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A Tutorial on Default LogicsGRIGORIS ANTONIOU

Griffith University

Default Logic is one of the most prominent approaches to nonmonotonic reasoning,and allows one to make plausible conjectures when faced with incompleteinformation about the problem at hand. Default rules prevail in many applicationdomains such as medical diagnosis and legal reasoning.

Several variants have been developed over the past years, either to overcome someperceived deficiencies of the original presentation, or to realize somewhat differentintuitions. This paper provides a tutorial-style introduction to some importantapproaches of Default Logic. The presentation is based on operational models forthese approaches, thus making them more easily accessible to a broader audience,and more easily usable in practical applications.

Categories and Subject Descriptors: I.2.3 [Artificial Intelligence]: Deduction andTheorem Proving—Nonmonotonic reasoning and belief revision; I.2.4 [ArtificialIntelligence]: Knowledge Representation Formalisms andMethods—Representation languages

General Terms: Languages, Theory

Additional Key Words and Phrases: Default Logic, nonmonotonic reasoning,operational models

1. INTRODUCTION: DEFAULT REASONING

When an intelligent system (either com-puter-based or human) tries to solve aproblem, it may be able to rely on com-plete information about this problem,and its main task is to draw the correctconclusions using classical reasoning. Insuch cases, classical predicate logic maybe sufficient.

However, in many situations the sys-tem has only incomplete information athand, be it because some pieces of infor-mation are unavailable, or because ithas to respond quickly and does nothave the time to collect all relevant

data. Classical logic indeed has the ca-pacity to represent and reason with cer-tain aspects of incomplete information.But there are occasions where addi-tional information needs to be “filled in”to overcome the incompleteness, be-cause certain decisions must be made.In such cases the system has to makesome plausible conjectures, which in thecase of default reasoning are based onrules of thumb, called defaults. For ex-ample, an emergency doctor has tomake some conjectures about the mostprobable causes of the symptoms ob-served. Obviously, it would be inappro-priate to await the results of possibly

Author’s address: School of Computing & Information Technology, Griffith University, Brisbane, QLD4111, Australia; email: [email protected] to make digital / hard copy of part or all of this work for personal or classroom use is grantedwithout fee provided that the copies are not made or distributed for profit or commercial advantage, thecopyright notice, the title of the publication, and its date appear, and notice is given that copying is bypermission of the ACM, Inc. To copy otherwise, to republish, to post on servers, or to redistribute tolists, requires prior specific permission and / or a fee.© 2000 ACM 0360-0300/99/0900–0337 $5.00

ACM Computing Surveys, Vol. 31, No. 3, September 1999

extensive and time-consuming tests be-fore beginning with treatment.

When decisions are based on assump-tions, these may turn out to be wrong inthe face of additional information thatbecomes available; i.e., medical testsmay lead to a modified diagnosis. Thephenomenon of having to take backsome previous conclusions is called non-monotonicity; it says that if a statementw follows from a set of premises M andM # M9, w does not necessarily followfrom M9. Default Logic, originally pre-sented in Reiter [1980], provides formalmethods to support this kind of reason-ing.

Default Logic is perhaps the mostprominent method for nonmonotonicreasoning, basically because of the sim-plicity of the notion of a default, andbecause defaults prevail in many appli-cation areas. However, there exist sev-eral alternative design decisions whichhave led to variations of the initial idea;

actually, we can talk about a family ofdefault reasoning methods because theyshare the same foundations.

In this paper we present the motiva-tions and basic ideas of some of themost important default logic variants,and compare them both with respect tointerconnections and the fulfillment ofsome properties. When designing a tuto-rial, it is good to have some aimsagainst which the final product can betested. The particular aims of this tuto-rial paper are the following:

—To present the basic ideas of DefaultLogic to persons without prior knowl-edge in the area. Only basic under-standing of classical logic is required.

—To equip the reader with skills andmethods so that they can apply theconcepts of Default Logic to concretesituations. This is achieved by the useof operational models which can beapplied in a straightforward mannerto examples without having to makeany guesses, as is the case with theusual fixpoint or quasiinductive defi-nitions.

—To give the reader a sense of thediversity of the topic.

The paper is organized as follows: Sec-tion 2 presents the basics of Reiter’sDefault Logic. Section 3 discusses prop-erties and design decisions of this ap-proach, and outlines some alternativeintuitions on these issues. Sections 4–6describe some basic default logic vari-ants: Justified Default Logic [Lukasze-wicz 1988], Constrained Default Logic[Schaub 1992], and approaches to pri-oritization [Brewka 1994]. In Section 7we briefly discuss some further defaultlogics, namely Cumulative DefaultLogic [Brewka 1991], Rational DefaultLogic [Mikitiuk and Truszczynski 1995],Disjunctive Default Logic [Gelfond et al.1991], and weak extensions [Marek andTruszczynski 1993].

No prior knowledge of Default Logic isrequired since this is a tutorial paper.However, we assume that the reader is

CONTENTS

1. Introduction: Default Reasoning2. Default Logic

2.1 The Notion of a Default2.2 The Syntax of Default Logic2.3 Informal Discussion of the Semantics2.4 An Operational Definition of Extensions2.5 Some Examples2.6 Reiter’s Original Definition of Extensions

3. Properties of Default Logic3.1 Existence of Extensions3.2 Joint Consistency of Justifications3.3 Cumulativity and Lemmas

4. Justified Default Logic4.1 Motivation and Formal Presentation4.2 Lukaszewicz’ Original Definition

5. Constrained Default Logic5.1 Motivation and Definition5.2 A Fixpoint Characterization5.3 Interconnections

6. Priorities on Defaults6.1 Motivation6.2 Static Priorities6.3 Dynamic Priorities

7. Other Variants of Default Logic7.1 Rational Default Logic7.2 Cumulative Default Logic7.3 Disjunctive Default Logic7.4 Weak Extensions

8. Conclusion

338 • G. Antoniou


familiar with notation and the basicconcepts of classical logic.

2. DEFAULT LOGIC

2.1 The Notion of a Default

A rule used by football organizers inGermany might be: “A football gameshall take place, unless there is snow inthe stadium.” This rule of thumb is rep-resented by the default

football : ¬snow

takesPlace.

The interpretation of the default is asfollows: If there is no information thatthere will be snow in the stadium, it isreasonable to assume ¬snow and con-clude that the game will take place (sopreparations can proceed). But if thereis a heavy snowfall during the nightbefore the game is scheduled, then thisassumption can no longer be made. Nowwe have definite information that thereis snow, so we cannot assume ¬snow,therefore the default cannot be applied.In this case we need to refrain from theprevious conclusion (the game will takeplace), so the reasoning is nonmono-tonic.

Before proceeding with more exam-ples, let us first explain why classicallogic is not appropriate to model thissituation. Of course, we could use therule

football ∧ ¬snow 3 takesPlace.

The problem with this rule is that wehave to definitively establish that therewill be no snow in the stadium beforeapplying the rule. But that would meanthat no game could be scheduled in thewinter, which would create a revolutionin Germany! It is important to under-stand the difference between having toknow that it will not snow, and beingable to assume that it will snow. De-faults support the drawing of conclu-sions based upon assumptions.

The same example could have beenrepresented by the default

football : takesPlace

takesPlace,

together with the classical rule snow3 ¬takesPlace. In case we know snowthen we can deduce ¬takesPlace in clas-sical logic, therefore we cannot assumetakesPlace, as required by the default.In this representation, the default says“Football matches usually takes place,”and exceptions to this rule are repre-sented by classical rules such as theabove one.

Defaults can be used to model proto-typical reasoning, which means thatmost instances of a concept have someproperty. One example is the statement“Typically, children have (living) par-ents” which may be expressed by thedefault

child~X! : hasParents~X!

hasParents~X!.

