Nonmonotonic Inheritance, Argumentation and Logic Programming

Nonmonotonic Inheritance, Argumentation andLogic ProgrammingPhan Minh Dung and Tran Cao SonComputer Science ProgramSchool of Advanced TechnologyAsian Institute of TechnologyG.P.O Box 2754, Bangkok 10501, ThailandEmail: [email protected], [email protected]. We study the conceptual relationship between the semanticsof nonmonotonic inheritance reasoning and argumentation. We show thatthe credulous semantics of nonmonotonic inheritance network can be cap-tured by the stable semantics of argumentation. We present a transfor-mation of nonmonotonic inheritance networks into equivalent extendedlogic programs.1 IntroductionArgument-based approaches to nonmonotonic reasoning have been intensivelystudied and became prominent in AI and Logic Programming [6, 21, 24, 1, 20]just recently. But reasoning based on arguments represented as paths, has beenstudied in nonmonotonic inheritance reasoning, a speci�c �eld of nonmonotonicreasoning, from the very �rst day [30] and then in [13, 15, 27, 28, 29, 26, 25,8]. Path-based reasoning approaches to nonmonotonic inheritance networks arewidely accepted because they are intuitive and easy to implement.The interesting and surprising problem here is that the argument-based se-mantics of nonmonotonic inheritance network [13, 15, 27, 28, 29, 26, 25, 8] andthe general argumentation reasoning [6, 21, 24, 1, 20] seem to have conceptuallylittle in common. Touretzky et al. went so far to claim that one of the funda-mental principle of argumentation - the use of reinstater - can not be applied innonmonotonic inheritance reasoning [29].The relation between nonmonotonic inheritance reasoning and more generalframeworks to nonmonotonic reasoning like default logic, autoepistemic logic,logic programming has been intensively studied in [3, 4, 14, 11, 7, 23, 16]. Thebasic idea of these works is to �nd a way to translate a nonmonotonic inheritancenetwork into a \equivalent" theory of the respected nonmonotonic logic. But allof these transformations do not preserve the original semantics of nonmonotonicinheritance networks. Hence, conceptually, the relationship between the natu-ral path-based semantics of nonmonotonic inheritance networks and other moregeneral nonmonotonic logics such as default logic, autoepistemic logic, etc. re-mains unclear. The goal of this paper is to address this problem. We do that bystudying the relationship between the argumentation framework given in [1] andinheritance networks.

We show that each acyclic, consistent nonmonotonic inheritance network� can be viewed as an argumentation framework AF� so that the creduloussemantics of � \coincides" with the stable semantics of AF� . Further, we provethat the grounded semantics of AF� is contained in the skeptical semantics of� [13, 26]. Thus, we can say that grounded semantics provides the baseline ofskepticism in inheritance reasoning.We present a transformation of consistent nonmonotonic networks into ex-tended logic programs and show that the credulous semantics of the formercoincides with the answer set semantics [10] of the latter. To our knowledge,this is the �rst transformation of nonmonotonic inheritance network into othermore general nonmonotonic logics preserving the semantics of nonmontonic in-heritance networks.2 Preliminaries2.1 Inheritance networkA defeasible inheritance network � is de�ned here as a �nite collection of positiveand negative direct links between nodes. If x, y are nodes then x ! y (resp.x 6! y) represents a positive (resp. negative) direct link from x to y. A network� is consistent if there exist no two nodes x, y such that both x ! y andx 6! y belong to � . A positive path from x1 to xn through x2, ..., xn�1, denotedby �(x1; �; xn) or �(x1; x2; :::; xn�1; xn), is a sequence of direct links x1 ! x2,x2 ! x3, ... , xn�1 ! xn. Similarly, a negative path from x1 to xn through x2,..., xn�1, denoted by ��(x1; �; xn) or ��(x1; x2; :::; xn�1; xn), is a sequence of directlinks x1 ! x2, x2 ! x3, ... , xn�1 6! xn. A generalized path is a sequence ofdirect links (x1; x2), (x2; x3), ... , (xn�1; xn), where (xi; xi+1) denotes a positiveor negative direct link. � is acyclic if there is no generalized path (x1; x2),(x2; x3), ... , (xn�1; xn) with x1 = xn. The degree of the path � = �(x; �; y)(resp. � = ��(x; �; y)), denoted by deg� (�), is de�ned as the length (number ofedges) of the longest generalized path from x to y. Furthermore, we also use thenotation �(x1; �; xn�1) ! xn (resp. �(x1; �; xn�1) 6! xn) to denote the path�(x1; x2; :::; xn�1; xn) (resp. ��(x1; x2; :::; xn�1; xn)).From now on we will use � to denote an arbitrary but �xed network and � todenote a set of paths in � if no confusion is possible. The notion of inheritabilitypresented here relies on three concepts: constructibility, con ict, and preemption.Their de�nitions are taken from [15, 29, 13, 25, 26].De�nition1. A positive path �(x; �; u)! y is constructible in � i� �(x; �; u) 2� and u ! y 2 � . A negative path �(x; �; u) 6! y is constructible in � i��(x; �; u) 2 � and u 6! y 2 � .De�nition2. �(x; �; y) con icts with any path of the form ��(x; �; y) and viceversa. A path � is con icted in � i� � contains a path that con icts with �.Di�erent ways have been proposed to de�ne defeasible preemption in � [7,15, 13, 28, 26, 25]. Here, we follow the o�-path preemption given in [13].2

