Laboratoire d'informatique Fondamentale de Lille

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1 Laboratoire d'informatique Fondamentale de Lille Publication IT { 314 Operators with Memory for Iterated Revision Sebastien Konieczny mai 1998 c L.I.F.L. { U.S.T.L. LABORATOIRE D'INFORMATIQUE FONDAMENTALE DE LILLE U.R.A. 369 C.N.R.S. UNIVERSIT E DES SCIENCES ET TECHNOLOGIES DE LILLE U.F.R. d'i.e.e.a. B^at. M3 { VILLENEUVE D'ASCQ CEDEX Tel. (+33) { Telecopie (+33) { direction@li.fr

2 Operators with Memory for Iterated Revision Sebastien Konieczny Laboratoire d'informatique Fondamentale de Lille - CNRS URA 369 Universite de Lille Villeneuve d'ascq - FRANCE konieczn@li.fr Abstract Following the work of Darwiche and Pearl (TARK V), and of Nayak and al. (TARK VI), we examine in this work the problem of the dynamic of belief revision and its link with iteration. We propose a family of belief revision operators called revision with memory operators and we give a logical characerization of these operators. They obey what we call the strong principle of primacy of update, i.e. when one revise by a new evidence, then all the possible worlds satisfying that new evidence become more reliable than those that not. This approach can be viewed as a generalization of Boutilier's natural revision (IJCAI93). We show in particular that there is a unique revision operator with memory that satisies postulate (C2) proposed by Darwiche and Pearl. 1 Introduction Modeling belief change is a central topic in articial intelligence, psychology and databases. One of the predominant approach is the AGM (after its authors Alchourron, Gardenfors and Makinson) framework [AGM85, Gar88, KM91]. The main requirements imposed by AGM postulates are rst the so called principle of minimal change saying that we have to keep as much of the old information as possible. Second is the principle of primacy of update that demands the new information to be true in the new knowledge base. But this denition of revision is a static one. It means that with the AGM denition of revision operators one can have a rational one step revision but the conditions for the iteration of the process are very weak. The problem is that AGM postulates state conditions only between the initial knowledge base, the new evidence and the resulting knowledge base. But the way to perform new revisions on the new knowledge base does not depend on the way the old knowledge base was revised. In particular, conditionals of the old knowledge base can be totally lost in the revision process. We argue that in order to have a rational behaviour concerning the iteration of the revision process one has to care about keeping conditionals of the old knowledge bases. It has already been pointed out by Darwiche and Pearl [DP94]. They show that AGM postulates are too weak to ensure a good behaviour concerning the iteration. We dene an epistemic state as a knowledge base (K i ) and a conditional set ( Ki ), the conditional set codes the agent condence in alternative worlds. An epistemic input (') is a new information. So an epistemic input induce an epistemic change: an epistemic change is a transition between epistemic states. With this description of the revision process we can immediately see the lack of AGM postulates concerning iteration: (K 1 ; K1 ) '?! (K 2 ;?) What we want to show here is that an epistemic state is not representable only by a knowledge base. We need an additional information, the conditional set K1 (represented either by a selection function, or by an epistemic entrenchment, or by a faithful assignment, etc.). An epistemic state is not only what the agent believes but also what he is willing to believe. AGM postulates 1

