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On cumulative default logics
Artificial Intelligence 66 (1994) 161-179 Elsevier
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ARTINT 1148
On cumulative default logics* Laura Giordano and Alberto Martelli Dipartimento di Informatica, Universitgt di Torino, C.so Svizzera 185, 10149 Torino, Italy Received June 1993 Revised November 1993
Abstract Giordano, L. and A. Martelli, On cumulative default logics, Artificial Intelligence 66 (1994) 161-179.
Recently cumulative variants of Reiter's default logic have been proposed. All of them happen to be semimonotonic and to commit to assumptions, i.e. to require consistency of justifications. In this paper we define two new cumulative variants of default logic in the style of Brewka's CDL, which show that the above two properties are not required to achieve cumulativity. In fact, the first variant commits to assumptions, like Brewka's CDL, but it is not semimonotonic, and hence it allows priorities among defaults to be represented. The other cumulative variant does not even commit to assumptions, and thus it is very close to Reiter's default logic.
I. Introduction A number of variants of Reiter's default logic (DL) [9] have recently been proposed [2, 3, 6, 11]. Such variants are motivated by the aim of getting some desirable properties that DL does not satisfy. For instance, it is well known that DL is not cumulative, i.e. the addition of theorems to the set of premises may change the derivable formulas, thus preventing the use of lemmas in proofs. Furthermore DL does not commit to assumptions, i.e. it is possible to have inconsistencies among justifications of nonnormal defaults, and this may lead to unintuitive results [8]. To overcome these problems, Brewka [2] has defined a Cumulative Default Logic (CDL) which is cumulative and commits to assumptions. To achieve this, Brewka introduces assertions, i.e. formulas labelled with a support, and strengthens the applicability conditions for defaults. However, CDL happens to be semimonotonic, which is often regarded as a desirable property from a Correspondence to: A. Martelli, Dipartimento di Informatica, Universit~ di Torino, C.so Svizzera 185, 10149 Torino, Italy. * Originally submitted as a Research Note. 0004-3702/94/$07.00 © 1994--Elsevier Science B.V. All rights reserved SSDI 0004-3702(93)E0095-4
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computational point of view [6], but as a drawback from a representational point of view. In fact, as observed in [2], "semimonotonicity destroys part of the additional expressiveness of nonnormal defaults", for instance, it makes it impossible to represent priorities among defaults. To regain this ability, Brewka shows how some unwanted C D L extensions can be filtered out. Other cumulative variants of default logic, such as Constrained Default Logic [11] and J-Default Logic [3], turn out to be equivalent. These variants do not make use of assertions as CDL does, but (in the style of Lukaszewicz's default logic [6]) they allow extensions as pairs of sets of formulas (E, C), where C is the context supporting the beliefs in E. In spite of this difference, these logics are very similar to C D L and an equivalence result between Constrained Default Logic and CDL is given in [12]. In this paper, we define two variants of Reiter's default logic that are cumulative like CDL, but closer to DL. We compare these variants to CDL and D L by analysing their properties of semimonotonicity and commitment to
assumptions. The first variant we propose is called CA default logic or C A D L , Jr Commitment to Assumptions Default Logic). C A D L is cumulative and commits to assumptions, like CDL, but it is not semimonotonic. To achieve this, in C A D L only justifications are recorded and checked for consistency, and not consequents of defaults as in CDL. Since C A D L is not semimonotonic, there is no loss in the expressiveness of nonnormal defaults, and priorities among defaults can be represented. For this reason, filtering out solutions as in [2] is not needed in this case. The other cumulative variant we define is even closer to Reiter's DL, since it does not require joint consistency of justifications. We will call it Q D L (Quasi-Default Logic). Like DL and C A D L , but differently from CDL, Q D L is not semimonotonic. Moreover, like DL, but differently from CDL and C A D L , it does not commit to assumptions. In Sections 2 and 3 we recall the definition of DL and CDL, respectively, and discuss some examples. In Section 4 we define C A D L and prove that it is cumulative, and in Section 5 we show how priorities among defaults can be represented in this logic. In Section 6 we introduce the other variant, QDL.
2. Reiter's default logic In this section we recall the definition of Reiter's D L and we discuss the properties of cumulativity, semimonotonicity, and commitment to assumptions that will be used throughout the paper. For simplicity, in this and the following sections, we will restrict our attention to the propositional case. A default theory is defined as a pair (D, W), where D is a set of default rules of the form A : B/C and W is a set of well-formed-formulas (wffs). A is called
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the prerequisite, B the justification, and C the consequent of the default rule. We consider default rules with a single justification. The notion of extension of a default theory is defined in the following way. Definition 2.1 (Reiter [9]). A DL-extension of a default theory (D, W) is a fixpoint of the operator F which, given the set of wffs S, produces the smallest set of wffs S' such that: (1) WC_S', (2) S' is deductively closed, i.e., Th(S') = S', (3) if (A : B / C ) @ D, A E S', and --nB~S, then C C S'. It is well known that, with this definition of default extension, mutually inconsistent justifications can be applied together to get an extension. Consider the following example from [8]. Example 2.2. The default theory (D, W) with D =
: left-hand-usable/x 7left-hand-broken left-hand-usable ' • right-hand-usable/x 7 right-hand-broken~ right-hand-usable J
W = {left-hand-broken v right-hand-broken} ,
has a single (Reiter's) default extension E = Th({left-hand-broken v right-hand-broken, left-hand-usable, right-hand-usable}),
which is obtained by applying both defaults, though their justifications are mutually inconsistent, given W. The problem here is that in default logic assumptions can be made implicitly, without considering their possible side-effects, i.e. given the assumption --nlefthand-broken (which is used by the first default) and the set of formulas W, we cannot conclude right-hand-broken, so as to block the second default• In essence, there is no commitment to assumptions. For a more detailed discussion on this, see [8]. It has been pointed out by Makinson [7] that DL is not cumulative. Cumulativity is one of the general properties that a nonmonotonic inference relation is expected to satisfy [4, 7]. A nonmonotonic inference relation ~ is said to be cumulative if it satisfies the following condition: If A ~ x , then (A ~ y iff A U {x} ~ y ) . This property is quite useful since it allows for the generation of lemmas. For
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default logic, when a non-skeptical notion of derivability is used, Brewka formulates cumulativity as follows. Let (D, W) be a default theory, if there is an extension F of (D, W) containing y, then E is an extension of (D, W) containing y iff E is an extension of ( D, W tO {y}). Reiter's DL is not cumulative. Example 2.3. (Makinson [7]). D={: p pvq'-qp} P ~P
W={}
(D, W) has a unique (Reiter's) extension E = Th({p}). Hence, p v q E E. However, if p v q is added to W, the resulting default theory has two extensions: E 1 = Th({p}) and E 2 = Th({~p, q}). As regards semimonotonicity, a default logic is said to be semimonotonic if, given a default theory (D, W) and a set of default rules D', such that D C D', for every extension E of (D, W) there is an extension E' of (D', W) that contains it, i.e. an extension E' such that E C E'. In essence, no extension of (D, W) can be ruled out by the addition of new default rules. While semimonotonicity holds for normal default theories (see [9]), whose default rules have the form A : B/B, it does not hold in the general case. Example 2.4.
D=
:TB}
---5-'
W={}
(D, W) has a unique DL extension E = Th({C}). Let us assume that an additional default is added, so that D , = { : ~ B :--qA} C ' B " (D', W) has a unique DL extension E ' = Th({B}), obtained by applying the second default. Since E is not contained in E', DL is not semimonotonic. In the next section, we will come back to these examples to see how Brewka's CDL commits to assumptions, is cumulative and semimonotonic.
3. Brewka's cumulative default logic
In Brewka's CDL, formulas are associated with supports by allowing assertions of the form ( p : {r 1. . . . . rn} >, where r I . . . . . r n are formulas supporting the belief in p. Given a set of assertions W, Form(W) denotes the asserted
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formulas of W, i.e., Form(W) = {p : ( p : { r l , . . . , rn}) E W } , and Supp(W) denotes the supports of W, i.e.,
Supp(W) = ( r : ( p : ( r l , . . . ,
r,...,
rn} ) E W}.
Since we consider only the propositional case, we assume that only propositional formulas can occur in the set of assertions W and in default rules. An assertion default theory is a pair (D, W), where D is a set of defaults in the sense of Reiter and W is a set of assertions. In order to define CDL extensions, Brewka generalizes the notion of classical deductive closure to sets of assertions. Let A be a set of assertions. Ths(A), the set of supported theorems of A, is the smallest set such that (1) A C__Ths(A), (2) if ( p , : J1) . . . . , (Pn: Jn) E Ths(A ) and p ~ , . . . , • ." O J~) E Ths(A),
p,,kq, then ( q : J~ U
where k is derivability in classical logic. In the following we will omit the subscript S in Ths(A ) when it is clear that A is a set of assertions. The notion of extension of an assertion default theory is then defined in the following way. Definition 3.1 (Brewka [2]). A CDL extension of an assertion default theory (D, W) is a fixpoint of the operator F which, given the set of assertions S, produces the smallest set of assertions S' such that: (1) W C S ' , (2) S' is deductively closed, i.e., Ths(S' ) = S', (3) if ( A : B / C ) ~ D , ( A : { J 1 . . . . . J k } ) E S ' , and ( B , C } U F o r m ( S ) U Supp(S) is consistent, then (C : (J1 . . . . , Jk, B, C}) C S'. Notice that in the definition above both the justification and the consequent of each applied default are checked for consistency and then recorded in the support of the added assertion. Since to apply a default its justification is checked for consistency with all the supports of the extension, default rules with mutually inconsistent justifications cannot be applied together to get an extension, i.e. CDL commits to assumptions. Example 3.2. The assertion default theory (D, W) with
D
(
: left-hand-usable ^ mleft-hand-broken left-hand-usable : right-hand-usable ^ -7right-hand-broken
W = { (left-hand-broken v right-hand-broken : { } )} ,
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has two CDL extensions---El, obtained by applying the first default rule and containing the assertion
( left-hand-usable : {left-hand-usable ^ ~left-hand-broken, left-hand-usable} ) , and E2, obtained by applying the first default rule and containing the assertion
( right-hand-usable : { right-hand-usable ^ --1right-hand-broken, right-hand-usable } ) . Brewka proves that CDL is cumulative (see [2, Proposition 2.13]): if there is an extension F of (D, W) containing ( p : J), then E is a CDL extension of (D, W) containing ( p : J ) iff E is a CDL extension of (D, W U { ( p : J ) ) ) . Consider again Example 2.3, on which DL fails cumulativity.
