cumulative default logic: in defense of nonmonotonic inference rules

Artificial Intelligence 50 (1991) 183-205 183 Elsevier

Cumulative Default Logic: in defense of nonmonotonic inference rules*

G e r h a r d B r e w k a

GMD, Postfach 12 40, D 5205 Sankt Augustin, Germany

Received March 1990 Revised July 1990

Abstract

Brewka, G., Cumulative Default Logic: in defense of nonmonotonic inference rules, Artificial Intelligence 50 (1991) 183-205.

Two problems of Reiter's default logic have recently been discussed in the literature: first, inconsistencies between justifications of nonnormal defaults may lead to unintuitive results, and second, default logic is not cumulative, i.e., the addition of theorems to the set of premises may change the derivable formulas. To solve these two problems we strengthen the applicability condition for defaults and make the reasons for believing something an explicit part of the derived formulas. The resulting new logic turns out to be semimonotonic. If the additional expressiveness of nonnormal defaults is to be retained only some of the extensions of this logic are to be taken as acceptable sets of beliefs, however: those preserving priorities between defaults.

1. Introduction

Reiter's default logic, DL, [12] is one of the most prominent formaliza- tions of nonmonotonic reasoning. One of the reasons for its attractiveness certainly is the simplicity and naturalness of its underlying idea, namely to represent defaults as a certain type of inference rules whose applicability does not only depend on the derivability, but also on the underivability of some formulas.

In DL the user has to specify a set D of defaults of the form A:B/C with the

* This research was supported by the German Ministry for Research and Technology within project TASSO (Grant No. ITW8900A7).

0004-3702/91/$03.50 © 1991 - - Elsevier Science Publishers B.V.

184 G. Brewka

intuitive meaning "if A is provable and ~ B is not, then derive C". t A is called the prerequisite, B the justification, and C the consequent of the default. A default is called normal if B and C are logically equivalent, seminormal if B logically implies C. A set of defaults D together with a set of first-order formulas W form a default theory (D, W). Default theories induce sets of "acceptable beliefs", so-called extensions. In this paper we present only the definitions for closed default theories where the defaults contain no free variables. The generalization to arbitrary default theories is possible as de- scribed in [12]. The extensions of closed default theories are defined as fixed points of an operator F:

Definition 1.1 (Reiter) [12]. Let (D, W) be a (closed) default theory, S a set of formulas. F(S) is the smallest set such that

(D1) W C_ F(S), (D2) F(S) is deductively closed, (D3) if (A:B/C) E D, A ~ F(S), and ~ B ~ S , then C E F(S).

E is an extension of (D, W) iff F(E) = E, i.e., E is a fixed point of F.

According to Reiter each arbitrarily chosen extension can be seen as an acceptable set of beliefs. Extensions, however, can also be used to define a skeptical notion of inference where a formula is derivable iff it is contained in all extensions.

Many commonsense examples can be handled adequately in DL. There are, however, some cases discussed in the literature where DL does not produce the expected answers. Poole [11] gives the following example: 2

Example 1.2.

:USABLE(X) A - q B R O K E N ( X ) / U S A B L E ( X ) ,

BROKEN(LEFTARM) V BROKEN(RIOHTARM) .

(1) (2)

This default theory has exactly one extension containing both USABLE(RIGHT-

ARM) and USABLE(LEFTARM) although we know that at least one of the arms is broken. The reason is that nothing in the definition of extensions forces the justifications of all applied defaults to be consistent with each other and what is believed. This results in conclusions which are too strong. This problem only

1For sake of simplicity we only consider defaults with one justification in this paper. These defaults have been advocated in [13]. It should be noted, however, that for the "equivalence" between DL and autoepistemic logic [5, 9] multiple justifications are needed. Generalizing the logic to be presented in this paper to defaults with multiple justifications turns out to be a trivial matter: multiple justifications simply can be treated as their conjunction.

2 Since it is clear from the syntax which parts of the examples belong to D and which to W we leave these sets implicit.

Cumulative Default Logic: in defense of nonmonotonic inference rules 185

arises in case of nonnormal defaults, i.e., defaults where the justification is not equivalent to the consequent.

Some people have argued against the above example claiming that implicit knowledge is not made explicit here: if we add the information

VX.BROKEN(X) D --3USABLE(X) , (3)

then of course two extensions are generated, as intended. In this case, nonnormal defaults are not needed at all and we can simply replace (1) by the normal default :USABLE(X)/USABLE(X). This type of "hard" exceptions to a default, i.e., exceptions for which the negation of the default's consequent can be proven, are handled by normal defaults without any problem.

The basic assumption underlying the broken-arms example, however, is that (3) does no t hold: we want to make (1) inapplicable if we know that a given x is broken, but without stating that x is not usable in this case. This is a weaker type of exception to a default: we want to block a default but we don't assert the negation of its consequent. Nonnormal defaults, which can give rise to the problem mentioned above of conclusions that are too strong, are needed precisely to express this weaker type of exceptions. Given this intended use of nonnormal defaults it seems unreasonable to conclude that both arms are usable in the above example. We entirely agree with Poole here who writes [11, p. 3341:

I would argue that this is definitely a bug, being able to conclude both arms are usable given we know one of his arms is broken. The problem is we have implicitly made an assumption, but have been prevented from considering what other assumptions we made as a side effect of this assumption.

In Section 3 we will show that even weak exceptions can be handled, with a simple trick, by normal defaults provided the exception itself can be derived monotonically. In other words, we need the seminormal default (1) only if we want to make the commonsense default "typically, objects are usable" inapplicable to broken objects even if it has been derived by default that they are broken. This amounts to giving the default deriving broken priority over the default deriving usability.