A further form of default reasoning isno-risk reasoning. It concerns situationswhere we draw a conclusion even if it isnot the most probable, because anotherdecision could lead to a disaster. Per-haps the best example is the followingmain principle of justice in the Westerncultures: “In the absence of evidence tothe contrary assume that the accused isinnocent.” In default form:

accused~X! : innocent~X!

innocent~X!.

Defaults naturally occur in many ap-plication domains. Let us give an exam-ple from legal reasoning. According toGerman law, a foreigner is usually ex-pelled if they have committed a crime.One of the exceptions to this rule con-cerns political refugees. This informa-tion is expressed by the default

A Tutorial on Default Logics • 339


criminal~X! ∧ foreigner~X! : expel~X!

expel~X!

in combination with the rule

politicalRefugee~X! 3 ¬expel~X!.

Hierarchies with exceptions are com-monly used in biology. Here is a stan-dard example:

Typically, molluscs are shell-bearers.Cephalopods are molluscs.Cephalopods are not shell-bearers.

It is represented by the default

mollusc~X! : shellBearer~X!

shellBearer~X!

together with the rule

cephalopod~X! 3 mollusc~X!

∧ ¬shellBearer~X!.

Defaults can be used naturally to modelthe Closed World Assumption [Reiter1978], which is used in database theory,algebraic specification, and logic pro-gramming. According to this assump-tion, an application domain is describedby certain axioms (in form of relationalfacts, equations, rules, etc.) with thefollowing understanding: a ground fact(that is, a nonparameterized statementabout single objects) is taken to be falsein the problem domain if it does notfollow from the axioms. The closedworld assumption has the simple de-fault representation

true : ¬w

¬w

for each ground atom w. The explana-tion of the default is: if it is consistentto assume ¬w (which is equivalent tonot having a proof for w) then conclude¬w.

Further examples of defaults can befound in, say, Besnard [1989]; Ethering-

ton [1987b]; Lukaszewicz [1990]; andPoole [1994].

2.2 The Syntax of Default Logic

A default theory T is a pair ~W, D!consisting of a set W of predicate logicformulae (called the facts or axioms ofT) and a countable set D of defaults. Adefault d has the form

w : c1, . . . , cn

x

where w, c1, . . . , cn, x are closedpredicate logic formulae, and n . 0.The formula w is called the prerequisite,c1, . . . , cn the justifications, and x theconsequent of d. Sometimes w is denotedby pre~d!, $c1, . . . , cn% by just~d!, andx by cons~d!. For a set D of defaults,cons~D! denotes the set of consequentsof the defaults in D. A default is callednormal iff it has the form w : c / c.

One point that needs some discussionis the requirement that the formulae ina default be ground. This implies that

bird~X! : flies~X!

flies~X!

is not a default according to the defini-tion above. Let us call such rules ofinference open defaults. An open defaultis interpreted as a default schema,meaning that it represents a set of de-faults (this set may be infinite).

A default schema looks like a default,the only difference being that w, c1,. . . , cn, x are arbitrary predicate logicformulae (i.e., they may contain freevariables). A default schema defines aset of defaults, namely

ws : c1s, . . . , cns

xs

for all ground substitutions s that as-sign values to all free variables occur-

340 • G. Antoniou


ring in the schema. That means freevariables are interpreted as being uni-versally quantified over the whole de-fault schema. Given a default schema

bird~X! : flies~X!

flies~X!

and the facts bird~tweety! andbird~sam!, the default theory repre-sented is ~$bird~tweety!, bird~sam!%,$bird~tweety! : flies~tweety!/flies~tweety!,bird~sam! : flies~sam!/flies~sam!%!.

2.3 Informal Discussion of the Semantics

Given a default w : c1, . . . , cn / x, itsinformal meaning is the following:

If w is known, and if it is consistent toassume c1, . . . , cn, then conclude x.

In order to formalize this interpretationwe must say in which context w shouldbe known, and with what c1, . . . , cnshould be consistent. A first guesswould be the set of facts, but this turnsout to be inappropriate. Consider thedefault schema

friend~X, Y! ∧ friend~Y, Z! : friend~X, Z!

friend~X, Z!

which says “Usually my friends’ friendsare also my friends”. Given the infor-mation friend~tom, bob!, friend~bob,sally! and friend~sally, tina!, we wouldlike to conclude friend~tom,tina!. But this is only possible if weapply the appropriate instance of thedefault schema to friend~sally, tina!and friend~tom,! sally}. The latter for-mula stems from a previous applicationof the default schema.1 If we did notadmit this intermediate step and usedthe original facts only, then we couldnot get the expected result.

Another example is the default theory

T 5 ~W, D! with W 5 $green,aaaMember% and D 5 $d1, d2% with

d1 5green : ¬likesCars

¬likesCars,

d2 5aaaMember : likesCars

likesCars.

If consistency of the justifications wastested against the set of facts, then bothdefaults could be subsequently applied.But then we would conclude bothlikesCars and ¬likesCars, which is acontradiction. It is unintuitive to let theapplication of default rules lead to aninconsistency, even if they contradicteach other. Instead, if we applied thefirst default, and then checked applica-tion of the second with respect to thecurrent knowledge collected so far, thesecond default would be blocked: fromthe application of the first default weknow ¬likesCars, so it is not consis-tent to assume likesCars. After theseexamples, here is the formal definition:

d 5 w : c1, . . . , cn / x is applicableto a deductively closed set of formulaeE iff w [ E and ¬c1 [y E, . . . , ¬cn

[y E.

The example of Greens and AAA mem-bers indicates that there can be severalcompeting current knowledge baseswhich may be inconsistent with one an-other. The semantics of Default Logicwill be given in terms of extensions thatwill be defined as the current knowl-edge bases satisfying some conditions.Intuitively, extensions represent possi-ble world views which are based on thegiven default theories; they seek to ex-tend the set of known facts with “rea-sonable” conjectures based on the avail-able defaults. The formal definition willbe given in the next subsection. Here wejust collect some desirable properties ofextensions.1With other instantiations, of course.



—An extension E should include the setW of facts since W contains the cer-tain information available: W # E.

—An extension E should be deductivelyclosed because we do not want to pre-vent classical logical reasoning. Actu-ally, we want to draw more conclu-sions, and that is why we applydefault rules in addition. Formally: E5 Th~E!, where Th denotes the de-ductive closure.

—An extension E should be closed un-der the application of defaults in D(formally: if w : c1, . . . , cn / x [ D,w [ E and ¬c1 [y E, . . . , ¬cn [y Ethen x [ E). That is, we do not stopapplying defaults until we are forcedto. The explanation is that there is noreason to stop at some particularstage if more defaults might be ap-plied; extensions are maximal possi-ble world views.

These properties are certainly insuffi-cient because they do not include any“upper bound,” that is, they don’t pro-vide any information about which for-mulae should be excluded from an ex-tension. So we should require that anextension E be minimal with respect tothese properties. Unfortunately, this re-quirement is still insufficient. To seethis, consider the default theory T 5

~W, D! with W 5 $aussie% and D 5

$aussie : drinksBeer/drinksBeer%.Let E 5 Th~$aussie, ¬drinksBeer%!.It is easily checked that E is minimalwith the three properties mentionedabove, but it would be highly unintui-tive to accept it as an extension, sincethat would support the following argu-ment: “If Aussies usually drink Beerand if somebody is an Aussie, then as-sume that she does not drink Beer.”