Example 3.3. D={: p pvq:~p) p -np
W={}
(D, W) has a unique CDL extension E ' = T h ( { ( p : { p } ) ) ) . Hence, ( p v q : {p}> E E'. Adding ( p v q : {p}) to W gives a new assertion default theory still having E' as the unique extension. Indeed, the default p v q : 7 p / - ~ p cannot be applied, since the justification -np is inconsistent with the support {p} o f p v q. Since, in CDL, consequents of default are recorded and tested for consistency, defaults are implicitly regarded as being seminormal: a default A : B / C can be equivalently replaced by the default A : B ^ C/C. This also leads to semimonotonicity. In fact, Brewka proves (see [2, Proposition 2.11]) that CDL is semimonotonic. If we go back to Example 2.4, we can see that it is not a counterexample to monotonicity for CDL.
Example 3.4. D=
:-IB
W=(}
(D, W) has a unique CDL extension E = Th({( C : {-riB, C})}). Let us assume that an additional default is added so that D' = { : T B / C , :-~A/B}. (D', W) has two CDL extensions E 1 = T h ( { ( C : {--riB, C}>}) and E 2 = T h ( { ( B : {-hA, B}}}). Since E I coincides with E, in particular, it contains E. Semimonotonicity is recognized by Brewka as being a drawback rather than an advantage, since it limits the expressiveness of nonnormal defaults. The same can be said with regard to having implicit seminormal default rules. This matter will be discussed in detail in Section 5.
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4. CA default logic In this section, we define a cumulative variant of default logic, Commitment to Assumption Default Logic (CADL), which is not semimonotonic, and in which defaults are not implicitly regarded as seminormal. Like CDL, CADL also requires mutually consistent justifications. We will show in Section 5 that this variant allows priorities among defaults to be represented, so that filtering out extensions as in [2] is not needed. In defining CA default logic (CADL) we follow the approach proposed by Brewka. We refer to the notions of assertion default theory and set of supported theorems of A (Ths(A)) defined in [2] and recalled in Section 3. CADL extensions are defined as follows. Definition 4.1. A C A D L extension of an assertion default theory (D, W) is a fixpoint of the operator F which, given the set of assertions S, produces the smallest set of assertions S' such that:
(1) WC_S', (2) S' is deductively closed, i.e., Ths(S' ) -- S', (3) if ( A : B / C ) E D , (A:(J1,...,Jk})ES' , and ( B } U F o r m ( S ) U Supp(S) is consistent, then ( C : {Jl . . . . . J~, B}) E S'.
Note that in the definition above only the justifications of applied defaults are recorded. Accordingly, only the justifications, and not consequents of defaults, are tested for consistency in order to apply defaults, similarly to Reiter's default logic. Hence, defaults are not implicitly regarded as seminormal. When we want defaults to behave as seminormal, then we must explicitly write them as seminormal. By comparing the above definition with the definition of CDL extension (Definition 3.1), it comes out that, if seminormal defaults are used, then CADL obviously collapses to CDL, i.e. the CADL extensions of a seminormal theory coincide with its CDL extensions. Though in [2] it is observed that recording consequents of applied defaults is actually needed to have cumulativity, this variant of default logic can be proved to be cumulative. The reason for this is that, in the definition of CADL extension, consequents are neither recorded nor tested for consistency. As usual, an equivalent quasi-inductive definition of CADL extensions can be given (the proof of the equivalence is quite the same as the one for CDL in [21),
Definition 4.2. Let (D, W) be an assertion default theory and E a set of assertions. Define E 0 = W and, for i t> 0,
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Ei+ 1 =
Ths(Ei) U
((C: {J1,..-, Jk, B}) I(A : B/C)~D, ( A : { J 1 , . . . , Jk}) ~ Ei and {B} U Form(E) U Supp(E) is consistent}. E is a CADL extension of (D, W) iff E = Ui=0,~ Ei. By making use of this definition it can be proved that CADL is cumulative. Proposition 4.3. CADL is cumulative. lh'o~ff. The proof of this proposition is quite the same as the proof of cumulativity for CDL (see [2, Proposition 2.13]). It makes use of the notion of well-based assertion default theory defined in [2, Definition 2.6]) and of the Lemmas 2.7 and 2.8 in [2]. These lemmas also hold for CADL and have the same proof as given in [2]. [] For instance, the default theory (D, W) in Example 3.3, which contains default rules which are all normal, has a unique CADL extension E ' = T h ( { ( p : { p } ) ) , which coincides with its unique CDL extension. Hence, ( p v q : { p } ) E E ' . As in CDL, adding ( p v q : { p } ) to W gives a new assertion default theory still having E' as the unique extension. It is easy to see that CA default logic is not semimonotonic. As for Reiter's DL, a counterexample to semimonotonicity of CADL is given by Example 3.4. (D, W), with D = {: 7B/C} and W = (}, has a unique CADL extension E = Th({(C: ( ~ B } ) } ) . If the additional default : ~ A / B is added, the default theory (D', W), with D, = { :--qB : ~ Z } C ' B ' has a unique CADL extension E' = Th({(B : {--qA})}), obtained by applying the second default. Since E is not contained in E', CADL is not semimonotonic. From this example, it is quite clear that CADL is more similar to Reiter's DL than Brewka's CDL and other cumulative variants of DL proposed in the literature. CADL is cumulative but, like default logic, it is not semimonotonic. In the next section we will show how this allows priorities among defaults to be expressed. Another similarity of CADL with default logic is that a default theory is not guaranteed to have a CADL extension. For instance, the theory (D, W), with D = {:A/-qA,:A/B} and W = { }, has no CADL extension (and no DL extension), while it has a unique CDL extension E = Th({ (B : {A, B } )}). Of course, differently from default logic but similarly to CDL, in CADL defaults with mutually inconsistent justifications cannot be jointly applied, i.e. CADL
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commits to assumptions. The assertion default theory (D, W) in Example 3.2 has two CADL extensions: E 1 containing (left-hand-usable: {left-handusable ^ -nleft-hand-broken} ) and E 2 containing (right-hand-usable : {righthand-usable ^ -7 right-hand-broken} ).
5. Representing priorities among defaults In this section we want to show that CADL recovers the ability to represent priorities among defaults which is lost in CDL (see [2, Section 3]). Of course, priorities among defaults cannot be represented in CADL by making use of seminormal defaults as usual in Reiter's default logic [10], since for seminormal theories CADL extensions are the same as CDL ones. Another way to represent priorities is used. In this section we will also see that in CADL we do not get the same extensions obtained by Brewka by filtering out CDL extensions. Consider the following example, taken from [2] and here slightly modified.
Example 5.1. D=
(
student : -7 married 1) -nmarried ,
(3)
adult : married (2) married '
living-in-college : student student ,
(4)
beard : adult -adu~ )'
W = {living-in-college, beard} .
In Reiter's DL (D, W) has two different extensions, E 1 = Th({living-in-college, beard, student, adult, married}), E 2 = Th({living-in-college, beard, student, adult, ~ m a r r i e d } ) .
Let us assume that we want both student and adult to be concluded, but, since defaults (1) and (2) lead to mutually inconsistent conclusions, we want default (1) to have a higher priority than default (2). To achieve this purpose, default (2) can be replaced with the seminormal default (see [10]): (2')
adult : married ^ --qstudent married
The resulting theory has a single DL extension, E2, as wanted, since applying default (3) blocks (2'). This same theory, however, has two CDL extensions: F 1 containing (adult : ( a d u l t } ) , (married : (adult, married ^ -qstudent} ) and F2 containing (student: {student}), (adult : {adult}), (~married : {student,-nmarried}), since in CDL (3) blocks (2'), but (2') blocks (3) in turn. Hence, in CDL, using seminormal defaults does not allow to enforce priorities among default rules (1) and (2), as in Reiter's default logic.
