Preferred extensions as stable models

Under  onsider ation for publi ation in The ory and Pr ati e of L o gi Pr o gr amming 1 Pr eferr e d extensions as stable mo dels ∗ JUAN CARLOS NIEVES, ULISES COR TÉS Universitat Politè ni a de Catalunya Softwar e Dep artment (LSI) /Jor di Gir ona 1-3, E08034, Bar  elona, Sp ain ( e-mail: {jnieves,ia}lsi.up.ed u ) MA URICIO OSORIO Universidad de las A méri as - Puebla CENTIA Sta. Catarina Mártir, Cholula, Puebla, 72820 Méxi o ( e-mail: osoriomaurigooglemail. om ) submitte d 6 June 2007; r evise d 17 De  emb er 2007; a  epte d 29 F ebruary 2008 Abstrat Giv en an argumen tation framew ork AF , w e in tro due a mapping funtion that onstruts a disjuntiv e logi program P , su h that the preferred extensions of AF orresp ond to the stable mo dels of P , after in terseting ea h stable mo del with the relev an t atoms. The giv en mapping funtion is of p olynomial size w.r.t. AF . In partiular, w e iden tify that there is a diret relationship b et w een the minimal mo dels of a prop ositional form ula and the preferred extensions of an argumen tation framew ork b y w orking on represen ting the defeated argumen ts. Then w e sho w ho w to infer the preferred extensions of an argumen tation framew ork b y using UNSA T algorithms and disjuntiv e stable mo del solv ers. The relev ane of this result is that w e dene a diret relationship b et w een one of the most satisfatory argumen tation seman tis and one of the most su- essful approa h of non-monotoni reasoning i.e. logi programming with the stable mo del seman tis. KEYW ORDS : preferred seman tis, abstrat argumen tation seman tis, stable mo del se- man tis, minimal mo dels. 1 In tro dution Dung's approa h, presen ted in (Dung 1995 ), is a unifying framew ork whi h has pla y ed an inuen tial role on argumen tation resear h and Artiial In telligene (AI). In fat, Dung's approa h has inuened subsequen t prop osals for argumen tation systems, e.g., ( Ben h-Cap on 2002 ). Besides, Dung's approa h is mainly relev an t in elds where onit managemen t pla ys a en tral role. F or instane, Dung sho w ed ∗ This is a revised and impro v ed v ersion of the pap er Inferring pr eferr e d extensions by minimal mo dels whi h app eared in Guillermo R. Simari and P aolo T orroni (Eds), pro eedings of the w orkshop Argumen tation and Non-Monotoni Reasoning (LPNMR-07 W orkshop). 2 J. C. Nieves, M. Osorio, and U. Cortés that his theory naturally aptures the solutions of the theory of n-p erson games and the w ell-kno wn stable marriage problem. Dung dened four argumen tation seman tis: stable semantis , pr eferr e d seman- tis , gr ounde d semantis , and  omplete semantis . The en tral notion of these se- man tis is the a  eptability of the ar guments . The main argumen tation seman tis for olletiv e aeptabilit y are the grounded seman tis and the preferred seman- tis (Prakk en and V reeswijk 2002 ; ASPIC:Pro jet 2005 ). The rst one represen ts a sk eptial approa h and the seond one represen ts a redulous approa h. Dung sho w ed that argumen tation an b e view ed as logi programming with ne ga- tion as failur e . Sp eially , he sho w ed that the grounded seman tis an b e  harater- ized b y the w ell-founded seman tis (Gelder et al. 1991 ), and the stable seman tis b y the stable mo del seman tis ( Gelfond and Lifs hitz 1991 ). This result is of great imp ortane b eause it in tro dues a general metho d for generating metain terpreters for argumen tation systems ( Dung 1995 ). F ollo wing this issue, w e will pro v e that it is p ossible to  haraterize the preferred seman tis based on the minimal mo dels of a prop ositional form ula (Theorem 1). W e will also sho w that the preferred seman tis an b e  haraterized b y the stable mo dels of a p ositiv e disjuntiv e logi program (Theorem 3). The imp ortane of this  haraterization is that w e are dening a di- ret relationship b et w een one of the most satisfatory argumen tation seman tis and ma y b e the most suessful approa h of non-monotoni reasoning of the last t w o deades i.e. logi programming with the stable mo del seman tis. As a natural onsequene of our result, w e presen t t w o easy-to-use forms for inferring the preferred extensions of an argumen tation framew ork ( AF ). The rst one is based on a mapping funtion whi h is quadrati size w.r.t. the n um b er of argumen ts of AF and UNSA T algorithms. The seond one is also based on a map- ping funtion whi h is quadrati size w.r.t. the n um b er of argumen ts of AF and disjuntiv e stable mo del solv ers. It is w orth men tioning that the deision problem of the preferred seman tis is hard sine it is o-NP-Complete (Dunne and Ben h-Cap on 2004 ). In fat, w e an nd dif- feren t strategies for omputing the preferred seman tis (Besnard and Doutre 2004 ; Ca yrol et al. 2003 ; Dung et al. 2006 ; Dung et al. 2007 ). Ho w ev er, w e an nd re- ally