A second problem, probably even more serious, has been pointed out by David Makinson [8]. He studied general properties of nonmonotonic inference relations. One of the properties which arguably can be seen as constitutive for an inference relation is cumulativity. Intuitively, cumulativity says that adding a theorem of a set of premises to these premises does not change the derivable formulas. More formally cumulativity can be expressed as the condition:

If W ~-. y then W ~-. x iff W U { y } ~-- x ,

where ~-- denotes an arbitrary inference relation.

186 G. Brewka

This usual formulation of the cumulativity condition is adequate for default logic only if the skeptical notion of derivability is used. For the arbitrary choice notion of derivability we need a reformulation. Assume a default theory has an extension containing a formula y. Adding y to W certainly should give us all former extensions containing y, but it also should not produce any new extensions. The natural formulation of choice cumulativity therefore is:

If there is at least one extension o f A = (D, W) containing y, then E is an extension of A containing y iff E is an extension of A ' = (D, W U {y}).

Note that cumulativity in the choice sense implies skeptical cumulativity if the existence of extensions is guaranteed.

It turns out that Reiter's default logic is not cumulative, neither in the skeptical nor in the choice sense. Makinson gives the following example:

Example 1.3.

: p / p , (4)

p v q : T p / ~ p . (5)

From these defaults ( W = {}) we get the single extension Th({p}). This extension clearly contains p v q. But adding p v q to the premises gives rise to an additional extension Th({-Tp, q}).

Let 's first discuss a similar "real life" example to better understand what happens here. Take the following example:

Example 1.4.

DOG V BIRD D PET, (6)

DOG D 7 B I R D , (7)

P E T : D O G / D O G , (8)

S INGS:BIRD/BIRD, (9)

S1NGS . (10)

From this default theory we obtain the single extension Th(W tO (BIRD}) .

This extension contains PET. Adding PET tO the premises, however, makes (8) applicable and gives rise to an additional extension where the object is a DOG.

Why was this default not applicable before? The only reason to believe PET was that BIRD was believed. This, of course, is inconsistent with DOG. If we add PET to the premises this implicit information is lost. There now might be independent information that PET is true. It is far from obvious that these cases


actually should yield the same results if the reasons for believing something are taken into account. Default logic distinguishes these cases and hence is more a logic of reasoned belief than just plain belief. It will turn out that cumulativity of a version of default logic can be obtained if this implicit reasoning about reasons is made explicit)

In Section 2 we define cumulative default logic (CDL), a new version of default logic. Instead of simple first-order formulas this logic will use more complex structures, called assertions, which contain the justifications and consequents of defaults used to derive a belief. This allows to distinguish between believing PET because BIRD is consistent (and hence believed), and just believing PET independently. Moreover, since the consistency conditions are part of the formulas the applicability condition can easily be strengthened to handle mutually inconsistent justifications adequately. We prove some properties of the logic, in particular existence of extensions, semimonotonicity, and cumulativity.

In Section 3 we argue, however, that semimonotonicity is not a desired property from the representational point of view. Nonnormal defaults are needed to represent priorities between defaults, but part of this expressiveness is destroyed if the logic is semimonotonic. We show how this problem can be solved by filtering out some of the generated extensions, those which are priority-preserving. Section 4, finally, discusses some related work.

2. Making default logic cumulative

We now define CDL, our new version of default logic. As motivated in the introduction the formulas of our logic, which we will call assertions, consist of a classical first-order formula together with reasons to believe this formula. It will be sufficient to keep track of the justifications and consequents of the defaults which lead to the derivation of the formula.

Definition 2.1. Let p, r t . . . . , r , be first-order formulas. {p:{r~ . . . . . r ,}) is called an assertion, and { r l , . . . , r,} the support of this assertion.

The intuitive meaning of the assertion is: p is believed, since r~ ^ . . . ^ r n is consistent with what is believed and the consistency conditions of other believed formulas.

3 There certainly is an argument for preferring default (9) to (8) in this example even if there is independent information that PET holds: (9) is more specific than (8). The idea of preferring the most specific information is missing from default logic. But this also holds for most of the other nonmonotonic formalisms, e,g. circumscription.

188 G. flrewka

Definition 2.2. Let W be a set of assertions.

• Fo rm(W) , the asserted formulas of W, is the set

{p l (p:{r , . . . . . r,,}) e W}.

• Supp(W), the support o f W, is the set { r l (p : {r 1 . . . . . r . . . . . r , } ) E W}.

Definition 2.3. 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.

We will also simply speak of a default theory if it is clear from context that we mean an assertion default theory. As the next step the classical inference relation of first-order logic has to be extended to assertions in an obvious way:

Definition 2.4. 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 (px: J~) . . . . . ( p k : J k ) E T h s ( A ) a n d p , . . . . . Pk~-q , then (q : J~ U . . . UJk ) E T h s ( A ) .

~- here stands for classical derivability in first-order logic. When it is clear from context that A is a set of assertions we will omit the subscript from Th s.

We are now in a position to define the extensions of a default theory:

Definition 2.5. An extension of an assertion default theory (D, W) is a fixed point of the operator F which, given a 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) i f ( A : B / C ) E D , ( A : { J , . . . . . Jk ) )ES ' , and {B, C} U Form(S) U Supp(S) is consistent, then (C:{J, . . . . . Jk, B, C)) E S'

There are two differences between Reiter 's original logic and our modified version:

(1) In every derivation for an asserted formula the justification and consequent of every default needed for the derivation are recorded.

(2) The applicability condition for a default requires its justification and consequent to be consistent not only with what is believed but also with the support of believed formulas, i.e., the set of justifications and consequents of all other applied defaults.


We first show that the examples from the introduction are handled correctly in the new logic.

