Updating Knowledge Bases with Disjunctive Information Yan Zhang Department of Computing University of Western Sydney, Nepean Kingswood, NSW 2747 Australia E-mail:
[email protected] Norman Y. Foo Department of Computer Science University of Sydney NSW 2006, Australia E-mail:
[email protected] Abstract
It is well known that the minimal change principle was widely used in knowledge base updates. However, recent research has shown that the conventional minimal change methods, eg. the PMA [11], were generally problematic for updating knowledge bases with disjunctive information. In this paper, we propose two di erent approaches to deal with this problem { one is called the minimal change with exceptions (the MCE), the other is called the minimal change with maximally disjunctive inclusions (the MCD). The rst method is syntax-dependent, while the second is model-based. We show that these two approaches are equivalent for propositional knowledge base updates, and the second method is also appropriate for the rst order knowledge base update. We then prove that our new update approaches still satisfy the standard Katsuno and Mendelzon's update postulates.
Key words: knowledge base upate, knowledge representation, minimal change, disjunctive information
1 Introduction The knowledge base update has been widely studied in AI. It generally addresses the following question: given a knowledge base (i.e. a set of logical formulas as a description of the world), what changes may be caused by an occurrence of new knowledge 1
and how to specify the new knowledge base when the old one has changed? It is well known that the minimal change principle was employed in most formalizations of knowledge base updates [2, 11, 7, 1, 4]. However, recent research has revealed that such minimal change was generally inappropriate for representing knowledge change with disjunctive information [6, 12]. This paper is concerned with the problem of updating knowledge bases with disjunctive information. In particular, we propose two di erent approaches to solve this problem { one is called the minimal change with exceptions, the other is called the minimal change with maximally disjunctive inclusions. The rst method is syntaxdependent, while the second is model-based. We show that these two approaches are equivalent for propositional knowledge base updates, and the second method is also appropriate for the rst order knowledge base update. We then prove that our new approaches still the standard Katsuno-Mendelzon update postulates. The paper is organized as follows. Next section rst reviews the PMA { the classical minimal change approach for knowledge base updates, and shows how the PMA fails to deal with the update with disjunctive information. Section 3 proposes a new method called the minimal change with exceptions, which overcomes the problem of the PMA but is syntax-dependent. Section 4 proposes an alternative method called the minimal change with maximally disjunctive inclusions that is model-based. It is observed that this approach is also suitable for the rst order knowledge base updates. Section 5 shows that these two approaches are equivalent for propositional knowledge base updates. Section 6 investigates the relationship between our approaches and the standard Katsuno-Mendelzon update theory. Finally, section 7 discusses related work and concludes the paper.
2 The Minimal Change Approach: A Review In this section we rst introduce some preliminary concepts and review the PMA (possible models approach [11]) { a classical minimal change approach for update. Consider a nite propositional language 1. We represent a knowledge base by a propositional formula . A propositional formula is complete if is consistent and for any propositional formula , = or = . Models( ) denotes the set of all models of , i.e., all interpretations of in which is true. 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 = C . Let I be an interpretation of . We say that I is a state of the world if I = C . 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 L
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Note that the PMA was originally based on a rst order language. Here we rst consider propositional knowledge base update and will discuss the rst order case in section 4. 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. 1
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regarded as a new knowledge (information) about the world. Then, informally, the general question of updating with is how to specify the new knowledge base after combining the new knowledge (information) into the current knowledge base (we also call as the update e ect). 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 the principle of minimal change on models. Formally, let I1 and I2 be two interpretations of . We say that I1 and I2 di ers on a propositional letter l if l appears in exactly one of I1 and I2. Diff (I1; I2) denotes the set of all di erent propositional letters between I1 and I2. Let I be an interpretation and a set of interpretations. We de ne the set of all minimally di erent interpretations of with respect to I as follows: Min(I , ) = I I , and there does not exist other I such that Diff (I; I ) Diff (I; I ) . Then we can present the formal de nition of the state update in the PMA as follows. L
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De nition 1 Let C be the state constraint, S a state of the world, i.e., S = 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: j
Res(S; ) = Min(S; Models(C )):
(1)
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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. denotes the update of
with by the PMA3, where
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is inconsistent, then pma , otherwise S 2. Models( pma ) = S Models( ) Res(S; ).