2.4 An Operational Definition ofExtensions

For a given default theory T 5 ~W, D!let P 5 ~d0, d1, . . . ! be a finite or infi-nite sequence of defaults from D with-out multiple occurrences. Think of P asa possible order in which we apply somedefaults from D. Of course, we don’twant to apply a default more than oncewithin such a reasoning chain becauseno additional information would begained by doing so. We denote the ini-tial segment of P of length k by P@k#,provided the length of P is at least k(from now on, this assumption is alwaysmade when referring to P@k#). Witheach such sequence P we associate twosets of first-order formulae, In~P! andOut~P!:

—In~P! is Th~W ø $cons~d!?d occursin P}. So, In~P! collects the informa-tion gained by the application of thedefaults in P and represents the cur-rent knowledge base after the defaultsin P have been applied.

—Out~P! 5 $¬c?c [ just~d! for somed occurring in P}. So, Out~P! collectsformulae that should not turn out tobe true, i.e., that should not becomepart of the current knowledge baseeven after subsequent application ofother defaults.

Let us give a simple example. Considerthe default theory T 5 ~W, D! with W5 $a% and D containing the followingdefaults:

d1 5a : ¬b

¬b, d2 5

b : c

c.

For P 5 ~d1! we have In~P! 5 Th~$a,¬b%! and Out~P! 5 $b%. For P 5 ~d2,d1! we have In~P! 5 Th~$a, c, ¬b%!and Out~P! 5 $¬c, b%.

Up to now we have not assured that

342 • G. Antoniou


the defaults in P can be applied in theorder given. In the example above, ~d2,d1! cannot be applied in this order (ap-plied according to the definition in theprevious subsection). To be more spe-cific, d2 cannot be applied, since b [yIn~~!! 5 Th~W! 5 Th~$a%! which isthe current knowledge before we at-tempt to apply d2. On the other hand,there is no problem with P 5 ~d1!; inthis case we say that P is a process of T.Here is the formal definition:

—P is called a process of T iff dk isapplicable to In~P@k#!, for every ksuch that dk occurs in P.

Given a process P of T we define thefollowing:

—P is successful iff In~P! ù Out~P!5 À, otherwise it is failed.

—P is closed iff every d [ D that isapplicable to In~P! already occurs inP. Closed processes correspond to thedesired property of an extension Ebeing closed under application of de-faults in D.

Consider the default theory T 5 ~W,D! with W 5 $a% and D containing thefollowing defaults

d1 5a : ¬ b

d, d2 5

true : c

b.

P1 5 ~d1! is successful but not closedsince d2 may be applied to In~P1! 5Th~$a, d%!. P2 5 ~d1, d2! is closed butnot successful: both In~P2! 5 Th~$a,d, b%! and Out~P2! 5 $b, ¬c% containb. On the other hand, P3 5 ~d2! is aclosed and successful process of T.According to the following definition,which was first introduced in Antoniouand Sperschneider [1994], In~P3! 5

Th~$a, b%! is an extension of T, in factits single extension.

Definition 1. A set of formulae E isan extension of the default theory T iffthere is some closed and successful pro-cess P of T such that E 5 In~P!.

In examples, it is often useful to ar-range all possible processes in a canoni-cal manner within a tree, called theprocess tree of the given default theoryT. The nodes of the tree are labeledwith two sets of formulae, an In-set (tothe left of the node) and an Out-set (tothe right of the node). The edges corre-spond to default applications, and arelabeled with the default that is beingapplied. The paths of the process treestarting at the root correspond to pro-cesses of T.

2.5 Some Examples

Let T 5 ~W, D! with W 5 À and D 5$true : a/¬a%. The process tree in Fig-ure 1 shows that T has no extensions.Indeed, the default may be applied be-cause there is nothing preventing usfrom assuming a. But when the defaultis applied, the negation of a is added tothe current knowledge base, so the de-fault invalidates its own application be-cause both the In and the Out-set con-tain ¬a. This example demonstratesthat there need not always be an exten-sion of a default theory.

Figure 1. A processtree.



Let T 5 ~W, D! be the default theorywith W 5 À and D 5 $d1, d2% with

d1 5true : p

¬q, d2 5

true : q

r.

The process tree of T is found in Figure2 and shows that T has exactly oneextension, namely Th~$¬q%!. The rightpath of the tree shows an examplewhere application of a default destroysthe applicability of a previous default:d1 can be applied after d2, but then ¬qbecomes part of the In-set, whilst it isalso included in the Out-set (as thenegation of the justification of d2).

Let T 5 ~W, D! with W 5 $green,aaaMember% and D 5 $d1, d2% with

d1 5green : ¬likesCars

¬likesCars,

d2 5aaaMember : likesCars

likesCars.

The process tree in Figure 3 showsthat T has exactly two extensions(where g stands for green, a foraaaMember, and l for likesCars).

2.6 Reiter’s Original Definition ofExtensions

In this subsection we present Reiter’soriginal definition of extensions [Reiter1980]. In subsection 2.3 we briefly ex-plained that the most difficult problemin describing the meaning of a default isto determine the appropriate set withwhich the justifications of the defaultsmust be consistent. The approachadopted by Reiter is to use some theorybeforehand. That is, choose a theorywhich plays the role of a context or beliefset and always check consistencyagainst this context. Let us formalizethis notion:

—A default d 5 w : c1, . . . , cn / x isapplicable to a deductively closed setof formulae F with respect to belief setE (the aforementioned context) iff w

[ F, and ¬c1 [y E, . . . , ¬cn [y E(that is, each c i is consistent with E).

Note that the concept “ d is applicable toE” used so far is a special case whereE 5 F. The next question that arises iswhich context to use. First, note thatwhen a belief set E has been estab-lished, some formulae will become partof the knowledge base by applying de-

Figure 2. A process tree.

344 • G. Antoniou


faults with respect to E. Therefore, theyshould be believed, i.e., be members ofE. On the other hand, what would be ajustification for a belief if it were notobtained from default application w.r.t.E? We require that E contain only for-mulae that can be derived from the axi-oms by default application w.r.t. E.

Let us now give a formal presentationof these ideas. For a set D of defaults,we say that F is closed under D withrespect to belief set E iff, for every de-fault d in D that is applicable to F withrespect to belief set E, its consequent x

is also contained in F.Given a default theory T 5 ~W, D!

and a set of formulae E, let LT~E! bethe least set of formulae that containsW, is closed under logical conclusion(i.e., first-order deduction), and closedunder D with respect to E. Informallyspeaking, LT~E! is the set of formulaethat are sanctioned by the default the-ory T with respect to the belief set E.

Now, according to Reiter’s definition,E is an extension of T iff E 5 LT~E!.This fixpoint definition says that E isan extension iff by deciding to use E asa belief set, exactly the formulae in Ewill be obtained from default applica-tion. But please note the difficulty inapplying this definition: we have toguess E and subsequently check for thefulfillment of the fixpoint equation.Having to guess is one of the most seri-

ous obstacles in understanding the con-cepts of Default Logic and in being ableto apply them to concrete cases.

The following theorem shows that Re-iter’s extension concept is equivalent tothe definition in subsection 2.4.

THEOREM 1. Let T 5 ~W, D! be a de-fault theory. E is an extension of T (inthe sense of definition 2.1) iff E 5LT~E!.

We conclude by giving a quasiinduc-tive characterization of extensions, alsodue to Reiter: Given a default theory T5 ~W, D!, we say that E has a quasi-inductive definition in T iff E 5 ø

iEi,

where E0 5 Th~W! and Ei11 5 Th~Ei

ø $cons~d!%d [ D! is applicable to Ei

w.r.t. belief set E}.

THEOREM 2. E is an extension of T iffE has a quasiinductive definition in T.

This characterization replaces theLT-operator by a construction, both ofthem using the set E as context or beliefset. Given a set of formulae E, thischaracterization is intuitively appeal-ing. But notice that still it is necessaryto first guess E before checking whetherit is an extension. In this sense, thecharacterization is not as easy to applyas the process model from subsection2.4.