It is not difficult to see that Example 1.2 yields the desired results. Let 's assume that BROKEN(LEFTARM)v BROKEN(RIGHTARM) is believed for no further reason, i.e. with empty support. The example becomes:

:USABLE(X) /x -7 BROKEN(X) / USABLE(X), (1')

(BROKEN(LEFTARM) V BROKEN(RIGHTARM): { } ) . (2')

We get two extensions. One extension contains

(USABLE(LEFFARM): {USABLE(LEFTARM) A 7BROKEN(LEFrARM)}) .4

The support of this formula blocks the application of the default instance with x = RIGHTARM since 7BROKEN(RIGHTARM) is not consistent with the support of the above formula and BROKEN(LEFTARM) v BROKEN(RIGHTARM).

The other extension contains

( USABLE(RIGHTARM): {USABLE(RIGHTARM) A -TBROKEN(RIGHTARM)}) .

For similar reasons the default instance with x = LEFTARM is blocked in this case. As intended in none of the extensions both arms are usable.

Here is the new version of Example 1.4:

(DOG V BIRD D PET:(}), (6')

(DOG ~ 7BIRD: {} ) , (7')

PET:DOG/DOG, (8')

SINGS:BIRD / BIRD, (9')

(SINGS:{}). (10')

We derive (BIRD:{BIRD}) and, via (6'), (PET:{BIRD}). Adding this last formula to the premises makes default (8') no longer applicable, unlike the case in the original version of DL, since DOG is inconsistent with BIRD. On the other hand, adding to the premises (PET:{}) (which is not a theorem of the former theory) in fact does change the results. This is reasonable since the premises now state that there is information that the object at hand is a PET independent from its being a BIRD.

The question might arise why in CDL justifications and consequents of applied defaults are recorded. It is obvious that we need the default justifica-

4 We omit subsumed formulas from supports throughout the rest of the paper.

190 (;. Brewka

tions since we want their joint consistency. Recording consequents is necessary if we want to make a default inapplicable whenever its consequent contradicts what is already believed. Assume we would change condition (3) in Definition 2.5 in such a way that only default justifications become part of the assertions, i.e., the third occurrence of C is deleted. The following simple example shows noncumulativity of this modification:

TRUE: r / p , (11)

p v q: -Tp/-Tp . (12)

With the above modification this default theory has one extension containing (p: {r}> and hence (p v q:{r}>. However, adding this last formula to the premises leads to an extension containing (-Tp: {r, ~p}>. This shows that it actually is necessary to record consequents as in our Definition 2.5. In CDL (p v q: {r, p}> is contained in the single extension, and adding this formula causes no problem.

It should be noted that with our definition of CDL extensions it makes no difference whether two formulas A and B or their conjunction A/x B are contained in the support of a formula. This observation and the fact that consequents of defaults are part of the generated supports of formulas show that all defaults implicitly become seminormal. A default A : B / C can equival- ently be replaced by A : B / x C/C, or--the other way around--a normal default A : B / B can be replaced by A:TRUE/B.

We now prove some formal properties of CDL, in particular that the logic deserves its name. We first define an interesting subclass of assertion default theories and prove two useful lemmas:

Definition 2.6. An assertion default theory (D,W) is well-based iff Form(W) U Supp(W) is consistent.

Lemma 2.7. Let A = (D, W) be a well-based assertion default theory and E an extension o f A. Then Form(E)U Supp(E) is consistent.

Lemma 2.8. Let A = (D, W) be a well-based assertion default theory. Let E be

an extension o f A containing (p:J>. Then A' = (D, W U {(p:J>}) is a well-

based assertion default theory.

Our first proposition provides us with a quasi-inductive characterization of CDL similar to the one of [12, Theorem 2.1].

Proposition 2.9. E is a CDL extension o f an assertion default theory (D, W) iff

E = U ~o Ei where


E o = W and for i>>-O

Ei+ I = Ths(Ei)

u {<c:{J, , . . . , L, 8, c}}l A : B / C ~ D, { a : { J , , . . . , J ,}) E Ei ,

and {B, C} U Form(E) U Supp(E) is consistent} .

The next proposition shows that the results of CDL and DL are equivalent (modulo supports) if all defaults are normal, i.e., of the form A:B/B, and all formulas in W have empty supports.

Proposition 2.10. Let D be a set of normal defaults and W a set of assertions with empty supports. I f E is a CDL extension of (D, W), then Form(E) is a DL extension of (D, Form(W)). Conversely, if E' is a DL extension of (D, Form(W)), then there exists a CDL extension E of (D, W) such that Form(E) = E' (i.e., Form is a surjective mapping from the set of CDL extensions of (D, W) to the set of DL extensions of (D, Form(W))).

The last proposition can be seen as a partial rehabilitation of Reiter's logic. Another property of CDL is semimonotonicity.

Proposition 2.11. CDL is semimonotonic, i.e., for every extension E' of an arbitrary assertion default theory (D', W) and every set of defaults D such that D' C D there is an extension E of (D, W) such that E' C_ E.

We will argue in Section 3 that semimonotonicity is not desirable from a representational point of view since this property makes it impossible to use seminormal defaults for representing priorities (which is the only reasonable use we can think of). This problem will be solved by filtering out some of the extensions, those preserving priorities.

An immediate consequence of Proposition 2.11 is the existence of extensions.

Proposition 2.12. Every CDL default theory has an extension.

We now present our main result: CDL actually is cumulative.

Proposition 2.13. I f 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, wu((p:J)}).