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Indeed, since CDL is semimonotonic it is not possible that the introduction of a new default rule in a theory (D, W) can rule out some of its previous extensions. Hence, as noted in [2], it is not possible to define a default rule with higher priority than others and this destroys the additional expressive power of nonnormal defaults. Since, on the contrary, C A D L is not semimonotonic, it is possible that a new default can rule out some of the previous extensions and, then, it is possible to define priorities among defaults. However, this cannot be done by making use of seminormal defaults as in DL. In fact, as told above, for seminormal defaults C A D L gives the same results as CDL. For instance, C A D L extensions of the default theory in Example 5.1 are exactly the same as CDL extensions. In order to express priorities among defaults in C A D L , defaults which are not seminormal must be used. The default theory in Example 5.1 can be reformulated in the following way: ( D=
1) (3')
student : -7 married 7married ,
(2")
living-in-college: student student A ab-adult ,
adult : married ^ -7 ab-adult married ' (4)
beard - adult adult J'
with W = {, (beard : { })}. Notice that default (3') is not seminormal. The atom ab-adult is used to denote the fact that a student is an abnormal adult. When (3') is applied, the atom ab-adult is added to the extension and default (2") is indeed blocked, since it is not consistent to assume Tab-adult. On the contrary, applying default (2") does not block default (3') (consistency of ab-adult is not checked in C A D L to apply (3')). Hence the default theory (D, W) has a unique extension E containing the assertions (student: {student}), ( ab-adult : {student}), (adult: {adult}), (~married : {Tmarried, student}), which corresponds to the CDL extension F 2. In [2] theability to represent priorities is restored by filtering out some of the C D L extensions. To this purpose, Brewka defines the notion of prioritypreserving C D L extension as follows. Given an assertion default theory (D, W), let GD(E) be the set of generating defaults of E in (D, W), i.e. the set of defaults applied to construct E. A C D L extension E of (D, W) is called priority-preserving if for no ( A : B/C) E D\GD(E), A E Form(E), {B, C} U Form(E) is consistent and {C} U Form(E) U Supp( E ) is inconsistent. For instance, extension F 1 in Example 5.1 is filtered out (it is not prioritypreserving due to default (3)) and F 2 is regarded as the unique prioritypreserving extention. By Example 5.1 itself it is clear that C A D L extensions do not coincide with priority-preserving CDL extensions: since all defaults are seminormal, there are two C A D L extensions that are precisely the C D L extensions, while there is a single priority-preserving C D L extension.
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CDL v, i.e. CDL with the additional filter on extensions, is not semimonotonic, but commits to assumptions and is cumulative like CDL. Thus it has the same properties as CADL. However, it is difficult to compare the two proposals because in CDL F the property of being nonsemimonotonic is achieved "a posteriori", by filtering out some extensions, while in CADL it is a direct consequence of the applicability criterion for defaults. We have shown in Example 5.1 that CDL F is closer to DL than CADL, but it is easy to find other cases where the opposite holds. For instance, the default theory (D, W) with D=
[:B :A} B '--7B '
W={},
has a unique CADL extension E 1 = Th({(-TB:{A})}), corresponding to the unique DL extension. Instead, CDL v has also the extension E2= Th({ (B : {B} )}), which is priority-preserving and, thus, it is not filtered out in CDL v. This happens because applicability of defaults in CDL requires consistency of consequents, differently from CADL and DL.
6. Getting closer to Reiter's default logic
In this section we propose another cumulative variant of default logic which is even closer to Reiter's default logic than CADL. This variant, like CADL, is not semimonotonic, but, differently from CADL, does not require joint consistency of justifications. An alternative variant of default logic, which is also cumulative and does not require joint consistency of justifications, has been proposed by Makinson (and referred to in [2]). This variant, however, requires to modify the notion of assertion, since both justifications and consequents of applied defaults must be recorded and kept separate. On the contrary, in this section we will show that a cumulative variant of DL which does not require joint consistency of justifications can be defined by only recording justifications (and thus keeping the notion of assertion as defined in [2]). Since the definition of extension for this variant, apart from remembering supports, is very close to the one for DL, we will call this variant QDL (Quasi-Default Logic). Let us define the notion of QDL extension. Definition 6.1. A set S of assertions is well-defined if there is no formula a such that a C Supp(S) and -7a ~ Form(S). Definition 6.2. Let (D, W) be an assertion default theory and let F be an operator that, given the set of assertions S, produces the smallest set of assertions S' such that:
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(1) W C S ' , (2) S' is deductively closed, i.e., Ths(S' ) = S', (3) if ( A : B / C ) ~ D , ( A : { J 1 , . . . ,Jk}) E S ' , and { B } U F o r m ( S ) is consistent, then ( C : {J1, • • •, Jk, B}> E S'. E is a Q D L extension of (D, W) iff E is a fixpoint of the operator F and E is well-defined. Note that, as a difference with CDL and CADL, recorded supports are not used to determine the applicability of default rules, but only to guarantee that the extension is well-defined. An equivalent quasi-inductive definition of Q D L extensions is the following (the proof of equivalence is the same as the one for DL in [2]).
Definition 6.3. Let (D, W) be an assertion default theory and E a set of assertions. Define E 0 = W and, for i > 0, Ei+l = Ths(Ei) U
{ ( C : {J1 . . . . , Jk, B}> I(A : B / C ) E D , ( A : {J1,. . " , J~}> E E i and {B} U Form(E) is consistent} . E is a Q D L extension of (D, W) iff E = Ui=0,~ Ei and E is well-defined. In QDL, joint consistency of justifications is not required. Consider again Example 3.2. The assertion default theory (D, W) has a single Q D L extension E 1 containing both the assertions (left-hand-usable : Cleft-hand-usable ^ ~lefthand-broken } > and ( right-hand-usable : { right-hand-usable A --1right-handbroken}) corresponding to the single (Reiter's) default extension E = Th({lefthand-usable, right-hand-usable}). This variant of default logic can be proved to be cumulative.
Proposition 6.4. Q D L is cumulative. The proof can be found in Appendix A. Let us now compare Q D L with Reiter's default logic. If in the definition of Q D L extension the condition of well-definedness were omitted, then Q D L extensions would coincide with Reiter's extensions, but for the presence of (unused) supports of formulas. Requiring a Q D L extension E to be well-defined is essential to get cumulativity. Consider again Example 2.3 which shows that Reiter's default logic fails cumulativity. The default theory (D, W) with D = { :p p
pvq!-'qPl "qp J
W={}
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has a unique QDL extension E = Th({(p: {p})}). Hence, ( p v q : {p}) E E. Adding ( p v q : {p}) to W gives a new assertion default theory (D, W'), having E as its unique QDL extension. E ' = Th({(p v q: {p}), (rap : {p,-np})}) is not a QDL extension since it does not satisfy the well-definedness condition. However, if this condition is omitted, E' is an extension of (D, W') and this is a counterexample to cumulativity. The well-definedness condition allows to get rid of unwanted extensions in which some of the derived formulas are the negation of formulas in the supports. Note however that this condition is only introduced to deal properly with initial assertions; for the assertion default theories in which W has empty supports this condition is not really needed. The next proposition shows that the results of QDL and DL are equivalent if all the formulas in W have empty supports, that is, QDL extensions of an assertion default theory (D, W) with empty supports coincide (apart from the supports) with Reiter's default extensions of the default theory (D, Form(W)).
Proposition 6.5. Let (D, W) be an assertion default theory such that W contains only assertions of the form (a :0). Then, E is a QDL extension of(D, W) iff Form(E) is a Reiter's default extension of the default theory (D, Form(W)), i.e. Form is a surjective mapping from the set of QDL extensions of (D, W) to the set of DL extensions of (D, Form(W)).
The proof can be found in Appendix A. The fact that QDL is cumulative is in perfect agreement with the result in [12] that also for Reiter's default logic suitable lemma default rules can be defined, which can be added to the set of defaults without changing the extensions of the default theory. Then, in a sense, DL itself can be considered to be cumulative. Indeed, if a belongs to an extension E of a default theory (D, W), then by Proposition 6.5 above there must be an extension E' (with Form(E') = E) of the corresponding assertion default theory (D, W') with empty supports such that ( a : {Jx . . . . , Jn}) belongs to E', for some J l , . . . , Jn. A lemma default rule 6~ for a, as defined by Schaub in [12], can be simply given as follows: :Jl''''' OL
Jn
As noted by Schaub, the addition of an assertion is stronger than the addition of a lemma default rule. While adding the assertion ( a : (jl . . . . . Jn } ) eliminates all the extensions that are inconsistent with the assertion, adding the lemma default rule does not change the extensions of the theory.
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7. Conclusions In this paper we have defined a cumulative variant of default logic, called CADL, which is not semimonotonic and requires consistency of justifications. We have shown that priorities among defaults can be represented in CADL, while, as shown in [2], they cannot in CDL. In CADL defaults are not implicitly regarded as seminormal. For seminormal default theories CADL and CDL give the same results. We have also presented an additional cumulative variant, QDL, which is very close to DL itself and does not require consistency of justifications. In this paper, we have not addressed the problem of defining a semantics for these variants. Recently, a uniform semantic framework has been defined by Besnard and Schaub [1], by giving the semantics of various default logics in terms of Kripke structures. It is possible to see that, within their framework, Kripke semantics for CADL and QDL can be defined too. Also, in [5] a uniform syntactic characterization of the different variants of Reiter's DL has been defined by making use of a modal approach. A characterization for CADL and QDL can also be accommodated in this framework. To conclude, we want to point out that, while in this paper we have followed the approach proposed by Brewka in defining these variants, we also could have defined them in the style of Schaub, Delgrande and Jackson, and Lucaszewicz. That is, instead of allowing assertions in the language, we could have defined extensions as pairs (E, C), where C is a set of formulas supporting the beliefs in E.
Appendix A. Proofs In order to prove cumulativity for QDL, we need the following lemmas. Lemma A.1. Let A = (D, W) be an assertion default theory. If (D, W) has a Q D L extension, then W is well-defined. Proof. If (D, W) has a QDL extension E, then W C_E. If W is not well-defined E is not well-defined. This contradicts the fact E is a QDL extension. [] Lemma A.2. Let A = (D, W) be an assertion default theory. The following holds. (a) Let Form(W) be inconsistent. If W is well-defined and Supp( W) = ~), then (D, W) has a unique QDL extension E = Ths(W ) such that Form(E) is inconsistent; otherwise, (D, W) has no QDL extension. (b) If (D, W) has a QDL extension E such that Form(E) is inconsistent, then Form(W) is inconsistent.
