %(A) u E +pref q in the sense of possibilistic logic. Clearly, adding p to E corresponds to revising X ( A ) u E in the sense of Section V1.A. In other words the AGM-like revision of propositional ordered belief bases captures factual revision, which at the syntactic level of the conditional language can be achieved by a simple addition to the factual evidence set. This operation exactly corresponds to the notion of focusing when the cognitive state is described by means of belief functions or a family of probabilities, as in Section V.

Contrastedly, the cognitive state @,A) can be modified by changing the generic knowledge, turning A into another set of conditional sentences A'. The most obvious way is to add a new conditional sentence, say p + q, to A. This problem starts receiving attention (e.g. Boutilier and Gold~zmidt~~). A particular case of such a generic belief change is when the input information claims that some sentence p should ever be true. This can be encoded as a strict rule in A. It comes down to


reasoning only from the models of p, all other situations being ruled out forever. This kind of drastic generic belief revision is very much akin to Dempster rule of conditioning in belief function theory or in the setting of order 2-superadditive lower probabilities, whereby the constraint P([p]) = 1 is enforced. Note that in numerical frameworks, the problem of revising a probability distribution by means of a conditional input of the form P(A I B) = CL has been recently considered into account in the probabilistic literature by van Fraassen.

VII. CONCLUSION

The main thrust of this chapter is to suggest, as G&denfors* already did in more restricted frameworks, that beyond the diversity of formal models of cognitive states, and their corresponding belief change rules, there is some agreement upon what revision and updating mean. The complementarity of semantic and syntactic representations of cognitive states has been emphasized: the use of numbers and orderings in the representation greatly facilitates the calculation of belief revision, be it by making the revision rule unique in a given setting. However semantic representations are poorer than syntactic ones because pieces of information contained in the cognitive state are lumped into a single uncertainty measure.

This paper has considered different numerical settings for the representation of epistemic states and has surveyed the existing rules for taking into account a new information either in a belief revision perspective in the spirit of Alchourrdn, Gadenfors and Makinson postulates, or in an updating perspective in the sense of Katsuno and Mendelzon' postulates. Interestingly enough the two kinds of epistemic change can be defined through conditioning and imaging respectively both in the probabilistic and in the possibilistic settings. However the possibilistic framework leads to a more complete agreement with the two sets of postulates (first stated for propositional logic) than the probabilistic setting. The paper also has tried to highlight that revising an uncertainty distribution on a set of possible worlds is not necessarily equivalent to revising the corresponding knowledge base made of uncertain logical formulas. More work is needed to relate probabilistic rules to the axiomatic approaches to belief change in the logical framework, despite the existing bridges for probability and possibility theories. Especially the justification of the different rules in evidence theory is in its infancy.

This paper has also pointed out that the distinction between focusing and revision discovered in generalized probabilistic setting corresponds to factual and generic revision respectively, and that this distinction looks crucial for discussing belief change in conditional knowledge bases. Lastly, we have not considered the problem of syntax-based methods for updating possibilistic belief bases. This question seems not to have been addressed, but in the setting of Bayes networks, for the local computation of Bayes rule. This is one more topic for further investigations.

This work has been partially supported by the European ESPRIT Basic Research Action no 6156 entitled "Defeasible Reasoning and Uncertainty Management Systems (DRUMS-II).

94 DUBOIS AND PRADE

APPENDIX A

Proof of n ( B I * A u A') = m a x ( a * n ( B I* A ) , a' * n ( B I * A ' ) ) with max(a,cx') = 1

where

conditioning is based on operation *, i.e., QB, VA, n(B A A) = n(B /* A) * n(A) a = n(A') *+ n(A) and a' = UiA) *+ I'I(A'L x *+ y = supjz E [ O , l ] , x * z I y )

Then, using the minimal specificity principle

since x *+ y is increasing with respect to y. For notational simplicity we omit the subscript * attached to the conditioning bar in the following.

For * = min, l i f x < y y if x > y. x * + y f x m + y =

n ( B I A u A ' ) 1 if n(A) 2 n(A) and n ( B I A) = 1 1 if n(A) 2 n(A) and n ( B I A) = 1 min(n(B I A), n(A)) otherwise min(n(B I A'), n(A)) otherwise '

Observing that if n ( B I A) f 1, n ( B I A) = n ( A n B) I n(A), it can be checked that the left side of the max can also be written:

min(n(A) m+ ll(A),Il(B I A)) 1 if n(A) I n ( A ) and n(B I A) = 1

if n(A) > IRA) n ( B I A) = n ( B n A) = min(n(B I A),ll(A)) if n ( B I A) # 1.

= (min(n(B I A),n(A))


Clearly max(n(A') m-+ n(A), n ( A ) m+ n(A')) = 1. This establishes the result.

For * = prod

x *+ y 4 x p' y =

= max((n(A) p+ TI(A)) . n(B I A), (n(A) p+ IXA')) * n ( B I A')).

If n(A) = n(A) = 0, n ( A u A') = 0, n ( B I A) = ll(B I A ) = 1 = n ( B I A u A'). This establishes the result.

APPENDIX B

Possibilistic Updating Satisfies the Updating Axioms

To prove the result we assume that V w E R, V A # 0, A c R, there exists a non-empty subset A(w) c A of possible worlds that are the closest to w in A. Moreover if w E A then A(o) = (01. Lastly, for any subset B c R, if w' E A(w) n B then 0' E [A n Bl(o), i.e. if the elements that are the closest ones to w in A lie also in B then they are also the closest elements to 61 in A n B. This is natural if we assume that R is equipped with a proximity structure, i.e. to each w E R is attached a complete partial ordering relation 4,, where w' I, w" means that w' is at least as close to w as w". We also assume that if 0' 61 then w' = w. Then A(,) = {o' E A, V w" E A, o' I, a"). If w E A, A(w) = ( w ) stems from w Sw o', V w' E R and the assumption that o is the closest element to itself. A(@) n B c [A n B](o) is due to the fact that lf a' E A(o) n B, if w' w" for all w" E A, then this is a fortiori true for all w" E A n B. The updating rule is defined by equation (18)

It is obvious in (18)

since for all w such that a(w) > 0, w E A so that A(w) = (w) ,

U3) If A # 0 and C(x) f 0 then C(T~'A) f 0, where C denotes the core of possibility distributions.

96 DUBOIS AND PRADE

Indeed if o is such that z(w) = 1, then K'A(o') = 1 for all w' E A(@) # 0.

It is trivially satisfied with a possible world approach.

This is because A(w) n B E [A n Bl(o).

. U6)

If ICOA B. Similarly, ICOB I p~ implies Vw, B(w) c_ A. Hence A(w) u B(w) cz A n B since A(w) E A, B(w) E B. Hence updating x: by A or B is equivalent to updating by A n B.

U7) If IC is maximally specific then min(x:'A,x:"B) I ~ : ' A ~ B .

""A I pg, IC'B I =$ IC'A = "'B.

p~g, it means that V W s.t. K ( O ) > 0 then A(o)

If 3wo, n(w0) = 1, x(w) = 0, V w f 00, then min(z'A,x'B) I ~ : O A ~ B , as it can be checked on (18). This property is exactly the same as in the classical logic setting.

It is obvious due to the commutation between the max-operations in U8 and in (18). This is a particular case of the invariance under weighted max-combination pointed out in the text, which is as easy to check.

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a survey of belief revision and updating rules in various uncertainty models

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