The relationship of processes to thequasiinductive definition is that the tra-

Figure 3. A process tree.



versal of the process tree operational-izes the idea of guessing. More formally:if a branch of the process tree leads to aclosed and successful process P, thenthe quasiinductive construction usingIn~P! as a belief set yields the sameresult. But some branches of the processtree can lead to failed processes; this isthe price we have to pay if we wish toavoid guessing.

3. PROPERTIES OF DEFAULT LOGIC

In this section we discuss some proper-ties of Default Logic. Some of theseproperties can be interpreted as defi-ciencies, or they highlight some of Reit-er’s original “design decisions” and showalternative ideas that could be followedinstead. In this sense, the discussion inthis section motivates alternative ap-proaches that will be presented in sub-sequent sections. One point that shouldbe stressed is that there is not a “cor-rect” default logic approach, but ratherthe most appropriate for the concreteproblem at hand. Different intuitionslead to different approaches that maywork better for some applications andworse for others.

3.1 Existence of Extensions

We saw that a default theory may nothave any extensions. Is this a shortcom-ing of Default Logic? One might holdthe view that if the default theory in-cludes “nonsense” (for example true :p / ¬p), then the logic should indeed beallowed to provide no answer. Accordingto this view, it is up to the user toprovide meaningful information in theform of meaningful facts and defaults;after all, if a program contains an error,we don’t blame the programming lan-guage.

The opposite view regards nonexist-ence of extensions as a drawback, andwould prefer a more “fault-tolerant” logic;one which works even if some pieces ofinformation are deficient. This view-point is supported by the trend towards

heterogeneous information sources,where it is not easy to identify whichsource is responsible for the deficiency,or where the single pieces of informa-tion are meaningful, but lead to prob-lems when put together.

A more technical argument in favor ofthe second view is the concept of semi-monotonicity. Default Logic is a methodfor performing nonmonotonic reasoning,so we cannot expect it to be monotonicwhen new knowledge is added to the setof facts. However, we might expect thatthe addition of new defaults would yieldmore, and not less information.2 For-mally, semi-monotonicity means the fol-lowing:

Let T 5 ~W, D! and T9 5 ~W, D9! bedefault theories such that D # D9.Then for every extension E of T thereis an extension E9 of T9 such that E# E9.

Default Logic violates this property. Forexample, T 5 ~À, $true : p/p%! has thesingle extension E 5 Th~$p%!, but T95 ~À, $true : p/p, true : q/¬q%! has noextension. So nonexistence of extensionsleads to the violation of semi-monotonic-ity. Even though the concept of semi-monotonicity is not equivalent to theexistence of extensions, these two prop-erties usually come together (for a moreformal support of this claim see Anto-niou et al. [1996]).

If we adopt the view that the possiblenonexistence of extensions is a problem,then there are two alternative solutions.The first one consists in restricting at-tention to those classes of default theo-ries for which the existence of exten-sions is guaranteed. In his classicalpaper [Reiter 1980], Reiter alreadyshowed that if all defaults in a theory Tare normal (in which case T is called anormal default theory), then T has at

2Some researchers would disagree with this viewand regard semi-monotonicity as not desirable;see, for example, Brewka [1991].

346 • G. Antoniou


least one extension. Essentially this isbecause all processes are successful, ascan be easily seen.

THEOREM 3. Normal default theoriesalways have extensions. Furthermore,they satisfy semi-monotonicity.

One problem with the restriction tonormal default theories is that theirexpressiveness is limited. In general, itcan be shown that normal default theo-ries are strictly less expressive thangeneral default theories. Normal de-faults have limitations, particularly re-garding the interaction among defaults.Consider the example

Bill is a high school dropout.Typically, high school dropouts areadults.Typically, adults are employed.

These facts are naturally representedby the normal default theory T 5~$dropout~bill!%, $dropout~X! : adult~X!/adult~X!, adult~X! : employed~X!/employed~X!%!. T has the single exten-sion Th~$dropout~bill!, adult~bill!,employed~bill!}!. It is acceptable to as-sume that Bill is adult, but it is counter-intuitive to assume that Bill is em-ployed! That is, whereas the seconddefault on its own is accurate, we wantto prevent its application in case theadult X is a high school dropout. Thiscan be achieved if we change the seconddefault to

adult~X! : employed~X! ∧ ¬dropout~X!

employed~X!.

But this default is not normal.3 Defaultsof this form are called semi-normal;Etherington [1987a] studied this classof default theories, and gave a sufficientcondition for the existence of exten-

sions. Another way of expressing inter-actions among defaults is the use ofexplicit priorities; this approach will befurther discussed in Section 6.

Instead of imposing restrictions onthe form of defaults in order to guaran-tee the existence of extensions, theother principal way is to modify theconcept of an extension in such a waythat all default theories have at leastone extension, and that semi-monoto-nicity is guaranteed. In Sections 4 and 5we will discuss two important variantswith these properties, Lukaszewicz’Justified Default Logic and Schaub’sConstrained Default Logic.

3.2 Joint Consistency of Justifications

It is easy to see that the default theoryconsisting of the defaults true : p / qand true : ¬p / r has the single exten-sion Th~$q, r%!. This shows that thejoint consistency of justifications is notrequired. Justifications are not sup-posed to form a consistent set of beliefs,rather they are used to sanction “jump-ing” to some conclusions.

This design decision is natural andmakes sense for many cases, but canalso lead to unintuitive results. As anexample consider the default theory,due to Poole, which says that, by de-fault, a robot’s arm (say a or b) is us-able unless it is broken; further, weknow that either a or b is broken. Giventhis information, we would not expectboth a and b to be usable.

Let us see how Default Logic treatsthis example. Consider the default the-ory T 5 ~W, D! with W 5 $broken~a!∨ broken~b!% and D consisting of thedefaults

true : usable~a! ∧ ¬broken~a!

usable~a!,

true : usable~b! ∧ ¬broken~b!

usable~b!.

3Note that it is unreasonable to add ¬dropout~X!to the prerequisite of the default to keep it nor-mal, because then we would have to definitelyknow that an adult is not a high school dropoutbefore concluding that the person is employed.



Since we do not have definite informa-tion that a is broken we may apply thefirst default and obtain E9 5 Th~W ø

$usable~a!%!. Since E9 does not includebroken~b! we may apply the second de-fault and get Th~W ø $usable~a!,usable~b!%! as an extension of T. Thisresult is undesirable, as we know thateither a or b is broken.

In Section 5 we shall discuss Con-strained Default Logic as a prototypicalDefault Logic approach that enforcesjoint consistency of justifications of de-faults involved in an extension.

The joint consistency property givesup part of the expressive power of de-fault theories: under this property anydefault with several justifications w :c1, . . . , cn / x is equivalent to the mod-ified default w : c1 ∧ . . . ∧ cn / xwhich has one justification. This is incontrast to a result in Besnard [1989],which shows that in Default Logic, de-faults with several justifications arestrictly more expressive than defaultswith just one justification. Essentially,in default logics adopting joint consis-tency it is impossible to express defaultrules of the form “In case I am ignorantabout p (meaning that I know neither pnor ¬p) I conclude q”. The natural rep-resentation in default form would betrue : p, ¬p / q, but this default cannever be applied if joint consistency isrequired, because its justifications con-tradict one another; on the other hand itcan be applicable in the sense of DefaultLogic.

Another example for which joint con-sistency of justifications is undesirableis the following.4 When I prepare for atrip then I use the following defaultrules:

If I may assume that the weather will bebad I’ll take my sweater.If I may assume that the weather will begood then I’ll take my swimsuit.