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Proof. (a) If Form(W) is inconsistent, then, since W must be contained in any Q D L extension of (D, W), no default is applicable. Hence, if E = Th(W) is well-defined, it is the unique QDL extension of (D, W) and Form(E) is inconsistent. Note that, when Form(W) is inconsistent, E = Th(W) is welldefined if and only if W is well-defined and Supp(W) = O. (b) If (D, W) has a QDL extension E such that Form(E) is inconsistent, then no default has been applied to get E. Hence, E = Th(W), and, in particular, Form(E)= Th(Form(W)). Thus, if Form(E) is inconsistent, also Form(W) must be inconsistent. []
Proposition 6.4. QDL is cumulative. Proof. We have to prove that, given an assertion default theory (D, W), if there is a QDL extension F of (D, W) containing ( p : J ) , then E is a Q D L extension of (D, W) containing ( p : J ) iff E is a QDL extension of (D, W U
{(p:J))). As hypothesis there is a QDL extension F of (D, W) containing ( p : J ) . Hence, by Lemma A.1, W must be well-defined. Two cases are considered: the one in which Form(W) is inconsistent, and the other in which it is consistent. Let us assume that Form(W) is inconsistent. Since (D, W) has an extension F, by Lemma A.2(a), it holds that F = Ths(W ) and Supp(W)= 0. In particular, since ( p : J ) is in F, we have that J = 0 and W U { ( p : J } } is well-defined. Moreover, given that Form(W U { ( p : J)}) is inconsistent, W U { ( p : J)} is well-defined and Supp(W)U J = 0, we can conclude, by Lemma A.2(a), that the theory (D, WU { ( p : J)}) has a single extension Ths(WU { ( p : J)}). Finally, from the hypothesis (p : J) E Ths(W), we get Ths(W ) = Ths(W U { ( p : J)}). Hence, the thesis holds. Let us assume that Form(W) is consistent. We prove the two parts of the equivalence separately. ( ~ ) A QDL extension E of (D, W) is a fixpoint of the operator F, i.e. E is the smallest set of assertions S such that:
(1) w c s , (2) S is deductively closed, i.e., Ths(S ) = S, (3) if (A : B/C)~ D, CA : { J 1 , . . . , J~}) c S, and {B} U Form(E) is consistent, then ( C : {J1,- • •, Jk, B}) E S. Moreover, E is well-defined. Since E contains ( p : J ), it is also the smallest set S such that the conditions (2) and (3) and the stronger condition (1') W U { ( p : J ) } C _ S hold. Hence, since E is well-defined, E is a QDL extension of (D, W U
{(p: J)}).
( ~ ) Assume that E is a QDL extension of (D, W U { ( p : J)}). According to the quasi-inductive definition of extension, there is a sequence E0, E 1. . . .
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such that
(1)
Eo=WU{(p:J)}
(2)
for i > 0,
,
El+ 1 = T h s ( E i ) U
{ ( C : { J 1 , . . . , J k , B})
(3)
( A : {J1 . . . . . Jk}) ~ Ei, {B} U Form(E) is consistent}, E = Ui=0,o~ Ei and E is well-defined. !
i
We define a sequence E 0, E l , . . . (1) (2)
I(A:B/C)eD,
as follows:
E 0 = W, for i > 0, E;+ 1 = Ths(E;) U
{(C: {J, . . . . , Jk, B}) ] ( a : B/C) E D, ( a : { J 1 , . . . , Jk}) E E ; ,
( B ) U Form(E) is consistent}, Moreover, let E ' = Ui=0,~ E~. We will prove that E is an extension of (D, W) by proving that E ' = E (of course, if this equality holds, E ' is also well-defined). To show that E' C_ E we prove by induction that E; C_ Ei for all i/> 0. The basis is immediate, since W C _ W U { ( p : J ) } , and the induction step is straightforward. To show that E C_ E' we prove by induction that E i C_E' for all i t> 0. The induction step is trivial. The difficult part is the basis, where we need to show W U { ( p : J)} C_ E'. Since W C_ E', it suffices to show ( p : J ) E E t"
By hypothesis, there is an extension F of (D, W) containing ( p : J ) . Hence, there is a sequence F 0, F 1 , . . . such that (1) (2)
Fo=W, for i > 0, Fi+ 1 = Ths(Fi) 0 {(C:
(3)
Jk, B}) I (A : B/C)e D,
(J1,..-,
( A : ( J , , . . . , Jk}) E F/, {B} tO Form(F) is consistent}, F = Ui=o,o, Fi and F is well-defined. t
t
We define a sequence Fo, F~ . . . . (1) (2)
as follows:
F o = W, for i > 0, F;+ 1 =
Ths(F;)
U
((C : {J,,...,
Moreover, let F ' = Ui=0.o, F;.
Jk, B } ) I ( A : B / C ) e D , ( A : {J, . . . . , Jk}) • F ; , {B} U Form(F) is consistent, B E J}.
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To show ( p : J) ~ E' and thereby to complete the proof, it will suffice to show that ( p : J) E F} for some j >/0 and that F; C_ E; for all i/> 0. To show that ( p : J) CF~ note that ( p : J) EFj for some j~>0. A simple induction on i shows that, whenever ( q : I) E F / a n d I C C_J, then ( q : I) E F ; . Thus, since ( p : J) E F j , also ( p : J) E F ; . It remains to show that F; C_ E~ for all i ~>0. The basis is trivial, since F 0 = E 0 = W. For the induction step, suppose ( q I ) E F i + ~. We want to prove that ( q : I ) E Ei+ ~. Given that I C_J, it suffices to show that, for all B E J, {B} U Form(E) is consistent. Since we are considering the case in which Form(W) is consistent, by Lemma A.2(b), Form(E) must be consistent. Let us assume that, for some B E J, {B} U Form(E) is inconsistent. We show that this leads to a contradiction. In fact, since Form(E) is deductively closed, we have that --riBE Form(E). Moreover, since {p : J) E E, B C J C Supp(E). Hence, B E Supp(E) and --riBE Form(E). This contradicts the hypothesis that E, as any Q D L extension, is well-defined. Therefore, for all B E J, {B} U Form(E) is consistent. [] t
t
.
t
t
Proposition 6.5. Let (D, W) be an assertion default theory such that W contains
only assertions of the form ( a : 0). Then, E is a QDL extension of (D, W) iff Form(E) is a Reiter' s default extension of the default theory (D, Form(W)), i.e. Form is a surjective mapping from the set of QDL extensions of (D, W) to the set of DL extensions of (D, Form(W)). Proof. We give a sketch of the proof. According to the quasi-inductive definition of Q D L extension, if E is a QDL extension of (D, W), there is a sequence E0, E 1. . . . such that (1) (2)
E o = W, for i > 0, El+ 1 = T h s ( E i ) U
(3)
{(C : { J , , . . . ,Jk, B } ) I ( A : B / C ) E D , ( A : {gl . . . . . Jk}) ~ Ei, {B} U Form(E) is consistent}, E = (.-Ji=0,o~ Ei and E is well-defined.
On the other hand, if F is a DL extension of the default theory (D, Form(W)), then (by the quasi-inductive definition of DL extension in [9, Theorem 2.1]) there is a sequence F0, F 1 , . . . such that (1') (2')
F 0 = Form(W), for i > 0,
Fi+1 = Th(Fi) O {C [ (A : B/C) @ D, A @ Fi, {B} U F is consistent}, (3')
F = [-Ji=0,o Fi.
We prove the two directions separately. ( ~ ) Assume that E is a Q D L extension of (D, W). We can easily show that
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F = F o r m ( E ) is a Reiter's default extension of the default theory (D, Form(W)). It suffices to define a sequence F0, F 1. . . . by taking F, = Form(Ei). It is not difficult to prove that this sequence satisfies conditions (1'), (2'), and (3') above. ( ~ ) Assume that F is a Reiter's default extension of the default theory (D, Form(W)), so that there is a sequence Fo, F 1 , . . . as defined above. We want to prove that there is a Q D L extension E of ( D , W ) , such that
F = Form(E). Let GDi(D , F) = {(A : B/C) E D ] A E Fi and {B} U F i s consistent) be the set of default rules applicable at the step i. We define the sequence E0, E1 . . . . of sets of assertions as follows: E 0 = W, for i > 0 , Ei+ 1 = T h s ( E i ) U
{(C: (J1 . . . . . Jk, B})I (A : B/C)~ GDi(D, F), (A : { J , , . . . , Jk}) E E,}. Moreover, let E = (--J~=0.o, Ei- By construction, it is clear that Form(Eg) = Fg. Hence, Form(E) = F. To prove that E is a Q D L extension of (D, W), we need to show that the sequence E0, E1 . . . . satisfies conditions (1), (2) and (3). (1) holds by construction. To see that (2) holds, we observe the following: on one hand, if (A : B/C) E GDi(D, F) then {B} U F is consistent, i.e., {B} U Form(E) is consistent; on the other hand, if {B} U Form(E) is consistent and (A : {J1 . . . . , Jk}) E E~, it holds that { B ) U F is consistent and A E E i, and thus ( A : B / C ) E
GD~(D, F). To see that (3) holds, it suffices to show that E is well-defined. This follows from the assumption that each assertion in W has the form ( a : 0 ) . All supports in the Q D L extension E come from justifications of applied default rules; moreover, the consistency of justifications of applied defaults with Form(E) is guaranteed by the applicability condition. []
Acknowledgement This work has been partially supported by the ESPRIT Basic Research Project 6471, Medlar II.
References [1] P. Besnard and T. Schaub, Possible worlds semantics for default logic, in: Proceedings Ninth
Biennial Conference of the Canadian Society for Computational Studies of Intelligence, Vancouver, BC (1992).