1. If
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In the above de nition, condition 1 says that if entails , then nothing is changed since the knowledge has been represented by knowledge base ; or if is inconsistent, then any update can not change it into a consistent knowledge base [7]. Condition 2 says that if is consistent and does not imply , then should be changed, and this change is made by updating every model of with as de ned in De nition 1. It has been shown that under many circumstances, the PMA is powerful and e ective for representing knowledge updates and reasoning about action [11, 7]. However, recent research reveals that the minimal change principle, and so the PMA, is problematic to deal with disjunctive information. The following example illustrates the diculty with the PMA. C.
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Example 1. The dropping-box problem4. Suppose a table is painted with one part white and one part black. Therefore, a box on the table implies that it may be entirely within the white region, or within the black region, or touching the both regions. This constraint can be expressed by the following formula:
Ontable(Box) Inwhite(Box) Inblack(Box):
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Now suppose the current knowledge base is Ontable(Box) Inwhite(Box) Inblack(Box) (2) to express the fact that the box is not on the table. Consider the update of with Ontable(Box) (i.e., the box is dropped on the table). Using the PMA, we have the result: Inblack(Box)) pma Ontable(Box) (Inwhite(Box) ( Inwhite(Box) Inblack(Box)) (2), which says that the box must be only within one of white or black region. Obviously, this result is not quite plausible from our intuition. 2 In the above example, although the update has a simple e ect Ontable(Box), together with the constraint (2), it implies an indirect disjunctive e ect Inwhite(Box) Inblack(Box). Therefore, according to the minimal change principle of the PMA, only Inwhite(Box) or Inblack(Box) should be true after this update, but not both, and this leads to an unintuitive solution. :
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3 Minimal Change with Exceptions To overcome the problem with the PMA, in this section we propose an approach for update based on the principle of Minimal Change with Exceptions, which we call 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 update5. 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 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 e ect and not satis ed in the initial knowledge base (or state). Consider the dropping-box example presented in last section where the constraint is (2) and the update e ect is Ontable(Box). As both Inwhite(Box) and Inblack(Box) are not true in the initial knowledge base but the disjunction This example was suggested by Ray Reiter and discussed in [6]. Similarly to the PMA, in our approach, updating a knowledge base is achieved by updating every model of the knowledge base. 4 5
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Inwhite(Box) Inblack(Box) is entailed by (2) and Ontable(Box), we know that Inwhite(Box) Inblack(Box) should be true in the resulting knowledge base but we can not determine the truth values of Inwhite(Box) and Inblack(Box) exactly. In this case, we say literals Inwhite(Box) and Inblack(Box) are logically inde nite with respect to the update. According to our idea described above, Inwhite(Box) and Inblack(Box) should be regarded as exceptions to the minimal change principle. Thus, 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 both 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 di erent propositional letters, which are not in EXC , between I1 and I2. That is, l Diff (I1 ; I2)EXC i [ ]l I1 I2 and l EXC , where notation [ ] means that the negation sign may or may not occur. For example, let I1 = a; b; c; d , I2 = a; b; c; d and EXC = a; b . Then Diff (I1 ; I2)EXC = c . Let I be an interpretation and a set of interpretations. We de ne the set of all minimally di erent interpretations of with respect to I with the exception EXC as follows: Min(I , )EXC = I I , and there does not exist other I such that Diff (I; I )EXC Diff (I; I )EXC . Let C be a propositional formula used to Wrepresent the state constraintWand a propositional formula. We say a disjunction ni=1[ ]li satisfying C = ni=1 [ ]li, where li is a propositional letter (1 i n), Wis a non-trivial disjunction entailed by C if for any M 1; ; n , C = j M [ ]lj .WWe denote the set of all non-trivial disjunctions entailed by C as D(). If d ni=1[ ]li in D(), then we denote d = l1; ; ln . In the dropping-box example presented previously, as (2) Ontable(Box) = Inwhite(Box) Inblack(Box), and (2) Ontable(Box) = Inwhite(Box) and (2) Ontable(Box) = Inlack(Box), we have D(Ontable(Box)) = d = Inwhite(Box) Inblack(Box) , and d = Inwhite(Box); Inblack(Box) . Now we give the de nition of state update in the MCE as follows. _
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De nition 3 Let C be the state constraint, S a state of the world, i.e., S = C , a j
propositional formula, and D() the set of non-trivial disjunctions entailed by C ^ . We de ne the exceptional letters with respect to S and as follows: [ EXC (S; ) = jdj: (3) d D();S =d 2
6j
Then the set of possible states resulting from updating S with by the MCE, denoted as Res(S; )EXC(S;) , is de ned as
Res(S; )EXC(S;) = Min(S; Moldes(C ))EXC(S;): ^
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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 principle during the state update. If a d 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 = a; b; c; d and D() = a b; b c , then EXC (S; ) = a; b while c is not in EXC (S; ) as S = 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 d6, then we get Res(S; )EXC(S;) = S1; S2; S3 , where S1 = a; b; c; d , S2 = a; b; c; d , and S3 = a; b; c; d . 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 1. If entails or is inconsistent, then mce , otherwise 2. Models( mce ) = S Models( ) Res(S; )EXC(S;) . Comparing 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 Moldes( ), then the MCE reduces to the PMA. Consider the droppingbox example once again. As the knowledge base corresponds to a unique state S = Ontable(Box); Inwhite(Box); Inblack(Box) , we have EXC (S; Ontable(Box)) = Inwhite(Box); Inblack(Box) , then from De nition 3 and 4, we get the desired result: mce Ontable(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 derive every non-trivial disjunction d from the constraint C and update e ect , and then verify if S = d for each S Moldes( ). Obviously, the rst step is a syntactic procedure. Hence, we say that the MCE is syntax-dependent. 2
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4 Minimal Change with Maximally Disjunctive Inclusions In this section, we propose an alternative method which solves the problem of updating knowledge bases with disjunctive information and is model-based. 6
This follows that D() = fa _ b; b _ cg.
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Red Black
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Figure 1: The extended dropping-box domain.
4.1 The Approach
Our approach is constructed based on the idea of the Minimal Change with maximally Disjunctive inclusions, that we call as the MCD in short. To illustrate this idea clearly, we rst consider the following example.
