Update with Disjunctive Information: From ... - Semantic Scholar

Update with Disjunctive Information: From Syntactical and Semantical Perspectives Yan Zhang Department of Computing University of Western Sydney, Nepean Kingswood, NSW 2747 Australia E-mail: [email protected] Phone: (country code 61) 2 47 360 746 Fax: (country code 61) 2 47 360 770

Norman Y. Foo Department of Arti cial Intelligence School of Computer Science and Engineering University of New South Wales NSW 2052, Australia E-mail: [email protected]



This paper is an extended and revised version of \Updating knowledge bases with disjunctive infor-

mation" in Proceedings of AAAI-96.

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Abstract The possible models approach (PMA) is a classical minimal change semantics for knowledge base update, which provides an exclusive interpretation for disjunctive information in update. It has been recognized that the exclusive interpretation for disjunction may be problematic under some circumstances. In this paper, we investigate inclusive interpretations for disjunctions in update from both syntactical and semantical viewpoints. In particular, we propose two approaches for disjunctive update { the minimal change with exceptions (MCE) and the minimal change with maximal disjunctive inclusions (MCD). Both approaches provide inclusive interpre-

tations for disjunctions in update, while the rst one is syntax-based and the second is model-theoretic. We then characterize the MCE and MCD in terms of alternative minimal change criteria and relate them to traditional Katsuno and Mendelzon's update postulates. Key Words: knowledge base update, knowledge representation, disjunctive information, minimal change

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1 Introduction Minimal change plays a fundamental role in formalizing reasoning about change in Arti cial Intelligence. In one way or another, the principle of minimal change has been widely employed in formulations of reasoning about action, belief revision and knowledge base (belief) update (Dalal 1988; Gardenfors 1988; Katsuno and Mendelzon 1991a; Katsuno and Mendelzon 1991b; Winslett 1988). This paper concerns the issue of knowledge base update under the minimal change principle. The general question addressed by knowledge base update is as follows: given a initial knowledge base (or logical database), e.g. a set of logical formulas, what changes may be caused by an occurrence of new knowledge and how to specify the new knowledge base when the old one has changed? Let us rst consider the following scenario. Peter is a second year computer science student. When the semester begins, he enrolls in three computer science courses CS201, CS202 and CS301. After two weeks, he decides to drop course CS301. A department enrollment rule requires that a second year computer science student must enroll in at least two computer science courses with level 200 or 300. If the department maintains a knowledge base of Peter's course enrollment information, it is quite straightforward that the information about CS301 enrollment should be deleted from the knowledge base and other information should be kept unchanged as the resulting knowledge base still satis es the department enrollment rule. The situation described above presents an intuitive idea of the minimal change principle used in knowledge base update. In general, the minimal change principle says that

during a state transition, the change between states should be as small as possible. In Peter's course-enrollment example presented above, after dropping course CS301, according to the minimal change principle, this change should not aect other information about Peter's enrollment. That is, Peter will still keep his enrollment for course CS201 3

and CS202. One of the classical minimal change approaches for update is Winslett's possible models approach (PMA) (Winslett 1988), which has an elegant model semantics with

a simple minimal change criterion between interpretations. However, it has been also recognized that it might be problematic for the PMA to deal with disjunctive information under some circumstances since it provides exclusive interpretations for disjunctions in update (we will discuss this issue in section 2). In other words, it is also important to have inclusive interpretations for disjunctions in update under other situations. Various suggestions were proposed to overcome problems of the PMA, e.g. (Brewka 1993; Cordier and Siegel 1992; Kartha and Lifschitz 1994; Zhang and Foo 1993). These methods and theories, more or less, present some desired solutions to update with disjunctive information (or simply called disjunctive update) in some speci c domains. Nevertheless, it is not clear yet how to characterize these variations of the PMA in terms of the minimal change principle in general. The main aim of this paper is to address this problem from both syntactical and semantical perspectives. To provide inclusive interpretations for disjunctions in update, the central point we will argue in this paper is twofold: (1) from a syntactical point of view, the PMA-like minimal change criterion should be weaken with some syntax-based exceptions; (2) from a semantical point of view, on the other hand, an alternative minimal change criterion based on a partial ordering on models should be used to characterize disjunctive update. The rest of this paper is organized as follows. The next section reviews the PMA { the classical minimal change approach for knowledge base update, and shows how the PMA fails to deal with the update with disjunctive information in some situations. Section 3 proposes a new method called minimal change with exceptions, which provides inclusive interpretations for disjunctions in update but is syntax-based. Section 4 pro4

poses an alternative method called minimal change with maximal disjunctive inclusions that is model-based, and illustrates the dierence between the MCE and MCD. Section 5 then characterizes the MCE and MCD in terms of alternative minimal change criteria and relates them to traditional Katsuno and Mendelzon's update postulates. Section 6 discusses some related issues, and nally section 7 concludes the paper.

2 PMA: A Review 2.1 The Language Consider a nite propositional language L. We represent a knowledge base by a propositional formula . A propositional formula  is complete if  is consistent and for any propositional formula ,  j=  or  j= :. Models( ) denotes the set of all models of , i.e. all interpretations of L in which

is true1 . We also consider state constraints

about the world. Let C be a satis able propositional formula that represents all state constraints about the world2 . Thus, for any knowledge base , we require j= C . Let

I be an interpretation of L. We say that I is a state of the world if C is satis ed in I . Since I is a nite set of ground literals, I can be also viewed as a conjunction of all these literals. Therefore, without confusion, we may also denote I j= C if C is satis ed in I . A knowledge base

can be treated as a description of the world, where Models( ) is

the set of all possible states of the world with respect to . Let be the current knowledge base and  a propositional formula which is regarded as new knowledge (information) about the world. Then, informally, the general question of updating 1

with  is how to specify the new knowledge base after combining the

We take an interpretation to be a set of propositional literals (i.e. including propositional letters

and/or negated propositional letters). 2 Usually, we use a set of formulas to represent state constraints. In this case, C can be viewed as a conjunction of all such formulas.

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new knowledge (information)  with the current knowledge base . In Peter's courseenrollment example, if courses CS201, CS202, CS301 and CS303 are available for second year students, then the constraint stating that a second year student must enroll in at least two CS courses with level 200 or 300 may be expressed by a formula C : (Enrolled(CS 201) _ Enrolled(CS 202)_ Enrolled(CS 301))^ (Enrolled(CS 201) _ Enrolled(CS 202)_ Enrolled(CS 303))^ (Enrolled(CS 201) _ Enrolled(CS 301)_ Enrolled(CS 303))^ (Enrolled(CS 202) _ Enrolled(CS 301)_ Enrolled(CS 303)). Therefore, a possible initial knowledge base of Peter's course-enrollment could be ex-

 Enrolled(CS 201) ^ Enrolled(CS 301) ^ C . Now suppose Peter hopes to drop CS301. Let  denote :Enrolled(CS 301). Then the question is: what is the new knowledge base of Peter's course-enrollment after updating with ? We use notation   to denote the resulting knowledge base after updating with . With dierent update approaches, we may obtain dierent answers to the above pressed by the formula

question. We rst review Winslett's PMA below.

2.2 Winslett's PMA The PMA (possible models approach) { a classical minimal change approach for update proposed by Winslett3 (Winslett 1988; Winslett 1990). In the PMA, the knowledge base update is achieved by updating every possible state of the world with respect to with

, and such state update is constructed based on minimal changes on the models. Formally, let I1 and I2 be two interpretations of L. We say that I1 and I2 diers on a propositional letter l if l appears in exactly one of I1 and I2 . Diff (I1; I2) denotes the 3

Note that PMA was originally based on a rst order language. Here we restrict it to the propositional

case.

