The matrices of argumentation frameworks

The matrices of argumen tation framew orks X u Y uming ∗ S chool of M athematics, S handong U niv e r sity , J inan, C hina Abstract W e introduce mat rix and its blo c k to the Dun g’s theory of argument ation framew orks. It is show ed that eac h argumen tation f ramew ork has a matrix represent ation, and the c ommon e xtension-based seman tics of argumen ta- tion framew ork can b e c haracteriz ed b y blo c ks of m atrix and their relatio ns. In cont rast with traditional m ethod of d irected graph , the matrix wa y has the adv an tage of computabilit y . Therefore, it has an extensive p ersp ectiv e to bring the theory of m atrix int o the researc h of argumen tation framewo rks and related areas. Keywor ds: Argumen tation f ramew ork; extension seman tics; matrix; b lock 1. In tro duction In recen t years, the area of argumen tation begins to b ecome increasingly cen t ral as a core study within Artificial In telligence. A n um b er of pap ers in v estigated and compared the prop erties of differen t seman tics whic h ha v e b een prop osed for arg umentation framew orks (AFs, for short) as introduced b y Dung [8, 4, 3, 9, 6]. In early time, man y of the analys is of argumen ts are expresse d in natural lang uage. Later on, a tr adition of using diagrams has b een dev elop ed to explicate the relations b et w een the comp onen ts of the argumen ts. No w, argumen tation framew o r ks are usually represen ted as directed graphs, whic h pla y a significant ro le in mo deling and analyzing the extension-based seman tics of AFs. F o r further not a tions and tec hniques of argumen tation, w e refer the reader to [8, 2, 15, 1]. Our aim is to intro duce matrix as a new mathematic to ol to the researc h ∗ Corresp onding a uthor. E- mail: xuyuming@s du.edu.cn 1 of argumen tation framew orks. First, w e assign a matrix of order n for eac h argumen tation framew ork with n argumen ts. Eac h elemen t of the matrix has only tw o p ossible v alues: one and zero, where one represe nts the at tac k relation and zero represen ts the non-attac k relation b et w een tw o argumen ts (they can b e the same o ne). Under this circumstance, the matrix can b e though t to b e a represen tation of the a rgumen t ation framew ork. Secondly , w e analysis the in ternal structure of the matrix corresp onding to v ario us extension-based seman tics of the ar g umen tat ion framew ork, and obtain the matrix approac hes t o determine the stable extension, admissible extension and complete extension, whic h can b e easily r ealized on computer. As will b e seen in later, the matrix of an argumentation f ramew or k is not only visualized as the directed gra ph, but also has another significan t adv an- tage on the asp ect of computation. W e shall study v arious extension-based seman t ics of the argumen tation framew ork by comparing a nd computing the matrix of the AF and its blo ck s. 2. D u ng’s theor y of argumen tation Argumen tation is a general approac h to mo del defeasible reasoning and justification in Artificial In telligence. So far, many theories of argumen tation ha v e b een established. Among them, Dung’s theory o f argumen tation frame- w ork is quite influence. In fact, it is abstract enough to manage without an y assumption o n the natur e o f argumen ts and t he at tac k r elation b et w een argumen ts. Let us first recall some basic notio n in Dung’s theory of argu- men tation framew ork. W e restrict them to finite argumen tation f ramew or ks. An argumen tation framew ork is a pair F = ( A, R ), where A is a finite set of argumen ts a nd R ⊂ A × A represen ts the att ac k-relation. F or S ⊂ A , we sa y t hat (1) S is conflict-free in ( A, R ) if there ar e no a, b ∈ S suc h that ( a, b ) ∈ R ; (2) a ∈ A is defeated b y S in ( A, R ) if there is b ∈ S suc h that ( b, a ) ∈ R ; (3) a ∈ A is defended b y S in ( A, R ) if fo r eac h b ∈ A with ( b, a ) ∈ R , w e ha v e b is defeated by S in ( A, R ). (4) a ∈ A is acceptable with resp ect to S if fo r eac h b ∈ A with ( b, a ) ∈ R , there is some c ∈ S suc h tha t ( c, b ) ∈ R . The conflict-freeness, as observ ed b y Baroni and Giacomin[1] in their study of ev aluativ e criteria for extens ion-ba sed seman tics, is view ed as a min- 2 imal requiremen t to b e satisfied within any computationally sensible notio n of ” collection of justified argumen ts”. Ho w ev er, it is to o we ak a condition to b e applied as a reasonable g uaran tor that a set of argumen ts is ”collectiv ely acceptable”. Seman tics for argumen tation framew orks can b e give n by a f unction σ whic h assigns eac h AF F = ( A, R ) a collection S ⊂ 2 A of extensions. Here, w e mainly f o cus on the seman tic σ ∈ { s, a, p, c, g , i, ss, e } for stable, admissi- ble, preferred, complete, grounded, ideal, semi-stable and eager extens ions, resp ectiv ely . Definition 1 [14 ] Let F = ( A, R ) b e an argumen tation framew ork and S ∈ A . (1) S is a stable extension of F , i.e. , S ∈ s ( F ), if S is conflict-free in F and eac h a ∈ A \ S is defeated b y S in F . (2) S is an admissible extension of F , i.e. , S ∈ a ( F ), if S is conflict-free in F a nd eac h a ∈ A \ S is defended b y S in F . (3) S is a preferred extension of F , i.e. , S ∈ p ( F ), if S ∈ a ( F ) a nd for eac h T ∈ a ( F ), w e hav e S 6⊂ T . (4) S is a complete extension of F , i .e. , S ∈ c ( F ), if S ∈ a ( F ) and fo r eac h a ∈ A defended b y S in F , we hav e a ∈ S . (5) S is a grounded extension of F , i.e. , S ∈ g ( F ), if S ∈ c ( F ) and for eac h T ∈ c ( F ), w e hav e T 6⊂ S . (6) S is an ideal extension of F , i.e. , S ∈ i ( F ), if S ∈ a ( F ), S ⊂ ∩{ T : T ∈ p ( F ) } and for each U ∈ a ( F ) suc h that U ⊂ ∩{ T : T ∈ p ( F ) } , w e hav e S 6⊂ U . (7) S is a semi-stable extension of F , i.e. , S ∈ ss ( F ), if S ∈ a ( F ) and for eac h T ∈ a ( F ), w e ha v e R + ( S ) 6⊂ R + ( T ), where R + ( U ) = { U ∩ { b : ( a, b ) ∈ R , A ∈ U } } . (8) S is a eager extension of F , i.e. , S ∈ e ( F ), if S ∈ c ( F ), S ⊂ ∩{ T : T ∈ ss ( F ) } and for each U ∈ a ( F ) suc h tha t U ⊂ ∩{ T : T ∈ ss ( F ) } , w e hav e S 6⊂ T . Note that, there a re some elemen tary prop erties for an y argumentation framew ork F = ( A, R ) a nd seman tic σ . If σ ∈ { a, p, c, g } , then we hav e 3 σ ( F ) 6 = ∅ . And if σ ∈ { g , i, e } , then σ ( F ) c ontains exactly one extension. F urthermore, the followin g relatio ns hold fo r eac h argumen tation fra mework F = ( A, R ) : s ( F ) ⊆ p ( F ) ⊆ c ( F ) ⊆ a ( F ). Since ev ery extension of a n AF under the standard seman tics (stable, preferred, complete and grounded extensions) introduced b y Dung is an ad- missible set, the concept of a dmissible extensions pla ys an imp or t a n t role in the study of ar g umen tat io n fra mew orks. 3. The matrix of an argumen tatio n framework W e kno w that the directed graph is a traditional to o l in the researc h of argumen tation framew orks, and has the feature of visualization [7, 10, 11]. It is widely used for mo deling and ana lyzing argumen tation framew orks. In this section, w e shall in t r o duce the matrix represen tation of argumen tation framew orks. Except for the visualization, the matrix also ha s the adv antage of computabilit y in analyzing arg umen tat io n framew o rks and computing v ar- ious extension semantic s. An m × n m atrix A is a rectangular arra y of nu mbers, consisting of m ro ws and n columns, denoted b y A =      a 1 , 1 a 1 , 2 . . . a 1 ,n a 2 , 1 a 2 , 2 . . . a 2 ,n . . . . . . a m, 1 a m, 2 . . . a m,n      . The m × n n um b ers a 1 , 1 , a 1 , 2 , ..., a m,n are the elemen ts of the matrix A . W e often called a i,j the ( i, j ) t h elemen t, a nd write A = ( a i,j ) for short. It is imp ortan t to remem b er that t he first suffix of a i,j indicates the ro w and the second the column of a i,j . A column matrix is an n × 1 matrix, and a ro w matrix is an 1 × n matrix, denoted by           x 1 x 2 . . . x n           ,  x 1 x 2 . . . x n  4 resp ectiv ely . Matrices of b oth the se t yp es can b e regarded as v ectors and referred to resp ectiv ely as column v ectors and ro w v ectors. Usually , the i th ro w o f a matrix A is denoted b y A i, ∗ , and the j th column of A is denoted b y A ∗ ,j . Definition 2 In an n × m matrix A = ( a i,j ), w e sp ecify an y k ( ≤ min { n, m } ) differen t ro ws i 1 , i 2 , ..., i k and the same num b er of differen t columns i 1 , i 2 , ..., i k . The elemen ts app earing at the in tersections of these row s a nd columns form a square matrix of order k . W e call this matrix a principal blo c k of order k of the orig inal matrix A ; it is denoted b y M =      a i 1 ,i 1 a i 1 ,i 2 . . . a i 1 ,i k a i 2 ,i 1 a i 2 ,i 2 . . . a i 2 ,i k . . . . . . a i k ,i 1 a i k ,i 2 . . . a i k ,i k      , or M = M i 1 ,i 2 ,...,i k i 1 ,i 2 ,...,i k for short. Definition 3 If in the original n × m matrix A = ( a i,j ), w e delete the rows and columns whic h mak e up the blo c k M = M i 1 ,i 2 ,...,i k i 1 ,i 2 ,...,i k , then the remaining elemen ts form a n ( n − k ) × ( m − k ) matrix. W e call this matrix the comple- men tary blo ck of M , and is denoted by the sym b ol M = M i 1 ,i 2 ,...,i k i 1 ,i 2 ,...,i k . Definition 4 In an n × m matr ix A , w e specify an y k ( ≤ n ) differen t row s i 1 , i 2 , ..., i k and h ( ≤ m ) differen t columns j 1 , j 2 , ..., j h . The elemen ts app ear- ing at the interse ctions of these rows and columns form a k × h matrix. W e call this mat r ix a k × h blo c k of the original matrix A ; it is denoted by M =      a i 1 ,j 1 a i 1 ,j 2 . . . a i 1 ,j h a i 2 ,j 1 a i 2 ,j 2 . . . a i 2 ,j h . . . . . . a i k ,j 1 a i k ,j 2 . . . a i k ,j h      , or M = M j 1 ,j 2 ,...,j h i 1 ,i 2 ,...,i k for short. F or the underlying set A of an argumen tation framew ork F = ( A, R ), there is no o rdering in nature. But, in man y cases the ordering set can benefit us a lo t. Con trasting with t he fo rm A = { a, b, ... } , it is more con v enience to put A = { 1 , 2 , ..., n } while the cardina lity of A is larg e. In particular, w e can 5 map eac h argumen t to the correspo nding row and column of a matrix. W e will fo llo w t his arr a ngemen t in the b elow discussion. Definition 5 Let F = ( A, R ) b e an argumen tation framew ork with A = { 1 , 2 , ..., n } . The matrix of F , denoted by M ( F ), is a Bo olean matrix of order n , its elemen t is determined by the following rules: (1) a i,j = 1 iff ( i, j ) ∈ R ; (2) a i,j = 0 iff ( i, j ) / ∈ R . Example 6 Considering the argumen tation framew ork F = ( A, R ) , where A = { 1 , 2 , 3 } and R = { ( 1 , 2 ) , ( 2 , 3 ) , ( 3 , 1 ) } . By the definition, w e hav e the follo wing matrix of F : M ( F ) =    0 1 0 0 0 1 1 0 0    Example 7 Give n an argumen tation fra mework F = ( A, R ), where A = { 1 , 2 , 3 , 4 } and R = { (1 , 2 ) , ( 1 , 3 ) , (2 , 1) , (2 , 3 ) , (3 , 4) } . The mat r ix of F is a s follo ws: M ( F ) =      0 1 1 0 1 0 1 0 0 0 0 1 0 0 0 0      In comparison with graph-theoretic w a y a nd mathematical logic wa y , the matrix of an argumen tation fra mew ork has man y excellen t features. First, it p ossess a concise mathematical format. Secondly , it contains all infor mation of the AF by combin ing the argumen ts with attac k relation in a sp