Example 2. The extended dropping-box problem. Suppose a round table is painted with three equal parts of red color, white color and black color respectively. 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. Also, a constraint to formalize this domain is speci ed as follows:
Ontab(Box) Inred(Box) Inwhite(Box) Inblack(Box):
_
(5)
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Now suppose the current knowledge base is Ontable(Box) Inred(Box) Inwhite(Box) Inblack(Box) (5), which corresponds to a unique state: S = Ontable(Box); Inred(Box); Inwhite(Box); Inblack(Box) : 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; ) = S1; S2; S3 , where S1 = Ontable(Box); Inred(Box); Inwhite(Box); Inblack(Box) , S2 = Ontable(Box); Inred(Box); Inwhite(Box); Inblack(Box) , S3 = Ontable(Box); Inred(Box); Inwhite(Box);Inblack(Box) . :
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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 = Ontable(Box); Inred(Box); Inwhite(Box); Inblack(Box) , S5 = Ontable(Box); Inred(Box); Inwhite(Box); Inblack(Box) , S6 = Ontable(Box); Inred(Box); Inwhite(Box); Inblack(Box) , S7 = Ontable(Box); Inred(Box); Inwhite(Box); Inblack(Box) . In fact, states S4; S5; S6 and S7 can be generated from Res(S; ). Let P be any nonempty subset of Res(S; ). Then, there always exists a model S in Models(Ontable(Box) (5))7, such that S satis es: (i) for each Si P , Diff (S; Si ) Diff (S; S ), and (ii) there does not exist another S in Models(Ontable(Box) (5)) satisfying condition (i) but Diff (S; S ) Diff (S; S ). Now we claim that S is also a desired resulting state. For instance, let P be S1 . Then from conditions (i) and (ii), the corresponding S is S1 itself. On the other hand, if P = S1; S2 , we get S = Ontable(Box); Inred(Box); Inwhite(Box); Inblack(Box) = S4. Similarly, we can get S5, S6, and S7 from S1; S3 , S2; S3 and S1; S2; S3 respectively. Therefore, every desired resulting state can be generated from the corresponding subset of Res(S; ) by using the above procedure. 2 Before we explain the above procedure in detail, we rst introduce a useful concept. Let I1 and I2 be two interpretations, and a1 ak a disjunction that is satis ed in I1 and I2. We say that I1 includes I2's interpretation for this disjunction i for any disjunct ai, I2 = ai implies I1 = ai. In the above example, as the disjunction Inred(Box) Inwhite(Box) Inblack(Box) is a logical consequence of the update e ect Ontable(Box) and the constraint (5), different states in Res(S; ) also represent di erent 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 possible resulting states S1; S2 and S3 which only represent three possible interpretations for this disjunction. To describe the update with disjunctive e ect 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 is the state which includes Si's interpretation for the disjunction, while condition (ii) restricts this S to obey the minimal change on other information. For every subset SP of Res(S; ), we can get the corresponding S . So, we take all such S s to be the possible resulting states that include maximally possible interpretations f
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Clearly, Models(Ontable(Box) ^ (5)) is the set of all possible states of the world in which Ontable(Box) is true. 7
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of the disjunction without losing the minimal change criterion on other information. Therefore, this approach is so called the minimal change with maximally disjunctive inclusions.
4.2 Formal Descriptions
Base on the above discussions, now we are ready to develop our method formally. Similarly to the previous presentations, we rst de ne the state update in the MCD as follows.
De nition 5 Let C be the state constraint, S a state of the world, i.e., S = C , and j
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 = Dis(S; P )8; where (6) S
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Dis(S; P ) = S S Models(C ) such that (i) for each Si in P , Diff (S; Si ) Diff (S; S ), and (ii) there does not exist other S in Models(C ) satisfying Diff (S; S ) Diff (S; S ) . S
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Note that in De nition 5, we consider the power set of Res(S; ), while any element P of 2Res(S;) is a subset of Res(S; ). Dis(S; P ) represents the set of states that include the interpretations of the disjunctive information represented by all states of P without losing the minimal change criterion on other information. Clearly, if there is only one element S in P , then Dis(S; P ) = P = S . After specifying the state update, we then de ne the MCD update operator mcd for knowledge bases as follows. S
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De nition 6 Let be a knowledge base and a propositional formula. denotes the update of with by the MCD, where 1. If entails or is inconsistent, then mcd , otherwise 2. Models( mcd ) = SS Models( ) Res(S; )mcd .
mcd
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Example 3. Continue considering the extended dropping-box example. The initial knowledge base is :Ontable(Box) ^ :Inred(Box) ^ :Inwhite(Box) ^ :Inblack (Box) ^ (5). Now, we update with Ontable(Box). From De nition 5, we have S mcd = fS1 , S2 ; S3; S4 ; S5; S6 ; S7g, S Models( ) Res(S; ) 2
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Recall that Res(S; ) is de ned by (1) in section 2.