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set of all dierent propositional letters between I1 and I2 . Let I be an interpretation and

I a set of interpretations. We de ne the set of all minimally dierent interpretations of I with respect to I as follows: Min(I , I ) = fI 0 j I 0 2 I , and there does not exist other I 00 2 I such that Diff (I; I 00)  Diff (I; I 0)g. Then we can present the formal de nition of the state update in the PMA as follows.

De nition 1 Let C be the state constraint, S a state of the world, i.e. S j= C , and  a propositional formula. Then the set of all possible states of the world resulting from updating S with  by the PMA, denoted as Res(S; ), is de ned as follows:

Res(S; ) = Min(S; Models(C ^ )):

(1)

Based on the de nition of state update, we can then de ne the PMA update operator

pma for knowledge bases.

De nition 2 Let be a knowledge base and  a propositional formula. pma  denotes S the update of with  by the PMA4, where Models( pma ) = S 2Models( ) Res(S; ). The above de nition simply says that the update of a knowledge base made by updating every model of trivial case occurs whenever

with  is

with  as de ned in De nition 1. Note that a

entails , i.e. no change of

will be made. It has been

shown that in many circumstances, the PMA is powerful and eective for representing knowledge update and reasoning about action (Kastuno and Mendelzon 1991a; Winslett 1988). However, as will be shown next, the PMA may also be sometimes problematic for update with disjunctive information. 4

Here we only consider the well-de ned update, that is,  is consistent with the state constraint C .

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Red Black

White

Figure 1: The extended dropping-box domain.

Example 1 Suppose a round table is painted with three equal parts of red color, white color and black color respectively5 . Intuitively, a box on the table implies that it may be entirely within one of these three regions, or touching any two of these three regions, or touching all of these three regions. This situation can be described by Figure 1.

A constraint to formalize this domain is speci ed as:

Ontable(Box)  Inred(Box) _ Inwhite(Box) _ Inblack(Box):

(2)

Now suppose the current knowledge base is

 :Ontable(Box) ^:Inred(Box)^ :Inwhite(Box) ^:Inblack(Box) ^ (2), which corresponds to a unique state:

S = f:Ontable(Box); :Inred(Box), :Inwhite(Box); :Inblack(Box)g: Consider updating state S with   Ontable(Box) (i.e. the box is dropped on the table). Using the PMA, it is not dicult to see that from De nitions 1 and 2, the resulting knowledge base is: 5

This is an extension of the dropping-box example originally raised by Reiter (Kartha and Lifschitz

1994).

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pma   Ontable(Box)^ (Inred(Box) ^:Inwhite(Box) ^:Inblack(Box)_ :Inred(Box) ^ Inwhite(Box) ^:Inblack(Box)_ :Inred(Box) ^ :Inwhite(Box)^ Inblack(Box)) ^ (2), which means that the box will only be in one of these three regions. Obviously, this solution is not intuitively reasonable. It could be argued that the PMA may work well with the disjunctive information under other situations. For instance, suppose that a robot is asked to maintain a goal of

Ontable(Box1 )_Ontable(Box2 ). Then the PMA does the right thing by just maintaining one of Box1 and Box2 on the table because we believe that the robot usually prefers to do minimal work to achieve its goal6. Nevertheless, the essential issue here is that the PMA provides an exclusive or interpretation for a disjunction in update under its minimal change semantics. What we would like to know is that: Are there any syntactical and semantical methods which provide an inclusive or interpretation for a disjunction under some alternative minimal change semantics?

3 MCE: A Syntactical Approach In this section we propose an approach for update based on an idea of Minimal Change with Exceptions, which we abbreviate as the MCE. In fact, our approach is based on

the PMA but with some modi cations. The idea is described as follows. Consider the state update7 . Generally, during the update, the truth value of any literal in the state changes minimally by default. But if the truth value of a literal is logically inde nite 6 7

This example was due to Fangzhen Lin from personal communications in 1994. Similar to the PMA, in our approach, updating a knowledge base is achieved by updating every

possible model of the knowledge base.

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with respect to the update, then this literal is treated as an exception to the minimal change principle. In this case, the change of this literal's truth value will not obey the rule of minimal change. Informally, we say that the truth value of a literal is logically inde nite with respect to an update, if this literal occurs in a disjunction which is entailed by the constraint and the update eect, and not satis ed in the initial knowledge base (or the state of initial knowledge base). Consider the extended dropping-box example presented in section 2 where the constraint is (2) and the update eect is Ontable(Box). As

Inred(Box), Inwhite(Box) and Inblack(Box) are not true in the initial knowledge base but the disjunction Inred(Box) _ Inwhite(Box) _ Inblack(Box) is entailed by (2) and

Ontable(Box), we know that Inred(Box) _ Inwhite(Box) _ Inblack(Box) should be true in the resulting knowledge base but we can not determine the truth values of

Inred(Box), Inwhite(Box) and Inblack(Box) exactly. In this case, we say literals Inred(Box), Inwhite(Box) and Inblack(Box) are logically inde nite with respect to the update. According to our idea described above, Inred(Box), Inwhite(Box) and

Inblack(Box) should be regarded as exceptions to the minimal change principle. Thus, Inred(Box), Inwhite(Box) and Inblack(Box) are not forced to change minimally during the update, from which we get the desired solution including the case that all of

Inred(Box), Inwhite(Box) and Inblack(Box) may be true after updating the knowledge base with Ontable(Box). Formally, let EXC be a set of propositional letters that we represent to be exceptional to the minimal change, I1 and I2 two interpretations. Diff (I1; I2)EXC denotes the set of all dierent propositional letters, which are not in EXC , between I1 and I2. That is, l 2 Diff (I1 ; I2)EXC i l 2 Diff (I1; I2) and l 62 EXC . For example, let

I1 = fa; b; :c; :dg, I2 = f:a; b; c; :dg and EXC = fa; bg. Then Diff (I1; I2)EXC = fcg. 10

Let I be an interpretation and I a set of interpretations. We de ne the set of all minimal dierent interpretations of I with respect to I with the exception EXC as follows:

Min(I , I )EXC = fI 0 j I 0 2 I , and there does not exist other I 00 2 I such that Diff (I; I 00)EXC  Diff (I; I 0)EXC g. Let C be a propositional formula used to represent the state constraint and  a

W

propositional formula. We say a disjunction ni=1 [:]li (1 < n) satisfying C ^  j= Wn [:]l , where l is a propositional letter (1  i  n) and notation [:] means that i i=1 i the negation sign : may or may not occur, is a prime implicate of C ^ 8 , if for any

M  f1;


; ng, C ^  6j= Wj2M [:]lj . We denote the set of all prime implicates of C ^  W as D(). If d  n [:]l in D(), then we denote jdj = fl ;


; l g. i=1

1

i

n

In the extended dropping-box example presented in section 2, as we have (2) ^

Ontable(Box) j= Inred(Box) _ Inwhite(Box) _ Inblack(Box), (2) ^ Ontable(Box) 6j= Inred(Box), (2)^Ontable(Box) 6j= Inwhite(Box), and (2)^Ontable(Box) 6j= Inblack(Box), we then get D(Ontable(Box)) = fdg = fInred(Box); Inwhite(Box), Inblack(Box)g, and jdj = fInred(Box); Inwhite(Box), Inblack(Box)g. Now we give the de nition of state update in the MCE as follows.