Our definition of CDL is intended to remove both shortcomings of DL mentioned in Section 1. For those who do not share Poole's (and the present

I 9 2 G. Brewka

author's) view regarding joint consistency of justifications it might be interesting to know whether there is a way to obtain cumulativity without requiring joint consistency. David Makinson (personal communication) meanwhile has adapted the methods of this paper to show that this actually is the case. However, a more complicated structure of formulas is needed for that purpose: instead of our assertions { p:J} Makinson uses structures of the form ( p: l : J } , called affirmations, where I is the set of default consequents used to derive p and J the set of default justifications. Let W be a set of affirmations, then we define

Form(W) = {p l ( p : l : J } ~ W} ,

Cons(W) = { q E I I ( p : l : J ) E W} ,

Just(W) = ( q E J I ( p : I :J} E W} .

With the obvious modifications of our definitions of default theory and justified theorem Makinson changes condition (3) of Definition 2.5 to

(3') i f A : B / C E D , { A : { C I . . . . . Ct}:{Jj . . . . . ] k } ) E S ' , a n d

Form(S) U Cons(W) U {j} is consistent for all j E Just(W) U {B}

then (C:{CI . . . . . 6" l, C}:{J, . . . . , Jk, B}) E S ' .

Makinson has shown that also this logic is cumulative, provided we define the skeptical theorems of (D, W) to be Ths(W ) whenever no extension exists. The proof is similar to the proof of our Proposition 2.13. However, as intended by Makinson, the broken-arms example yields one extension where both arms are usable.

Schaub [15] describes an adaptation of Etherington's default logic semantics for CDL together with correctness and completeness proofs. Etherington [3] defines a preference relation on sets of models to characterize extensions semantically. For CDL a similar preference relation can be defined on pairs (H, F) where H is a set of models and F a set of formulas. (11, F) is preferred to (H', F ' ) whenever there is a default d = A : B / C E D such that

(a) H = H ' \{p C H ' I p ~ - 7 C } ,

(b) F = F ' U {B, C},

(e) Vp ~ n'.p ~ A,

(d) 3 p ~ U ' . p p F ' U { B , C } .

Let MOD(S) denote the set of all models of S. It is shown in [15] that E is an extension of (D, W) iff (MOD(Form(E)), Supp(E)) is minimal with respect to the preference relation and preferred or equal to (MOD(Form(W)), Supp(W)).


3. Why seminormal defaults?

Contrary to other authors, e.g. Lukaszewicz [7], we see semimonotonicity as a disadvantage: semimonotonicity destroys part of the additional expressiveness of nonnormal defaults (remember that for normal defaults the problem with inconsistent justifications doesn't exist at all). Consider a normal default d = A : B / B . We will distinguish three types of exceptions E to d in this section:

(1) hard exceptions for which it is explicitly known that -TB holds, (2) weak exceptions for which it is open whether B holds or not, but where

the applicability of d has to be blocked if (a) the exceptional condition E can be derived monotonically from W,

or

(b) the exceptional condition E can be derived using defaults.

Type (1) poses no problems for normal defaults. We will see that also type (2a) can be handled by normal defaults using a simple trick involving named defaults. Only for type (2b) do we need seminormal defaults, i.e., defaults where the single justification logically implies the consequent. As we shall also see, exceptions of this last type, however, can only be handled if the logic is not semimonotonic.

As discussed in [14], seminormal defaults can be used to represent priorities between defaults. This is achieved by making the prerequisite of the preferred default a weak exception of the other default. Consider the following example.5

Example 3ol .

STUDENT:--q MARR1ED/-'7 MARR1ED, (13)

ADULT:MARRIED/MARRIED, (14)

STUDENT, (15)

ADULT. (16)

Defaults (13) and (14) are conflicting given (15) and (16). We probably want to give the more specific first default priority over the second one. Reiter and Criscuolo [14] propose to achieve this by changing the second default to:

ADULT:MARRIED A --qSTUDENT/MARRIED. (14')

In terms of exceptions STUDENT becomes a weak exception of default (14). With this representation in DL as well as in CDL only the first default can be applied, as intended.

5 In the rest of this section, when we use examples without explicit supports as examples for CDL, then the supports are empty and premises have to be read as the corresponding assertions, e.g. STUDENT as (STUDENT:(}>.

194 G. Brewka

For the above example, these results can be achieved with a normal representation, however. As proposed in [1] we can introduce named defaults and an applicability predicate APPL and represent (14) as follows: ~

APPL(R1) A A D U L T : M A R R I E D / M A R R I E D . (14")

R1 here is used as a unique name for the default itself. Additionally we add a "metadefault" stating that the default is typically applicable to adults. Note that also this default is normal:

ADULT:APPL(RI) /APPL(R1 ) . (MD)

We now can block the applicability of (14") to students by adding the blocking formula

STUDENT ~ --qAPPL(R1). (BF)

This shows that seminormal defaults are not necessarily needed to block the applicability of defaults if the condition for blocking, the weak exception (here: STUDENT) , c a n be derived monotonically. Exceptions of type (2a) can be

• 7 handled by a normal representation. The situation changes, however, if the weak exception itself is derived via a

default. In this case the naming technique does not work and we have to use the seminormal representation. Consider the following sight modification of Example 3.1.

Example 3.2.

STUDENT:-n MARRIED / --q M A R R I E D , (13)

A D U L T : M A R R I E D / M A R R I E D , (14)

B E A R D : S T U D E N T / S T U D E N T , (17)

B E A R D : A D U L T / A D U L T , (is)

BEARD. (19)

Again we want to give (13) priority over (14). Let's first see how this can be achieved in DL. Our naming technique with normal defaults no longer works in this example: if we replace (14) with (14") and add (MD) and (BF) then we get--via the contraposition of (BF)--an additional DL extension where the person at hand is not a student, and hence married. Our representation of a priority between two defaults which makes the prerequisite of the preferred default a weak exception of the other one fails since it affects a default used to

~A similar naming technique has been proposed in [10]. 7 Note that as a side effect of this representation the negation of the week exception becomes

derivable by default. In our example we obtain -nSTUDENT if premise (15) is removed.


derive the weak exception. If we want to avoid this we need to use Reiter's seminormal representation with (14') replacing (14).