few implemen tations of them (ASPIC:Pro jet 2006 ; Gaertner and T oni 2007 ). One of the relev an t p oin ts of our result is that w e an tak e adv ane of eien t disjuntiv e stable mo del solv ers, e.g., the DL V System (DL V 1996 ), for inferring the preferred seman tis. The DL V System is a suessful stable mo del solv er that inludes dedutiv e database optimization te hniques, and non-monotoni reason- ing optimization te hniques in order to impro v e its p erformane ( Leone et al. 2002 ; Gebser et al. 2007 ). In fat, w e an implemen t the preferred seman tis inside ob jet- orien ted programs based on our  haraterization and the DL V JA V A W rapp er (Ria 2003 ). The rest of the pap er is divided as follo ws: In 2, w e presen t some basi onepts of logi programs and argumen tation theory . In  3 , w e presen t a  haraterization of the preferred seman tis b y minimal mo dels. In 4, w e presen t ho w to ompute the preferred seman tis b y using the minimal mo dels of a p ositiv e disjuntiv e logi program. Finally in the last setion, w e presen t our onlusions. Pr eferr e d extensions as stable mo dels 3 2 Ba kground In this setion, w e presen t the syn tax of a v alid logi program, the denition of the stable mo del seman tis, and the denition of the preferred seman tis. W e will use basi w ell-kno wn denitions in omplexit y theory su h as that of o-NP-omplete problem. 2.1 L o gi Pr o gr ams: Syntax The language of a prop ositional logi has an alphab et onsisting of (i) A signature L that is a nite set of elemen ts that w e all atoms, denoted usually as p 0 , p 1 , ... (ii) onnetiv es : ∨ , ∧ , ← , ¬ , ⊥ , ⊤ (iii) auxiliary sym b ols : ( , ). where ∨ , ∧ , ← are 2-plae onnetiv es, ¬ is 1-plae onnetiv e and ⊥ , ⊤ are 0-plae onnetiv es or onstan t sym b ols. A literal is an atom, a , or the negation of an atom ¬ a . Giv en a set of atoms { a 1 , ..., a n } , w e write ¬{ a 1 , ..., a n } to denote the set of literals {¬ a 1 , ..., ¬ a n } . F orm ulæ are onstruted as usual in logi. A theory T is a nite set of form ulæ. By L T , w e denote the signature of T , namely the set of atoms that o ur in T . A general lause, C , is denoted b y a 1 ∨ . . . ∨ a m ← l 1 , . . . , l n , 1 where m ≥ 0 , n ≥ 0 , m + n > 0 , ea h a i is an atom, and ea h l i is a literal. When n = 0 and m > 0 the lause is an abbreviation of a 1 ∨ . . . ∨ a m ← ⊤ . When m = 0 the lause is an abbreviation of ⊥ ← l 1 , . . . , l n . Clauses of this form are alled onstrain ts (the rest, non-onstrain t lauses). A general program, P , is a nite set of general lauses. Giv en a univ erse U , w e dene the  omplement of a set S ⊆ U as e S = U \ S . W e p oin t out that whenev er w e onsider logi programs our negation ¬ orre- sp onds to the default negation not used in Logi Programming. Also, it is on v enien t to remark that in this pap er w e are not using at all the so alled str ong ne gation used in ASP . 2.2 Stable Mo del Semantis First, to dene the stable mo del seman tis, let us dene some relev an t onepts. Denition 1 Let T b e a theory , an in terpretation I is a mapping from L T to { 0 , 1 } meeting the onditions: 1. I ( a ∧ b ) = min { I ( a ) , I ( b ) } , 2. I ( a ∨ b ) = max { I ( a ) , I ( b ) } , 3. I ( a ← b ) = 0 i I ( b ) = 1 and I ( a ) = 0 , 4. I ( ¬ a ) = 1 − I ( a ) , 1 l 1 , . . . , l n represen ts the form ula l 1 ∧ · · · ∧ l n . 4 J. C. Nieves, M. Osorio, and U. Cortés 5. I ( ⊥ ) = 0 . 6. I ( ⊤ ) = 1 . It is standard to pro vide in terpretations only in terms of a mapping from L T to { 0 , 1 } . Moreo v er it is easy to pro v e that this mapping is unique b y virtue of the denition b y reursion (v an Dalen 1994 ). An in terpretation I is alled a mo del of P i for ea h lause c ∈ P , I ( c ) = 1 . A theory is onsisten t if it admits a mo del, otherwise it is alled inonsisten t. Giv en a theory T and a form ula α , w e sa y that α is a logial onsequene of T , denoted b y T | = α , if for ev ery mo del I of T it holds that I ( α ) = 1 . It is a w ell kno wn result that T | = α i T ∪ {¬ α } is inonsisten t. It is p ossible to iden tify an in terpretation with a subset of a giv en signature. F or an y in terpretation, the orresp onding subset of the signature is the set of all atoms that are true w.r.t. the in terpretation. Con v ersely , giv en an arbitrary subset of the signature, there is a orresp onding in terpretation dened b y sp eifying that the mapping assigned to an atom in the subset is equal to 1 and otherwise to 0. W e use this view of in terpretations freely in the rest of the pap er. W e sa y that a mo del I of a theory T is a minimal mo del if there do es not exist a mo del I ′ of T dieren t from I su h that I ′ ⊂ I . Maximal mo dels are dened in the analogous form. By using logi programming with stable mo del seman tis, it is p ossible to desrib e a omputational problem as a logi program whose stable mo dels