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[2] G. Brewka, Cumulative Default Logic: in defense of nonmonotonic inference rules, Artif. Intell. 50 (1991) 183-205. ]3] J. Delgrande and W. Jackson, Default logic revised, in: J.F. Allen, R.E. Fikes and E. Sandewall, eds., Proceedings 2nd International Conference on the Principles of Knowledge Representation and Reasoning, Cambridge, MA (1991) 118-127. [4] D. Gabbay, Theoretical foundations for non-monotonic reasoning in expert systems, in: K.R. Apt, ed., Logics and Models of Concurrent Systems (Springer, Berlin, 1985). [5] L. Giordano, Defining variants of default logic: a modal approach, in: Proceedings ISMIS-93 Seventh International Symposium on Methodologies for Intelligent Systems, Trondheim, Norway, Lecture Notes in Computer Science 689 (Springer, Berlin, 1993) 59-68. [6] W. Lucaszewicz, Considerations on default logic--an alternative approach, Comput. Intell. 4 (1988) 1-16. [7] D. Makinson, General theory of cumulative inference, in: M. Reinfrank, ed., Proceedings International Workshop on Non-Monotonic Reasoning, Lecture Notes in Artificial Intelligence 346 (Springer, Berlin, 1989) 1-18. [8] D. Poole, What the Lottery Paradox tells us about default reasoning, in: R.J. Brachman, H.J. Levesque and R. Reiter, eds., First International Conference on the Principles of Knowledge Representation and Reasoning, Toronto, Ont. (Morgan Kaufmann, San Mateo, CA, 1989) 333-340. [9] R. Reiter, A logic for default reasoning, Artif. lntell. 13 (1980) 81-132. [10] R. Reiter and G. Criscuolo, On interacting defaults, in: Proceedings IJCAI-81, Vancouver, BC (1981) 270-276. [11] T. Schaub, On commitment and cumulativity in default logics, in: R. Kruse, ed., Proceedings European Conference on Symbolic and Quantitative Approaches to Uncertainty (Springer, Berlin, 1991) 304-309. [12] T. Schaub, On constrained default theories, in: Proceedings ECAI-92, Vienna, Austria (1992) 304-308.
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ARTINT 1148
On cumulative default logics* Laura Giordano and Alberto Martelli Dipartimento di Informatica, Universitgt di Torino, C.so Svizzera 185, 10149 Torino, Italy Received June 1993 Revised November 1993
Abstract Giordano, L. and A. Martelli, On cumulative default logics, Artificial Intelligence 66 (1994) 161-179.
Recently cumulative variants of Reiter's default logic have been proposed. All of them happen to be semimonotonic and to commit to assumptions, i.e. to require consistency of justifications. In this paper we define two new cumulative variants of default logic in the style of Brewka's CDL, which show that the above two properties are not required to achieve cumulativity. In fact, the first variant commits to assumptions, like Brewka's CDL, but it is not semimonotonic, and hence it allows priorities among defaults to be represented. The other cumulative variant does not even commit to assumptions, and thus it is very close to Reiter's default logic.
I. Introduction A number of variants of Reiter's default logic (DL) [9] have recently been proposed [2, 3, 6, 11]. Such variants are motivated by the aim of getting some desirable properties that DL does not satisfy. For instance, it is well known that DL is not cumulative, i.e. the addition of theorems to the set of premises may change the derivable formulas, thus preventing the use of lemmas in proofs. Furthermore DL does not commit to assumptions, i.e. it is possible to have inconsistencies among justifications of nonnormal defaults, and this may lead to unintuitive results [8]. To overcome these problems, Brewka [2] has defined a Cumulative Default Logic (CDL) which is cumulative and commits to assumptions. To achieve this, Brewka introduces assertions, i.e. formulas labelled with a support, and strengthens the applicability conditions for defaults. However, CDL happens to be semimonotonic, which is often regarded as a desirable property from a Correspondence to: A. Martelli, Dipartimento di Informatica, Universit~ di Torino, C.so Svizzera 185, 10149 Torino, Italy. * Originally submitted as a Research Note. 0004-3702/94/$07.00 © 1994--Elsevier Science B.V. All rights reserved SSDI 0004-3702(93)E0095-4
162
L. Giordano, A. Martelli
computational point of view [6], but as a drawback from a representational point of view. In fact, as observed in [2], "semimonotonicity destroys part of the additional expressiveness of nonnormal defaults", for instance, it makes it impossible to represent priorities among defaults. To regain this ability, Brewka shows how some unwanted C D L extensions can be filtered out. Other cumulative variants of default logic, such as Constrained Default Logic [11] and J-Default Logic [3], turn out to be equivalent. These variants do not make use of assertions as CDL does, but (in the style of Lukaszewicz's default logic [6]) they allow extensions as pairs of sets of formulas (E, C), where C is the context supporting the beliefs in E. In spite of this difference, these logics are very similar to C D L and an equivalence result between Constrained Default Logic and CDL is given in [12]. In this paper, we define two variants of Reiter's default logic that are cumulative like CDL, but closer to DL. We compare these variants to CDL and D L by analysing their properties of semimonotonicity and commitment to
assumptions. The first variant we propose is called CA default logic or C A D L , Jr Commitment to Assumptions Default Logic). C A D L is cumulative and commits to assumptions, like CDL, but it is not semimonotonic. To achieve this, in C A D L only justifications are recorded and checked for consistency, and not consequents of defaults as in CDL. Since C A D L is not semimonotonic, there is no loss in the expressiveness of nonnormal defaults, and priorities among defaults can be represented. For this reason, filtering out solutions as in [2] is not needed in this case. The other cumulative variant we define is even closer to Reiter's DL, since it does not require joint consistency of justifications. We will call it Q D L (Quasi-Default Logic). Like DL and C A D L , but differently from CDL, Q D L is not semimonotonic. Moreover, like DL, but differently from CDL and C A D L , it does not commit to assumptions. In Sections 2 and 3 we recall the definition of DL and CDL, respectively, and discuss some examples. In Section 4 we define C A D L and prove that it is cumulative, and in Section 5 we show how priorities among defaults can be represented in this logic. In Section 6 we introduce the other variant, QDL.
2. Reiter's default logic In this section we recall the definition of Reiter's D L and we discuss the properties of cumulativity, semimonotonicity, and commitment to assumptions that will be used throughout the paper. For simplicity, in this and the following sections, we will restrict our attention to the propositional case. A default theory is defined as a pair (D, W), where D is a set of default rules of the form A : B/C and W is a set of well-formed-formulas (wffs). A is called
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the prerequisite, B the justification, and C the consequent of the default rule. We consider default rules with a single justification. The notion of extension of a default theory is defined in the following way. Definition 2.1 (Reiter [9]). A DL-extension of a default theory (D, W) is a fixpoint of the operator F which, given the set of wffs S, produces the smallest set of wffs S' such that: (1) WC_S', (2) S' is deductively closed, i.e., Th(S') = S', (3) if (A : B / C ) @ D, A E S', and --nB~S, then C C S'. It is well known that, with this definition of default extension, mutually inconsistent justifications can be applied together to get an extension. Consider the following example from [8]. Example 2.2. The default theory (D, W) with D =
: left-hand-usable/x 7left-hand-broken left-hand-usable ' • right-hand-usable/x 7 right-hand-broken~ right-hand-usable J
W = {left-hand-broken v right-hand-broken} ,
has a single (Reiter's) default extension E = Th({left-hand-broken v right-hand-broken, left-hand-usable, right-hand-usable}),
which is obtained by applying both defaults, though their justifications are mutually inconsistent, given W. The problem here is that in default logic assumptions can be made implicitly, without considering their possible side-effects, i.e. given the assumption --nlefthand-broken (which is used by the first default) and the set of formulas W, we cannot conclude right-hand-broken, so as to block the second default• In essence, there is no commitment to assumptions. For a more detailed discussion on this, see [8]. It has been pointed out by Makinson [7] that DL is not cumulative. Cumulativity is one of the general properties that a nonmonotonic inference relation is expected to satisfy [4, 7]. A nonmonotonic inference relation ~ is said to be cumulative if it satisfies the following condition: If A ~ x , then (A ~ y iff A U {x} ~ y ) . This property is quite useful since it allows for the generation of lemmas. For
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default logic, when a non-skeptical notion of derivability is used, Brewka formulates cumulativity as follows. Let (D, W) be a default theory, if there is an extension F of (D, W) containing y, then E is an extension of (D, W) containing y iff E is an extension of ( D, W tO {y}). Reiter's DL is not cumulative. Example 2.3. (Makinson [7]). D={: p pvq'-qp} P ~P
W={}
(D, W) has a unique (Reiter's) extension E = Th({p}). Hence, p v q E E. However, if p v q is added to W, the resulting default theory has two extensions: E 1 = Th({p}) and E 2 = Th({~p, q}). As regards semimonotonicity, a default logic is said to be semimonotonic if, given a default theory (D, W) and a set of default rules D', such that D C D', for every extension E of (D, W) there is an extension E' of (D', W) that contains it, i.e. an extension E' such that E C E'. In essence, no extension of (D, W) can be ruled out by the addition of new default rules. While semimonotonicity holds for normal default theories (see [9]), whose default rules have the form A : B/B, it does not hold in the general case. Example 2.4.
D=
:TB}
---5-'
W={}
(D, W) has a unique DL extension E = Th({C}). Let us assume that an additional default is added, so that D , = { : ~ B :--qA} C ' B " (D', W) has a unique DL extension E ' = Th({B}), obtained by applying the second default. Since E is not contained in E', DL is not semimonotonic. In the next section, we will come back to these examples to see how Brewka's CDL commits to assumptions, is cumulative and semimonotonic.
3. Brewka's cumulative default logic
In Brewka's CDL, formulas are associated with supports by allowing assertions of the form ( p : {r 1. . . . . rn} >, where r I . . . . . r n are formulas supporting the belief in p. Given a set of assertions W, Form(W) denotes the asserted
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165
formulas of W, i.e., Form(W) = {p : ( p : { r l , . . . , rn}) E W } , and Supp(W) denotes the supports of W, i.e.,
Supp(W) = ( r : ( p : ( r l , . . . ,
r,...,
rn} ) E W}.