It is crucial them however, that--as in DL--(17) overrides the seminormal default (14') and not vice versa, i.e., that the logic is not semimonotonic.

CDL generates two extensions for the seminormal representation of Exam- pie 3.2. Since the justification of (14') can block (17) from being applied we run into the same problems as with the normal representation. This shows that it is not possible in CDL to handle exceptions of type (2b). In other words: CDL's "solution" to the problem with nonnormal defaults destroys the additional expressiveness of these defaults.

The question now is: can CDL be changed in a way such that our problem is solved without giving up the expressive power of nonnormal defaults? The answer is yes: what we have to do is consider only some of the generated extensions, those respecting the intended priorities. If, for example, we have defaults :B/B and :TB ^ C/C, then we are interested only in the extension containing B, i.e. we expect the logic to behave more like DL in this case. To achieve this we have to treat consequents and justifications differently: if there is a default d inapplicable with respect to an extension E, but only because its consequent contradicts the justifications of applied defaults, then we know that this default should have been applied and simply reject E. This motivates the following definitions:

Definition 3.3. Let E be a CDL extension of (D, W). GD(E), the set of generating defaults of E, is the set

{ A : B / C C D I

A E Form(E), {B, C} U Form(E) U Supp(E) is consistent} .

Definition 3.4. Let E be a CDL extension of (D, W). E 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.

Example 3.2 with the seminormal representation, i.e., where (14) is replaced by (14'), now yields the intended results in CDL: the fixed point with generating defaults {(18), (14')} is not priority-preserving (consider default (17) to see this) and hence rejected. The other fixed point generated by {(18), (17), (13)} is priority-preserving and we derive <'reMARRIED: {STUDENT, --qMARRIED} > as intended.

This shows that priorities can be expressed in the way proposed by Reiter. In this respect CDL F, i.e., CDL with the additional filter on extensions, is closer to Reiter's original logic than CDL with no filter. On the other hand, since all CDL v extensions are CDL extensions, we keep the property of joint con-

196 G. Brewka

sistency of justifications, i.e., DL and CDL F still differ in their treatment of the broken-arms example.

As our Example 3.2 shows CDL v is not semimonotonic. Moreover, the existence of priority-preserving extensions is not guaranteed: consider {B:A/A, :--qA A B/B} which has the single non-priority-preserving CDL extension Ths({(B:{-aA A B})}) as an example. This is the price we have to pay if we want to use seminormal defaults to represent priorities. Proposition 2.10 still holds, since the property of being priority-preserving trivially holds for all extensions of normal theories: if E is an extension of a normal default theory, then F o r m ( E ) U S u p p ( E ) = Form(E) and hence the consequent of a default cannot at the same time be consistent with Form(E) but inconsistent with Form(E) USupp(E) . Also the cumulativity property (Proposition 2.13) is obviously not affected.

We defined CDL v extensions above in terms of CDL extensions plus an additional test. It is an open question whether these extensions can directly be defined in terms of an adequate applicability criterion for defaults.

4. Related work and conclusion

Lukaszewicz [7] also presented a modified version of DL. His logic is based on a two-place fixed point operator. The second argument of the operator is used to keep track of justifications of applied defaults. Lukaszewicz describes his applicability criterion for defaults as follows:

If the prerequisite of a default is believed (its justification is consistent with what is believed), and adding its consequent to the set of beliefs neither leads to inconsistency nor contradicts the justification of this or any other already applied default, then the consequent of the default is to be believed [7, p. 3].

It is not difficult to see why this applicability condition fails to handle the broken-arms example correctly. Lukaszewicz is only concerned with conflicts between a consequent and a justification, now with conflicts between justifications (we argued in Section 3 that the consequent should win in this case). His applicability condition is too weak to guarantee consistency between the justifications of all applied defaults. His logic is semimonotonic. Lukaszewicz does not consider this as a problem, however, and hence loses part of the additional expressiveness of nonnormal defaults, the very reason for using them at all. Cumulativity is not discussed in his paper.

Since we had to make the justifications and consequents used for derivations an explicit part of the assertions in order to obtain cumulativity the second argument of the fixed point operator was not necessary in our approach. With a stronger applicability condition we were able to define a cumulative default


logic which handles the problem of inconsistent justifications adequately. CDL also turned out to be semimonotonic, but the ability to represent priorities between defaults could be restored by filtering out some of the generated extensions, those preserving the intended priorities.

There is an interesting relation between our two-step definition of priority- preserving extensions and some recent approaches by Dressier [2] and Junker [4] to make the ATMS nonmonotonic. Both authors encode defaults with explicit Out-assumptions. The resulting ATMS labelings correspond to extensions of the underlying default theory. As in CDL there are too many extensions generated this way and an additional test checks, among other things, whether priority is preserved or not. This test clearly corresponds to our priority criterion. It is a topic of further research whether the existing nonmonotonic ATMS can be modified such that they can be used for computing CDL extensions.

There are certainly other useful variations on default logic one could think of. The representation of priorities with seminormal defaults is not fully satisfactory. We could, for instance, apply the ideas underlying prioritized circumscription [6] to a default logic: we might split the set' of defaults into sets with different priority and define E to be an extension of a default theory (D 1, . . . , D n, W) iff there exists sets of formulas El . . . . . En such that E~ is an extension of ( D I , W ) , E 2 is an extension of (D2, E ~ ) , . . . , E = E , is an extension of (D~, E,_~). Such further modifications are certainly important topics of future research.