orresp ond to the solutions of the giv en problem. The follo wing denition of a stable mo del for general programs w as presen ted in (Gelfond and Lifs hitz 1991 ). Let P b e an y general program. F or an y set S ⊆ L P , let P S b e the general program obtained from P b y deleting (i) ea h rule that has a form ula ¬ l in its b o dy with l ∈ S , and then (ii) all form ulæ of the form ¬ l in the b o dies of the remaining rules. Clearly P S do es not on tain ¬ . Hene S is a stable mo del of P i S is a minimal mo del of P S . In order to illustrate this denition let us onsider the follo wing example: Example 1 Let S = { b } and P b e the follo wing logi program: b ← ¬ a . b ← ⊤ . c ← ¬ b . c ← a . W e an see that P S is: b ← ⊤ . c ← a . Notie that P S has t w o mo dels: { b } and { a, b, c } . Sine the minimal mo del amongst these mo dels is { b } , w e an sa y that S is a stable mo del of P . 2.3 A r gumentation the ory No w, w e dene some basi onepts of Dung's argumen tation approa h. The rst one is that of an argumen tation framew ork. An argumen tation framew ork aptures Pr eferr e d extensions as stable mo dels 5 the relationships b et w een the argumen ts (All the denitions of this subsetion w ere tak en from the seminal pap er (Dung 1995 )). Denition 2 An argumen tation framew ork is a pair AF = h AR, attack s i , where AR is a nite set of argumen ts, and attaks is a binary relation on AR , i.e. attaks ⊆ AR × AR . F or t w o argumen ts a and b , w e sa y that a attaks b (or b is atta k ed b y a ) if attack s ( a, b ) holds. Notie that the relation attaks do es not y et tell us with whi h argumen ts a dispute an b e w on; it only tells us the relation of t w o oniting argumen ts. It is w orth men tioning that an y argumen tation framew ork an b e regarded as a direted graph. F or instane, if AF = h{ a, b, c } , { ( a, b ) , ( b, c ) }i , then AF an b e represen ted as sho wn in Fig. 1 . Fig. 1. Graph represen tation of the argumen tation framew ork AF = h{ a, b, c } , { ( a, b ) , ( b, c ) }i . Denition 3 A set S of argumen ts is said to b e onit-free if there are no argumen ts a, b in S su h that a attaks b . A en tral notion of Dung's framew ork is a  eptability . It aptures ho w an argu- men t that annot defend itself, an b e proteted b y a set of argumen ts. Denition 4 (1) An argumen t a ∈ AR is a  eptable w.r.t. a set S of argumen ts i for ea h argumen t b ∈ AR : If b atta ks a then b is atta k ed b y an argumen t in S . (2) A onit-free set of argumen ts S is admissible i ea h argumen t in S is aeptable w.r.t. S . Let us onsider the argumen tation framew ork AF of Fig. 1 . W e an see that AF has three admissible sets: {} , { a } and { a, c } . In tuitiv ely , an admissible set is a oheren t p oin t of view. Sine an argumen tation framew ork ould ha v e sev eral oheren t p oin t of views, one an tak e the maxim um admissible sets in order to get maxim um oheren t p oin t of views of an argumen tation framew ork. This idea is aptured b y Dung's framew ork with the onept of pr eferr e d extension . Denition 5 A preferred extension of an argumen tation framew ork AF is a maximal ( w.r.t. inlusion) admissible set of AF . Sine an argumen tation framew ork ould ha v e more than one preferred extension, the preferred seman tis is alled redulous. The argumen tation framew ork of Fig. 1 has just one preferred extension whi h is { a, c } . 6 J. C. Nieves, M. Osorio, and U. Cortés R emark 1 By denition, it is lear that an y argumen t whi h b elongs to a preferred extension E is aeptable w.r.t. E . Hene w e will sa y that an y argumen t whi h do es not b elong to some preferred extension is a defe ate d ar gument . 3 Preferred extensions and UNSA T problem In this setion, w e will dene a mapping funtion that onstruts a prop ositional form ula, su h that its minimal mo dels  haraterize the preferred extensions of an ar- gumen tation framew ork. This  haraterization will pro vide a metho d for omputing preferred extensions based on Mo del Che king and Unsatisabilit y (UNSA T). In order to  haraterize the preferred seman tis in terms of minimal mo dels, w e will in tro due some onepts. Denition 6 Let T b e a theory with signature L . W e sa y that L ′ is a op y-signature of L i • L ∩ L ′ = ∅ , • the ardinalit y of L ′ is the same to L and • there is a bijetiv e funtion f from L to L ′ . It is w ell kno wn that there exists a bijetiv e funtion from one set to another if b oth sets ha v e the same ardinalit y . No w one an establish an imp ortan t relation- ship b et w een maximal and minimal mo dels. Pr op osition 1 Let T b e a theory with signature L T . Let L ′ b e a op y-signature of L T . By g ( T ) w e denote the theory obtained from T b y replaing ev ery o urrene of an atom x in T b y ¬ f ( x ) . Then M is a maximal mo del of T i f ( L T \ M ) is a minimal mo del of g ( T ) . Pr o of See App endix