De nition 3 Let C be the state constraint, S a state of the world, i.e. S j= C ,  a propositional formula, and D() the set of all prime implicates of C ^ . We de ne the exceptional letters with respect to S and  as follows:

EXC (S; ) =

[ d2D();S 6j=d

jdj:

(3)

Then the set of possible states resulting from updating S with  by the MCE, denoted as 8

Precisely, from the de nition of prime implicate (Reiter and de Kleer 1987), C ^  should be rewritten

as a set of clauses. Since any propositional formula can always be translated into a clause form, we will not explicitly state that C ^  is a set of clauses.

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Res(S; )EXC(S;), is de ned as Res(S; )EXC(S;) = Min(S; Models(C ^ ))EXC(S;) :

(4)

Let us examine De nition 3 in detail. Firstly, (3) de nes a set of propositional letters that should be viewed as exceptions to the minimal change during the state update. If

d 2 D() is already satis ed in S , then any letters which or whose negations occur in d will not be speci ed in EXC (S; ), otherwise the letters should be in EXC (S; ). For instance, suppose S = f:a; :b; c; :dg and D() = fa _ b; b _ cg, then EXC (S; ) = fa; bg while c is not in EXC (S; ) as S j= b _ c. Secondly, (4) de nes the set of possible resulting states after updating S with . Note that any literals in S whose corresponding letters are in EXC (S; ) will not obey the minimal change principle during the update. In the above example, if C  (d  a _ b) ^ (d  b _ c) and   d9 , then we get

Res(S; )EXC(S;) = fS1 ; S2; S3g, where S1 = fa; :b; c; dg, S2 = f:a; b; c; dg, and S3 = fa; b; c; dg. Based on De nition 3, we de ne the knowledge base update in the MCE as follows.

De nition 4 Let be a knowledge base,  a propositional formula. mce  denotes the update of with  by the MCE, where Models( mce) = [S 2Models( ) Res(S; )EXC (S;). Compared with De nition 1 and 2, it is easy to see that the MCE is de ned based on the PMA but with exception EXC (S; ). Clearly, if EXC (S; ) = ; for any

S 2 Models( ), then the MCE reduces to the PMA. Let us consider the extended dropping-box example once again. As the knowledge base 9

This implies that D() = fa _ b; b _ cg.

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corresponds to a unique

state S = f:Ontable(Box), :Inred(Box), :Inwhite(Box), :Inblack(Box)g, we have

EXC (S; Ontable(Box)) = fInred(Box), Inwhite(Box), Inblack(Box)g, then from Definition 3 and 4, we get the desired result:

mce   Ontable(Box)^ (Inred(Box) _ Inwhite(Box) _ Inblack(Box)) ^ (2). From the above discussion, we can see that the MCE overcomes the problem with the PMA of updating knowledge bases with disjunctive information. However, to specify the exceptional letters EXC (S; ), we need to compute every prime implicate d of constraints

C and update eect , and then verify if S j= d for each S 2 Models( ). Obviously, such computation is a purely syntactical procedure10 . Hence, we also say that the MCE is syntax-based.

4 MCD: A Model-theoretic Approach In this section, we propose an alternative method which also provides inclusive interpretations for disjunctions in update and is model-based.

4.1 The Approach Our approach is based on an idea so-called Minimal Change with maximal Disjunctive inclusions, that we call the MCD for short. To illustrate this idea clearly, we consider

the extended dropping-box example once again.

Example 2 Example 1 revisited. We still suppose the current knowledge base is  :Ontable(Box) ^:Inred(Box)^ :Inwhite(Box)^ :Inblack(Box) ^ (2), 10

We will discuss this issue more closely in section 7.

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which corresponds to a unique state:

S = f:Ontable(Box); :Inred(Box), :Inwhite(Box); :Inblack(Box)g: Consider updating state S with   Ontable(Box) (i.e. the box is dropped on the table). The question is: how can we get the desired possible states? Our idea is described as follows. Firstly, using the PMA we get the set of possible resulting states:

Res(S; ) = fS1; S2; S3g, where S1 = fOntable(Box); Inred(Box), :Inwhite(Box); :Inblack(Box)g, S2 = fOntable(Box); :Inred(Box), Inwhite(Box); :Inblack(Box)g, S3 = fOntable(Box); :Inred(Box), :Inwhite(Box); Inblack(Box)g. Obviously, S1; S2 and S3 are the desired possible resulting states. But, we know that the following states are also our desired resulting states:

S4 = fOntable(Box); Inred(Box), Inwhite(Box); :Inblack(Box)g, S5 = fOntable(Box); Inred(Box), :Inwhite(Box); Inblack(Box)g, S6 = fOntable(Box); :Inred(Box), Inwhite(Box); Inblack(Box)g, S7 = fOntable(Box); Inred(Box), Inwhite(Box); Inblack(Box)g. In fact, states S4 ; S5; S6 and S7 can be generated from Res(S; ). Let SP be any nonempty subset of Res(S; ). Then, there always exists a model S 0 in Models(Ontable(Box)^ (2))11, such that S 0 satis es: (i) for each Si 2 SP , Diff (S; Si)  Diff (S; S 0), and (ii) there does not exist another S 00 in Models(Ontable(Box) ^ (2)) satisfying condition (i) but Diff (S; S 00)  Diff (S; S 0). 11

Clearly, Models(Ontable(Box) ^ (2)) is the set of all possible states of the world in which

Ontable(Box) is true.

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Now we claim that S 0 is also a desired resulting state. For instance, let SP be fS1 g. Then from conditions (i) and (ii), the corresponding S 0 is S1 itself. On the other hand, if SP = fS1; S2g, we get

S 0 = fOntable(Box); Inred(Box), Inwhite(Box); :Inblack(Box)g = S4 . Similarly, we can get S5 , S6 , and S7 from fS1; S3g, fS2; S3g and fS1; S2; S3g respectively. Therefore, every desired resulting state can be generated from the corresponding subset of Res(S; ) by using the above procedure. Let us look at the above more closely. As the disjunction Inred(Box)_Inwhite(Box)_

Inblack(Box) is a logical consequence of the update eect Ontable(Box) and the constraint (2), dierent states in Res(S; ) also represent dierent interpretations for this disjunction. But, because of the minimal change principle, Res(S; ) may only include partial interpretations for the disjunction. For instance, using the PMA, we get the pos-

sible resulting states S1 ; S2 and S3 which only represent three possible interpretations for this disjunction. To describe the update with disjunctive eect properly, as we have observed previously, we need to represent every possible interpretation of the disjunction but without losing the minimal change criterion for other information. It is not hard to see that conditions (i) and (ii) above achieve this purpose. In particular, given a subset SP of Res(S; ), condition (i) states that for every state Si in SP , S 0 is the state satisfying this disjunction (i.e. S 0 is an interpretation of the disjunction) but dierent from Si , while condition (ii) restricts this S 0 to minimal change on other literals with respect to S . For every subset SP of Res(S; ), we can get the corresponding S 0. So, we take all such S 0s to be the possible resulting states that include maximal possible interpretations of the disjunction without losing the minimal change

criterion on other information. Hence, we call this approach the minimal change with maximal disjunctive inclusions.

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4.2 Formal Descriptions Based on the above discussion, we now develop our method formally. Similar to the previous presentation, we rst de ne the state update in the MCD.

De nition 5 Let C be the state constraint, S a state of the world, i.e., S j= C , and  a propositional formula. Then the set of all possible states of the world resulting from updating S with  by the MCD, denoted as Res(S; )mcd , is de ned as follows:

Res(S; )mcd =

[ SP 22Res(S;)

Dis(S; SP )12;

(5)

where

Dis(S; SP ) = fS 0 j S 0 2 Models(C ^ ) such that (i) for each Si in SP , Diff (S; Si)  Diff (S; S 0), and (ii) there does not exist other S 00 in Models(C ^ ) satisfying (i) but Diff (S; S 00)  Diff (S; S 0)g.