De�nition3. A positive path �(x; �; u)! y (see �gure 1) is preempted in � i�there is a node v such that (i) v 6! y 2 � and (ii) either v=x or there is a pathof the form �(x; �1; v; �2; u) 2 �. A negative path �(x; �; u) 6! y is preempted in� i� there is a node v such that (i) either v=x or (ii) there is a path of the form�(x; �1; v; �2; u) 2 � and v ! y 2 � .������ @@@@@I ����� �@@IJJ] �� ` ` ` ` ` ` ` ` ` ``````````` ````````````` yu vx� �1�2Fig. 1. �(x; �; u)! y is preemptedThe credulous semantics of an inheritance network is given in the followingde�nition.De�nition4. [13] � is defeasibly inheritable in �, written as � j� �, i�either � is a direct linkor � is a compound path, �=�(x; �; y) (likewise for negative path) such that(i) � is constructible in �, and(ii) � is not con icted in �, and(iii) � is not preempted in �.De�nition5. A set � of paths is a credulous extension of the net � i� � = f�: � j� � g.The skeptical semantics for inheritance network is de�ned by the notion ofideally skeptical extension and is de�ned as follows.De�nition6. [26, 25] The intersection of all credulous extensions of � is calledthe ideally skeptical extension of � .2.2 Argumentation frameworkIn the following section, the basics of the abstract theory of argumentation frame-work of Dung [1] is recalled.De�nition7. An argumentation framework is a pair AF=<AR; attacks>,where AR is a set of arguments, and attacks 2 AR� AR.3

If (A;B) 2 attacks we say A attacks B or B is attacked by A. A set ofarguments S is said to be attacked by an argument A if there is B 2 S suchthat (A;B) 2 attacks. Similarly, we say S attacks A if there is B 2 S such that(B;A) 2 attacks.De�nition8. A set of arguments S is said to be con ict-free if there exist notwo arguments A,B in S such that (A;B) 2 attacks.The stable semantics of AF is de�ned as follows.De�nition9. A con ict-free set of arguments S is called a stable extension ofAF if S attacks every argument which does not belong to S.It is easy to see that the following proposition holds.Proposition. S is stable i� S=fA j A is not attacked by Sg.The stable semantics of argumentation framework captures the semantics ofmany other mainstreamapproaches to nonmonotonic reasoning such as extensionof Reiter's Default Logic [22], stable expansion of Autoepicstemic Logic [17], andstable model of Logic Programming [9].We will see in section 3 that the creduloussemantics of an inheritance network coincides with the stable semantics of acorresponding argumentation framework.Often, a more restricted form of skeptical semantics is advocated in manyapproaches to nonmonotonic reasoning [21, 5]. This form of skeptical semanticsis de�ned in the argumentation framework by the notion of grounded extensionde�ned as the least �xpoint of the following operator.FAF : 2AR ! 2AR, whereFAF (S) = fA j 8B, if B attacks A, then 9C 2 S such that C attacks Bg.The idea behind this operator will become clear in the next de�nition.De�nition10. An argument A is defendable wrt S i� for every argument B, ifB attacks A, then there is an argument C in S such that C attacks B.So, we can rede�ne FAF by FAF (S) = fA j A is defendable wrt Sg.The grounded extension of an argumentation framework is de�ned next.De�nition11. The grounded extension of an argumentation framework AF de-noted by GEAF is the least �xpoint of FAF .It has been pointed out in [1] that both the semantics, Pollock's InductiveDefeasible Logic [21], and well-founded semantics of Logic Programming [5] arecaptured by the grounded extension of argumentation framework.The maximal �xpoint of FAF are called the preferred extension of AF. Ingeneral, stable extension are preferred extension but not vice versa. But as wewill see later, for any argumentation framework corresponding to inheritancenetworks, stable semantics and preferred semantics coincide. So it is enough forus to work only with stable semantic. 4