Our main interest in this paper was to show that the basic idea underlying default logic, the representation of defaults as nonstandard inference rules, is still alive and not undermined by some recent criticisms. Default logic should be seen as a logic of reasoned plausible belief. If this is made explicit, then the logic can be made cumulative and the "misbehaviour" disappears.

Appendix A. Proofs of lemmas and propositions

Lemma 2.7. Let A = (D, W) be a well-based assertion default theory and E an extension of A. Then Form(E)U Supp(E) is consistent.

Proof. Assume there is an extension E such that Form(E)U Supp(E) is inconsistent. Applying the operator F to E yields Ths(W), since no default can be applied according to Definition 2.5. Hence either E is no extension, which contradicts our assumption, or E = Ths(W ). It remains to show that also the latter case leads to a contradiction. It is obvious from Definition 2.4 that

Supp(Ths(W)) = Supp(W) and Form(Ths(W)) = Th(Form(W)).

198 G. Brewka

Therefore, since Form(W) USupp(W) is consistent, Form(E)USupp(E) is also consistent, contrary to our assumption. This concludes the proof of Lemma 2.7 []

Lemma 2.8. Let A = (D, W) be a well-based assertion default theory. Let E be an extension of A containing ( p: J ) . Then A' = (D, W U { ( p: J) }) is a well- based assertion default theory.

Proof. According to Lemma 2.7 Form(E)U Supp(E) is consistent. Since E must contain W,

Form(W) C_ Form(E) and Supp(W) C_ Supp(E).

Moreover, since ( p : J ) C E, p E Form(E) and J C_ Supp(E). From this the lemma follows immediately. []

Proposition 2.9. E is a CDL extension of a default theory (D, W) iff E =

U i~o Ei where

E o = W and zfbr i >~ 0

El+ 1 = Ths(Ei)

U { ( C : { Q , . . . . . Q,,,B,C}) I A : B / C E D, ( A : { Q , . . . . . Q,,} ) E E, ,

and {B, C} U Form(E) U Supp(E) is consistent}.

Proof. We define an operator 12 as follows:

n(s) = 0 s,, i = 0

where

S 0 = W a n d f o r i / > 0

Si+ i = Ths(Si)

U {(C:{Q, . . . . . Qn, B, C})I

A : B / C E O, ( A:{ Q I . . . . . Q,} ) E S i ,

and {B, C} U Form(S) U Supp(S) is consistent} .

Obviously E = U i=0 Ei iff O ( E ) = E. Since E is an extension of (D, W) iff F(E) = E, it is sufficient to show that, for an arbitrary set of assertions S, O(S) = r ( S ) .


We first show that /2(S)C_ F(S) proving inductively that S~ C F(S) for all i 1> O. The base is trivial, since S O = W and W C_ F(S). Assume S i C_ F(S). Then Ths(Si) C_ F(S), since F(S) is deductively closed. Moreover, if

A : B / C E D , (A:{Q, . . . . , O n } ) ~ S i ,

and {B, C} U Form(S) U Supp(S)

is consistent, then (C:{Q 1 . . . . . Q,,, B, C}) E F(S), since according to the induction hypothesis S~ C_ F(S). Hence Si+ l C_ F(S).

It remains to show F(S)C II(S). Recalling the definition of F, F(S) is the smallest set S' such that

(1) W c _ S ' ,

(2) S' is closed with respect to Ths, and

(3) i f A : B / C E D , ( A : { J 1 . . . . . J , } ) E S ' , and {B, C} U Form(S) U Supp(S) is consistent, then (C :{J l . . . . . Jk, B, C } ) E S ' .

We first show that I2(S) satisfies conditions (1)-(3). Since S O = W we have W C_ I2(S). Moreover, O(S) is closed with respect to Th s. Finally, if A:B/ C E D, (A:{ J1,. • •, J , } ) E I2(S), and { B, C} U Form(S) tO Supp(S) is consistent, then (C:{Jj . . . . , Jk, B, C}) E O(S). Since F(S) is the smallest set such that these three conditions hold, F(S)C_ O(S). This completes the proof. []

Proposition 2.10. Let D be a set of normal defaults and W a set of assertions with empty supports. I f E is a CDL extension of (D, W), then Form(E) is a DL extension of (D, Form(W)). Conversely, if E' is a DL extension of (D, Form(W)), then there exists a CDL extension E of (D, W) such that Form(E) = E' (i.e., Form is a surjective mapping from the set of CDL extensions of (D, W) to the set of DL extensions of (D, Form(W))).

Proof. ( ~ ) Assume E is a CDL extension of (D, W). Then, by Proposition 2.9, E = U ~=0 Ei. According to [12, Theorem 2.1] Form(E) is a DL extension of (D, Form(W)) iff Form(E) = U~=0 E'i where

E o = Form(W) and for i >~ 0

E; +, = Th(E'i) U { C J A : B / C E D, A E E i , - a B ~ F o r m ( E ) } .

Since Fo rm(E)= U,=0Form(Ei) it suffices to show that, for i>~0, Form(E~) = E'~. The base is trivial. Assume Form(Ei) = E'~. Then clearly

Form(Ths(E,) ) = Th(Form(E~) = Th(E;) .

Moreover, since all defaults in D are normal and all formulas in W have empty

200 (3. Brewka

supports, {B, C} U Form(E) U Supp(E) is consistent iff {B} U Form(E) is consistent and hence, since Form(E) is deductively closed, i f f -qB ~ F o r m ( E ) . Therefore C C Form(Ei+ t) iff C E El + ~. This finishes the first half of the proof.