Note that in De nition 5, we consider the power set of Res(S; ), so that any element

SP of 2Res(S;) is a subset of Res(S; ). Dis(S; SP ) represents the set of states that include the interpretations of the disjunctive information represented by all states of SP without losing the minimal change criterion on other information. Clearly, if there is only one element S 0 in SP , then Dis(S; SP ) = SP = fS 0g. After specifying the state update, we then de ne the MCD update operator mcd for knowledge bases as follows.

De nition 6 Let be a knowledge base and  a propositional formula. mcd  denotes S the update of with  by the MCD, where Models( mcd) = S 2Models( ) Res(S; )mcd . 12

Recall that Res(S; ) is de ned by (1) in section 2.

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Example 3 Example 2 continued. The initial knowledge base is  :Ontable(Box)^:Inred(Box)^ :Inwhite(Box) ^ :Inblack(Box)^(2). Now, we update with   Ontable(Box). From De nition 5, we have

S

S 2Models( ) Res(S; )

mcd =

fS1; S2; S3; S4; S5; S6; S7g,

where S1 ;


; S7 have been given in section 4.1 previously. Finally, from De nition 6, we get the desired resulting knowledge base

mcd   Ontable(Box) ^ (Inred(Box)_ Inwhite(Box) _ Inblack(Box)) ^ (2).

4.3 Dierence from MCE From previous descriptions, we have seen that both the MCE and MCD present the same and desired solution for the dropping box example. However, in general these two methods have dierent ways to provide inclusive interpretations for disjunctions in update. Let us consider the following example.

Example 4 We reconsider Peter's enrollment scenario described in section 1 but with a bit more complex situation. The set of constraints C includes the following formulas:

Enrolled(CS 303)  Prestudy(CS 401) _ Prestudy(CS 403);

(6)

Enrolled(CS 305)  Prestudy(CS 403) _ Prestudy(CS 405);

(7)

where (6) and (7) state that if someone enrolls in CS303 (or CS305 respectively), then he/she is preparing to study course CS401 or CS403 (or CS403 or CS405 respectively). 17

We assume that currently Peter enrolls in CS301 but does not enroll in CS303 and CS305. Further, Peter does not prepare to study courses CS401, CS403 and CS405. So, the knowledge base of Peter's course enrollment is

 Enrolled(CS 301)^ :Enrolled(CS 303) ^ :Enrolled(CS 305)^ :Prestudy(CS 401) ^ :Prestudy(CS 403)^ :Prestudy(CS 405) ^ C . Now suppose that after two weeks the semester begins, Peter decides to enroll in courses CS303 and CS305. Hence, we need to update with   Enrolled(CS 303)^Enrolled(CS 305). Let us rst use the MCE to derive the resulting knowledge base. Obviously, has a unique model13 :

S = fEnrolled(CS301); :Enrolled(CS 303); :Enrolled(CS 305), :Prestudy(CS 401),

:Prestudy(CS 403); :Prestudy(CS 405)g. Clearly, we have

D() = fPrestudy(CS 401)_Prestudy(CS 403), Prestudy(CS 403)_Prestudy(CS 405)g. Since S 6j= Prestudy (CS 401)_Prestudy (CS 403) and S 6j= Prestudy (CS 403)_Prestudy (CS 405), the set of exceptional letters with respect to S and  is

EXC (S; ) = fPrestudy(CS 401); Prestudy(CS 403); Prestudy(CS 405)g. Therefore, we have

Res(S; )EXC(S;) = fS1 ; S2; S3; S4; S5g, where S1 = fEnrolled(CS301); Enrolled(CS303); Enrolled(CS305); Prestudy(CS 401),

:Prestudy(CS 403); Prestudy(CS 405)g, S2 = fEnrolled(CS301); Enrolled(CS303); Enrolled(CS305); :Prestudy(CS 401), 13

we assume that there is no more propositional letters in our language

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Prestudy(CS 403); :Prestudy(CS 405)g, S3 = fEnrolled(CS301); Enrolled(CS303); Enrolled(CS305); Prestudy(CS 401), Prestudy(CS 403); Prestudy(CS 405)g, S4 = fEnrolled(CS301); Enrolled(CS303); Enrolled(CS305); Prestudy(CS 401), Prestudy(CS 403); :Prestudy(CS 405)g, S5 = fEnrolled(CS301); Enrolled(CS303); Enrolled(CS305); :Prestudy(CS 401), Prestudy(CS 403); Prestudy(CS 405)g. Finally, the resulting knowledge is

mce   Enrolled(CS 301)^ Enrolled(CS 303) ^ Enrolled(CS 305)^ (Prestudy (CS 401) ^ :Prestudy (CS 403)^ Prestudy (CS 405)_ :Prestudy(CS 401) ^ Prestudy(CS 403) ^ :Prestudy(CS 405)_ Prestudy(CS 401) ^ Prestudy(CS 403) ^ Prestudy(CS 405)_ Prestudy(CS 401) ^ Prestudy(CS 403) ^ :Prestudy(CS 405)_ :Prestudy(CS 401) ^ Prestudy(CS 403) ^ Prestudy(CS 405)) ^C , From which we conclude that by enrolling in CS303 and CS305, Peter may also prepare to study one of the following collections of subjects: CS401 and CS405; CS403; CS401, CS403 and CS405; CS401 and CS403; or CS403 and CS405. Now, let us use the MCD to deal with this example. We rst get Res(S; ) by the PMA as follows:

Res(S; ) = fS1 ; S2g. From Res(S; ) we generate every (non-empty) subset of Res(S; ) as follows:

SP 1 = fS1g, SP 2 = fS2g, SP 3 = fS1; S2g. 19

Then from De nition 5, we have

Dis(S; SP 1) = fS1g, Dis(S; SP 2) = fS2g, Dis(S; SP 3) = fS3g, and then Res(S; )mcd = fS1 ; S2; S3g. Hence, the nal result is:

mcd   Enrolled(CS301)^ Enrolled(CS 303) ^ Enrolled(CS 305)^ (Prestudy (CS 401) ^ :Prestudy (CS 403)^ Prestudy (CS 405)_ :Prestudy(CS 401) ^ Prestudy(CS 403) ^ :Prestudy(CS 405)_ Prestudy(CS 401) ^ Prestudy(CS 403) ^ Prestudy(CS 405)) ^ C . Compared with mce , we can see that the solution obtained by the MCD is less conservative (or more speci c) than that obtained from MCE, which implies that by enrolling in CS303 and CS305, Peter may prepare to study one of the following three possible collections of subjects: CS401 and CS405; CS403; or CS401, CS403 and CS405.

Example 4 describes a situation where the MCE produces a more conservative result than the MCD does for the same update problem. However, this is not a general case. The next example further presents a situation where the MCD produces a more conservative solution than the MCE does14.

Example 5 Let  :A ^ :B ^ :C ^ :D be a knowledge base. Consider the update of with   ((A 6= B ) ^ :C ^ :D) _ (A ^ B ^ C ). Ignoring the detail, it is not dicult to conclude that mce  (A = 6 B) ^ :C ^ :D, and mcd   ^ :D. In this case, it is quite clear that mce j= mcd. 14

This example is due to Liberatore's discussion on the MCE and MCD in (Liberatore 1997).