3 fail to model iterated revision because they induce no condition on conditional sets. Some non-prioritized revisions have been explored recently [Han98, Mak98, FH, Sch98]. However if one accept the principle of primacy of update, stating that the new information must be true in the revised knowledge base, one can drastically apply it, considering that one has full condence in the new information. We will adopt this point of view in this paper and so applying what we call the principle of strong primacy of update. That is, when one learns a new evidence in which one has full condence, all possible worlds satisfying this new evidence become more credible than the worlds that don't. This can be viewed as close in spirit to the approach of Spohn [Spo87]. For us, this is close to the problem of revision history. The point is that one cannot model an agent belief only by its present belief but has to care about how the agent has got its belief. So an epistemic state can't be only represented by a knowledge base but by the history of the knowledge bases accepted by the agent: for example let us consider the two following series K1 = a (a! b) and K2 = (a! b) a, we have then the same knowledge base as result that is a ^ b. But if we learn later than b is not true, do we have to obtain K1 :b = K2 :b? We don't think so. In K1 the information a is less reliable than a! b so when we learn :b we are willing to accept the falsity of the older information and so obtain K1 :b = :a ^ :b. Whereas in K2, it is a! b that is less reliable, then learning :b will lead to K2 :b = a ^ :b. So we claim that in order to have a rational behaviour concerning the iteration we have to model epistemic state by a knowledge base history rather than by a sole knowledge base. Maintain a knowledge base history is a mean for dening the conditional sets we talk above. We have seen in the second example that dierences between knowledge base and epistemic state reside in conditionals and so motivate the name of conditionals set. K1 :b = :a ^ :b means that in K 1 there was the conditional \If :b then :a", whereas in K 2 the conditional was \If :b then a". The paper is organized as follows: in Section 2 we give some previous approaches to iterated revision. In Section 3 we dene revision operators with memory and give a logical characterization of these operators. We provide two examples of revision operators with memory in Section 4. Finally, in Section 5, we give some conclusions and discuss an ontology for these operators. 2 Background We will work in a nite propositional language L. A pre-order is a reexive and transitive relation and < is its strict counterpart, i.e I < J if and only if I J and J 6 I. As usual, ' is dened as I ' J i I J and J I. Let ' be a formula, Mod(') denotes the set of models of '. And let M be a set of interpretations, form(m) denotes a formula whose set of models is M. When M = fi; Jg we will use the notation f orm(i; J) for reading convenience. 2.1 AGM Postulates We give here the AGM postulates for belief revision [AGM85, Gar88, KM91]. More exactly we give a formulation of these postulates in terms of epistemic states that is a more general case than the one with knowledge bases. We will motivate this change later. To each epistemic state S is associated a knowledge base Bel(S) which is a sentence and which represents the objective (logical) part of S. The models of S are the models of its associated knowledge base M od(s) = M od(bel(s)). Let S be an epistemic state and be a sentence denoting the new information. S denotes the epistemic state result of revision of S by. The operator is a revision operator if it satises the following postulates: (R*1) Bel(S ) implies 2

4 (R*2) If Bel(S) ^ is satisable then Bel(S ) = Bel(S) ^ (R*3) If is satisable then Bel(S ) is satisable (R*4) If S 1 = S 2 and 1 2 then Bel(S 1 1 ) Bel(S 2 2 ) (R*5) Bel(S ) ^ implies Bel(S ( ^ )) (R*6) If Bel(S ) ^ is satisable then Bel(S ( ^ )) implies Bel(S ) ^ This is nearly the Katsuno Mendelzon (KM) postulates [KM91], the only dierences is that we work with epistemic states instead of knowledge bases and that postulates (R*4) is weaker than its KM counterpart, in fact it is the only postulate in which we use explicitly the epistemic nature of the knowledge. See [DP97, FH96] for full motivation of this denition. We have the same representation theorem as usual, let's rst dene the faithful assignment on epistemic states Definition 1 A function that maps each epistemic state S to a total pre-order S on interpretations is called a faithful assignment if and only if: 1. If I 2 Mod(S) and J 2 Mod(S), then I ' S J. 2. If I 2 Mod(S) and J =2 Mod(S), then I < S J. 3. If S 1 S 2, then S1 = S2. We have a reformulation of Katsuno and Mendelzon representation theorem [KM91] in terms of epistemic states: Theorem 1 A revision operator satises postulates (R1-R6) if and only if there exists a faithful assignment that maps each