ecific man- ner in the matrix M ( F ). Also, it can b e deal with by program on computer. The most imp o r tan t is tha t we can imp ort the kno wledge of matrix to the researc h of argumen tation fra mew orks. 4. D e termination of t he confl i c t-free se ts As w e know , there is no efficien t metho d for us to decide a conflict set in an argumen tation framework, ev en w e can draw up the directed g raph o f the AF. After we in tro duce the mat rix of the AF, the situation will b e c hanged completely . By c hec king the matrix of the argumen tation framew ork, w e can 6 easily find out all the conflict-free sets of t he AF. Let us see an example, firstly . Example 8 Give n an argumen tation fra mework F = ( A, R ), where A = { 1 , 2 , 3 , 4 , 5 } and R = { (1 , 2) , (2 , 3 ) , (2 , 5 ) , (4 , 3 ) , (5 , 4 ) } . Then, we can easily to show that the collection of conflict-free sets of F is {∅ , { 1 } , { 2 } , { 3 } , { 4 } , { 5 }{ 1 , 3 } , { 1 , 4 } , { 1 , 5 } , { 2 , 4 } , { 3 , 5 } , { 1 , 3 , 5 }} , b y the routine metho d of directed graph. On the other ha nd, w e consider the matrix of F = ( A, R ) and study its structure from t he leve l of blo c ks. F irst, we write out t he matrix of F : M ( F ) =         0 1 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0         . By observing the principal blo c ks of the ab ov e matrix, w e find that there are fiv e zero principal blo c ks of order 1 M 1 1 =  0  , M 2 2 =  0  , M 3 3 =  0  , M 4 4 =  0  , M 5 5 =  0  corresp onding to the conflict-free sets { 1 } , { 2 } , { 3 } , { 4 } , { 5 } , resp ectiv ely . There are fiv e zero principal blo c ks of or der 2 M 1 , 3 1 , 3 = 0 0 0 0 ! , M 1 , 4 1 , 4 = 0 0 0 0 ! , M 1 , 5 1 , 5 = 0 0 0 0 ! , M 2 , 4 2 , 4 = 0 0 0 0 ! , M 3 , 5 3 , 5 = 0 0 0 0 ! corresp onding to the conflict-free sets { 1 , 3 } , { 1 , 4 } , { 1 , 5 } , { 2 , 4 } , { 3 , 5 } , resp ectiv ely . Also, t here is a zero principal blo ck of order 3 M 1 , 3 , 5 1 , 3 , 5 =    0 0 0 0 0 0 0 0 0    corresp onding t o the conflict-free sets { 1 , 3 , 5 } . Note that, the ab o v e blo ck s a re a ll principal blo ck s which are zero in the matrix M ( F ), and there is a one to one corresp ond b et w een the collection of all conflict-free sets of F and the set o f all zero principal blo c ks of M ( F ). In fact, for an y arg umen tat ion framew ork F there exists suc h corresp onding 7 relation b etw een the collection o f all conflict-free sets of F and the set of all zero principal blo c ks of M ( F ). Since it is easy to find out the zero principal blo ck s in the matrix of an argumen tation framew or k, we obtain a g o o d w a y to decide the conflict-free sets of the AF thro ug h its matr ix. Certainly , this w a y can b e carried out on the computer readily . Definition 9 Let F = ( A, R ) b e an argumen tation framew ork with A = { 1 , 2 , ..., n } , and S = { i 1 , i 2 , ..., i k } ⊂ A . The principal blo ck M i 1 ,i 2 ,...,i k i 1 ,i 2 ,...,i k =      a i 1 ,i 1 a i 1 ,i 2 . . . a i 1 ,i k a i 2 ,i 1 a i 2 ,i 2 . . . a i 2 ,i k . . . . . . a i k ,i 1 a i k ,i 2 . . . a i k ,i k      of order k in the matrix M ( F ) is called the cf -blo c k of S , a nd denoted b y M cf . Theorem 10 Giv en a n argumen tation framew o r k F = ( A, R ) with A = { 1 , 2 , ..., n } , then S = { i 1 , i 2 , ..., i k } ⊂ A is a conflict-free set in F iff the cf -blo c k M i 1 ,i 2 ,...,i k i 1 ,i 2 ,...,i k of S is zero. Pro of Assume that M i 1 ,i 2 ,...,i k i 1 ,i 2 ,...,i k = 0, then for arbitrary 1 ≤ s, t ≤ k w e hav e a i s ,i t = 0, i.e. , ( i s , i t ) / ∈ R . Th us, S = { i 1 , i 2 , ..., i k } is a conflict-free set in F . Supp ose S = { i 1 , i 2 , ..., i k } ⊂ A is a conflict-fr ee set in F , then for arbi- trary 1 ≤ s, t ≤ k w e ha ve that ( i s , i t ) / ∈ R , i.e. , a i s ,i t = 0. Therefore, we ha v e M i 1 ,i 2 ,...,i k i 1 ,i 2 ,...,i k = 0. 