3 Inheritance Networks as Argumentation FrameworksOur goal is to clarify the conceptual relationship between the semantics of non-monotonic inheritance networks and the semantics of argumentation frameworks.This will also help to illuminate the conceptual relationship between the se-mantics of nonmonotonic inheritance networks and other general approaches tononmonotonic reasoning due to a result of Dung [1] showing that many generalapproaches to nonmonotonic reasoning [22, 17, 5] can be seen as special cases ofargumentation frameworks.We will show that every nonmonotonic inheritance network � can beconsidered as an argumentation framework AF�=<AR� ; attacks�> suchthat the credulous semantics of � coincides with the stable semantics ofAF�=<AR� ; attacks�> in the sense that every credulous extension of � isa stable extension of AF� and vice versa.A path � = �(x; �; u) is called a pre�x of path � = �(x; �; u; �; v) in � . Theset of all pre�xes of � is denoted by pre(�). � is a proper pre�x of � i� � 2 pre(�)and � 6= �.First, it is intuitive to view any path of � as an argument. So, we haveAR� = f�j� is a path in �g.As next we de�ne the attacks relation of AF� . The underlying principlein de�ning the attacks relation is that more speci�c information overrides lessspeci�c one. For two con icted paths, � and � , there are following cases:(i) � is a direct link. It is clear that we should let � attack � , but not viceversa.(ii) � and � are compound paths. In this case, neither � nor � are morespeci�c than the other path. Thus, we have: � attacks � and vice versa.Further, it should be also clear that if � attacks a pre�x � of � then � attacks� . We now consider another kind of attack.�����@@@@I����� @@@@I ��fb px�1 �����@@@@I����� @@@@I ��� fb px�2Fig. 2. Motivation of Attack De�nition5

Example 1. Consider the inheritance network �1 in �gure 2. The paths � =�(x; b; f) and � = ��(x; p; f) are in con ict. Thus, �(x; b; f) attacks ��(x; p; f)and vice versa as in case (ii). So, AF�1=<AR�1 ; attacks�1> with attacks�1 =f(�; � ); (�; �)g. Hence AF�1 has two stable extensions corresponding to two cred-ulous extensions of �1 : E1 = � [ f�g and E2 = � [ f�g.The network �2 in �gure 2 is received from the network �1 by adding thepositive link p ! b. This is the well- known Penguin-Bird-Fly example. It isclear that in AF�2=<AR�2 ; attacks�2>, � attacks � and � attacks � as in caseof �1. Adding the link p ! b into �1 makes the argument � more speci�c thanthe argument �. Thus, due to the principle that more speci�c information canoverride less speci�c one, we can say that adding p! b to �1 creates a attack ofnew kind against � = �(x; b; f). We can represent this by viewing the argument� = �(x; p; b) in the presence of the link p 6! f as an attack against the path� = �(x; b; f).These motivations lead to the following de�nition of attacks.������ @@@@@I ����� �@@IJJ] �� ` ` ` ` ` ` ` ` ` ``````````` ````````````` yu vx� � Fig. 3. �(x;�; u)! y is attacked by �(x; �; v; ; u) in presence of v 6! yDe�nition12. A path � attacks path � i�(a) � is a direct link x! y (resp. x 6! y) and � = ��(x; �; y) (resp. � =�(x; �; y)) or(b) � is in con ict with some compound path � 2 pre(� ) or(c) �; � are compound paths where there exists a pre�x � = �(x; �; u)! y of� (resp. �(x; �; u) 6! y) and � = �(x; �; v; ; u) with v 6! y 2 � (resp. v ! y 2 � )(see �gure 3).Remark. From now on we will refer to three types of attacks (a), (b), and (c)de�ned above as attack by direct link, by con ict, and by preemption, respec-tively. 6