( ~ ) A s s u m e E' is a DL extension of (D, Form(W)), i.e., E ' = U , - 0 El where El is defined as above. We construct a sequence Go, Gt . . . . as follows:

G 0 = W a n d fo r i~>0

Gi+ 1 ~ Ths(Gi)

u {{C:{Q, . . . . , Q. , 8, c}} l

A : B / C E O (A:{Q, . . . . , Q,,}) E G, ,

and {B} U E ' is consistent}.

Define E = U~=0 G~. We have to show that (1) E is a CDL extension of (D, W), i.e., that E - - U ~,~ E~, and that (2) Form(E) = E'. We start with the proof of (2).

To prove F o r m ( E ) = E ' we show by induction on i that, for i~>0, Form(Gi) = E~. The base is trivial, since Form(G0) = Form(W) = E 0. For the induction assume F o r m ( G i ) = E i. Clearly

Form(Ths(Gi) = Th(Form(Gi) ) = Th(EI ) .

Moreover, since according to the induction hypothesis ( A: { Ql . . . . . Q,,} ) E G i for some Q1 . . . . . Qn iff A E E i , C ~ F o r m ( E i + l ) iff C E E'~+~.

It remains to show (1), i.e., that E is a CDL extension of W. It suffices to prove that, for i i> 0, G i = E i. The base is trivial. For the induction assume G~ = Ei. A simple induction shows that Supp(E)C_ Form(E). For this reason and since all defaults A : B / C in D are normal, {B, C} U Form(E) U Supp(E) is consistent iff {B} U Form(E) is consistent. Moreover, we have already proven that F o r m ( E ) = E'. From this it follows immediately that an assertion ( C: J) ~ Gi+ 1 iff ( C : J ) ~ E~+ 1' This finishes the proof of Proposition 2.10. []

Proposition 2.11. CDL is semimonotonic, i.e., for every extension E' of an arbitrary default theory (D', W) and every set of defaults D such that D' C D there is an extension E of (D, W) such that E' C_ E.

Proof. 8 We first consider the case that (D', W) and (D, W) are not well-based. In this case Ths(W ) is the single extension of both default theories and Proposition 2.11 holds.

We now consider the principle case that (D', W) and (D, W) are well-based. Let d 1 = A i : B 1 / C I , d 2 = A 2 : B 2 / C 2 . . . . be a fixed enumeration of the

The proof is an adaptat ion of a corresponding proof in [7].

Cumulative Default Logic: in defense of nonmonotonic inference rules 2 0 1

defaults of D. We construct a sequence of sets of assertions So, S 1 . . . . as follows:

S O = E', and given S~, a sequence S~,0, S~,1 . . . . is defined such that

Si, o = S i, and for j i> 0

Si,j + I = i f ( A / J ) ~ S i for some J

and {Bj, Ci} U Form(S~4 ) U Supp(Sm) is consistent

then S~4 U {( Cj:{P, . . . . . Pk, Bi, C j } ) I ( A i : { P , , . . . , Pk} ) E S,}

else Si4.

We now define ~c

Si+ 1 =Ths(S~)U G S m and E = U Si. j=O i = 0

We note the following.

L e m m a A.I. Form(E) O Supp(E) is consistent.

Proof. Form(E ' ) U Supp(E') is consistent according to Lemma 2.7. The induction is straightforward. []

Clearly, E ' C E. It remains to be shown that E is an extension of (D, W). We define a sequence Go, G ~ , . . . in the following way:

G 0 = W a n d f o r i / > 0

Gi+l = Ths(Gi)

U { ( C : { Q 1 , . . . , O,, B, C}) I

A : B / C E D, ( A : { Q 1 , . . . , Q ,} ) ~ G i ,

and {B, C} U Form(E) U Supp(E) is consistent}.

According to Proposition 2.9 it is sufficient to show E = U i~-0 Gi. We prove both halfs of the equation separately.

We first show that

0 G i C _ E = O Si. i=O i=O

It suffices to show by induction that, for i/> 0, G~ c_ S~. The base is trivial, since G O = W and W C E ' = S 0. Assume G i C_ S i. Then Ths(Gi) C_ Si+ 1. Let (C:{Q1, • • . , Q,, B, C}) ~ Gi+l\Ths(Gi). Then there is a default dj = A:B/ C ~ D such that (A:{Q~ . . . . . Q,}) E Gi, and {B, C} U Form(E) U Supp(E) is consistent. Since E ' C E and, by induction hypothesis, G iCSi , (C:{Q, . . . . . Q,, B , C } ) ~_. Si,j+ 1 C Si+ 1 .

202 (;, Brewka

To finish the proof of Proposition 2.11 it remains to show that E C U/-~ Gi. We first prove that E' C_ U ,?o Gi. This is done by constructing a sequence H~, H~ . . . . as defined in Proposition 2.9 for E' and (D', W). From Proposition 2.9 it follows that E' = U ~ Hi. We prove by induction that, for i ~> 0, H i C_ Gi, and hence

E'= 0 Ci i=O i=O

The base is trivial, since H 0 = W= G 0. Also the induction is straightforward, since D' C D.

With this last result as induction base we can prove that, for j />0, S/C_ U ~_,, G,. Assume S/C U [-,, Gi. We show Sj+, C U ~-,, Gi by proving Sj, k C_ U ~=0 Gi for all k/> 0. The base is a trivial consequence of the induction hypothesis S~ C U ~-,, Gi. Assume S/, k _C U ~_,, G,. If S/, k +1 :~ Sj,k it follows immediately that S/, k +~ C_ U ~L0 Gi, since E = U ~--0 Si, i.e., Form(E) contains Form(S/,k) U {Ck} and Supp(E) contains Supp(S/,~)U {Bk}, and Form(E)U Supp(E) is consistent (Lemma A.1). Hence EC_ U~- . Gi and the proof of Proposition 2.11 is complete. []

Proposition 2.12. Every CDL default theory has an extension.