Since we consider only the propositional case, we assume that only propositional formulas can occur in the set of assertions W and in default rules. An assertion default theory is a pair (D, W), where D is a set of defaults in the sense of Reiter and W is a set of assertions. In order to define CDL extensions, Brewka generalizes the notion of classical deductive closure to sets of assertions. Let A be a set of assertions. Ths(A), the set of supported theorems of A, is the smallest set such that (1) A C__Ths(A), (2) if ( p , : J1) . . . . , (Pn: Jn) E Ths(A ) and p ~ , . . . , • ." O J~) E Ths(A),
p,,kq, then ( q : J~ U
where k is derivability in classical logic. In the following we will omit the subscript S in Ths(A ) when it is clear that A is a set of assertions. The notion of extension of an assertion default theory is then defined in the following way. Definition 3.1 (Brewka [2]). A CDL extension of an assertion default theory (D, W) is a fixpoint of the operator F which, given the set of assertions S, produces the smallest set of assertions S' such that: (1) W C S ' , (2) S' is deductively closed, i.e., Ths(S' ) = S', (3) if ( A : B / C ) ~ D , ( A : { J 1 . . . . . J k } ) E S ' , and ( B , C } U F o r m ( S ) U Supp(S) is consistent, then (C : (J1 . . . . , Jk, B, C}) C S'. Notice that in the definition above both the justification and the consequent of each applied default are checked for consistency and then recorded in the support of the added assertion. Since to apply a default its justification is checked for consistency with all the supports of the extension, default rules with mutually inconsistent justifications cannot be applied together to get an extension, i.e. CDL commits to assumptions. Example 3.2. The assertion default theory (D, W) with
D
(
: left-hand-usable ^ mleft-hand-broken left-hand-usable : right-hand-usable ^ -7right-hand-broken
W = { (left-hand-broken v right-hand-broken : { } )} ,
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has two CDL extensions---El, obtained by applying the first default rule and containing the assertion
( left-hand-usable : {left-hand-usable ^ ~left-hand-broken, left-hand-usable} ) , and E2, obtained by applying the first default rule and containing the assertion
( right-hand-usable : { right-hand-usable ^ --1right-hand-broken, right-hand-usable } ) . Brewka proves that CDL is cumulative (see [2, Proposition 2.13]): if there is an extension F of (D, W) containing ( p : J), then E is a CDL extension of (D, W) containing ( p : J ) iff E is a CDL extension of (D, W U { ( p : J ) ) ) . Consider again Example 2.3, on which DL fails cumulativity.
Example 3.3. D={: p pvq:~p) p -np
W={}
(D, W) has a unique CDL extension E ' = T h ( { ( p : { p } ) ) ) . Hence, ( p v q : {p}> E E'. Adding ( p v q : {p}) to W gives a new assertion default theory still having E' as the unique extension. Indeed, the default p v q : 7 p / - ~ p cannot be applied, since the justification -np is inconsistent with the support {p} o f p v q. Since, in CDL, consequents of default are recorded and tested for consistency, defaults are implicitly regarded as being seminormal: a default A : B / C can be equivalently replaced by the default A : B ^ C/C. This also leads to semimonotonicity. In fact, Brewka proves (see [2, Proposition 2.11]) that CDL is semimonotonic. If we go back to Example 2.4, we can see that it is not a counterexample to monotonicity for CDL.
Example 3.4. D=
:-IB
W=(}
(D, W) has a unique CDL extension E = Th({( C : {-riB, C})}). Let us assume that an additional default is added so that D' = { : T B / C , :-~A/B}. (D', W) has two CDL extensions E 1 = T h ( { ( C : {--riB, C}>}) and E 2 = T h ( { ( B : {-hA, B}}}). Since E I coincides with E, in particular, it contains E. Semimonotonicity is recognized by Brewka as being a drawback rather than an advantage, since it limits the expressiveness of nonnormal defaults. The same can be said with regard to having implicit seminormal default rules. This matter will be discussed in detail in Section 5.
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4. CA default logic In this section, we define a cumulative variant of default logic, Commitment to Assumption Default Logic (CADL), which is not semimonotonic, and in which defaults are not implicitly regarded as seminormal. Like CDL, CADL also requires mutually consistent justifications. We will show in Section 5 that this variant allows priorities among defaults to be represented, so that filtering out extensions as in [2] is not needed. In defining CA default logic (CADL) we follow the approach proposed by Brewka. We refer to the notions of assertion default theory and set of supported theorems of A (Ths(A)) defined in [2] and recalled in Section 3. CADL extensions are defined as follows. Definition 4.1. A C A D L extension of an assertion default theory (D, W) is a fixpoint of the operator F which, given the set of assertions S, produces the smallest set of assertions S' such that:
(1) WC_S', (2) S' is deductively closed, i.e., Ths(S' ) -- S', (3) if ( A : B / C ) E D , (A:(J1,...,Jk})ES' , and ( B } U F o r m ( S ) U Supp(S) is consistent, then ( C : {Jl . . . . . J~, B}) E S'.
Note that in the definition above only the justifications of applied defaults are recorded. Accordingly, only the justifications, and not consequents of defaults, are tested for consistency in order to apply defaults, similarly to Reiter's default logic. Hence, defaults are not implicitly regarded as seminormal. When we want defaults to behave as seminormal, then we must explicitly write them as seminormal. By comparing the above definition with the definition of CDL extension (Definition 3.1), it comes out that, if seminormal defaults are used, then CADL obviously collapses to CDL, i.e. the CADL extensions of a seminormal theory coincide with its CDL extensions. Though in [2] it is observed that recording consequents of applied defaults is actually needed to have cumulativity, this variant of default logic can be proved to be cumulative. The reason for this is that, in the definition of CADL extension, consequents are neither recorded nor tested for consistency. As usual, an equivalent quasi-inductive definition of CADL extensions can be given (the proof of the equivalence is quite the same as the one for CDL in [21),
Definition 4.2. Let (D, W) be an assertion default theory and E a set of assertions. Define E 0 = W and, for i t> 0,
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Ei+ 1 =
Ths(Ei) U
((C: {J1,..-, Jk, B}) I(A : B/C)~D, ( A : { J 1 , . . . , Jk}) ~ Ei and {B} U Form(E) U Supp(E) is consistent}. E is a CADL extension of (D, W) iff E = Ui=0,~ Ei. By making use of this definition it can be proved that CADL is cumulative. Proposition 4.3. CADL is cumulative. lh'o~ff. The proof of this proposition is quite the same as the proof of cumulativity for CDL (see [2, Proposition 2.13]). It makes use of the notion of well-based assertion default theory defined in [2, Definition 2.6]) and of the Lemmas 2.7 and 2.8 in [2]. These lemmas also hold for CADL and have the same proof as given in [2]. [] For instance, the default theory (D, W) in Example 3.3, which contains default rules which are all normal, has a unique CADL extension E ' = T h ( { ( p : { p } ) ) , which coincides with its unique CDL extension. Hence, ( p v q : { p } ) E E ' . As in CDL, adding ( p v q : { p } ) to W gives a new assertion default theory still having E' as the unique extension. It is easy to see that CA default logic is not semimonotonic. As for Reiter's DL, a counterexample to semimonotonicity of CADL is given by Example 3.4. (D, W), with D = {: 7B/C} and W = (}, has a unique CADL extension E = Th({(C: ( ~ B } ) } ) . If the additional default : ~ A / B is added, the default theory (D', W), with D, = { :--qB : ~ Z } C ' B ' has a unique CADL extension E' = Th({(B : {--qA})}), obtained by applying the second default. Since E is not contained in E', CADL is not semimonotonic. From this example, it is quite clear that CADL is more similar to Reiter's DL than Brewka's CDL and other cumulative variants of DL proposed in the literature. CADL is cumulative but, like default logic, it is not semimonotonic. In the next section we will show how this allows priorities among defaults to be expressed. Another similarity of CADL with default logic is that a default theory is not guaranteed to have a CADL extension. For instance, the theory (D, W), with D = {:A/-qA,:A/B} and W = { }, has no CADL extension (and no DL extension), while it has a unique CDL extension E = Th({ (B : {A, B } )}). Of course, differently from default logic but similarly to CDL, in CADL defaults with mutually inconsistent justifications cannot be jointly applied, i.e. CADL
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commits to assumptions. The assertion default theory (D, W) in Example 3.2 has two CADL extensions: E 1 containing (left-hand-usable: {left-handusable ^ -nleft-hand-broken} ) and E 2 containing (right-hand-usable : {righthand-usable ^ -7 right-hand-broken} ).
5. Representing priorities among defaults In this section we want to show that CADL recovers the ability to represent priorities among defaults which is lost in CDL (see [2, Section 3]). Of course, priorities among defaults cannot be represented in CADL by making use of seminormal defaults as usual in Reiter's default logic [10], since for seminormal theories CADL extensions are the same as CDL ones. Another way to represent priorities is used. In this section we will also see that in CADL we do not get the same extensions obtained by Brewka by filtering out CDL extensions. Consider the following example, taken from [2] and here slightly modified.
Example 5.1. D=
(
student : -7 married 1) -nmarried ,
(3)
adult : married (2) married '
living-in-college : student student ,
(4)
beard : adult -adu~ )'
W = {living-in-college, beard} .
In Reiter's DL (D, W) has two different extensions, E 1 = Th({living-in-college, beard, student, adult, married}), E 2 = Th({living-in-college, beard, student, adult, ~ m a r r i e d } ) .