So, in our point of view there are two kinds of attacks between two paths �and � , symmetric (� attacks � and � attacks � ) and asymmetric (� attacks �but not vice versa). Symmetric attacks are equivalent to con ictor in [29] whileasymmetric attacks have some similarity to preemptor in [29] but not identical.Example 2. (Continuation of example 1)For �2 in �gure 2 we have AF�2=<AR�2 ; attacks�2> withAR�2 = �2 [ f�; �; �; � �g andattacks�2 = f(p 6! f; �)g [ f(�; � ); (�; �); (�; �); (�; � )g[ f(�; �); (�; �)g with� = �(x; b; f), � = ��(x; p; f), � = �(x; p; b), � = �(x; p; b; f), and � = �(p; b; f).Here, f(p 6! f; �)g is the set of attacks by direct link, f(�; � ); (�; �); (�; �); (�; � )gis the set of attacks by con ict, and f(�; �); (�; �)g is the set of attacks bypreemption. utWe now prove the coincidence between the credulous extension of � and thestable extension of AF� .Theorem13. Let E be a set of paths in � . Then, E is a stable extension ofAF� i� E is a credulous extension of � .Proof. See Appendix. utWe give now the de�nition of the grounded skeptical semantics for an inher-itance network.De�nition14. The grounded skeptical semantics of the inheritance network �is de�ned as the grounded extension GEAF� of the corresponding argumentationframework AF� of � .Since GEAF� is contained in every stable extension of AF� we have thefollowing theorem.Theorem15. The grounded extension GEAF� of AF� is a subset of the ideallyskeptical extension of � .Proof. Since GEAF� is the smallest complete extension of AF� and the com-plete extensions form a complete semilattice wrt set conclusions [1] we have thatGEAF� is contained in every stable extension of AF� . Thus, GEAF� is containedin their intersection which is the ideally skeptical extension. ut4 Transforming Inheritance Network into Logic ProgramThe coincidence between the credulous semantics of an inheritance network �and the stable semantics of the corresponding argumentation frameworkAF� to-gether with the results in [1] stating that argumentation frameworks in principlecan be represented as logic programs, points out that an inheritance network �can be transformed into an equivalent logic program P� . Thus, proof procedures7

based on negation-as-failure can be applied to P� to compute the creduloussemantics of � .In this section we transform an inheritance network � into an extended logicprogram P� and show that the credulous semantics of � coincides with theanswer set semantics of P� 1.In following we assume that the readers are family with the answer set se-mantics of Logic Programs [10].The set of nodes of � is the union of two disjoint sets, the set of individualsof � , denotes by I� , consists of all nodes x of � such that there exists no directlink y ! x or y 6! x in � , and the set of predicate (or properties) nodes. Forexample, in the example 3 (Nixon-Diamond), a denotes the individual Nixon, andp, r and q denote the predicates Paci�st, Republican and Quaker, respectively.In following, a, b, c, ... will represent the individuals of � and p, q, r, ... are thepredicate nodes if not otherwise speci�ed.We �rst assign an unique natural number j 2 N to each direct link p! q 2 �(resp. p 6! q 2 � ) of � , p 62 I� , written as p!jq (resp. p6!jq), and introduce anew predicate abj representing the abnormal-literal at the edge j. The link p!jq(or p6!jq) is then referred simply as the link j. Based on the attack relationshipof AF� the inheritance network � can be transformed into an extended logicprogram P� as follows.As in case of attack by direct link, any direct link a ! p (resp. a 6! p)beginning from a fact node a can be transformed directly into a fact of P�because of there is no arguments which attack a ! p (resp. a 6! p). Hence, wehave:(i) For each a 2 I� if a! p (resp. a 6! p) is in � thenp(a) (resp. :p(a) )is a clause of P� .(ii) For p 62 I� and each direct link p!jq 2 � , the two clausesq(x) p(x); not abj(x) andabj(x) :q(x)belong to P� .Similarly, we have two clauses of P� for a negative direct link p6!jq 2 � asfollows:(iii) For p 62 I� and each direct link p6!jq 2 �:q(x) p(x); not abj(x) andabj(x) q(x)are clauses of P� .1 Gr�egoire [7] presented an algorithm for transformation of an inheritance network intoa logic program but in our view this could hardly be considered as a transformationbecause according to the algorithm we �rst have to compute the extensions of thenetwork and then de�ne a logic program specifying this extension.8