In the absence of any reliable infor-mation about the weather I am cautiousenough to take both with me. But notethat I am not building a consistent be-lief set upon which I make these deci-sions; obviously the assumptions of thedefault rules contradict each other. SoDefault Logic will treat this example inthe intended way, whereas joint consis-tency of justifications will prevent mefrom taking both my sweater and myswimsuit with me.

3.3 Cumulativity and Lemmas

Cumulativity is, informally speaking,the property that allows for the safe useof lemmas. Formally: Let D be a fixed,countable set of defaults. For a formulaw and a set of formulae W we define W£ Dw iff w is included in all extensions ofthe default theory ~W, D!. Now, cumu-lativity is the following property:

If W £ Dw, then for all c :

W £ Dc N W ø $w% £ Dc.

If we interpret w as a lemma, cumula-tivity says that the same formulae canbe obtained from W as from W ø $w%.This is the standard basis of using lem-mas in, say, mathematics. Default Logicdoes not respect cumulativity: considerT 5 ~W, D! with W 5 À and D consist-ing of the defaults

true : a

a,

a ∨ b : ¬a

¬a

(this example is due to Makinson). Theonly extension of T is Th~$a%!. Obvi-ously, W £ Da. From a ∨ b [ Th~$a%!we get W £ Da ∨ b. If we take W9 5 $a∨ b%, then the default theory ~W9, D! hastwo extensions, Th~$a%! and Th~$¬a,b%!; therefore W ø $a ∨ b% £y Da.

An analysis of cumulativity and otherabstract properties of nonmonotonic in-ference is found in Makinson [1994]. A4My thanks go to an anonymous referee.

348 • G. Antoniou


lot of work has been invested in devel-oping default logics that possess thecumulativity property, one notable ap-proach being Brewka’s Cumulative De-fault Logic [Brewka 1991]. But it isdoubtful whether this is the right wayto go, since it has additional conceptualand computational load, due to the useof assertions rather than plain formulae(see Section 7.2 for more information).

One might argue that semimonotonic-ity is rather unintuitive because it re-quires a defeasible conclusion whichwas based on some assumptions to berepresented by a certain piece of infor-mation, that means a fact, and yet ex-hibit the same behaviour.

From the practical point of view thereally important issue is whether weare able to represent and use lemmas ina safe way. How can we do this inDefault Logic? Schaub [1992] proposedthe representation of a lemma by a cor-responding lemma default whichrecords in its justifications the assump-tions on which a conclusion was based.The formal definition of a lemma de-fault is as follows.

Let Px be a nonempty, successful pro-cess of T, minimal with the property x

[ In~Px!. A lemma default dx corre-sponding to x is the default

true : c1, . . . , cn

x

where $c1, . . . , cn% 5 $c?c [ just~d!%for a d occurring in Px. This defaultcollects all assumptions that were usedin order to derive x.

THEOREM 4. Let x be included in anextension of T and dx a correspondinglemma default. Then every extension ofT is an extension of T ’ 5 ~W, D ø

$dx%!, and conversely.

So it is indeed possible to representlemmas in Default Logic, not as facts(as required by cumulativity) but ratheras defaults, which appears more natural

anyway, since it highlights the natureof a lemma as having been proven de-feasibly and thus as being open to dis-putation.

4. JUSTIFIED DEFAULT LOGIC

4.1 Motivation and Formal Presentation

Lukaszewicz considered the possiblenonexistence of extensions as a repre-sentational shortcoming of the originalDefault Logic, and presented a variant,Justified Default Logic [Lukaszewicz1988] which avoids this problem. Theessence of his approach is the following:If we have a successful but not yetclosed process, and all ways of expand-ing it by applying a new default lead toa failed process, then we stop and ac-cept the current In-set as an extension.In other words, we take back the final,“fatal” step that causes failure.

Consider the default theory T 5 ~W,D! with W 5 $holidays, sunday% andD consisting of the defaults

d1 5sunday : goFishing ∧ ¬ wakeUpLate

goFishing,

d2 5holidays : wakeUpLate

wakeUpLate.

It is easily seen that T has only oneextension (in the sense of Section 2),namely Th({holidays, sunday, wakeUp-Late}). But if we apply d1 first, then d2can be applied and leads to a failedprocess. In this sense we lose the inter-mediate information Th({holidays, sun-day, goFishing}). On the other hand, inJustified Default Logic we would stopafter the application of d1 instead ofapplying d2 and running into failure;therefore we accept Th({holidays, sun-day, goFishing}) as an additional (modi-fied) extension.

Technically, this is achieved by pay-ing attention to maximally successfulprocesses. Let T be a default theory,



and let P and G be processes of T. Wedefine P , G iff the set of defaults oc-curring in P is a proper subset of thedefaults occurring in G. P is called amaximal process of T, iff P is successfuland there is no successful process Gsuch that P , G. A set of formulae E iscalled a modified extension of T iff thereis a maximal process P of T such thatE 5 In~P!.

In the example above, P 5 ~d1! isa maximal process: the only processthat strictly includes P is G 5 ~d1, d2!which is not successful. ThereforeTh~$holidays, sunday, goFishing%! is amodified extension of T. T has anothermodified extension, which is the singleextension Th~$holidays, sunday, wake-UpLate%! of T. Obviously every closedand successful process is a maximal pro-cess (since no new default can be ap-plied). Therefore we have the followingresult:

THEOREM 5. Every extension of a de-fault theory T is a modified extension of T.

In the process of a default theory Tmaximal processes correspond either toclosed and successful nodes, or to nodesn such that all immediate children of nare failed.

It is instructive to look at a defaulttheory without an extension, for exam-ple T 5 ~W, D! with W 5 À and D 5$true : p/¬p%. The empty process ismaximal (though not closed), becausethe application of the “strange default”would lead to a failed process, thereforeTh~À! is a modified extension of T.Since any branch of the process tree canbe extended successfully to a modifiedextension, the following result can beshown.

THEOREM 6. Every default theory hasat least one modified extension. Further-more, Justified Default Logic satisfiessemi-monotonicity.

4.2 Lukaszewicz’ Original Definition

The original definition given inLukaszewicz [1988] was based on fix-point equations. Let T 5 ~W, D! be adefault theory, and E, F, E9 and F9 setsof formulae. We say that a default d 5

w : c1, . . . , cn / x is applicable to E9

and F9 with respect to E and F iff w [

E9 and E ø $x% ?5 ¬c for all c [ Fø $c1, . . . , cn%.

E9 and F9 are closed under the appli-cation of defaults in D with respect to Eand F iff, whenever a default d 5 w :c1, . . . , cn / x in D is applicable to E9

and F9 with respect to E and F, x [

E9 and $c1, . . . , cn% # F9.Define LT

1 ~E, F! and LT2 ~E, F! to be

the smallest sets of formulae such thatLT

1 ~E, F! is deductively closed, W #

LT1 ~E, F!, and LT

1 ~E, F! and LT2 ~E, F!

are closed under D with respect to Eand F.

The following theorem shows thatmodified extensions correspond exactlyto sets E and F satisfying the fixed-point equations E 5 LT

1 ~E, F! and F5 LT

2 ~E, F!. This is not surprising: in-tuitively, the idea behind the compli-cated definition of the L-operators is tomaintain the set of justifications of de-faults that have been applied (i.e., thesets F and F9 which, in fact, correspondto ¬Out~P!), and to avoid applicationsof defaults if they lead to an inconsis-tency with one of these justifications.

THEOREM 7. Let T be a default the-ory. For every modified extension E of Tthere is a set of formulae F such that E5 LT

1 ~E, F! and F 5 LT2 ~E, F!.

Conversely, let E and F be sets offormulae such that E 5 LT

1 ~E, F! andF 5 LT

2 ~E, F!. Then E is a modifiedextension of T.