Proof. Every default theory without defaults has an extension, namely Ths(W ). From this and Proposition 2.11 the result follows immediately. []

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:]>}).

Proof. First consider the case that (D, W) is not well-based. According to condition (3) of Definition 2.5 no default is applicable in this case and the single extension is Ths(W ). Then,

Form(W U { (p : J )} ) U Just(W U { (p : J )} )

must also be inconsistent and, for the same reasons, Ths(W U { (p: J)}) is the single extension of (D, WU {(p :J )}) . But since ( p : J ) ~ T h s ( W ) we have that Ths(W ) = Yhs(W U {(p : J )} ) , and hence the proposition holds.

We now consider the principal case, that (D, W) is well-based. We prove the two parts of the equivalence separately.

( ~ ) This is the easy half of the proof. E is an extension of (D, W) iff it is a fixed point of F as defined in Definition 2.5, i.e., iff E is the smallest set S such that the following conditions hold:

(1) w c s ,


(2)

(3)

Since E (3), and

Ths(S ) = S ,

i f A : B / C E D , ( a : { J , , . . . , J k } ) e S , and{B, C} U Form(E) tO Supp(E) is consistent, then (C:{J, . . . . , Jk, B, C}) E S .

contains ( p : J ) it is also the smallest set S such that conditions (2), the stronger condition

(1') W t O { ( p : J ) } C _ S

hold, and hence E is an extension of (D, W U { ( p : J ) } ) . ( ~ ) This is the difficult half. Suppose E is an extension of (D, W U

{ ( p : J ) } ) . Then, according to Proposition 2.9, there is a sequence E0, E ~ , . . . such that

(1) E o = W U { ( p : J ) } ) ,

(2) for i/> 0

E/+I = Ths(Ei)

U {(C:{Q, . . . . , Q, , B, C})[

A : B / C E D, ( a : { Q l , . . . , Q,} ) c Eg ,

and {B, C} U Form(E) U Supp(E) is consistent},

(3) E = G E i . i = 0

r ! Now define a sequence E 0, E 1 . . . . as follows:

!

(1) E 0 = W ,

(2) for i >/0

E'i+ , = Ths(E',)

U { ( C : { Q , , . . . , Q, , B, C}) [

A : B / C ~ D, ( A : { Q l . . . . . Q,} ) E E l ,

and {B, C} tO Form(E) U Supp(E) is consistent} .

~c t

Moreover, let E ' = U ~=0 E~ . We will show that E is an extension of (D, W) by proving that E' = E. To show E' C_ E we prove by induction that E'~ C Ei for all i/> 0. The basis is

immediate, since W C W t O { ( p : J ) } ) , and the induction step is straightforward.

To show EC_E' we prove by induction that E~C_E' for all i/>0. The induction step is trivial. The difficult part is the basis, where we need to show W tO { ( p: J ) } C_ E'. Since W C_ E' it suffices to show { ( p: J ) } E E'.

By hypothesis, there is an extension F of (D, W) containing ( p : J ) . Hence there is a sequence F 0, F I , . . . such that

204 G. Brewka

(1)

(2)

(3)

We define a sequence F[~, FI, . .

(1) F(, = W ,

(2) for i/> 0

El + i

for i ~> 0

Fi+, = Ths(F~)

U ({C:{Q, . . . . . Q,,, B, C}) I A:B/C E D, {A:{Q, . . . . . Q,} ) E F , ,

and { B, C} U Form(F) U Supp(F) is consistent} ,

i--[1

as follows:

= Ths(Fi )

U {( C:{Q, . . . . , Q , , B, C}} I

A : B / C E D, {A:{Q, . . . . . Q,}) E F~ ,

{ B, C} U Form(F) tO Supp(F) is consistent,

and {B, C} C_ J}.

Moreover, we define F ' = U ~=0 F~. To show that { p: J) E E' and thereby complete the proof, it will suffice to

show that (p:J)~_ F~ for some j~>0 and that F I C_ E~ for all i~>0. To show ( p: J ) E F~ note that ( p: J ) E ~ for some j ~> 0. A simple induction

on i shows that whenever {q : I )@F i and IC_J then ( q : I ) ~ F I. Thus since {p:J} E Fj, also {p:J} ~ F~.

It remains to show F~ C E' i for all i/> 0. The basis is trivial, since F0 = E; = W. For the induction step, suppose {q : I )E F~+~. We want to show {q:I)~_ El + 1- It suffices to show that Form(E) tO Supp(E) U J is consistent. By hypothesis, ( p : J ) E F. Since (D, W) is well-based also (D, WtO { ( p : J ) } ) is well- based according to Lemma 2.8. Hence, since E is an extension of (D, W U { (p:J)}) , we may apply Lemma 2.7 to conclude that F o r m ( E ) U Supp(E) is consistent. But since ( p : J ) U_ E, J U_ Supp(E). Hence

Form(E) U Supp(E) U J is consistent

and the proof is complete. []

Acknowledgement

Parts of this paper have been written during a stay at ICOT in summer 1989. Thanks to K. Fuchi for inviting me to Tokyo. I had valuable discussions on the


pape r with J. A r i m a , N. Helf t , J. He r t zbe rg and especially U. Junker . I would

part icular ly like to thank D. Makinson . H e not only drew at tent ion to the

defect of defaul t logic which mot iva ted the whole paper , he also p rov ided

coun te rexamples in early stages of research which saved me a lot of work and

he lped to find C D L ' s final form. Moreove r , he checked and improved the

proofs . All of his suggest ions were ext remely helpful.

References

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cumulative default logic: in defense of nonmonotonic inference rules

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