Let us assume that we want both student and adult to be concluded, but, since defaults (1) and (2) lead to mutually inconsistent conclusions, we want default (1) to have a higher priority than default (2). To achieve this purpose, default (2) can be replaced with the seminormal default (see [10]): (2')
adult : married ^ --qstudent married
The resulting theory has a single DL extension, E2, as wanted, since applying default (3) blocks (2'). This same theory, however, has two CDL extensions: F 1 containing (adult : ( a d u l t } ) , (married : (adult, married ^ -qstudent} ) and F2 containing (student: {student}), (adult : {adult}), (~married : {student,-nmarried}), since in CDL (3) blocks (2'), but (2') blocks (3) in turn. Hence, in CDL, using seminormal defaults does not allow to enforce priorities among default rules (1) and (2), as in Reiter's default logic.
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Indeed, since CDL is semimonotonic it is not possible that the introduction of a new default rule in a theory (D, W) can rule out some of its previous extensions. Hence, as noted in [2], it is not possible to define a default rule with higher priority than others and this destroys the additional expressive power of nonnormal defaults. Since, on the contrary, C A D L is not semimonotonic, it is possible that a new default can rule out some of the previous extensions and, then, it is possible to define priorities among defaults. However, this cannot be done by making use of seminormal defaults as in DL. In fact, as told above, for seminormal defaults C A D L gives the same results as CDL. For instance, C A D L extensions of the default theory in Example 5.1 are exactly the same as CDL extensions. In order to express priorities among defaults in C A D L , defaults which are not seminormal must be used. The default theory in Example 5.1 can be reformulated in the following way: ( D=
1) (3')
student : -7 married 7married ,
(2")
living-in-college: student student A ab-adult ,
adult : married ^ -7 ab-adult married ' (4)
beard - adult adult J'
with W = {
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CDL v, i.e. CDL with the additional filter on extensions, is not semimonotonic, but commits to assumptions and is cumulative like CDL. Thus it has the same properties as CADL. However, it is difficult to compare the two proposals because in CDL F the property of being nonsemimonotonic is achieved "a posteriori", by filtering out some extensions, while in CADL it is a direct consequence of the applicability criterion for defaults. We have shown in Example 5.1 that CDL F is closer to DL than CADL, but it is easy to find other cases where the opposite holds. For instance, the default theory (D, W) with D=
[:B :A} B '--7B '
W={},
has a unique CADL extension E 1 = Th({(-TB:{A})}), corresponding to the unique DL extension. Instead, CDL v has also the extension E2= Th({ (B : {B} )}), which is priority-preserving and, thus, it is not filtered out in CDL v. This happens because applicability of defaults in CDL requires consistency of consequents, differently from CADL and DL.
6. Getting closer to Reiter's default logic
In this section we propose another cumulative variant of default logic which is even closer to Reiter's default logic than CADL. This variant, like CADL, is not semimonotonic, but, differently from CADL, does not require joint consistency of justifications. An alternative variant of default logic, which is also cumulative and does not require joint consistency of justifications, has been proposed by Makinson (and referred to in [2]). This variant, however, requires to modify the notion of assertion, since both justifications and consequents of applied defaults must be recorded and kept separate. On the contrary, in this section we will show that a cumulative variant of DL which does not require joint consistency of justifications can be defined by only recording justifications (and thus keeping the notion of assertion as defined in [2]). Since the definition of extension for this variant, apart from remembering supports, is very close to the one for DL, we will call this variant QDL (Quasi-Default Logic). Let us define the notion of QDL extension. Definition 6.1. A set S of assertions is well-defined if there is no formula a such that a C Supp(S) and -7a ~ Form(S). Definition 6.2. Let (D, W) be an assertion default theory and let F be an operator that, given the set of assertions S, produces the smallest set of assertions S' such that:
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(1) W C S ' , (2) S' is deductively closed, i.e., Ths(S' ) = S', (3) if ( A : B / C ) ~ D , ( A : { J 1 , . . . ,Jk}) E S ' , and { B } U F o r m ( S ) is consistent, then ( C : {J1, • • •, Jk, B}> E S'. E is a Q D L extension of (D, W) iff E is a fixpoint of the operator F and E is well-defined. Note that, as a difference with CDL and CADL, recorded supports are not used to determine the applicability of default rules, but only to guarantee that the extension is well-defined. An equivalent quasi-inductive definition of Q D L extensions is the following (the proof of equivalence is the same as the one for DL in [2]).
Definition 6.3. Let (D, W) be an assertion default theory and E a set of assertions. Define E 0 = W and, for i > 0, Ei+l = Ths(Ei) U
{ ( C : {J1 . . . . , Jk, B}> I(A : B / C ) E D , ( A : {J1,. . " , J~}> E E i and {B} U Form(E) is consistent} . E is a Q D L extension of (D, W) iff E = Ui=0,~ Ei and E is well-defined. In QDL, joint consistency of justifications is not required. Consider again Example 3.2. The assertion default theory (D, W) has a single Q D L extension E 1 containing both the assertions (left-hand-usable : Cleft-hand-usable ^ ~lefthand-broken } > and ( right-hand-usable : { right-hand-usable A --1right-handbroken}) corresponding to the single (Reiter's) default extension E = Th({lefthand-usable, right-hand-usable}). This variant of default logic can be proved to be cumulative.
Proposition 6.4. Q D L is cumulative. The proof can be found in Appendix A. Let us now compare Q D L with Reiter's default logic. If in the definition of Q D L extension the condition of well-definedness were omitted, then Q D L extensions would coincide with Reiter's extensions, but for the presence of (unused) supports of formulas. Requiring a Q D L extension E to be well-defined is essential to get cumulativity. Consider again Example 2.3 which shows that Reiter's default logic fails cumulativity. The default theory (D, W) with D = { :p p
pvq!-'qPl "qp J
W={}
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has a unique QDL extension E = Th({(p: {p})}). Hence, ( p v q : {p}) E E. Adding ( p v q : {p}) to W gives a new assertion default theory (D, W'), having E as its unique QDL extension. E ' = Th({(p v q: {p}), (rap : {p,-np})}) is not a QDL extension since it does not satisfy the well-definedness condition. However, if this condition is omitted, E' is an extension of (D, W') and this is a counterexample to cumulativity. The well-definedness condition allows to get rid of unwanted extensions in which some of the derived formulas are the negation of formulas in the supports. Note however that this condition is only introduced to deal properly with initial assertions; for the assertion default theories in which W has empty supports this condition is not really needed. The next proposition shows that the results of QDL and DL are equivalent if all the formulas in W have empty supports, that is, QDL extensions of an assertion default theory (D, W) with empty supports coincide (apart from the supports) with Reiter's default extensions of the default theory (D, Form(W)).
Proposition 6.5. Let (D, W) be an assertion default theory such that W contains only assertions of the form (a :0). Then, E is a QDL extension of(D, W) iff Form(E) is a Reiter's default extension of the default theory (D, Form(W)), i.e. Form is a surjective mapping from the set of QDL extensions of (D, W) to the set of DL extensions of (D, Form(W)).
The proof can be found in Appendix A. The fact that QDL is cumulative is in perfect agreement with the result in [12] that also for Reiter's default logic suitable lemma default rules can be defined, which can be added to the set of defaults without changing the extensions of the default theory. Then, in a sense, DL itself can be considered to be cumulative. Indeed, if a belongs to an extension E of a default theory (D, W), then by Proposition 6.5 above there must be an extension E' (with Form(E') = E) of the corresponding assertion default theory (D, W') with empty supports such that ( a : {Jx . . . . , Jn}) belongs to E', for some J l , . . . , Jn. A lemma default rule 6~ for a, as defined by Schaub in [12], can be simply given as follows: :Jl''''' OL
Jn
As noted by Schaub, the addition of an assertion is stronger than the addition of a lemma default rule. While adding the assertion ( a : (jl . . . . . Jn } ) eliminates all the extensions that are inconsistent with the assertion, adding the lemma default rule does not change the extensions of the theory.
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7. Conclusions In this paper we have defined a cumulative variant of default logic, called CADL, which is not semimonotonic and requires consistency of justifications. We have shown that priorities among defaults can be represented in CADL, while, as shown in [2], they cannot in CDL. In CADL defaults are not implicitly regarded as seminormal. For seminormal default theories CADL and CDL give the same results. We have also presented an additional cumulative variant, QDL, which is very close to DL itself and does not require consistency of justifications. In this paper, we have not addressed the problem of defining a semantics for these variants. Recently, a uniform semantic framework has been defined by Besnard and Schaub [1], by giving the semantics of various default logics in terms of Kripke structures. It is possible to see that, within their framework, Kripke semantics for CADL and QDL can be defined too. Also, in [5] a uniform syntactic characterization of the different variants of Reiter's DL has been defined by making use of a modal approach. A characterization for CADL and QDL can also be accommodated in this framework. To conclude, we want to point out that, while in this paper we have followed the approach proposed by Brewka in defining these variants, we also could have defined them in the style of Schaub, Delgrande and Jackson, and Lucaszewicz. That is, instead of allowing assertions in the language, we could have defined extensions as pairs (E, C), where C is a set of formulas supporting the beliefs in E.
Appendix A. Proofs In order to prove cumulativity for QDL, we need the following lemmas. Lemma A.1. Let A = (D, W) be an assertion default theory. If (D, W) has a Q D L extension, then W is well-defined. Proof. If (D, W) has a QDL extension E, then W C_E. If W is not well-defined E is not well-defined. This contradicts the fact E is a QDL extension. [] Lemma A.2. Let A = (D, W) be an assertion default theory. The following holds. (a) Let Form(W) be inconsistent. If W is well-defined and Supp( W) = ~), then (D, W) has a unique QDL extension E = Ths(W ) such that Form(E) is inconsistent; otherwise, (D, W) has no QDL extension. (b) If (D, W) has a QDL extension E such that Form(E) is inconsistent, then Form(W) is inconsistent.