epistemic state S to a total pre-order S such that: Mod(S ) = min(mod(); S ) Notice that this theorem give information only on the objective part of the resulting epistemic state. As we have seen in the introduction a strong limitation of revision postulates is that they impose very weak constraints on the iteration of the revision process. Darwiche and Pearl has proposed in [DP94, DP97] postulates for iterated revision. The aim of these postulates is to keep as much as possible of conditional beliefs of the old knowledge base. (C1) If j=, then (' ) '. (C2) If j= :, then (' ) '. (C3) If ' j=, then (' ) j=. (C4) If ' 6j= :, then (' ) 6j= :. The explanation of the postulates is: (C1) states that if two pieces of information arrive, the second being stronger than the rst, the second alone would give the same knowledge base. (C2) says that when two contradictory pieces of information arrive, the second alone would give the same knowledge base. (C3) states that an information should be retained after accommodating a more recent information that implies the rst given the current knowledge base. (C4) says that no evidence can contribute to its own denies. Darwiche and Pearl provide a representation theorem for these postulates. Theorem 2 Suppose that a revision operator satises (R*1-R*6). The operator satises (C1-C4) if and only if the operator and its corresponding faithful assignment satisfy: 3

5 (CR1) If I 2 Mod() and J 2 Mod(), then I ' J i I ' J. (CR2) If I 2 Mod(:) and J 2 Mod(:), then I ' J i I ' J. (CR3) If I 2 Mod() and J 2 Mod(:), then I < ' J only if I < ' J. (CR4) If I 2 Mod() and J 2 Mod(:), then I ' J only if I ' J. In the framework of knowledge bases, Freund and Lehmann have shown in [FL94] that (C2) is inconsistent with (R*1-R*6). Furthermore Lehmann has shown [Leh95] that, postulates (C1) and (R*1-R*6) implies (C3) and (C4). In [DP97] Darwiche and Pearl have rephrased their postulates in terms of epistemic state instead of knowledge base and therefore have removed these contradictions. Before the work of Darwiche and Pearl, Boutilier proposed in [Bou93, Bou95] a natural revision operator which aims to have good iteration properties. This operator can be seen as accomplishing a minimal change in the pre-order associated with the epistemic state. When one learns a new piece of information, he looks at the minimum models of this information according to the preorder of the old epistemic state. And the pre-order of the new epistemic state is exactly the same as the old one except that the minimum models of the new information are the new minimum of the pre-order. This principle has been called absolute minimization by Darwiche and Pearl and can be characterized by [Bou93]: (CB) If ' j= :, then (' ) ' Darwiche and Pearl show that applying this principle leads to questionable results [DP97] but natural revision even seems to be a simple and computable operator. The basic memory operator 4.1 can be viewed as an extension of Boutilier's natural revision. Lehmann proposed in [Leh95] postulates for revision operators that ensure good iteration properties. Its postulates are based on revision sequences. Let S and S 0 be two revision sequences, i.e. S = ' 1 : : : ' n, and let ' and be two sentences: (I1) S is a consistent theory. (I2) S ' ` '. (I3) If S ' `, then S ` '!. (I4) If S `, then S S 0 S S 0. (I5) If ' `, then S ' S 0 S ' S 0. (I6) If S ' 0 :, then S ' S 0 S ' ' ^ S 0. (I7) S ^ ' ` S :' '. These postulates capture the spirit of AGM postulates and give additional conditions on the iteration. They are self-explained, see [Leh95] for their full justication. 3 Revision operators with memory As in [Leh95] we consider here sequences of revisions. The dierence is that a sequence of revisions is not a knowledge base but an epistemic state. Intuitively, we pair each sequence of revisions with a pre-order over interpretations that represents the epistemic state. The objective part of the epistemic state is the sentence (up to logical equivalence) which models are the minimum interpretations according to this pre-order. Darwiche and Pearl do not dene precisely what they call epistemic state. For us, an epistemic state is a formula denoting the objective part of this epistemic state, and an additional information 4