(iv) For each pair of direct links p!jq and r 6!kq in �(a) If there exists a positive path from p to r over the links j1; :::; jn then theclause abk(x) p(x); not abj1(x); : : : ; not abjn(x)belongs to P� .(b) If there exists a positive path from r to p over the links j1; :::; jn then theclause abj(x) r(x); not abj1(x); : : : ; not abjn(x)belongs to P� . utWe demonstrate the transformation from � into P� in the next two examples.�����@@@@I����� @@@@I ��pq ra� 12 Fig. 4. Nixon-DiamondExample 3. The corresponding program P� of � in the �gure 4 is:a! r r(a) a! q q(a) r 6!1p :p(x) r(x); not ab1(x)ab1(x) p(x)q!2p p(x) q(x); not ab2(x)ab2(x) :p(x)It is easy to see that P� has only two answer sets fr(a); q(a); p(a); ab1(a)g andfr(a); q(a);:p(a); ab2(a)g. ut9

�����@@@@I����� @@@@I� ��fq pa� 23 1Fig. 5. Penguin-Bird-FlyExample 4. Let consider the net � in �gure 5. a denotes Tweety, p, q, and f arepenguin, bird, and y, respectively. The corresponding program P� consists ofa! p p(a) a! q q(a) p6!2f :f(x) p(x); not ab2(x)ab2(x) f(x)q!3f f(x) q(x); not ab3(x)ab3(x) :f(x)p!1q q(x) p(x); not ab1(x)ab1(x) :q(x)and ab3(x) p(x); not ab1(x)The unique answer set of P� is fp(a); q(a);:f(a); ab3(a)g. utThe relationship between the answer set semantics of P� and the creduloussemantics of � can be established in the following way. First, for any set of pathsin � , E, we de�ne:ME = fq(a) j there is a positive path from a 2 I� to q in E g [ f:q(a) jthere is a negative path from a 2 I� to q in E g.The next theorem point out the relationship between the answer set semanticsof the program P� and the credulous semantics of the inheritance network � .Theorem16. M is an answer set of P� i� there exists a credulous extension Eof � such that ME = MnAB where AB denotes the set of all grounded instanceof abnormal-predicates in P� . ut5 ConclusionWe have studied the relationship between semantical concepts of nonmonotonicinheritance and of argumentation framework. In chapter 3 we showed that argu-mentation framework, a general form of argument-based reasoning can be applied10

to specify nonmonotonic inheritance reasoning in a simple way. We proved thata consistent and acyclic network can be viewed as an argumentation frameworkso that the credulous semantics of the former coincides with the stable semanticsof the latter.The capturing of credulous semantics of nonmonotonic inheritance by argu-mentation framework shows that argumentation can be applied successfully toformulize nonmonotonic inheritance reasoning. It is interesting to notice thatmany new developed approaches to nonmonotonic reasoning [6, 24, 2] can notapplied directly to inheritance reasoning as we did for Dung's argumentationframework. Ge�ner and Pearl's conditional entailment [6] is too weak as wecan not draw the conclusion \Tweety is an animal with feather" if we replacethe rule \Bird y" by two rules \Bird are animals with feather" and \Animalwith feather y" in the Penguin-Bird-Fly example. Simary and Loui's defeasiblereasoning [24] gives unintuitive answer even in simple cases as in the Penguin-Bird-Fly example given that Tweety is a penguin and a bird. Delgrande andSchaub's general approach [2] cannot give proper answer in the example withR = fa ! r; a ! p; a ! :q; p ! q; q ! :s; q ! r; r ! sg (�gure 18, page 158[13]).AcknowledgementThe �rst author is partially supported by EEC Keep in Touch activity KIT011.References1. Dung, P.M.: On the acceptability of arguments and its fundamental role in non-monotonic reasoning and logic programming and N-person game. AI Vol. 76 2(1995) (An extended abstract of this paper can be found in the proceeding ofIJCAI, 1993)2. Delgrande, J.P., Schaub, T.H.: A General Approach to Speci�city in Default Rea-soning3. Etherington, D.: Reasoning with Incomplete Information, Morgan Kaufmann.4. Etherington, D., Reiter, R.: On Inheritance Hierarchies with Exceptions in Proc.of AAAI-83 (1983) 104{1085. Van Gelder A., Ross, K., Schlipf J.S.: Unfounded sets and well-founded semanticsfor general logic programs. Proceeding of PODS 1988.6. Ge�ner H., Pearl, J.: Conditional entailment: bridging two approaches to defaultreasoning, Elsevier, AI Vol. 53, (1992) 209{2447. Gr�egoire, E.: Skeptical Theories of Inheritance and Nonmonotonic Logics, Method-ologies for Intelligent System 4, (1989) 430{4388. Gr�egoire, E.: Skeptical Inheritance Can Be More Expressive, Proceeding of ECAI(1990)9. Gelfond, M., Lipschitz, V.: The stable model semantics for logic programs, Pro-ceeding of the 5th ICLP, MIT Press, (1988), 1070 { 107910. Gelfond, M., Lipschitz, V.: Logic Programs with Classical Negation, Proceeding ofthe 7th ICLP, MIT Press, (1990), 579 {59711