20

From Example 4 and Example 5, it is observed that the constraint C and the form of update eect  aect the handling of disjunctive information in knowledge base update with the MCE and MCD respectively. However, the following theorem reveals a relationship between the MCE and MCD.

Proposition 1 Let be a knowledge base without including state constraint C and   l1 _


_ lk be a disjunction where each li is a propositional literal (i = 1;


; k). Then mce   mcd .

5 Disjunctive Update and Minimal Change In this section, we discuss relationships between minimal change semantics and the two disjunctive update approaches proposed previously. In particular, we explore how our MCE and MCD can be characterized by alternative minimal change criteria and related to Katsuno and Mendelzon's update theory (Kastuno and Mendelzon 1991a; Kastuno and Mendelzon 1991b).

5.1 Katsuno and Mendelzon's Update Postulates The motivation of Katsuno and Mendelzon's proposal for update is an observation on the dierence between revision and update. In particular, Katsuno and Mendelzon argued that the original revision postulates proposed by Gardenfors el al. (Gardenfors 1988; Kastuno and Mendelzon 1991a) are not quite suitable for update, and ignoring such dierence may lead to unreasonable solutions (Kastuno and Mendelzon 1991a). Alternatively, they proposed the following postulates for any update operator .

  implies . (U2) If implies  then    . (U3) If both and  are satis able then   is also satis able.

(U1)

21

(U4) If 1  2 and 1  2 then 1  1  2  2 . (U5) (  ) ^  implies  ( ^ ). (U6) If  1 implies 2 and  2 implies 1 then  1   2 . (U7) If is complete then (  1 ) ^ (  2 ) implies  (1 _ 2 ). (U8) ( 1 _ 2 )    ( 1  ) _ ( 2  ). In fact, the Katsuno-Mendelzon's update postulates characterize the update semantics for a class of update operators that are based on the principle of minimal change. For instance, the PMA update operator pma satis es all postulates (U1) { (U8) (Katsuno and Mendelzon 1991a). As it has been shown in (Katsuno and Mendelzon 1991a), an update operator that satis es postulates (U1) - (U8) is actually characterized by a minimal change criterion in terms of a partial ordering on models. Let I be the set of all interpretations of the propositional language L. A preordering

 over I is re exive and transitive relation on I .  is called a partial ordering if it is an antisymmetric preordering. Let M  I . We denote min(M; ) to be the set of all interpretations in M that are minimal with respect to . For a given interpretation S , if we de ne a partial ordering S as S1 S S2 i Diff (S; S1)  Diff (S; S2) for any interpretations S1 and S2 , then the PMA can be actually characterized as follows: Models( pma ) =

[ S 2Models( )

min(Models(C ^ ); S ):

(8)

5.2 Minimal Change and the MCE As it has been mentioned earlier, the MCE is based on a principle so-called minimal change with exceptions. We can actually reconstruct the MCE in terms of a partial

ordering that captures the intuition of this minimal change principle. Let EXC be a set of propositional letters, S an interpretation. A ordering EXC S on interpretations with respect to S and EXC is de ned as follows: for any given 22

interpretations S1 and S2 , S1 EXC S2 i Diff (S; S1)EXC  Diff (S; S2)EXC . It is S obvious that EXC is a partial ordering. Then the MCE can be easily rede ned as S follows:

Models( mce ) =

[ S 2Models( )

min(Models(C ^ ); SEXC(S;) ):

(9)

As the MCE is syntax-based, where Katsuno-Mendelzon's update postulates were semantically characterized by some partial ordering on models without any syntactical features, it would not be surprising that the MCE may not be consistent with some of Katsuno-Mendelzon's update postulates. In fact, we have the following results showing the relationship between the MCE and Katsuno-Mendelzon's update postulates.

Theorem 1 The MCE update operator mce satis es Katsuno-Mendelzon's update postulates (U1) - (U4) and (U8).

Theorem 2 The MCE update operator mce satis es (U6') and (U7') as follows. mce 1 implies 2, mce 2 implies 1, and for each S 2 Models( ) EXC (S; 1) = EXC (S; 2), then mce 1  mce 2 .

(U6') If

(U7') If

is complete where Models( ) = fS g and EXC (S; 1) = EXC (S; 2) =

EXC (S; 1 _ 2 ), then ( mce 1) ^ ( mce 2 ) implies mce (1 _ 2). Clearly, (U6') and (U7') represent weak forms of Katsuno-Mendelzon's update postulates (U6) and (U7) respectively. Note that condition EXC (S; 1) = EXC (S; 2) in (U6') is not trivial. For instance, if C ^ 1 j= (A _ B ) ^ (B _ C ) and C ^ 2 j= A _ B _ C , and neither (A_B )^(B _C ) nor A_B _C is satis ed in S , then EXC (S; 1) = EXC (S; 2) =

fA; B; C g. Similarly, condition EXC (S; 1) = EXC (S; 2) = EXC (S; 1 _ 2) in (U7') is not trivial either. Consider the case where 1  (A _ B ) ^ (B _ C ), 2  (A _ B _ C ), 23

and there is no constraints in the domain. Then if f:A; :B; :C g  S , we have

EXC (S; 1) = EXC (S; 2) = EXC (S; 1 _ 2 ) = fA; B; C g. The above result shows that under some restrictions, the syntactical feature of the MCE will not aect its satisfaction on postulates (U6) and (U7). However, as will be illustrated next, the postulate (U5) represents a crucial property of exclusive interpretation of disjunction in update, which neither the MCE nor MCD satis es.

5.3 Minimal Change and the MCD Due to the complexity of the MCD construction, it seems more dicult to characterize the MCD in terms of an alternative minimal change semantics. We will still follow the similar idea of Katsuno and Mendelzon to investigate this issue: rst try to propose a partial ordering on models to characterize the MCD operator and then relate the MCD operator to Katsuno-Mendelzon's update postulates.

De nition 7 Let I be the set of all interpretations of L, I1  I , and S an interpretation of L. We de ne an ordering over I with respect to I1 and S as follows. For any two S1; S2 2 I , S1 I ;S S2 i S1 = S2; or (i) For all Si 2 I1 , Diff (S; Si)  Diff (S; S1), and (ii) If for all Si 2 I1 , Diff (S; Si)  Diff (S; S2), then Diff (S; S1)  Diff (S; S2). 1

Lemma 1 I ;S is a partial ordering. 1

Intuitively, I ;S represents a measure on Diff (S; S 0) for any state S 0 in I with 1

respect to I1. A trivial case is S1 = S2. Generally, if S1 I ;S S2, it means that (i) 1

Diff (S; S1) is an upper bound for the set fDiff (S; Si) j Si 2 I1g with respect to set inclusion , while Diff (S; S2) is not an upper bound for this set; or (ii) both Diff (S; S1) and Diff (S; S2) are upper bounds for set fDiff (S; Si) j Si 2 I1g, but Diff (S; S1) is a 24

smaller one than Diff (S; S2) is (i.e. Diff (S; S1)  Diff (S; S2)). We can extend the ordering I ;S to the general case. 1

De nition 8 Let  be a class of sets of interpretations. We de ne S1 ;S S2 i there exists some Ik 2  such that S1 Ik ;S S2 or there exists some S 0 such that S1 ;S S 0 ;S S2. Clearly, the ordering ;S is the union of Ik ;S for every Ik in . For instance, if  = fI1; I2g, then S1 ;S S2 if one of the following cases holds: (i) S1 I ;S S2; (ii) 1

S1 I ;S S2; or (iii) there is S 0 such that S1 I ;S S 0 (or S1 I ;S S 0 ) and S 0 I ;S S2 2

1

2

2

(or S 0 I ;S S2 ). 1

Lemma 2 ;S is a partial ordering. Lemma 3 Let 1 and 2 be two classes of sets of interpretations and 1  2. Then S1  ;S S2 implies S1  ;S S2. 1

2

The following lemma is particularly useful for us to prove Theorem 3.