A. Our represen tations of an argumen tation framew ork use the prediate d(x) , where the in tended meaning of d(x) is: the argumen t x is defeated. By onsidering the prediate d ( x ) , w e will dene a mapping funtion from an argumen tation framew ork to a prop ositional form ula. This prop ositional form ula aptures t w o basi onditions whi h mak e an argumen t to b e defeated. Denition 7 Let AF = h AR, attack s i b e an argumen tation framew ork, then α ( AF ) is dened as follo ws: α ( AF ) = ^ a ∈ AR (( ^ b :( b,a ) ∈ att acks d ( a ) ← ¬ d ( b )) ∧ ( ^ b :( b,a ) ∈ att acks d ( a ) ← ^ c :( c,b ) ∈ att acks d ( c ))) Pr eferr e d extensions as stable mo dels 7 1. The rst ondition of α ( AF ) ( V b :( b,a ) ∈ attacks d ( a ) ← ¬ d ( b )) suggests that the argumen t a is defeated when an y one of its adv ersaries is not defeated. 2. The seond ondition of α ( AF ) ( V b :( b,a ) ∈ attacks d ( a ) ← V c :( c,b ) ∈ attack s d ( c )) suggests that the argumen t a is defeated when all the argumen ts that defend 2 a are defeated. Sine α ( AF ) aptures onditions whi h mak e an argumen t to b e defeated, it is quite ob vious that an y argumen t whi h satises these onditions ould not b elong to an admissible set. Therefore these argumen ts also ould not b elong to a preferred extension. Notie that α ( AF ) is a nite gr ounde d formula , this means that it do es not on- tain prediates with v ariables; hene, α ( AF ) is essen tially a prop ositional form ula (just onsidering the atoms lik e d ( a ) as d _ a ) of prop ositional logi. In order to illustrate the prop ositional form ula α ( AF ) , let us onsider the follo wing example. Example 2 Let AF = h AR , attack s i b e the argumen tation framew ork of Fig. 1. W e an see that α ( AF ) is: ( d ( b ) ← ¬ d ( a )) ∧ ( d ( b ) ← ⊤ ) ∧ ( d ( c ) ← ¬ d ( b )) ∧ ( d ( c ) ← d ( a )) Observ e that α ( AF ) has no prop ositional lauses w.r.t. argumen t a . This is essen- tially b eause α ( AF ) is apturing the argumen ts whi h ould b e defeated and the argumen t a will b e alw a ys an aeptable argumen t. It is w orth men tioning that giv en an argumen tation framew ork AF , α ( AF ) will ha v e at most 2 n 2 prop ositional lauses su h that n is the n um b er of argumen ts in AR and the maxim um length 3 of ea h prop ositional lause is n + 1 . Hene, w e an sa y that α ( AF ) is quadrati size w.r.t. the n um b er of argumen ts of AF . Essen tially α ( AF ) is a prop ositional represen tation of the argumen tation frame- w ork AF . Ho w ev er α ( AF ) has the prop ert y that its minimal mo dels  haraterize AF 's preferred extensions. In order to formalize this prop ert y , let us onsider the fol- lo wing prop osition whi h w as pro v ed b y Besnard and Doutre in ( Besnard and Doutre 2004 ). Pr op osition 2 (Besnard and Doutre 2004 ) Let AF = h AR, attack s i b e an argumen tation frame- w ork. Let β ( AF ) b e the form ula: ^ a ∈ AR (( a → ^ b :( b,a ) ∈ attacks ¬ b ) ∧ ( a → ^ b :( b,a ) ∈ attacks ( _ c :( c,b ) ∈ attack s c ))) then, a set S ⊆ AR is a preferred extension i S is a maximal mo del of the form ula β ( AF ) . 2 W e sa y that c defends a if b atta ks a and c atta ks b . 3 The length of our prop ositional lauses C is giv en b y the n um b er of atoms in the head of C plus the n um b er of literals in the b o dy of C 8 J. C. Nieves, M. Osorio, and U. Cortés In on trast with α ( AF ) whi h aptures onditions whi h mak e an argumen t to b e defeated, β ( AF ) aptures onditions whi h mak e an argumen t aeptable. Ho w ev er, w e will pro v e that when the mapping f ( x ) of the theory g ( β ( AF )) orresp onds to d ( x ) su h that x ∈ AF , α ( AF ) is logially equiv alen t to g ( β ( AF )) (see the pro of of Theorem 1). F or instane, let us onsider the argumen tation framew ork AF of Example 2. The form ula β ( AF ) is: ( ¬ a ← b ) ∧ ( ⊥ ← b ) ∧ ( ¬ b ← c ) ∧ ( a ← c ) If w e replae ea h atom x b y the expression ¬ d ( x ) , w e get: ( ¬¬ d ( a ) ← ¬ d ( b )) ∧ ( ⊥ ← ¬ d ( b )) ∧ ( ¬¬ d ( b ) ← ¬ d ( c )) ∧ ( ¬ d ( a ) ← ¬ d ( c )) No w, if w e apply transp osition to ea h impliation, w e obtain: ( d ( b ) ← ¬ d ( a )) ∧ ( d ( b ) ← ⊤ ) ∧ ( d ( c ) ← ¬ d ( b )) ∧ ( d ( c ) ← d ( a )) The latter form ula orresp onds to α ( AF ) . The follo wing theorem is a straigh tfor- w ard onsequene of Prop osition 2 and Prop osition 1. Giv en an argumen tation framew ork AF = h AR, attack s i and E ⊆ AR , w e dene the set compl ( E ) as { d ( a ) | a ∈ AR \ E } . Essen tially , compl ( E ) expresses the omplemen t of E w.r.t. AR . The or em 1 Let AF = h AR, attack s i b e an argumen tation framew ork and S ⊆ AR . When the mapping f ( x ) of the theory g ( β ( AF )) orresp onds to d ( x ) su h that x ∈ AR , the follo wing ondition holds: S is a preferred extension of AF i compl ( S ) is a minimal mo del of α ( AF ) . Pr o of See App endix A. This theorem sho ws that it