6 that represents the preferences of the agent, i.e. what he is willing to accept. This additional information can take dierent forms, such as a pre-order over possible worlds, a set of conditional sentences, an epistemic entrenchment [NFPS96], a revision history, etc. We will consider in the rest of the paper that an epistemic state is a revision history (or equivalently a pre-order over possible worlds). We will denote revision history by list of formulae, hence [' 1 ' 2 : : : ' n ] = ' 1 ' 2 : : : ' n. We can make the assumption than the initial knowledge of an agent is > and so [' 1 ' 2 : : :' n ] = > ' 1 ' 2 : : : ' n and [] = >. Considering agent with initial knowledge ', amounts to consider an agent with no initial knowledge and with a rst revision by ', so starting from [']. Definition 2 An Epistemic State S is a list of knowledge bases (a revision history) so S = [' 1 ' 2 : : : ' n ]. We will see in theorem 4, in terms of pre-order on interpretations, the conditional set corresponding to this epistemic state. Two epistemic states S = [' 1 : : : ' n ] and S 0 = [' 0 : : : 1 '0 m] are equal, noted S = S 0, if and only if 8i ' i ' 0 i. Two epistemic states are equivalents, noted S S 0, if and only if their objective part are equivalent formulae Bel(S) Bel(S 0 ). As we consider epistemic states rather than knowledge bases, we slightly deviate from the usual AGM framework. We assign to each knowledge base ', a pre-order ', which represents the models of the knowledge base (at the lowest level) and the credibility a priori of \alternative worlds". It can be viewed as the epistemic state an agent would have if it had no prior knowledge and learn this evidence. We impose very few constraints on this assignment, assuming that it is faithful, that is: Definition 3 A function that maps each knowledge base ' to a total pre-order ' on interpretations is called a faithful assignment if and only if: 1. If I 2 Mod(') and J 2 Mod('), then I ' ' J. 2. If I 2 Mod(') and J =2 Mod('), then I < ' J. 3. If ' 1 ' 2, then '1 = '2. This assignment on knowledge bases induces an assignment on the epistemic states in the following way: Definition 4 Consider a faithful assignment that maps each knowledge base ' to a pre-order ' and an epistemic state S = [' 1 : : : ' n ]. We dene S as: If n = 1, then S = '1. Else I S J if I < 'n J or I 'n J and I '1:::'n?1 J. The pre-order S is called the conditional set of S and the objective part of S, denoted Bel(S), is the sentence (up to logical equivalence) such that Mod(Bel(S)) = min( S ). By denition Mod(S) = Mod(Bel(S)). So, the pre-order of an epistemic state is the (reversed) lexicographical order on the sequences of knowledge bases pre-orders. It is easy to see that: Theorem 3 If the function that assigns to each knowledge base ' a pre-order ' is a faithful assignment, then the function that assigns to each epistemic state S a pre-order S as dened in denition 4 is a faithful assignment. 5

7 Corollary 1 When S is dened as in denition 4, the revision operator dened by Mod(S ) = min(mod(); S ) satises (R*1-R*6). This corollary gives only the knowledge base result of the revision, that is the objective part of the epistemic state. But the revision process has to specify the full new epistemic state and we need some additional logical properties, so more exactly we dene revision memory operators in the following way: Definition 5 An operator mapping an epistemic state S = [' 1 ' 2 : : : ' n ] and a sentence ' to an epistemic state [S '] = [' 1 ' 2 : : : ' n '] is a revision operator with memory if and only if the following conditions hold: (H1) [S '] implies ' (H2) If [S] ^ ' is satisable then [S '] [S] ^ ' (H3) If ' is satisable then [S '] is satisable (H4) If S 1 = S 2 and ' 1 ' 2 then [S 1 ' 1 ] = [S 2 ' 2 ] (H5) [S '] ^ implies [S ' ^ ] (H6) If [S '] ^ is satisable then [S ' ^ ] implies [S '] ^ (H7) If ' ^ 0?, then [S ' ] [S ' ^ ] (H8) If ' ^ `? and [' ], then [S ' ] [S ] (H9) If [' ] `, then [S ' ] ` (H1-H6) are exactly the postulates (R*1-R*6). (H7) is inspired by a postulate proposed by Nayak and al. in [NFPS96] called Conjunction, that in our framework can be written as: (Conjunction) If ' ^ 0? then [S ' ] = [S ' ^ ] It states that when you revise successively by two consistent pieces of information, it amounts to revise by their conjunction. Clearly (H7) is weaker than Conjunction, since it require only the equivalence of the two epistemic states, not the equality. Thus, the two resulting epistemic states have the same objective part but can have dierent conditional sets. Postulates (H8) and (H9) express the strong condence in the new information. We dene now what is a conservative assignment: Definition 6 A function that maps each epistemic state S to a total pre-order S on interpretations is called a conservative assignment if and only if: 1. If I 2 Mod(S) and J 2 Mod(S), then I ' S J. 2. If I 2 Mod(S) and J =2 Mod(S), then I < S J. 3. If S 1 = S 2, then S1 = S2. 