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Proof. (a) If Form(W) is inconsistent, then, since W must be contained in any Q D L extension of (D, W), no default is applicable. Hence, if E = Th(W) is well-defined, it is the unique QDL extension of (D, W) and Form(E) is inconsistent. Note that, when Form(W) is inconsistent, E = Th(W) is welldefined if and only if W is well-defined and Supp(W) = O. (b) If (D, W) has a QDL extension E such that Form(E) is inconsistent, then no default has been applied to get E. Hence, E = Th(W), and, in particular, Form(E)= Th(Form(W)). Thus, if Form(E) is inconsistent, also Form(W) must be inconsistent. []
Proposition 6.4. QDL is cumulative. Proof. We have to prove that, given an assertion default theory (D, W), if there is a QDL extension F of (D, W) containing ( p : J ) , then E is a Q D L extension of (D, W) containing ( p : J ) iff E is a QDL extension of (D, W U
{(p:J))). As hypothesis there is a QDL extension F of (D, W) containing ( p : J ) . Hence, by Lemma A.1, W must be well-defined. Two cases are considered: the one in which Form(W) is inconsistent, and the other in which it is consistent. Let us assume that Form(W) is inconsistent. Since (D, W) has an extension F, by Lemma A.2(a), it holds that F = Ths(W ) and Supp(W)= 0. In particular, since ( p : J ) is in F, we have that J = 0 and W U { ( p : J } } is well-defined. Moreover, given that Form(W U { ( p : J)}) is inconsistent, W U { ( p : J)} is well-defined and Supp(W)U J = 0, we can conclude, by Lemma A.2(a), that the theory (D, WU { ( p : J)}) has a single extension Ths(WU { ( p : J)}). Finally, from the hypothesis (p : J) E Ths(W), we get Ths(W ) = Ths(W U { ( p : J)}). Hence, the thesis holds. Let us assume that Form(W) is consistent. We prove the two parts of the equivalence separately. ( ~ ) A QDL extension E of (D, W) is a fixpoint of the operator F, i.e. E is the smallest set of assertions S such that:
(1) w c s , (2) S is deductively closed, i.e., Ths(S ) = S, (3) if (A : B/C)~ D, CA : { J 1 , . . . , J~}) c S, and {B} U Form(E) is consistent, then ( C : {J1,- • •, Jk, B}) E S. Moreover, E is well-defined. Since E contains ( p : J ), it is also the smallest set S such that the conditions (2) and (3) and the stronger condition (1') W U { ( p : J ) } C _ S hold. Hence, since E is well-defined, E is a QDL extension of (D, W U
{(p: J)}).
( ~ ) Assume that E is a QDL extension of (D, W U { ( p : J)}). According to the quasi-inductive definition of extension, there is a sequence E0, E 1. . . .
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such that
(1)
Eo=WU{(p:J)}
(2)
for i > 0,
,
El+ 1 = T h s ( E i ) U
{ ( C : { J 1 , . . . , J k , B})
(3)
( A : {J1 . . . . . Jk}) ~ Ei, {B} U Form(E) is consistent}, E = Ui=0,o~ Ei and E is well-defined. !
i
We define a sequence E 0, E l , . . . (1) (2)
I(A:B/C)eD,
as follows:
E 0 = W, for i > 0, E;+ 1 = Ths(E;) U
{(C: {J, . . . . , Jk, B}) ] ( a : B/C) E D, ( a : { J 1 , . . . , Jk}) E E ; ,
( B ) U Form(E) is consistent}, Moreover, let E ' = Ui=0,~ E~. We will prove that E is an extension of (D, W) by proving that E ' = E (of course, if this equality holds, E ' is also well-defined). To show that E' C_ E we prove by induction that E; C_ Ei for all i/> 0. The basis is immediate, since W C _ W U { ( p : J ) } , and the induction step is straightforward. To show that E C_ E' we prove by induction that E i C_E' for all i t> 0. The induction step is trivial. The difficult part is the basis, where we need to show W U { ( p : J)} C_ E'. Since W C_ E', it suffices to show ( p : J ) E E t"
By hypothesis, there is an extension F of (D, W) containing ( p : J ) . Hence, there is a sequence F 0, F 1 , . . . such that (1) (2)
Fo=W, for i > 0, Fi+ 1 = Ths(Fi) 0 {(C:
(3)
Jk, B}) I (A : B/C)e D,
(J1,..-,
( A : ( J , , . . . , Jk}) E F/, {B} tO Form(F) is consistent}, F = Ui=o,o, Fi and F is well-defined. t
t
We define a sequence Fo, F~ . . . . (1) (2)
as follows:
F o = W, for i > 0, F;+ 1 =
Ths(F;)
U
((C : {J,,...,
Moreover, let F ' = Ui=0.o, F;.
Jk, B } ) I ( A : B / C ) e D , ( A : {J, . . . . , Jk}) • F ; , {B} U Form(F) is consistent, B E J}.
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To show ( p : J) ~ E' and thereby to complete the proof, it will suffice to show that ( p : J) E F} for some j >/0 and that F; C_ E; for all i/> 0. To show that ( p : J) CF~ note that ( p : J) EFj for some j~>0. A simple induction on i shows that, whenever ( q : I) E F / a n d I C C_J, then ( q : I) E F ; . Thus, since ( p : J) E F j , also ( p : J) E F ; . It remains to show that F; C_ E~ for all i ~>0. The basis is trivial, since F 0 = E 0 = W. For the induction step, suppose ( q I ) E F i + ~. We want to prove that ( q : I ) E Ei+ ~. Given that I C_J, it suffices to show that, for all B E J, {B} U Form(E) is consistent. Since we are considering the case in which Form(W) is consistent, by Lemma A.2(b), Form(E) must be consistent. Let us assume that, for some B E J, {B} U Form(E) is inconsistent. We show that this leads to a contradiction. In fact, since Form(E) is deductively closed, we have that --riBE Form(E). Moreover, since {p : J) E E, B C J C Supp(E). Hence, B E Supp(E) and --riBE Form(E). This contradicts the hypothesis that E, as any Q D L extension, is well-defined. Therefore, for all B E J, {B} U Form(E) is consistent. [] t
t
.
t
t
Proposition 6.5. Let (D, W) be an assertion default theory such that W contains
only assertions of the form ( a : 0). Then, E is a QDL extension of (D, W) iff Form(E) is a Reiter' s default extension of the default theory (D, Form(W)), i.e. Form is a surjective mapping from the set of QDL extensions of (D, W) to the set of DL extensions of (D, Form(W)). Proof. We give a sketch of the proof. According to the quasi-inductive definition of Q D L extension, if E is a QDL extension of (D, W), there is a sequence E0, E 1. . . . such that (1) (2)
E o = W, for i > 0, El+ 1 = T h s ( E i ) U
(3)
{(C : { J , , . . . ,Jk, B } ) I ( A : B / C ) E D , ( A : {gl . . . . . Jk}) ~ Ei, {B} U Form(E) is consistent}, E = (.-Ji=0,o~ Ei and E is well-defined.
On the other hand, if F is a DL extension of the default theory (D, Form(W)), then (by the quasi-inductive definition of DL extension in [9, Theorem 2.1]) there is a sequence F0, F 1 , . . . such that (1') (2')
F 0 = Form(W), for i > 0,
Fi+1 = Th(Fi) O {C [ (A : B/C) @ D, A @ Fi, {B} U F is consistent}, (3')
F = [-Ji=0,o Fi.
We prove the two directions separately. ( ~ ) Assume that E is a Q D L extension of (D, W). We can easily show that
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F = F o r m ( E ) is a Reiter's default extension of the default theory (D, Form(W)). It suffices to define a sequence F0, F 1. . . . by taking F, = Form(Ei). It is not difficult to prove that this sequence satisfies conditions (1'), (2'), and (3') above. ( ~ ) Assume that F is a Reiter's default extension of the default theory (D, Form(W)), so that there is a sequence Fo, F 1 , . . . as defined above. We want to prove that there is a Q D L extension E of ( D , W ) , such that
F = Form(E). Let GDi(D , F) = {(A : B/C) E D ] A E Fi and {B} U F i s consistent) be the set of default rules applicable at the step i. We define the sequence E0, E1 . . . . of sets of assertions as follows: E 0 = W, for i > 0 , Ei+ 1 = T h s ( E i ) U
{(C: (J1 . . . . . Jk, B})I (A : B/C)~ GDi(D, F), (A : { J , , . . . , Jk}) E E,}. Moreover, let E = (--J~=0.o, Ei- By construction, it is clear that Form(Eg) = Fg. Hence, Form(E) = F. To prove that E is a Q D L extension of (D, W), we need to show that the sequence E0, E1 . . . . satisfies conditions (1), (2) and (3). (1) holds by construction. To see that (2) holds, we observe the following: on one hand, if (A : B/C) E GDi(D, F) then {B} U F is consistent, i.e., {B} U Form(E) is consistent; on the other hand, if {B} U Form(E) is consistent and (A : {J1 . . . . , Jk}) E E~, it holds that { B ) U F is consistent and A E E i, and thus ( A : B / C ) E
GD~(D, F). To see that (3) holds, it suffices to show that E is well-defined. This follows from the assumption that each assertion in W has the form ( a : 0 ) . All supports in the Q D L extension E come from justifications of applied default rules; moreover, the consistency of justifications of applied defaults with Form(E) is guaranteed by the applicability condition. []
Acknowledgement This work has been partially supported by the ESPRIT Basic Research Project 6471, Medlar II.
References [1] P. Besnard and T. Schaub, Possible worlds semantics for default logic, in: Proceedings Ninth
Biennial Conference of the Canadian Society for Computational Studies of Intelligence, Vancouver, BC (1992).
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