350 • G. Antoniou


5. CONSTRAINED DEFAULT LOGIC

5.1 Motivation and Definition

Justified Default Logic avoids runninginto inconsistencies and can thereforeguarantee the existence of modified ex-tensions. On the other hand, it does notrequire joint consistency of default jus-tifications; for example, the default the-ory T 5 ~W, D! with W 5 À and D 5$true : p/q, true :¬p/r% has the singlemodified extension Th~$q, r%!. Con-strained Default Logic [Schaub 1992;Delgrande et al. 1994] is a Default Logicapproach which enforces joint consis-tency. In the example above, after theapplication of the first default the sec-ond default may not be applied becausep contradicts ¬p.

Furthermore, since the justificationsare consistent with each other, we testthe consistency of their conjunctionwith the current knowledge base. In theterminology of processes, we require theconsistency of In~P! ø ¬Out~P!.

Finally, we adopt the idea from theprevious section that namely a defaultmay only be applied if it does not lead toa contradiction (failure) a posteriori.That means, if w : c1, . . . , cn / x istested for application to a process P,then In~P! ø ¬Out~P! ø $c1, . . . ,cn, x% must be consistent. We note thatthe set Out no longer makes sense sincewe require joint consistency. Instead,we have to maintain the set of formulaewhich consists of W, all consequentsand all justifications of the defaultsthat have been applied.

—Given a default theory T 5 ~W, D!and a sequence P of defaults in Dwithout multiple occurrences, we de-fine Con~P! 5 Th~W ø $w%?w is theconsequent or a justification of a de-fault occurring in P}. Sometimes werefer to Con~P! as the set of con-straints or the set of supporting be-liefs.

Con~P! represents the set of beliefssupporting P. For the default theory T5 ~W, D! with W 5 À and D 5 $d1 5true : p/q, d2 5 true : ¬p/r% let P1 5

~d1!. Then Con~P1! 5 Th~$p, q%!.We say that a default d 5 w : c1,

. . . , cn / x is applicable to a pair ofdeductively closed sets of formulae ~E,C! iff w [ E and c1 ∧ . . . ∧ cn ∧ x isconsistent with C. A pair ~E, C! of de-ductively closed sets of formulae iscalled closed under D if, for every de-fault w : c1, . . . , cn / x [ D that is ap-plicable to ~E, C!, x [ E and $c1,. . . , cn, x% # C.

In the example above, d2 is not applica-ble to ~In~P1!, Con~P1!! 5 ~Th~$q%!,!Th~$p, q%! because $ ¬p ∧ r% ø Th~$p,q%! is inconsistent.

Let P 5 ~d0, d1, . . . ! be a sequenceof defaults in D without multiple occur-rences.

—P is a constrained process of the de-fault theory T 5 ~W, D! iff, for all ksuch that P@k# is defined, dk is appli-cable to ~In~P@k#!, Con~P@k#!!.

—A closed constrained process P is aconstrained process such that everydefault d which is applicable to~In~P!, Con~P!! already occurs in P.

—A pair of sets of formulae ~E, C! is aconstrained extension of T iff there isa closed constrained process P of Tsuch that ~E, C! 5 ~In~P!, Con~P!!.

Note that we do not need a concept ofsuccess here because of the definition ofdefault applicability we adopted: d isonly applicable to ~E, C! if it does notlead to a contradiction. Let us reconsiderthe “broken arms” example: T 5 ~W, D!with W 5 $broken~a! ∨ broken~b!%, andD consisting of the defaults



d1 5true : usable~a! ∧ ¬broken~a!

usable~a!,

d2 5true : usable~b! ∧ ¬broken~b!

usable~b!.

It is easily seen that there are twoclosed constrained processes, ~d1! and~d2!, leading to two constrained exten-sions:

~Th~W ø $usable~a!%!, Th~$broken~b!,

usable~a!, ¬broken~a!%!!,

and

~Th~W ø $usable~b!%!, Th~$broken~a!,

usable~b!, ¬broken~b!%!!

The effect of the definitions above isthat it is impossible to apply both de-faults together: after the application of,say, d1, ¬broken~a! is included in theCon-set; together with broken~a! ∨broken~b! it follows broken~b!, there-fore d2 is blocked. The two alternativeconstrained extensions describe the twopossible cases we would have intuitivelyexpected.

For another example consider T 5~W, D! with W 5 $p% and D 5 $p :¬r/q, p : r/r%. T has two constrainedextensions, ~Th~$p, q%!, Th~$p, q,¬r%!! and ~Th~$p, r%!, Th~$p, r%!!.Note that for both constrained exten-sions, the second component collects theassumptions supporting the first compo-nent.

5.2 A Fixpoint Characterization

Schaub’s original definition of con-strained extensions used a fixed-pointequation [Schaub 1992]: Let T 5 ~W,D! be a default theory. For a set C offormulae let QT~C! be the pair of small-est sets of formulae ~E9, C9! such that

(1) W # E9 # C9

(2) E9 and C9 are deductively closed

(3) For every w : c1, . . . , cn / x [ D, ifw [ E9 and C ø $c1, . . . , cn, x% isconsistent, then x [ E9 and $c1,. . . , cn, x% # C9.

The following result shows that this def-inition is equivalent to the definition ofconstrained extensions from the previ-ous subsection.

THEOREM 8. ~E, C! is a constrainedextension of T iff ~E, C! 5 QT~C!.

THEOREM 9. Every default theory hasat least one constrained extension. Fur-thermore Constrained Default Logic issemi-monotonic.

5.3 Interconnections

In the following we describe the rela-tionship among the default logic vari-ants presented so far.

THEOREM 10. Let T be a default the-ory and E 5 In~P! an extension of T,where P is a closed and successful pro-cess of T. If E ø ¬Out~P! is consistent,then ~E, Th~E ø ¬Out~P!!! is a con-strained extension of T.

The converse does not hold since theexistence of an extension is not guaran-teed. For example T 5 ~À, $true :p/¬p%! has the single constrained ex-tension ~Th~À!, Th~À!!, but no exten-sion.

THEOREM 11. Let T be a default the-ory and E 5 In~P! a modified exten-sion of T, where P is a maximal processof T. If E ø ¬Out~P! is consistent,then ~E, Th~E ø ¬Out~P!!! is a con-strained extension of T.

The example T 5 ~W, D! with W 5 Àand D 5 $true : p/q, true : ¬p/r% showsthat we cannot expect the first compo-

352 • G. Antoniou


nent of a constrained extension to be amodified extension: T has the singlemodified extension Th~$q, r%!, but pos-sesses two constrained extensions,~Th~$q%!, Th~$p, q%!! and ~Th~$r%!,Th~$¬p, r%!!. As the following resultdemonstrates, it is not accidental thatfor both constrained extensions, thefirst component is included in the modi-fied extension.

THEOREM 12. Let T be a default the-ory and ~E, C! a constrained extensionof T. Then there is a modified extensionF of T such that E # F.

The following examples illustrateswell the difference between the threeapproaches. Consider the default theoryT 5 ~W, D! with W 5 À and

D 5 Htrue : p

q,

true : ¬p

r,

true : ¬q, ¬r

s J.

T has the single extension

Th~$q, r%!,

two modified extensions,

Th~$q, r%!

Th~$s%!,

and three constrained extensions

~Th~$q%!, Th~$q, p%!!

~Th~$r%!, Th~$r, ¬p%!!

~Th~$s%!, Th~$s, ¬q, ¬r%!!.

This theory illustrates the essential dif-ferences of the three approaches dis-cussed. Default Logic does not careabout inconsistencies among justifica-tions, and may run into inconsistencies.Thus the first two defaults can be ap-plied together, while if the third defaultis applied first, then the process is notclosed and subsequent application of an-

other default leads to failure. JustifiedDefault Logic avoids the latter situa-tion, so we obtain an additional modi-fied extension. Constrained DefaultLogic avoids running into failure, too,but additionally requires joint consis-tency of justifications; therefore the twofirst defaults cannot be applied in con-junction, as in the other two ap-proaches. Thus we get three constrainedextensions.