4. If I < ' J, then I < [S'] J. 5. If I ' ' J, then I [S'] J i I S J. The pre-orders dened in denition 4 satises these properties. Conversely, when one has a conservative assignment one recover straightforwardly the lexicographical order dened in denition 4 when one starts from the faithful assignment ' 7! [']. We have the following representation theorem: 6

8 Theorem 4 A revision operator [] satises postulates (H1-H9) if and only if there exists a conservative assignment that maps each epistemic state S to a total pre-order S such that: Mod([S ]) = min( [S] ) Sketch of Proof: The if part follows from the fact that operators dened by denition 4 satisfy postulates (H1-H9). For the only if part we dene (as usual) I S J if and only if I 2 Mod(S) or I 2 Mod([S form(i; J)]). The proof that postulates (H1-H6) implies conditions 1-3 is the usual one. (H7) gives 4 when I 2 Mod(') and J =2 Mod('). The other case, when I =2 Mod(') and J =2 Mod('), is given by (H9). And (H7) gives 5 when I 2 Mod(') and J 2 Mod('), whereas the other case, when I =2 Mod(') and J =2 Mod('), is given by (H8). This theorem states that revision operators build from a lexicographical order as done in denition 4 are exactly characterized by postulates (H1-H9). So the conditions on the conditional sets are captured in these logical properties. Now, we will look at some logical properties of this family of revision operators. Theorem 5 A revision operator with memory satises postulates (C1), (C3) and (C4). But it doesn't, in general, satisfy (C2). The proofs for (C1),(C3),(C4) are obvious, and a counterexample for (C2) is given in section 4.2. Postulate (C2) has been shown to be inconsistent with AGM Postulates by Freund and Lehmann in [FL94]. So Darwiche and Pearl slightly modify AGM Postulates in [DP97] and then remove the contradiction. But we claim that (C2) is not generally desirable and we show that satisfying this postulate leads to counterintuitive results. Consider the following example with two atomic propositions, adder ok and multiplier ok, denoting respectively the fact that the adder and the multiplier are working. And consider a circuit containing this adder and this multiplier. We have initially no information about this circuit ' = > and we learn that the adder and the multiplier are working = adder ok ^ multiplier ok. So we know that the adder and the multiplier are working. Then someone tells us that the adder is bad = :adder ok. There is then no reason to \forget" that the multiplier is working, what is imposed by (C2) : j= : so by (C2) we have [' ] [' ] So, in some cases, postulate (C2) induces exactly the same kind of bad behaviour it tries to prevent. Theorem 6 A revision operator with memory satises postulates (I1),(I2),(I3), but it doesn't satisfy (I4), (I5), (I6) and (I7). Counterexamples of (I4),(I5),(I6) and (I7) will be given in section 4.1 and Some revision operators with memory 4.1 Basic memory operator Let's dene the assignment that maps each knowledge base to a pre-order in the following way: Definition 7 I < b ' J if and only if I 2 Mod(') and J =2 Mod('). I ' b ' J if and only if I 2 Mod(') and J 2 Mod(') or I =2 Mod(') and J =2 Mod('). So we have a two-level order, with the models of ' at the lower level and the other worlds at the higher level. It is easy to show that the assignment that maps each knowledge base ' to a pre-order b ' is a faithful assignment. And we dene b S with denition 4. 7