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where S1; ; S7 have been given in 4.1 previously. Finally, from De nition 6, we get the desired resulting knowledge base mcd Ontable(Box) (Inred(Box) Inwhite(Box) Inblack (Box)) (5).
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4.3 Existence as Disjunctive Information
So far, we have only considered the propositional knowledge base update, while a disjunctive information related to the update is simply a propositional disjunction. If we consider the rst order knowledge base update, on the other hand, a disjunctive information may be also represented by a formula with an existential quanti er. For instance, a formula Occupied(Table) x:On(x; Table); (7) represents a constraint that the table is occupied i there exists (or exist) some object (or objects) on the table, but we can not determine how many objects and what the exact object (or objects) is (or are). A natural question is: can the MCD also deal with this kind of disjunctive information? Consider a rst order language F with equality. We can extend the MCD in the following way. A knowledge base F is represented by a closed rst order formula. Let CF and F be two closed rst order formulas, where CF represents the state constraints (CF can be viewed as the conjunction of a set of closed rst order formulas)9, and F represents a new knowledge (information). Then the update of F with F is de ned exactly the same as the propositional knowledge base update as de ned in De nition 5 and 6, except that propositional formulas C , and in De nition 5 and 6 are replaced by formulas CF , F and F respectively. Suppose a knowledge base is F Occupied(Table) (7). Then, updating Occupied(table) implies an e ect x:On(x; Table), i.e., (7) F = F with F x:On(x; Table). Using the extended MCD as described above, the resulting knowledge base F mcd F will imply a consequence that there may be one or more objects on the table. That is, every possible interpretation of x:On(x; Table) will be represented by F mcd F . If we use the PMA, on the other hand, the solution implies that there is only one object on the table. This result seems too restricted from our intuition. 9
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5 Equivalence between MCE and MCD In this section, we discuss the relationship between the MCE and the MCD. We rst consider the following example. 9
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Example 4. Consider a propositional language L where there are ve propositional letters a; b; c; d and e. Assume that the state constraint C is the conjunction of the following formulas:
a b c, and d c e. Suppose that the current knowledge base is a b c d e C. Now, consider the update of with a d. We rst use the MCE to derive the resulting knowledge base. As is complete, it corresponds to a unique state S : S = a; b; c; d; e . Clearly, we have D() = b c; c e . As S = b c and S = c e, the set of exceptional letters with respect to S and is EXC (S; ) = c; e . Therefore, we have Res(S; )EXC(S;) = S1; S2; S3 , where S1 = a; b; c; d; e , S2 = a; b; c; d; e , and S3 = a; b; c; d; e . Finally, the resulting knowledge base is mce a b d (c e) C . Now, let us use the MCD to deal with this example. We rst get the Res(S; ) by the PMA as follows: Res(S; ) = S1; S2 . From Res(S; ) we generate every (non-empty) subset of Res(S; ) as follows: SP 1 = S1 , SP 2 = S2 , and SP 3 = S1; S2 . Then from De nition 5, we have Dis(S; SP 1) = S1 , Dis(S; SP 2) = S2 , and Dis(S; SP 3) = S3 , and then Res(S; )mcd = S1; S2; S3 . Hence, the nal result is mcd a b d (c e) C ,
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From Example 4, it seems that the MCE and MCD produce the same results for propositional knowledge base updates. In particular, we have the following theorem to reveal the equivalent relationship between the MCE and MCD. Lemma 1 Let C be the state constraint, S a state of the world, i.e., S = C , a propositional formula, and EXC1 and EXC2 two sets of propositional letters. If EXC1 EXC2 then Min(S; Models(C ))EXC Min(S; Models(C ))EXC . Lemma 2 Let C be the state constraint, S a state of the world, i.e., S = C , and a propositional formula. Then for any P Res(S; ), there exists some set of propositional letters EXC such that Dis(S; P ) = Min(S; Models(C ))EXC . Theorem 1 Let be a propositional knowledge base and a propositional formula. Then for any propositional formula , mce = i mcd = . As we mentioned earlier, the MCD is model-based while the MCE is syntaxdependent, we can think that the MCD also provides a semantics for the MCE. The following result further shows the relationship between the PMA and the MCD (MCE). Theorem 2 Let be a propositional knowledge base and a propositional formula. Then for any propositional formula , mcd = (or mce = ) implies pma = . j
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6 Relationship with KM Update Postulates In this section, we discuss how our update approaches presented previously relate to Katsuno and Mendelzon's update theory [8, 7]. As we have Theorem 1 stating that the MCE and MCD are equivalence for propositional knowledge base updates, in the following, we will only concern the relationship between the MCD and Katsuno and Mendelzon's update theory. The motivation of Katsuno and Mendelzon's proposal for update is an observation on the di erence between revision and update, as they discussed in [8, 7]. Let be a knowledge base and a formula. Then Katsuno and Mendelzon presented the following eight postulates that, as they argued, should be satis ed by any update operator . (U1) implies . (U2) If implies then . (U3) If both and are satis able then is also satis able. (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 ).