We conclude this section by notingthat for normal default theories, all de-fault logic approaches discussed areidentical. In other words, they coincidefor the “well-behaved” class of defaulttheories, and seek to extend it in differ-ent directions.

THEOREM 13. Let T be a normal de-fault theory, and E a set of formulae.The following statements are equivalent.

(a) E is an extension of T.

(b) E is a modified extension of T.

(c) There exists a set of formulae C suchthat ~E, C! is a constrained exten-sion of T.

6. PRIORITIES ON DEFAULTS

6.1 Motivation

Often it is desirable to have prioritiesamong defaults indicating a preferenceon which default to apply in situationswhere several defaults are applicable.One approach is to use implicit criteriaof preference such as specificity amongdefaults; systems in this category in-clude Baader and Hollunder [1993];Etherington and Reiter [1983]; Pearl[1990]; and Touretzky et al. [1987].

This family of approaches suffersfrom the problem that the criterion maynot be sufficient to model different ap-plication domains. For example, whilespecificity is appropriate for taxonomicinformation found in biology, it cannotcapture the legal principle that a morerecent law is preferred to an older one.



Therefore other kinds of approachesmay have to be used, in which the prior-ity information is represented explicitly.In the following we discuss two suchapproaches, which are based on the ideaof static priorities and dynamic priori-ties, respectively.

6.2 Static Priorities

In Prioritized Default Logic [Brewka1994], the user gives explicitly the prior-ity order in which defaults have to beapplied in situations where more thanone default is applicable. In the sim-plest case a strict well-ordering (thatmeans, an irreflexive total order inwhich every subset of D has a smallestelement) is used. Let us consider thedefault theory T 5 ~W, D! with W 5$bird, penguin% and D consisting of thedefaults

d1 5penguin : ¬flies

¬flies, d2 5

bird : flies

flies.

Suppose a total order ,, is givenaccording to which d1 is preferred overd2, i.e. d1 ,, d2. Then T together with,, should only admit the extensionTh~$bird, penguin, ¬flies%!.

Let T 5 ~W, D! be a normal defaulttheory, and ,, a strict well order onD. A process P 5 ~d0, d1, . . . ! of T isgenerated by ,, iff, for all i, d i is the ,,— minimal default from P 2 P@i# thatis applicable to In~P@i#!, provided thatsuch a d i exists. Clearly, P is closed; itis also successful since T is a normaldefault theory. We say that E is gener-ated by ,, iff E 5 In~P! for a processP that is generated by ,, .

P and E, as defined above, areuniquely determined. In the previousexample, ,, is defined by d1 ,, d2.~d1! is the process generated by ,,, andTh~$bird, penguin, ¬flies%! is the ex-tension generated by ,,.

Of course, we cannot always expectthe defaults to be ordered in a totalway. Often defaults should be left in-comparable with one another. In otherwords, requiring a total ordering of de-faults will often be a kind of overspecifi-cation. Here is a simple example frompolitics.

According to conservative politicians,taxes should be cut without cuttingspending; the latter is unnecessary be-cause reduced taxes are supposed tolead to increased economic growth. Rad-ical conservatives proclaim tax cuts andspending cuts because the governmentshould be as lean as possible. Accordingto social democrats, neither taxes norspending should be cut because theybelieve government should afford wel-fare programs and create additional de-mand. This information can be ex-pressed by the following defaults:

d1 5conservative : taxCut ∧ ¬spendingCut

taxCut ∧ ¬ spendingCut

d2 5conservative ∧ radical : taxCut ∧ spendingCut

taxCut ∧ spendingCut

d3 5socialDemocrat : ¬taxCut ∧ ¬spendingCut

¬taxCut ∧ ¬spendingCut

Intuitively it is clear that d2 should begiven higher priority than d1 becausethe information that somebody is a rad-ical conservative is more specific thanthe information that she is a conserva-tive. But there is no reason to prescribeany order between d1 and d3, or betweend2 and d3. Thus we only have d2 , d1.

Technically speaking, the priority in-formation will be given in the form of astrict partial order , on the set of de-faults. The definition of the semanticswill make use of all strict well ordersthat contain ,. This approach resem-bles the treatment of concurrency incomputer science, where meaning is as-signed by considering all possible lin-earizations.

354 • G. Antoniou


Definition 2. T 5 ~W, D, ,! is aprioritized default theory if ~W, D! is anormal default theory and , a strictpartial order on D. E is a PDL-extensionof T iff there is a strict well order ,, onD which contains , and generates E.

Consider the previous example, withthe set of facts W 5 $conservative,radical%, and , defined by d2 , d1.There are three strict well orders on Dthat contain ,, namely

d3 ,, d2 ,, d1,

d2 ,, d3 ,, d1,

and

d2 ,, d1 ,, d3.

It is easily seen that all of them lead tothe same PDL-extension, namely Th~$conservative, radical, taxCut, spend-ingCut%!.

By definition of a PDL-extension E,E 5 In~P! for a closed and successfulprocess P. Therefore we make the fol-lowing observation:

THEOREM 14. If E is a PDL-extensionof the prioritized default theory T 5~W, D, ,! then E is an extension of~W, D!.

The next result follows from the ob-servation that given a finite set M, ev-ery strict partial order on M is includedin a strict well order on M. A strictpartial order , defines some con-straints about which elements shouldprecede which other elements. A stricttotal order ,, respecting these con-straints can always be constructed.Since M is finite, a minimal elementexists. Therefore the extension gener-ated by ,, is a PDL-extension.

THEOREM 15. A prioritized defaulttheory T 5 ~W, D, ,! always has aPDL-extension if D is finite.

The above theorem is not true forinfinite sets of defaults. Let D 5 $d iui$ 0%, and , the strict partial order onD determined by d i , d0 for all i . 0.Then there is no strict well order of Dcontaining ,; there are, of course, stricttotal orders, but the existence of mini-mal elements cannot be established.

6.3 Dynamic Priorities

The main drawback of the previous ap-proach is that the user has to provideall priority information, which is sup-posed to be static and not change overtime. These requirements may be toostrict for some problems. Brewka [1994]proposed an approach in which priorityinformation is part of the logical lan-guage and can be derived like any otherstatement. For example, it is possible towrite

p : d1 , d2

d1 , d2

to express that “In cases p is true, usu-ally default d1 should be given prefer-ence over default d2”. Of course theremay be some exceptions in which d1 ,d2 cannot be assumed due to other in-formation being available.

As this example shows, the approachuses names to refer to defaults, and apreference relation , as part of thelanguage. Reasoning about priorities isvery flexible, but the high expressive-ness brings problems, too; for example,it is possible that even if all defaults arenormal there is no extension. Considerthe theory consisting of the followingtwo defaults:

d1 5true : d2 , d1

d2 , d1

, d2 5true : d1 , d2

d1 , d2

.



Default d1 says that usually you shouldprefer default d2 instead of d1, and viceversa. Therefore this theory has no ex-tension.

7. OTHER VARIANTS OF DEFAULT LOGIC

In this section we briefly review somefurther Default Logic approaches with-out going into technical details.

7.1 Rational Default Logic

Constrained Default Logic enforces jointconsistency of the justifications of de-faults that contribute to an extension,but goes one step further by requiringthat the consequent of a default be con-sistent with the current Con-set. Ratio-nal Default Logic [Mikitiuk and Truszc-zynski 1995] does not require the latterstep. Technically, a default w : c1,. . . , cn / x is rationally applicable to apair of deductively closed sets of formu-lae ~E, C! iff w [ E and $c1, . . . , cn%ø C is consistent.