9 Definition 8 Let S be an epistemic state and ' be a new information, the basic revision operator is dened as Mod([S ']) = min( b [S'] ). Even with this basic order on knowledge bases, one could build a very precise epistemic state, due to revision history. We illustrate the behaviour of this operator on some simple examples: Consider a language L with only two propositional letters a and b. Let's see some examples of epistemic states: We will denote interpretations simply by the truth assignment, i.e 10 denotes the interpretation mapping a to True and b to False. Two interpretations are equivalent, with respect to the pre-order, if they appear at the same level. An interpretation I is better than an other J (I J) if it appears at a lower level. [ab] = [a^b] = [a^ba] = [a^ba :b] = [a^b:b] = With this examples, one can show that revision operators with memory don't satisfy Lehmann's postulates (I4), (I5) and (I6). Let's take [a ^ b a :b] 6 [a ^ b :b] as a counterexample for (I4) and (I5). And let's take [a b :(a ^ b)] 6 [a a ^ b :(a ^ b)] as a counterexample for (I6). The basic operator satises (I7) since, as we show below, it satises (C2) (see section 4.2 for a counterexample for (I7)). It is easy to see that this basic operator and all the operators we dene in that paper yield to a linear order over interpretations after a sucient number of iterations. This can be viewed as the learning process: when one has a long revision history he keep preferences over possible worlds due to his past experience. Note that Boutilier's natural revision operator leads to the same kind of linear order. Theorem 7 The basic revision operator satises postulate (C2). The order paired with each epistemic state by the basic order is a conservative assignment that satises the following condition: 6. If I =2 Mod(') and J =2 Mod('), then I [S'] J i I S J. We call such an assignment a strong conservative assignment. It is easy to see that the sole revision operator with memory that satises the conditions of the strong conservative assignment is the basic operator. Theorem 8 A revision operator [] satises postulates (H1-H9) and (C2) if and only if there exists a strong conservative assignment that maps each epistemic state S to a total pre-order S such that: Mod([S ]) = min( [S] ) Hence the sole revision operator with memory that satises (H1-H9) and (C2) is the basic revision operator. 4.2 Dalal memory operator We use in this section the Dalal's distance [Dal88], dened in the following way: Definition 9 Let I and J be two interpretations, the Dalal's distance dist(i; J) is dened as the number of propositional letters the two interpretations dier. Let ' be a knowledge base, the Dalal's distance between this interpretation and the knowledge base is: d(i; ') = min(dist(i; J)) Jj=' Let's dene the assignment that maps each knowledge base to a pre-order in the following way: 8

10 Definition 10 I d ' J if and only if d(i; ') d(j; ') So we have a pre-order, with the models of ' at the lowest level and the other worlds in the higher levels. Definition Let S be an epistemic state and ' be a new information, the basic revision operator is dened as Mod([S ']) = min( d [S'] ). Theorem 9 Dalal memory operator doesn't satisfy (C2) This is easily show with the example of section 3. Let ' = >, = adder ok ^ multiplier ok and = :adder ok ' = = = ['] = ['] = So j= : but [' ] :adder ok ^ multiplier ok whereas [' ] :adder ok. Theorem 10 Dalal memory operator doesn't satisfy (I7) For example :> ^ :(a ^ b) 0 [> a ^ b :(a ^ b)] Conclusions We have shown in this paper the connection between the problem of the iteration of the revision process and the problem of revision history. Thus, revision history is a mean to warrant rational iteration properties. A representation theorem was provided, showing that one can impose in logical terms conditions on the conditional set of epistemic states. We have compared our approach to iterated revision to previous related work. We have shown, in particular, that the only revision with memory operator that satises postulates (C2) proposed by Darwiche and Pearl was the basic memory operator. We have adopted here a drastic strategy, applying the strong principle of primacy of update. If one adopt the principle of primacy of update as a reasonable requirement, that is that the new information is more reliable that the old epistemic state. Then we claim that he is surely committed to nd the possible worlds satisfying this new information more reliable that those that not. Leading to what we call the strong principle of primacy of update. The ontology for this family of