is p ossible to  haraterize the preferred extensions of an argumen tation framew ork AF b y onsidering the minimal mo dels of α ( AF ) . In order to illustrate Theorem 1, let us onsider again α ( AF ) of Example 2 . This form ula has three mo dels: { d ( b ) } , { d ( b ) , d ( c ) } and { d ( a ) , d ( b ) , d ( c ) } . Then, the only minimal mo del is { d ( b ) } , this implies that { a, c } is the only preferred extension of AF . In fat, ea h mo del of α ( AF ) implies an admissible set of AF , this means that { a, c } , { a } and {} are the admissible sets of AF . There is a w ell kno wn relationship b et w een minimal mo dels and logial onse- quene, see (Osorio et al. 2004 ). The follo wing prop osition is a diret onsequene of su h relationship. Let S b e a set of w ell formed form ulæ then w e dene S etT oF ormul a ( S ) = V c ∈ S c . Pr op osition 3 Let AF = h AR, attack s i b e an argumen tation framew ork and S ⊆ AR . S is a preferred extension of AF i compl ( S ) is a mo del of α ( AF ) and α ( AF ) ∧ S etT oF ormul a ( ¬ ^ compl ( S )) | = S etT oF ormu la ( compl ( S )) . Pr eferr e d extensions as stable mo dels 9 Pr o of See App endix A. There are sev eral w ell-kno wn approa hes for inferring minimal mo dels from a prop ositional form ula (Dimop oulos and T orres 1996 ; Ben-Eliy ah u-Zohary 2005 ). F or instane, it is p ossible to use UNSA T's algorithms for inferring minimal mo dels. Hene, it is lear that w e an use UNSA T's algorithms for omputing the pre- ferred extensions of an argumen tation framew ork. This idea is formalized with the follo wing prop osition. The or em 2 Let AF = h AR, attack s i b e an argumen tation framew ork and S ⊆ AR . S is a preferred extension of AF if and only if compl ( S ) is a mo del of α ( AF ) and α ( AF ) ∧ S etT oF ormul a ( ¬ ^ compl ( S )) ∧ ¬ S etT oF o r mul a ( compl ( S )) is unsatisable. Pr o of Diretly , b y Prop osition 3. In order to illustrate Theorem 2, let us onsider again the argumen tation frame- w ork AF of Example 2 . Let S = { a } , then compl ( S ) = { d ( b ) , d ( c ) } . W e ha v e already seen that { d ( b ) , d ( c ) } is a mo del of α ( AF ) , hene the form ula to v erify its unsatisabilit y is: ( d ( b ) ← ¬ d ( a )) ∧ ( d ( b ) ← ⊤ ) ∧ ( d ( c ) ← ¬ d ( b )) ∧ ( d ( c ) ← d ( a )) ∧ ¬ d ( a ) ∧ ( ¬ d ( b ) ∨ ¬ d ( c )) Ho w ev er, this form ula is satisable b y the mo del { d ( b ) } , then { a } is not a preferred extension. No w, let S = { a, c } , then compl ( S ) = { d ( b ) } . As seen b efore, { d ( b ) } is also a mo del of α ( AF ) , hene the form ula to v erify its unsatisabilit y is: ( d ( b ) ← ¬ d ( a )) ∧ ( d ( b ) ← ⊤ ) ∧ ( d ( c ) ← ¬ d ( b )) ∧ ( d ( c ) ← d ( a )) ∧ ¬ d ( a ) ∧ ¬ d ( c ) ∧ ¬ d ( b ) It is easy to see that this form ula is unsatisable, therefore { a, c } is a preferred extension. The relev ane of Theorem 2 is that UNSA T is the protot ypial and b est-resear hed o-NP-omplete problem. Hene, Theorem 2 op ens the p ossibilities for using a wide v ariet y of algorithms for inferring the preferred seman tis. 4 Preferred extensions and general programs W e ha v e seen that the minimal mo dels of α ( AF )  haraterize the preferred exten- sions of AF . One in teresting p oin t of α ( AF ) is that α ( AF ) is logially equiv alen t to the p ositiv e disjuntiv e logi program Γ AF (dened b elo w). It is w ell kno wn that giv en a p ositiv e disjuntiv e logi program P , all the minimal mo dels of P orre- sp ond to the stable mo dels of P . This prop ert y will b e enough for  haraterizing the preferred seman tis b y the stable mo dels of the p ositiv e disjuntiv e logi program Γ AF . 10 J. C. Nieves, M. Osorio, and U. Cortés W e start this setion b y dening a mapping funtion whi h is a v ariation of the mapping of Denition 7 . Denition 8 Let AF = h AR , attack s i b e an argumen tation framew ork and a ∈ AR . W e dene the transformation funtion Γ( a ) as follo ws: Γ( a ) = { [ b :( b,a ) ∈ attacks { d ( a ) ∨ d ( b ) }} ∪ { [ b :( b,a ) ∈ attacks { d ( a ) ← ^ c :( c,b ) ∈ attack s d ( c ) }} No w w e dene the funtion Γ in terms of an argumen tation framew ork. Denition 9 Let AF = h AR, attack s i b e an argumen tation framew ork. W e dene its asso iated general program as follo ws: Γ AF = [ a ∈ AR Γ( a ) R emark 2 Notie that α ( AF ) (see Denition 7) is similar to Γ AF . The main syn tati dier- ene of Γ AF w.r.t. α ( AF ) is the rst part of Γ AF whi h is ( V b :( b,a ) ∈ attacks ( d ( a ) ∨ d ( b ))) ; ho w ev er this part is logially equiv alen t to the rst part of α ( AF ) whi h is ( V b :( b,a ) ∈ attacks d ( a ) ← ¬ d ( b )) . In fat, the main dierene is their b eha vior w.r.t. stable mo del seman tis. In order to illustrate this dierene, let us onsider the argumen tation framew ork AF = h{ a } , { ( a, a ) }i . W e an see that Γ AF = { d ( a ) ∨ d ( a ) } ∪ { d ( a ) ← d ( a ) } and α ( AF ) = ( d ( a ) ← ¬ d ( a )) ∧ ( d ( a ) ← d ( a )) It is lear that b oth form ulæ ha v e a minimal mo del whi h is { d ( a ) } 4 ; ho w ev er α ( AF ) has no stable mo dels. This suggests that α ( AF ) is not a suitable represen tation for  haraterizing preferred extensions b y using stable mo dels. Nonetheless w e will see that the stable mo dels of Γ AF  haraterize the preferred extensions of AF . Ev en though, in this pap er w e are only in terested in the preferred seman tis, it is w orth men tioning that the stable mo dels of the rst part of the form ula α ( AF ) i.e. ( V b :( b,a ) ∈ attacks d ( a ) ← ¬ d ( b )) ,  haraterize the so alled stable seman tis in argumen tation theory (Dung 1995 ). It is also imp ortan t to p oin t out that α ( AF ) and Γ AF ha v e dieren t use. On the one hand, w e will see that Γ AF is a suitable mapping for inferring preferred extensions b y using stable mo del solv ers. On the other hand, α ( AF ) has sho wn to b e most suitable for studying abstrat argumen ta- tion seman tis. F or example in (Niev es et al. 2006 ), α ( AF ) w as used for dening an extension of the preferred seman tis. Also, sine the w ell-founded mo del of α ( AF ) 4 Notie that { d ( a ) } suggests that AF has a preferred extensions whi h is {} . Pr eferr e d extensions as stable mo dels 11  haraterizes the grounded seman tis of AF , α ( AF ) w as used for dening exten- sions of the grounded seman tis and to desrib e the in teration of argumen ts based on reasoning under the grounded seman tis (Niev es et al. 2008 ). In the follo wing theorem w e formalize a  haraterization of the preferred seman- tis in terms of p ositiv e disjuntiv e logi programs and stable mo del seman tis. The or em 3 Let AF = h AR, attack s i b e an argumen tation framew ork and S ⊆ AR . S is a preferred extension of AF if and only if compl ( S ) is a stable mo del of Γ AF . Pr o of See App endix A. Let us onsider the follo wing example. Example 3 Let AF b e the argumen tation framew ork of Fig. 2 . W e an see that Γ AF is: d ( a ) ∨ d ( b ) . d ( a ) ← d ( a ) . d ( b ) ∨ d ( a ) . d ( b ) ← d ( b ) . d ( c ) ∨ d ( b ) . d ( c ) ∨ d ( e ) . d ( c ) ← d ( a ) . d ( c ) ← d ( d ) . d ( d ) ∨ d ( c ) . d ( d ) ← d ( b ) , d ( e ) . d ( e ) ∨ d ( d ) . d ( e ) ← d ( c ) . Γ AF has t w o stable mo dels whi h are { d ( a ) , d ( c ) , d ( e ) } and { d ( b ) , d ( c ) , d ( e ) , d ( d )) } , therefore { b, d } and { a } are the preferred extensions of AF. Fig. 2. Graph represen tation of the argumen tation framew ork AF = h{ a, b, c, d, e } , { ( a, b ) , ( b, a ) , ( b, c ) , ( c, d ) , ( d, e ) , ( e, c ) } . 4.1 Default ne gation As w e ha v e ommen ted in whole pap er, ours mappings are inspired b y t w o basi onditions that mak e an argumen t to b e defeated. One of the adv an tages of  har- aterizing the preferred seman tis b y using a logi programming seman tis with default ne gation , is that w e an infer the aeptable argumen ts from the stable mo dels of Γ AF in a straigh tforw ard form. F or instane, let Λ AF b e the disjuntiv e logi program Γ AF of Example 3 plus the follo wing lauses: 12 J. C. Nieves, M. Osorio, and U. Cortés a ← ¬ d ( a ) . b ← ¬ d ( b ) . c ← ¬ d ( c ) . d ← ¬ d ( d ) . e ← ¬ d ( e ) . su h that the in tended meaning of ea h lause is: the argumen t x is aeptable if it is not defeated. Λ AF has t w o stable mo dels whi h are { d ( a ) , d ( c ) , d ( e ) , b, d } and { d ( b ) , d ( c ) , d ( e ) , d ( d ) , a } . By taking the in tersetion of ea h mo del of Λ AF with AR (the set of argumen ts of AF ), w e an see that { b, d } and { a } are the preferred extensions of AF . This idea is formalized b y Prop osition 4 b elo w. Denition 10 Let AF = h AR, attack s i b e an argumen tation framew ork. W e dene its asso iated general program as follo ws: Λ AF = [ a ∈ AR { Γ( a ) ∪ { a ← ¬ d ( a ) }} Notie that Γ( a ) and Λ( a ) are equiv alen t, the main dierene b et w een Γ AF and Λ AF is the rule a ← ¬ d ( a ) for ea h argumen t. Pr op osition 4 Let AF = h AR, attack s i b e an argumen tation framew ork and S ⊆ AR . S is a preferred extension of AF i there is a stable mo del M of Λ AF su h that S = M ∩ AR . Pr o of The pro of is straigh tforw ard from Theorem 3 and the seman tis of default negation. It is w orth men tioning that b y using the disjuntiv e logi program Λ AF and the DL V System, w e an p erform an y query w.r.t. s epti al and r e dulous r e asoning . F or instane let gamma-AF b e the le whi h on tains Λ AF su h that AF is the argumen tation framew ork of Fig. 2 . Let us supp ose w e w an t to kno w if the argumen t a b elongs to some preferred extension of AF . Hene, let query-1 b e the le: a ? Let us all DL V with the br ave/r e dulous r e asoning fron t-end and query-1 : $ dlv -brave gamma-AF query-1 a is bravely true, evidened by { d ( b ) , d ( c ) , d ( e ) , d ( d ) , a } This