As an example, consider the defaulttheory T 5 ~W, D! with W 5 À and D5 $true : b/c, true : ¬b/d, true : ¬c/e,true : ¬d/f%. T has the single extensionTh~$c, d%!, three constrained exten-sions,

~Th~$e, f%!, Th~$e, ¬c, f, ¬d%!!

~Th~$c, f%!, Th~$c, b, f, ¬d%!!

~Th~$d, e%!, Th~$d, ¬b, e, ¬c%!! (5)

but two rational extensions, Th~$c, f%!and Th~$d, e%!. The first constrainedextension is “lost” in Rational DefaultLogic because it is not closed underapplication of further defaults. true :b / c and true : ¬b / d are both ratio-nally applicable to Th~$e, f%!; but onceone of them is applied to Th~$e, f%! weget a failed situation.

Mikitiuk and Truszczynski [1995]show that if E is an extension of T inRational Default Logic, then ~E, C! is a

constrained extension of T for some setC. The converse is true for semi-normaldefault theories.

Rational Default Logic does not guar-antee the existence of extensions. Forexample, the default theory consistingof the single default true : p / ¬p doesnot have any extensions.

7.2 Cumulative Default Logic

As mentioned earlier, Cumulative De-fault Logic was introduced by Brewka toensure the property of cumulativity[Brewka 1991]. The solution he adoptedwas to use so-called assertions, pairs~w, J! of a formula w and a set of formu-lae J, which collects the assumptionsthat were used to deduce w. When adefault is applied to deduce w the justi-fications of that default are added to J.

We illustrate this approach by consid-ering the example from subsection 3.3which showed that Default Logic vio-lates cumulativity. Consider T 5 ~W,D! with W 5 À and D consisting of thedefaults

true : a

a,

a ∨ b : ¬a

¬a.

In the beginning we can apply only thefirst default and derive the assertion~a, $a%!, meaning that we derived abased on the assumption a. Obviouslythe second default is not applicable. Theviolation of cumulativity in DefaultLogic was caused by the addition of a∨ b as a new fact which opened the wayfor the application of the second defaultinstead of the first one. But in Cumula-tive Default Logic we are allowed to addthe assertion ~a ∨ b, $a%! to the defaulttheory (if a is derived based on a, thena ∨ b is also derived based on a), butnow the second default is still not appli-cable because ¬a is not consistent withthe set of supporting beliefs $a%.

Note that adding a to the default the-

356 • G. Antoniou


ory as we did in Default Logic corre-sponds to adding the assertion ~a, À!,which is different from ~a, $a%!. If wedisregard $a%, which is the assumptionupon which the deduction of a wasbased, then indeed we can get moreconclusions; this forgetting is the deeperreason for the failure of Default Logic tosatisfy cumulativity.

From the technical and practicalpoint of view, the use of assertions iscomplicated and causes practical prob-lems, for example with regard to imple-mentation; this is the price we have topay for cumulativity. And the gain isquestionable in the light of our discus-sion in subsection 3.3, which arguedthat lemmas can and should be repre-sented as defaults, rather than facts.Nevertheless, Cumulative Default Logicwas historically an important one.

7.3 Disjunctive Default Logic

The “broken arm” example from subsec-tion 3.2 shows that Default Logic has adeficiency with the correct treatment ofdisjunctive information. Gelfond et al.[1991] proposes a way out of these diffi-culties by the following analysis: if aformula w ∨ c becomes part of the cur-rent knowledge base (either as a fact oras a consequent of some default), itshould not be included as a predicatelogic formula. Instead, it should havethe effect that one of w and c becomespart of an extension. In other words, theexpression

broken~a!ubroken~b!

should have the effect that an extensioncontains one of the two disjuncts, ratherthan the disjunction broken~a! ∨broken~b!. To see another example, con-sider the default theory T 5 ~W, D!with W 5 $p ∨ q% and D 5 $p : r/r,q : r/r%. In Default Logic we know theformula p ∨ q but are unable to applyany of the two defaults; so we end upwith the single extension Th~$p ∨ q%!.On the other hand, Disjunctive Default

Logic leads to two extensions, one inwhich p is included, and one in which qis included. In the former case p : r / rbecomes applicable, in the latter caseq : r / r becomes applicable. So we endup with two extensions, Th~$p, r%! andTh~$q, r%!, which is intuitively moreappealing. For more details, see Gelfondet al. [1991].

7.4 Weak Extensions

All variants of Default Logic discussedso far share the same idea of treatingprerequisites of defaults: in order for adefault d to be applicable, its prerequi-site must be proven using the facts andthe consequents of defaults that wereapplied before d. For example, in orderfor the default p : true /p to be applica-ble, p must follow from the facts, or bethe consequent of another default etc. Adefault theory consisting only of thisdefault has the single extension Th~À!.

This has led researchers to refer toDefault Logic as being “stronglygrounded” in the given facts. In con-trast, Autoepistemic Logic [Moore 1985]provides more freedom in choosing whatone wants to believe in. Weak exten-sions of default theories were intro-duced to capture this intuition in thedefault logic framework [Marek andTruszczynski 1993].

In the framework of weak extensions,we can simply decide to believe in someformulae. The only requirement is thatthis decision can be justified using thefacts and default rules. Reconsider thedefault theory consisting of the singledefault p : true / p. We may decide tobelieve in p or not. Suppose we do be-lieve in p; then the default can be ap-plied and gives us p as a consequence.In this sense the default justifies thedecision to believe in p; Th~$p%! is thusa weak extension. Of course, we couldalso adopt the more cautious view anddecide not to believe in p; then thedefault is not applicable, so p cannot be



proved and our decision is again justi-fied. In general, extensions of a theoryT are also weak extensions of T. For atechnical discussion, see Marek andTruszczynski [1993].

8. CONCLUSION

Default Logic is an important method ofknowledge representation and reason-ing, because it supports reasoning withincomplete information, and because de-faults can be found naturally in manyapplication domains, such as diagnosticproblems, information retrieval, legalreasoning, regulations, specifications ofsystems and software, etc. Default Logiccan be used either to model reasoningwith incomplete information, which wasthe original motivation, or as a formal-ism which enables compact representa-tion of information [Cadoli et al. 1994].Important prerequisites for the develop-ment of successful applications in thesedomains include (i) the understandingof the basic concepts, and (ii) the exist-ence of powerful implementations.

The aim of this paper is to contributeto the first requirement. We have dis-cussed the basic concepts and ideas ofDefault Logic, and based the presenta-tion on operational interpretations,rather than on fixpoints, as usuallydone. The operational interpretationsallow learners to apply concepts to con-crete problems in a straightforwardway. This is an important point, be-cause the difficulty of understandingDefault Logic should not be underesti-mated. A survey among students at theUniversity of Toronto regarding theirranking of the difficulty of several non-monotonic reasoning formalisms re-sulted in Default Logic (presented basedon fixpoints) being perceived as the sec-ond most difficult one, surpassed onlyby the full version of Circumscription[McCarthy 1980], but well ahead of Au-toepistemic Logic [Moore 1985].

In this paper we have not discussedthe semantics of Default Logic (see Bes-nard and Schaub [1994]; Engelfriet andTreur [1996]; Teng [1996]), or complex-

ity issues (see Gottlob [1992]; Kautz andSelman [1989]). An argumentation-the-oretic study of default logics is found inBondarenko et al. [1997].

Implementation aspects were alsooutside the scope of this paper; someentry points to current work in the areainclude Cholewinski [1994]; Cholewin-ski et al. [1995]; Courtney et al. [1996];Linke and Schaub [1995]; Niemela[1995]; Risch and Schwind [1994]; andSchaub [1995].

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Received: June 1996; revised: May 1997; accepted: August 1998



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