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Appendix: Proof of selected theoremsIn this section we prove that the credulous semantics of � and the stable seman-tics of the corresponding argumentation framework coincide. At �rst, we provesome general properties of AF� .Lemma17. Let � be an acyclic, consistent inheritance network andAF�=<AR� ; attacks�> is the corresponding argumentation framework. Then,if � attacks �0 then deg� (�) � deg� (�0). utProof. Consider three cases:1. � attacks �0 by a direct link then it is clear that deg� (�) = deg� (�0) because�0 and � have the same begin and the same end nodes.2. � attacks �0 by con ict. Then, either � and �0 have the same begin andend and therefore deg� (�) � deg� (�0) or there is some pre�x of �0 which iscon ict with �, in this case we have deg� (�) < deg� (�0).3. � attacks �0 by preemption. Then, there is a pre�x � = �(x; �; u)! y (resp.�(x; �; u) 6! y) of �0 such that � = �(x; �; v; ; u) with v ! y (resp. v 6! y)in � . By de�nition of deg� we have deg� (�) < deg� (�0).The lemma is proved from these three cases. utLemma18. Let S be set of arguments and � be an argument defendable wrt S.Then, each � 2 pre(�) is defendable wrt S.Proof. If � 2 pre(�) and � is an argument which attacks � then � attacks �and therefore � is attacked by S. Hence, � is defendable wrt S. utLemma19. Let S be a set of argument in AF� . If S j� � then � is not attackedby S.Proof. Assume that there exists an argument � in S such that (�; �) 2 attacks.By de�nition of attack we have three cases:1. � attacks � by direct link. Then, � is preempted in S.2. � attacks � by con ict. Then, � is con icted or non constructible in S.3. � attacks � by preemption. Then, � is preempted or non constructible inS. From the three cases, we learn that if (�; �) 2 attacks then S j6� �.Contradictory !!! Thus, S does not attack �. utThe next lemma follows directly from lemma 19, and the de�nition of thestable extension.Lemma20. If E is a stable extension of AF� and � 62 E. Then, E j6� �. utFurther, it is easy to prove the next two lemmae.Lemma21. If E is a stable extension of AF� , then � � E, and for every path� 2 E, � is not preempted in E. ut13

Lemma22. Let E be a credulous extension of � . Then � is attacked by E if� 62 E .Proof. Obviously, � is a compound path. Without a lost of generality, assumethat � is a positive path �(x; �; u) ! y. Since � does not belong to E, E j6� �.Hence,1. � is con ict in E. It means, there is some path � in E that is con ict with�. Therefore, � is attacked by E.2. � is not constructible in E. There is a � 2 pre(�) such that all proper pre�xof � is contained in E but not �. Therefore E j6� �. Further, since � isconstructible in E. For we have two sub-cases:� is con icted in E. Similarly to the �rst case, � is attacked by E. Thus,� is attacked by E.� is preempted in E. See next case.3. � is preempted in E. That is, there is some node v with v 6! y 2 � and thereis some � = �(x; �; v; �; u) in E. Thus, � attacks �.From the three cases, we can conclude that � is attacked by E. utFrom the lemmae 17{22 we can prove the theorem 13:Proof of Theorem 13Proof. 1. ` �' Suppose that E is a stable extension of AF� . Let � 62 E. Fromlemma 19 we have E j6� � (i). Now, let � 2 E. Any pre�x of � is defendablewrt E (Lemma 18). Thus, all pre�xes of � are containing in E. So, � isconstructible and non-con icted in E. Further, � is not preempted in E(Lemma 21). Hence, � is defeasibly inheritable in E (ii). From (i) and (ii) wecan conclude that E is a credulous extensions of � .2. `�!' Now, if E is a credulous extension of � then E is con ict-free(Lemma 18). It is easy to see that E attacks every argument which doesnot belong to E (Lemma 22). Hence, E is a stable extension of AF� .The theorem is proved. ut14