Lemma 4 Let S be a state, C and  the state constraint and a propositional formula respectively, and SP a subset of Res(S; ). Then the following results hold: 1. If S 0 2 min(Models(C ^ ); SP ;S ), then for any S 00 2 Models(C ^ ), S 0 SP ;S S 00; 2.

min(Models(C ^ ); 2Res S; ;S ) = (

)

[ SP 22Res(S;)

min(Models(C ^ ); SP ;S ): (10)

Some comments on this lemma are necessary. The Result 1 presents a uniqueness of the minimal element of Models(C ^ ) with respect to SP ;S . That means, given a subset SP of Res(S; ), there exists only one state in Models(C ^ ) such that the 25

dierence between this state and S is the least upper bound of all dierences between all states in SP and S . Recall Example 2 and Example 3 we discussed in section 4, this intuition is consistent with the de nition of Dis(S; SP ). On the other hand, the Result 2 says that a minimal element of Models(C ^ ) with respect to 2Res S; ;S actually corresponds to the minimal element of Models(C ^ ) with respect to SP ;S for some subset SP of Res(S; ). This is also consistent with (

)

our intuition that 2Res S; ;S is de ned in terms of the union of SP ;S for each SP in (

)

2Res(S;) . Then we can prove the following theorem.

Theorem 3 Let be a propositional knowledge base, where C is the state constraint, and  a propositional formula. Then

Models( mcd ) = SS2Models( ) min(Models(C ^ ); 2Res S; ;S ). (

)

The important point revealed by Theorem 3 is that our MCD can be characterized by an alternative minimal change criterion based on a partial ordering on models. Of course, this partial ordering is dierent from the one de ned by Katsuno and Mendelzon in (Katsuno and Mendelzon 1991a), which also leads to the following results on the relation between our MCD update operator and Katsuno-Mendelzon's update postulates.

Theorem 4 The MCD update operator mcd satis es Katsuno-Mendelzon's postulates (U1) { (U4) and (U6) { (U8).

Proposition 2 The MCD update operator mcd violates Katsuno-Mendelzon's postulate (U5).

To show the above proposition, it is sucient to give a counterexample15 that does not satisfy (U5). Suppose  :A ^ :B . Consider the update of with A _ B . Then it 15

Proposition 1 is due to Andreas Herzig's investigation on an earlier draft of our paper (personal

communications in June 1997).

26

is not dicult to see that ( mcd (A _ B )) ^ A  A and mcd ((A _ B ) ^ A)  A ^ :B . Obviously, A 6j= A ^ :B . Note that this is also a counterexample to show that the MCE does not satisfy (U5).

6 Related Issues In this section, we discuss some related work. In fact, the motivation of the work presented in this paper came from authors' earlier work on reasoning about inde nite actions (Zhang and Foo 1993; Zhang and Foo 1995). In (Zhang and Foo 1993; Zhang and Foo 1995) we proposed a persistent set approach (PSA) to deal with actions with disjunctive eects. The PSA employed a principle of persistence, which, from a syntactical point, addressed the frame problem. However, the semantics of the PSA was totally left open, e.g. the persistent set can only be computed by a x-point procedure, and we did not know how to characterize the persistent set from a semantical side (Zhang 1995). As the classical minimal change approach of update, Winslett's PMA has been extensively analyzed by many researchers, e.g. (Herzig 1994; Marquis 1994; Peppa 1996). Among these analyses, the issue of disjunctive information of the PMA, in one way or another, has been addressed. The major dierence between our work presented in this paper and others' is that we do not try to solve the problem of disjunctive update by employing more complex logics or extra logical concepts in the formalism, e.g. using conditional logic (Herzig 1994) or introducing release uents (Kartha and Lifschitz 1994). To the contrary, we just provided two dierent inclusive interpretations for disjunctive update under a general principle of minimal change. The other issue mostly discussed by people is the problem of dealing with domain constraints in the PMA. Let us consider a domain including a constraint C : (Up1 

Up2 )  L indicating that the light L is on exactly when both switches Up1 and Up2 are 27

on the same position16 . Then, using the PMA, updating a knowledge base  Up1 ^

Up2 ^ L ^ C with :Up1 will lead to a resulting knowledge base 0  :Up1 ^ (Up2 ^ :L_

:Up2 ^ L) ^ C , which represents an inde nite eect of Up2 with respect to this update: turning o Up1 may or may not cause Up2 to change its position. Of course, this solution is not quite right from our intuition. What we wish to express here is that

Up1 and Up2 should act as a cause to turn on and turn o the light L. Unfortunately, this semantics is not re ected by constraint C . Dierent approaches were proposed to overcome this problem. One of the major ideas is to introduce causal rule to represent the domain causality instead of using the normal logical formulas Lin 1995; Lin 1996; McCain and Turner 1995; Zhang 1996). The other method is to use conditional logic to formalize domain constraints so that the undesired property of the logical implication can be avoided (Herzig 1996). People may argue that our approaches for update with disjunctive information do not solve the problem of domain constraints in the PMA. Indeed, using the MCE or MCD to the Light-Switch example just mentioned, a similar solution of the PMA will be obtained. However, as it has been mentioned earlier, the main purpose of this paper is to investigate the feature of inclusive interpretations for disjunctions in update under a general principle of minimal change. We think that the problem of the PMA caused by domain constraints should not be thought as a limitation of the PMA, because this case occurs in all classical logic based formulations of reasoning about change. We should also mention that there is a similarity between our MCD and the work presented by Eiter et al. in (Eiter, Gottlob and Gurevich 1993). Since our AAAI-96 paper (Zhang and Foo 1996) published, we have been pointed that Eiter et al. have also addressed the issue of disjunctive information in nonmonotonic reasoning. We notice 16

This example was originally examined by Lifschitz (Lifschitz 1990).

28

that although these two independent methods have very dierent formulations17, the underlying semantics is eventually analogous. In fact, Liberatore recently illustrated some interesting relationships between Eiter et al.'s update operator Curb1 and our

mcd operator (Liberatore 1997). But a dierent emphase of our work presented here lies in that we also characterized our MCD in terms of an alternative criterion of minimal change and the relationship to Katsuno and Mendelzon's update postulates. So far, we have not discussed the computational issue of our approaches. This issue actually was investigated by Liberatore recently (Liberatore 1997). Obviously, computational costs of the MCE and MCD are not optimistic. In general, both MCE and MCD are p2 complete (see (Liberatore 1997) for detail). However, due to the syntactical feature of MCE, based on previous work on the PMA and prime implicate generation, we can actually implement the MCE without too many diculties. From de nitions of the MCE, we observe that the essential part of the computational algorithms of MCE may be constructed from previous work about the PMA and prime implicates. Given a knowledge base and a formula . From De nition 1 and 3, we can see that the key part of the MCE, i.e. Res(S; )EXC (S;) = Min(S; Models(C ^ ))EXC (S;), can be implemented by the corresponding part of the PMA, i.e. Res(S; ) = Min(S; Models(C ^ )), with an extra computation of EXC (S; ). As details of the implementation of the PMA have been examined previously (Chou and Winslett 1991; del Val 1994), here we only consider the computation of EXC (S; ). From the de nition of EXC (S; ), we know that EXC (S; ) is formed from the set of all prime implicates ds of C ^  such that

S 6j= d. Therefore, it turns out that one of the key issues of implementing the MCE is to generate the set of all prime implicates of C ^ . The investigation of generating prime implicates has been extensively studied by many researchers in theorem proving 17

Eiter et al. used a circumscriptive approach to deal with disjunctive information in reasoning and a

third order logic is involved to formalize their theory (Eiter, Gottlob and Gurevich 1993).