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In fact, 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). In order to investigate the relationship between the MCD and Katsuno-Mendelzon's update postulates, we rst introduce the following useful notations and concepts. Let be the set of all interpretations of the propositional language . A preordering over is re exive and transitive relation on . is called a partial ordering if it is an antisymmetric preordering. Let . We denote min( ; ) to be the set of all interpretations in that are minimal with respect to . De nition 7 Let be the set of all interpretations of , 1 , and S an interpretation of . We de ne an ordering over with respect to 1 and S as follows. For any two S1; S2 , S1 ;S S2 i (i) For all Si 1 , Diff (S; Si ) Diff (S; S1 ), and (ii) If for all Si 1, Diff (S; Si ) Diff (S; S2 ), then Diff (S; S1 ) Diff (S; S2 ).
I
L
I
I
M I
M
M
I
L
L
I
2 I
I
I
I
I1
2 I
2 I
Lemma 3 Ordering
I1
;S
is a partial ordering.
We can extend the ordering
I1
;S
to the general case as follows.
De nition 8 Let be a class of sets of interpretations. We de ne S1 there exists some
S1
I
k
2
such that S1
Ik
;S
;S S2 i S2 or there exists some S such that 0
S ;S S2. Clearly, the ordering ;S is the union of k ;S for every k in . For instance, if = 1; 2 , then S1 ;S S2 if one of the following cases holds: (1) S1 ;S S2; (2) S1 ;S S2; or (3) there is S such that S1 ;S S (or S1 ;S S ) and S ;S S2 (or S ;S S2).
;S
0
fI
I g
I
I1
0
I2 0
I
I1
0
0
I2
0
I2
I1
Lemma 4 Ordering ;S is a partial ordering. Theorem 3 Let be a propositional knowledge base, where C is the state constraint,
and a propositional formula. Then Models( mcd ) = SS Models( ) min(Models(C ^ ); 2Res S; ;S ). (
2
)
The above theorem shows that our MCD (and the same as the MCE) can be characterized by an alternative minimal change criterion. Therefore, it is not surprising that we have the following solution.
Theorem 4 The MCD update operator
(U1) { (U8).