operators is the one given by this strong principle of primacy of update. Consider an agent that learns successively pieces of evidence about some world. As all these evidence are observation about this world it has full condence in the new evidence. More, when a new piece of evidence arrives, it increases the condence in possible worlds that best t this new evidence. For example if the new information is a conjunct of three atomic propositions, then the possible worlds satisfying two of these three disjunct become more reliable than possible worlds satisfying only one of them (it is the case with the Dalal memory operator). Then, when a new piece of information arrive the agent reconsider the relative reliability of all possible worlds according to its past experience and to the new piece of information. The principle of primacy of update is often criticized, since it can't be accepted in all circumstances. Sometimes we have more condence in our current beliefs than in the new information. Some non-prioritized revisions, denying this principle, have been explored recently 9

11 [Han98, Mak98, FH, Sch98]. If we accept this point of view, our framework can be used in an amazing way by reversing the revision arguments, i.e. by revising the new information by the old epistemic state ( ' = [' ]). Thus we obtain a family of \revision" operator which give a strong primacy to the current knowledge. Then, the new information is not accepted as true in the new epistemic state, but its condence is increased. For example if two possible worlds are equally plausible for an epistemic state and if the new information is satised by only one of these possible worlds, then this world will be more reliable than the other in the new epistemic state. So one can imagine applications when we use alternatively these two denitions of revision operators, according to the relative reliability of the current beliefs and the new information. References [AGM85] [Bou93] [Bou95] C. E. Alchourron, P. Gardenfors, and D. Makinson. On the logic of theory change: Partial meet contraction and revision functions. Journal of Symbolic Logic, 50:510{530, C. Boutilier. Revision sequences and nested conditionals. In Proceedings of Thirteenth International Joint Conference on Articial Intelligence (IJCAI-93), C. Boutilier. Iterated revision and minimal revision of conditional beliefs. Journal of Philosophical Logic, (25):262{305, [Dal88] M. Dalal. Updates in propositional databases. Technical report, Rutgers University, [DP94] A. Darwiche and J. Pearl. On the logic of iterated belief revision. In Morgan Kaufmann, editor, Theoretical Aspects of Reasoning about Knowledge: Proceedings of the 1994 Conference, pages 5{23, [DP97] A. Darwiche and J. Pearl. On the logic of iterated belief revision. Articial Intelligence, (89):1{29, [FH] [FH96] [FL94] E. L. Ferme and S. O. Hansson. Selective revision. Manuscript. N. Friedman and J.Y. Halpern. Belief revision: a critique. In Proceedings of the Fifth International Conference on Principles of Knowledge Representation and Reasoning, M. Freund and D. Lehmann. Belief revision and rational inference. Technical Report TR-94-16, Institute of Computer Science, The Hebrew University of Jerusalem, [Gar88] P. Gardenfors. Knowledge in ux. MIT Press, [Han98] S. O. Hansson. Semi-revision. Journal of applied non-classical logic, [KM91] H. Katsuno and A. O. Mendelzon. Propositional knowledge base revision and minimal change. Articial Intelligence, 52:263{294, [Leh95] D. Lehmann. Belief revision, revised. In Proceedings of the Fourteenth International Joint Conference on Articial Intelligence (IJCAI-95), pages 1534{1540, [Mak98] D. Makinson. Screened revision. Theoria, Special issue on non-prioritized belief revision. [NFPS96] A. C. Nayak, N. Y. Foo, M. Pagnucco, and A. Sattar. Changing conditional beliefs unconditionally. In Proceedings of the sixth conference of Theoretical Aspects of Rationality and Knowledge, pages 9{135. Morgan Kaufmann, [Sch98] K. Schlechta. Non-prioritized belief revision based on distances between models. Theoria, Special issue on non-prioritized belief revision. [Spo87] W. Spohn. Ordinal conditional functions: a dynamic theory of epistemic states. In W. L. Harper and B. Skyrms, editors, Causation in decision, belief change, and statistics, number 3, pages 105{