29

eld, and some approximate and ecient algorithms could be employed for our purpose from previous work, e.g. (de Kleer 1992; Jackson 1992; Ngair 1993; Reiter and de Kleer 1987)18.

7 Conclusions In this paper, we considered update with disjunctive information under the principle of minimal change from both syntactical and semantical viewpoints. In particular, we proposed two approaches named the MCE and MCD respectively, to provide inclusive interpretations for disjunctions in update. We then characterized the MCE and MCD via alternative minimal change criteria and related them to Katsuno and Mendelzon's update postulates. We also discussed related issues to these two approaches.

Acknowledgements Our special thanks are due to Andreas Herzig for pointing out an error in an earlier draft of our paper. We also thank Tyrone O'Neill for suggestion in simplifying our formalizations. This research is supported in part by a grant from the University of Western Sydney, Nepean and a grant from the Australian Research Council.

18

Readers are referred to these references for the detail of prime implicate generation.

30

Appendix A: Proofs of Lemmas and Theorems Proposition 1 Let be a knowledge base without including state constraint C and   l1 _


_ lk be a disjunction where each li is a propositional literal (i = 1;


; k). Then mce   mcd . Proof: To prove the result, we only need to show that for each S 2 Models( ), Res(S; )EXC(S;) = Res(S; )mcd. Since there is no state constraint C , from De nition 3, it is clear that D() = fg = fl1 _


_ lk g. We consider the following two cases: Case 1. S j= . In this case, EXC (S; ) = ; and Res(S; )EXC (S;) = Res(S; ) = S . From De nition 5, on the other hand, the only non-empty subset SP of Res(S; ) is

fS g, and then Res(S; )mcd = Dis(S; SP ) = Dis(S; fS g) = S . So Res(S; )EXC(S;) = Res(S; )mcd = S . Case 2. S 6j= . In this case, we can assume that S = f:l1 ;


; :lk ; h1;


; hmg and hj is a literal (j = 1;


; m). We also have EXC (S; ) = l1 _


_ lk . From the speci cation of Min(S; Models())EXC (S;) and De nition 3, it is easy to see that for any S 0 2 Res(S; )EXC (S;), S 0 has a form fl1;


; li; :li+1;


; :lk ; h1;


; hm g, where 1  i  k. Now we show that S 0 is also in Res(S; )mcd . We specify S1 = fl1; :l2;


; :lk ; h1;


; hmg, S2 = f:l1; l2; :l3;


; :lk ; h1;


; hmg,




, Si = f:l1;


; :li?1 ; li; :li+1 ;


; :lk , h1 ;


; hm g. It is obvious that each of S1;


; Si satis es   l1 _


_ lk and has a minimal dierence from S . So each of S1 ;


; Si is in Res(S; ). Let SP = fS1 ;


; Si g. From De nition 5, it is clear that S 0 = fl1;


; li; :li+1;


; :lk ; h1 ;


; hm g is in Dis(S; SP ). So S 0 2

Res(S; )mcd. This follows Res(S; )EXC(S;)  Res(S; )mcd . Similarly, we can prove Res(S; )mcd  Res(S; )EXC (S;). 31

Theorem 1 The MCE update operator mce satis es Katsuno-Mendelzon's update postulates (U1) - (U4) and (U8).

Proof: The proof is directly followed from the construction of MCE (i.e. De nition 3 and De nition 4).

Theorem 2 The MCE update operator mce satis es (U6') and (U7') as follows. mce 1 implies 2, mce 2 implies 1, and for each S 2 Models( ) EXC (S; 1) = EXC (S; 2), then mce 1  mce 2 .

(U6') If

(U7') If

is complete where Models( ) = fS g and EXC (S; 1) = EXC (S; 2) =

EXC (S; 1 _ 2 ), then ( mce 1) ^ ( mce 2 ) implies mce (1 _ 2).

Proof: We rst prove (U6'). Let S 0 2 Models( mce 1) but S 0 62 Models( mce 2). Since mce 1 implies 2 , S 0 is a model of C ^ 2 . Since it is assumed that S 0 62 Models( mce 2), for any S 2 Models( ), there exists a model S 00 of C ^ 2 such that S 00 SEXC(S; ) S 0 but S 0 6SEXC(S; ) S 00, and S 00 2 min(Models(C ^ 2 ); SEXC(S; )). Since mce 2 implies 1 , it follows that S 00 is also a model of C ^ 1 . From condition EXC (S; 1) = EXC (S; 2) for any S 2 Models( ), we have S 00 SEXC(S; ) S 0 but S 0 6SEXC(S; ) S 00, which means that S 0 is not a model of mce 1 . This contradicts that S 0 is model of mce 1 . Therefore, mce 1 implies mce 2 . Similarly, we can prove mce 2 implies mce 1 . 2

2

2

1

1

Now we prove (U7'). As

is complete, there is a unique model S of . Let

S 0 2 Models(( mce 1) ^ ( mce 2 )). Clearly, S 0 is also a model of C ^ (1 _ 2). We assume that S 0 is not a model of

mce (1 _ 2). Then, there is a model S 00 of mce (1 _ 2 ) such that S 00 SEXC (S; _ ) S 0 but S 0 6SEXC (S; _ ) S 00, and S 00 2 min(Models(C ^ (1 _ 2 )); SEXC(S; _ )). From condition EXC (S; 1) = 1

1

32

2

2

1

2

EXC (S; 2) = EXC (S; 1 _ 2 ), it follows that S 00 SEXC(S; ) S 0 but S 0 6SEXC(S; ) S 00, 1

1

6 SEXC(S; ) S 00. Now if S 00 is a model of C ^ 1, this and S 00 SEXC (S; ) S 0 but S 0  contradicts the minimality of S 0 in min(Models(C ^ 1 ); SEXC (S; ) ). If S 00 is a model 2

2

1

of C ^ 2 , this contradicts the minimality of S 0 in min(Models(C ^ 2 ); SEXC (S; ) ). 2

Lemma 1 I ;S is a partial ordering. Proof: Obviously, from the de nition, I ;S is re exive and antisymmetric. Now we prove I ;S is transitive. Let S1 I ;S S2 and S2 I ;S S3 . Clearly, if S1 = S2 or S2 = S3 , then it directly follows S1 I ;S S3 . Now let us consider other cases. From S1 I ;S S2 , we have (i) for all Si 2 I1, Diff (S; Si)  Diff (S; S1) and there exists some S 0 2 I1, Diff (S; S 0) 6 Diff (S; S2); or (ii) for all Si 2 I1 , Diff (S; Si)  Diff (S; S1) and for all Si 2 I1, Diff (S; Si)  Diff (S; S2), and Diff (S; S1)  Diff (S; S2). Similarly, there are also two cases for S2 I ;S S3. However, since S1 I ;S S2 and S2 I ;S S3 , from the de nition of I ;S , we know that for all Si 2 I1, we have Diff (S; Si)  Diff (S; S1) and Diff (S; Si)  Diff (S; S2). It follows that the case (i) described above can not occur. So from S1 I ;S S2 , it must be the case (ii) presented above. On the other hand, from S2 I ;S S3 , we have two possible cases: (1) there exists some S 0 2 I1 , such that Diff (S; S 0) 6 Diff (S; S3); or (2) for all Si 2 I1 , Diff (S; Si)  Diff (S; S3), and Diff (S; S2)  Diff (S; S3). For case (1), it follows S1 I ;S S3 from De nition 7. For case (2), on the other hand, we have Diff (S; S2)  Diff (S; S3). But we already know Diff (S; S1)  Diff (S; S2), so we then have Diff (S; S1)  Diff (S; S3). Therefore, S1 I ;S S3 . 1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