means that it is true that the argumen t a b elongs to a preferred extension and ev en more w e ha v e a preferred extension whi h on tains the argumen t a . No w let us supp ose that w e w an t to kno w if the argumen t a b elongs to all the preferred extensions of AF . Let us all DL V with the  autious/s epti al r e asoning fron t-end and query-1 : $ dlv -autious gamma-AF query-1 a is autiously false, evidened by { d ( a ) , d ( c ) , d ( e ) , b, d } This means that it is false that the argumen t a b elongs to all the preferred extensions of AF . In fat, w e ha v e a oun terexample. Pr eferr e d extensions as stable mo dels 13 5 Conlusions Sine Dung in tro dued his abstrat argumen tation approa h, he pro v ed that his approa h an b e regarded as a sp eial form of logi programming with ne gation as failur e . In fat, he sho w ed the grounded and stable seman tis an b e  haraterized b y the w ell-founded and stable mo dels seman tis resp etiv ely . This result is imp or- tan t b eause it dened a general metho d for generating metain terpreters for argu- men tation systems (Dung 1995 ). Conerning this issue, Dung did not giv e an y  har- aterization of the preferred seman tis in terms of logi programming seman tis. It is w orth men tioning that aording to the literature (Prakk en and V reeswijk 2002 ; ASPIC:Pro jet 2005 ; P ollo  k 1995 ; Bondarenk o et al. 1997 ; Dung 1995 ), the pre- ferred seman tis is regarded as one of the most satisfatory argumen tation seman- tis of Dung's argumen tation approa h. In this pap er, w e  haraterize the preferred seman tis in terms of minimal mo dels (see Theorem 1 ) and stable mo del seman tis (see Theorem 3 ). These  harateriza- tions are based on t w o mapping funtions that onstrut a prop ositional form ula and a disjuntiv e logi program resp etiv ely . These  haraterizations ha v e as main result the denition of a diret relationship b et w een one of the most satisfatory ar- gumen tation seman tis and ma y b e the most suessful approa h of non-monotoni reasoning of the last t w o deades i.e. logi programming with the stable mo del seman tis. Based on this fat, w e in tro due a no v el and easy-to-use metho d for implemen ting argumen tation systems whi h are based on the preferred seman tis. It is quite ob vious that our metho d will tak e adv an tage of the platform that has b een dev elop ed under stable mo del seman tis for generating argumen tation sys- tems. F or instane, w e an implemen t the preferred seman tis inside ob jet-orien ted programs based on our  haraterization (Theorem 3, Prop osition 4) and the DL V JA V A W rapp er (Ria 2003 ). W e an see that our approa h falls in the family of the mo del- he king meth- o ds for inferring the preferred seman tis. In fat, our approa h is losely related to the metho ds suggested in (Besnard and Doutre 2004 ; Egly and W oltran 2006 ). As seen in Theorem 1, our prop ositional form ula α ( AF ) is losely related to one of the prop ositional form ulæ (see Prop osition 2) whi h w ere suggested in ( Besnard and Doutre 2004 ). It is w orth men tioning that the prop ositional form ula suggested b y (Egly and W oltran 2006 ) for inferring the admissible sets of an argumen tation framew ork is the same to the prop ositional form ula of Prop osition 2. The main dierene b et w een the ap- proa hes suggested b y ( Besnard and Doutre 2004 ; Egly and W oltran 2006 ) and our approa h is the strategy for inferring the mo dels of a prop ositional form ula. Instead of using maximal mo dels for  haraterizing the preferred seman tis as it is done dy (Besnard and Doutre 2004 ), w e are using minimal mo dels/stable mo dels . Hene, w e an use an y system whi h ould ompute minimal mo dels/stable mo dels of a prop ositional form ula. Maximalit y in Egly and W oltran' approa h is  he k ed on the ob jet lev el, i.e. within the resulting Quan tied Bo olean form ula (QBF). An in teresting prop ert y of our approa h is that whenev er w e use stable mo del solv ers for omputing the preferred extensions of an argumen tation framew ork, w e an ompute all the preferred extensions in full. In deision-making systems, it is not 14 J. C. Nieves, M. Osorio, and U. Cortés strange to require all the p ossible oheren t p oin ts of view (preferred extensions) in a dispute b et w een argumen ts. F or instane, in the medial domain when a do tor has to giv e a diagnosis under inomplete information, he has to onsider all the p ossible alternativ es in his deisions (Cortés et al. 2005 ; T ol hinsky et al. 2005 ). A  kno wledgemen t W e are grateful to anon ymous referees for their useful ommen ts. J.C. Niev es thanks to CONA CyT for his PhD Gran t. J.C. Niev es and U. Cortés w ould lik e to a kno wl- edge supp ort from the EC funded pro jet SHARE-it: Supp orted Human Autonom y for Reo v ery and Enhanemen t of ognitiv e and motor abilities using information te hnologies (FP6-IST-045088). The views expressed in this pap er are not nees- sarily those of the SHARE-it onsortium. Referenes ASPIC:Pr ojet . 