mcd
satis es Katsuno-Mendelzon's postulates
Theorem 4 reveals an important fact that Katsuno-Mendelzon's postulates (U1) { (U8) still correctly capture the semantics of updating knowledge base with disjunctive information, which is quite di erent from the arguments of some previous work (eg. [12]). 13
7 Concluding Remarks In this paper, we proposed two di erent approaches, called the MCE and MCD respectively, to deal with the problem of updating knowledge bases with disjunctive information. In fact, the problem of dealing with the incomplete (disjunctive) information in dynamic systems has been studied by many researchers (eg. [3, 6, 9, 12]). However, most of their work concentrated on reasoning about action. Restricted by the formalizations, their methods, therefore, seemed not directly applicable for knowledge base updates as we discussed in this paper. On the other hand, some of their approaches only dealt with the direct disjunctive e ect [3]. Zhang and Foo did extend their work to deal with knowledge base updates [12]. However, as their method is syntax-based, there is no proper semantics to support their theory. There are some features of our approaches. First, our approaches can deal with both direct and indirect disjunctive e ects of updates. For instance, in the droppingbox problem (i.e. Example 1), the update e ect Ontable(Box) is de nite. Together with the constraint (2), however, it implies an indirect disjunctive e ect Inwhite(Box) Inblack(Box). In this case, our approaches can produce the desired result. Second, as the equivalence between the MCE and MCD for propositional knowledge base updates, the MCD provides a proper semantics for the MCE. Moreover, the MCD is also suitable for rst order knowledge base updates. Third, as the MCD (and so the MCE) satis es Katsuno and Mendelzon's postulates (U1) - (U8), we can see that in general, the principle of minimal change10 still captures the semantics of update with disjunctive information. Many related problems remain to explore. One is the computational tractability of our approaches. The other is the extension of our ideas to solve the problem of reasoning about actions with indeterministic e ects. A third is the connections of the principles of minimal change with exceptions and minimal change with maximally disjunctive inclusions to other nonmonotonic mechanisms such as default logic and circumscription. These issues are being addressed in our current work.
_
Acknowledgements This research is supported in part by a grant from the Australian Research Council. We thank Tyrone ONeill for good suggestions of simplifying our formalizations.
References [1] C. Baral, Rule based updates on simple knowledge bases. In Proceedings of the Eleventh National Conference on Arti cial Intelligence (AAAI'94). Morgan Kaufmann, Inc. (1994) 136{141. 10
Obviously, we should de ne an appropriate criterion for minimal change.
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[2] M. Dalal, Investigations into a theory of knowledge base revision: Preliminary report. In Proceedings of Seventh National Conference on Arti cial Intelligence (AAAI'88). Morgan Kaufmann Publisher, Inc. (1988) 475{479. [3] G. Brewka and J. Hertzberg, How do things with worlds: On formalizing actions and plans. Journal of Logic and Computation, 3(5) (1993). [4] N. Friedman and J.Y. Halpern, A knowledge-based framework for belief change, part II: revision and update. In Proceedings of the Fourth International Conference on Principles of Knowledge Representation and Reasoning (KR'94). Morgan Kaufmann, Inc. (1994) 190{201. [5] P. Gardenfors, Knowledge in Flux. MIT Press (1988). [6] G.N. Kartha and V. Lifschitz, Actions with indirect e ects. in Proceedings of the Fourth International Conference on Principles of Knowledge Representation and Reasoning (KR'94). Morgan Kaufmann, Inc. (1994) 341{350. [7] H. Katsuno and A.O. Mendelzon, On the di erence between updating a knowledge database and revising it. In Proceedings of Second International Conference on Principles of Knowledge Representation and Reasoning (KR'91). Morgan Kaufmann, Inc. (1991) 387{394. [8] H. Katsuno and A.O. Mendelzon, Propositional knowledge base revision and minimal change. Arti cial Intelligence 52 (1991) 263{294. [9] F. Lin, Abstract operators, indeterminate e ects, and the magic predicate. In The Third Symposium on Logical Formalizations of Commonsense Reasoning, Stanford, (1996) to appear. [10] R. Reiter, Equality and domain closure in rst-order databases. Journal of the Associate for Computing Machinery 27 (1980) 235{247. [11] M. Winslett, Reasoning about action using a possible models approach. In Proceedings of the Seventh National Conference on Arti cial Intelligence (AAAI'88). Morgan Kaufmann Publisher, Inc. (1988) 89{93. [12] Y. Zhang and N.Y. Foo, Applying the persistent set approach in temporal reasoning. Annals of Mathematics and Arti cial Intelligence 14 (1995) 75{98.
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