Lemma 2 ;S is a partial ordering. 33

Proof: The proof is directly followed from Lemma 1 and De nition 8. Lemma 3 Let 1 and 2 be two classes of sets of interpretations and 1  2. Then S1  ;S S2 implies S1  ;S S2. Proof: This is directly followed from De nition 8. 1

2

Lemma 4 Let S be a state, C and  the state constraint and a propositional formula respectively, and SP a subset of Res(S; ). Then the following results hold: 1. If S 0 2 min(Models(C ^ ); SP ;S ), then for any S 00 2 Models(C ^ ), S 0 SP ;S S 00; 2.

min(Models(C ^ ); 2Res S; ;S ) = (

)

[ SP 22Res(S;)

min(Models(C ^ ); SP ;S ): (10)

Proof: We rst prove Result 1 above. Let S 0 2 min(Models(C ^ ); SP ;S ). Then for any S 00 2 Models(C ^ ), if S 0 and S 00 are comparable with respect to SP ;S , then S 0 SP ;S S 00. From De nition 7, we know that the only possibility that S 0 and S 00 are not comparable with respect to SP ;S is that Diff (S; Si)  Diff (S; S 00) for each Si 2 SP , Diff (S; S 0) 6 Diff (S; S 00) and Diff (S; S 00) 6 Diff (S; S 0). This follows Diff (S; S 0) \

Diff (S; S 00)  Diff (S; S 0). Let L = fa1;


; ak g = Diff (S; S 0) ? Diff (S; S 00), where a1 ;


; ak are propositional letters. In this case, we can always nd some interpretation S  of L such that all propositional letters in S  have exactly the same truth values as those in S 0 except propositional letters occurring in L. Then we have S1 \ S2  S  and Diff (S; S ) =

Diff (S; S1) \ Diff (S; S2). Hence S1 \ S2  S  and both S1 and S2 are models of C ^ , it follows that S  is also a model of C ^ . On the other hand, From Diff (S; Si)  34

Diff (S; S1) and Diff (S; Si)  Diff (S; S2) for each Si in SP , it follows Diff (S; Si)  Diff (S; S1) \ Diff (S; S2) = Diff (S; S ) for each Si in SP . This turns out S  SP ;S S 0 where S  6= S 0 . Obviously, it is a contradiction with the minimum of S 0 with respect to

SP ;S . So the case that some S 00 in Models(C ^ ) is not comparable with S 0 will never occur.

S

Now we prove Result 2. Let S 0 2 SP 22Res S; min(Models(C ^ ); SP ;S ). Then there exists some SP 2 2Res(S;) such that S 0 2 min(Models(C ^ ); SP ;S ). From the (

)

Result 1 we just proved, it is clear that for all S 00 2 Models(C ^ ), S 0 SP ;S S 00, and then S 0 2Res S; ;S S 00. So S 0 2 min(Models(C ^ ); 2Res S; ;S ). (

)

(

)

On the ther hand, from Result 1 we know that for any S  2 Models(C ^ ), there always exists some SP 2 2Res(S;) and S 0 2 min(Models(C ^ ); SP ;S ) such that

S 0 SP ;S S  and S 0 = 6 S . This also follows S 0 2Res S; ;S S . Then we can con(

)

clude that any element of min(Models(C ^ ); 2Res S; ;S ) must be one of these S 0 s. So (

)

min(Models(C ^ ); 2Res S; ;S )  SSP 22Res S; min(Models(C ^ ); SP ;S ). (

)

(

)

Theorem 3 Let be a propositional knowledge base, where C is the state constraint, and  a propositional formula. Then

Models( mcd ) = SS2Models( ) min(Models(C ^ ); 2Res S; ;S ). (

)

Proof: From De nition 6, it is clear that we only need to prove the following condition: Res(S; )mcd = min(Models(C ^ ); 2Res S; ;S ): (11) S From De nition 5, we know that Res(S; )mcd = SP 22Res S; Dis(S; SP ). Then from (

)

(

)

de nition of I ;S (i.e. De nition 7) and Dis(S; SP ) (i.e. De nition 5), it follows that 1

Dis(S; SP ) = min(Models(C ^ ); SP ;S ). Therefore, (11) is directly followed from the Result 2 in Lemma 4. 35

Theorem 4 The MCD update operator mcd satis es Katsuno-Mendelzon's postulates (U1) { (U4) and (U6) { (U8).

Proof: From the construction of update operator mcd (i.e. De nition 5 and De nition 6), it is obvious that mcd satis es (U1), (U2), (U3), (U4) and (U8). We prove that mcd satis es (U6). Firstly, from De nitions 5 and 6, it is easy to see that for any propositional formula , mcd  j=  implies pma  j= . So from conditions mcd 1 j= 2 and mcd 2 j= 1 , we have pma 1 j= 2 and pma 2 j= 1. Secondly, since the PMA update operator pma satis es (U1) - (U8) (Kastuno and Mendelzon 1991a), it is not dicult to see Res(S; 1) = Res(S; 2) for any

S 2 Models( ). Recall that in De nition 5, Res(S; )mcd = SSP 22Res S; Dis(S; SP ). (

)

So it is clear that Res(S; 1)mcd = Res(S; 2)mcd . Therefore, Models( mcd 1 ) =

Models( mcd 2). So mcd satis es (U6). We prove that mcd satis es (U7). Since

is complete, there is a unique model of

, say S . Let S 0 2 Models(( mcd 1 )^ ( mcd 2 )). Then S 0 2 Models( mcd 1 ) and S 0 2 Models( mcd 2 ). From Theorem 4, it follows that S 0 2 min(Models(C ^

1); 2Res S; ;S ) and S 0 2 min(Models(C ^ 2 ); 2Res S; ;S ) respectively. Since (

1)

(

2)

Res(S; 1)  Res(S; 1 _ 2 ) and Res(S; 2)  Res(S; 1 _ 2 ), from Lemma 3, it is easy to see that S 0 2 min(Models(C ^ 1 ); 2Res S; _ ;S ) and S 0 2 min(Models(C ^ 2); 2Res S; _ ;S ). This follows that S 0 2 min(Models(C ^ 1 ) [ Models(C ^ 2 ), (

(

1

2)

1

2)

2Res S; _ ;S ) = min(Models(C ^ (1 _ 2)); 2Res S; _ ;S ). So S 0 2 Models( ^ (1 _ 2 )). The result holds. (

1

2)

(

36

1

2)

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41

List of symbols - j= -2 -^ -_ - - - Diff (I1; I2) - Min(I; I ) - Res(S; ) - pma - mce - mcd - - Models( ) - Models( pma ) - Models( mce ) - Models( mcd ) - Min(I; I )EXC 42

- jdj - EXC (S; ) - Res(S; )EXC (S;) - Min(I; I )EXC (S;) - Dis(S; SP ) - 2Res(S;) - - S - EXC S - SEXC (S;) - I ;S - ;S - 2Res S; ;S (

)

List of gure captions Figure 1. The extended dropping-box domain.

43