2005. Deliver able D2.2:F ormal semantis for infer en e and de ision- making . 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L o gi and strutur e , 3rd., aumen ted edition ed. Springer-V erlag, Berlin. App endix A Pr o of of Pr op osition 1 Pr o of First of all t w o observ ations: 1. Giv en M 1 , M 2 ⊆ L T , it is true that M 1 ⊂ M 2 i f ( L T \ M 2 ) ⊂ f ( L T \ M 1 ) . 2. Giv en a prop ositional form ula A , an in terpretation M from L T to { 0 , 1 } and x ∈ { 0 , 1 } . Then it is not diult to pro v e b y indution on A 's length 5 that M ( A ) = x i f ( L T \ M )( g ( A )) = x . 5 Sine A is a disjuntiv e lause, the length of A is giv en b y the n um b er of atoms in the head of A plus the n um b er of literals in the b o dy of A . 16 J. C. Nieves, M. Osorio, and U. Cortés => T o pro v e that if M is a maximal mo del of T then f ( L T \ M ) is a minimal mo del of g ( T ) . The pro of is b y on tradition. Let us supp ose that M is a maximal mo del of T but f ( L T \ M ) is a mo del of g ( T ) and is not minimal. Then if f ( L T \ M ) is not minimal then there exists M 2 su h that f ( L T \ M 2 ) is a mo del of g ( T ) and f ( L T \ M 2 ) ⊂ f ( L T \ M ) . Then b y observ ation 2, if f ( L T \ M 2 ) is a mo del of g ( T ) then M 2 is a mo del of T . By observ ation 1, if f ( L T \ M 2 ) ⊂ f ( L T \ M ) then M ⊂ M 2 . But this is a on tradition b eause M is a maximal mo del of T . <= T o pro v e that if f ( L T \ M ) is a minimal mo del of g ( T ) then M is a maximal mo del of T . The pro of is also b y on tradition. Let us supp ose that f ( L T \ M ) is a minimal mo del of g ( T ) but M is mo del of T and is not maximal. If M is not maximal, then exists a mo del M 2 of T su h that M ⊂ M 2 . Then b y observ ation 2, if M 2 is a mo del of T then f ( L T \ M 2 ) is a mo del of g ( T ) . By observ ation 1, if M ⊂ M 2 then f ( L T \ M 2 ) ⊂ f ( L T \ M ) . But this is a on tradition b eause f ( L T \ M ) is a minimal mo del of g ( T ) . Pr o of of The or em 1 Pr o of T w o observ ations: 1. Sine the mapping f ( x ) orresp onds to d ( x ) , then compl ( S ) = f ( AR \ S ) b eause compl ( S ) = { d ( a ) | a ∈ AR \ S } and f ( AR \ S ) = { f ( a ) | a ∈ AR \ S } . 2. α ( AF ) is logially equiv alen t to g ( β ( AF )) : g ( β ( AF )) = ^ a ∈ AR (( ¬ d ( a ) → ^ b :( b,a ) ∈ attacks d ( b )) ∧ ( ¬ d ( a ) → ^ b :( b,a ) ∈ attacks ( _ c :( c,b ) ∈ attack s ¬ d ( c )))) Sine a → V b ∈ S b ≡ V b ∈ S ( a → b ) , w e get: ^ a ∈ AR ( ^ b :( b,a ) ∈ attacks ( ¬ d ( a ) → d ( b )) ∧ ( ^ b :( b,a ) ∈ attacks ( ¬ d ( a ) → _ c :( c,b ) ∈ attack s ¬ d ( c )))) By applying transp osition and anelation of double negation in b oth impliations, w e get: ^ a ∈ AR ( ^ b :( b,a ) ∈ attacks ( ¬ d ( b ) → d ( a )) ∧ ( ^ b :( b,a ) ∈ attacks ( ¬ _ c :( c,b ) ∈ attack s ¬ d ( c ) → d ( a )))) No w, for the righ t hand side of the form ula w e need to apply Morgan la ws: ^ a ∈ AR ( ^ b :( b,a ) ∈ attacks ( ¬ d ( b ) → d ( a )) ∧ ( ^ b :( b,a ) ∈ attacks ( ^ c :( c,b ) ∈ attack s d ( c ) → d ( a )))) Finally b y  hanging → b y ← , w e get α ( AF ) . Pr eferr e d extensions as stable mo dels 17 ^ a ∈ AR ( ^ b :( b,a ) ∈ attacks ( d ( a ) ← ¬ d ( b )) ∧ ( ^ b :( b,a ) ∈ attacks ( d ( a ) ← ^ c :( c,b ) ∈ attack s d ( c )))) = α ( AF ) No w the main pro of: S is a preferred extension of AF i (b y Prop osition 2 ) S is a maximal mo del of β ( AF ) i (b y Prop osition 1) f ( AR \ S ) is a minimal mo del of g ( β ( AF )) i (b y observ ations 1 and 2) compl ( S ) is a minimal mo del of α ( AF ) . Pr o of of Pr op osition 3 First of all, let us in tro due the follo wing relationship b et w een minimal mo dels and logi onsequene. L emma 1 (Osorio et al. 2004 ) F or a giv en general program P , M is a mo del of P and P ∪ ¬ g M ) | = M i M is a minimal mo del of P . This lemma w as in tro dued in terms of augmen ted programs. Sine a general program is a partiular ase of an augmen ted program, w e write the lemma in terms of general programs (see (Osorio et al. 2004 ) for more details ab out augmen ted programs). Pr o of S is a preferred extension of AF i (b y Theorem 1 ) compl ( S ) is a minimal mo del of α ( AF ) i (b y lemma 1 ) compl ( S ) is a mo del of α ( AF ) and α ( AF ) ∧ S etT oF ormul a ( ¬ ^ compl ( S )) | = S etT oF ormu la ( compl ( S )) . Pr o of of The or em 3 Pr o of S is a preferred extension of AF i  ompl(S) is a minimal mo del of α ( AF ) (b y Theorem 1) i compl ( S ) is a minimal mo del of Γ AF (sine Γ AF is logially equiv alen t to α ( AF ) in lassial logi) i  ompl(S) is a stable mo del of Γ AF (sine Γ AF is a p ositiv e disjuntiv e logi program and for ev ery p ositiv e disjuntiv e logi program P , M is a stable mo del of P i M is a minimal mo del of P ).