5. D e termination of t he stable extension s Example 11 W e con tinuous to study the argumen tation framew ork F = ( A, R ), where A = { 1 , 2 , 3 , 4 , 5 } and R = { (1 , 2) , (2 , 3) , (2 , 5) , (4 , 3) , (5 , 4 ) } . Since the stable ex tension is firstly a conflict-free set, we can lo ok for the stable extension from t he collection {∅ , { 1 } , { 2 } , { 3 } , { 4 } , { 5 }{ 1 , 3 } , { 1 , 4 } , { 1 , 5 } , { 2 , 4 } , { 3 , 5 } , { 1 , 3 , 5 }} of conflict-free sets. In fact, the set S = { 1 , 3 , 5 } is the only stable extension in F by a simple discussion. Again, w e turn our attention to the ma t r ix of the F = ( A, R ): 8 M ( F ) =         0 1 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0         . Since S = { 1 , 3 , 5 } is a stable extension of F , the ar g umen ts 2 and 4 are defeated b y { 1 , 3 , 5 } . This f act is r eflected in the matrix M ( F ) of F as follo ws. In the column vec tor F ∗ , 2 (column 2), a 1 , 2 = 1 me ans that (1 , 2) ∈ R , and th us the argument 1 at tac ks the argumen t 2. In the column v ector F ∗ , 4 (column 4), a 5 , 4 = 1 means that (5 , 4) ∈ R , and thus the argumen t 5 attac ks the a rgumen t 4. F rom t he b eha vior o f the elemen ts a 1 , 2 = 1 and a 5 , 4 = 1 in the matrix M ( F ), we can extract a matrix approach to decide that the conflict-free set S = { 1 , 3 , 5 } is a stable extens ion: Corresp o nding to t he argumen ts 2 , 4 ∈ A \ S , w e firstly pic k o ut the column v ectors F ∗ , 2 and F ∗ , 4 in the matrix M ( F ), then chec k t he elemen ts a 1 , 2 , a 3 , 2 , a 5 , 2 of F ∗ , 2 , and the elemen ts a 1 , 4 , a 3 , 4 , a 5 , 4 of F ∗ , 4 . If there is one elemen t of { a 1 , 2 , a 3 , 2 , a 5 , 2 } whic h is non- zero, then the argumen t 2 is defeated by S . Similar result is hold f o r the argument 4. This pro cess leads to a blo c k of the matrix M ( F ) at the in tersection of columns 2 , 4 a nd rows 1 , 3 , 5. T o sum up, we can decide that the conflict set S = { 1 , 3 , 5 } is a stable extension b y the fact t ha t the tw o column vec tors of t he ab o v e blo ck of the matrix M ( F ) are all no n- zero. F urther analysis indicates that the conv erse is also t r ue. This motiv ation makes us to giv e the following definition. Definition 12 Let F = ( A, R ) b e an argumen tation fra mew ork with A = { 1 , 2 , ..., n } , and S = { i 1 , i 2 , ..., i k } ⊂ A is a stable extension of F . The k × h blo c k M i 1 ,i 2 ,...,i k j 1 ,j 2 ,...,j h =      a i 1 ,j 1 a i 1 ,j 2 . . . a i 1 ,j h a i 2 ,j 1 a i 2 ,j 2 . . . a i 2 ,j h . . . . . . a i k ,j 1 a i k ,j 2 . . . a i k ,j h      in the matrix M ( F ) is called the s -blo ck of S and denoted by M s , where { j 1 , j 2 , ..., j h } = A \ S . 9 In other w ords, the elemen ts app earing at the interse ctions of ro ws i 1 , i 2 , ..., i k and columns j 1 , j 2 , ..., j h in the matrix M ( F ) form the s -blo c k M i 1 ,i 2 ,...,i k j 1 ,j 2 ,...,j h of S . Theorem 13 Giv en a n argumen tation framew o r k F = ( A, R ) with A = { 1 , 2 , ..., n } , then S = { i 1 , i 2 , ..., i k } ⊂ A is a stable extension in F iff the follo wing conditions hold: (1) The cf -blo ck M i 1 ,i 2 ,...,i k i 1 ,i 2 ,...,i k of S is zero, (2) Ev ery column vec tor o f the s -blo c k M i 1 ,i 2 ,...,i k j 1 ,j 2 ,...,j h of S is non-zero, where A \ S = { j 1 , j 2 , ..., j h } . Pro of Let S b e a conflict-free set and A \ S = { j 1 , j 2 , ..., j h } , then w e need only to prov e that ev ery elemen t of A \ S (1 ≤ t ≤ h ) is defeated b y S in F iff all column ve ctors o f the s -blo ck M i 1 ,i 2 ,...,i k j 1 ,j 2 ,...,j h of S are non-zero. Assume that ev ery elemen t of A \ S (1 ≤ t ≤ h ) is defeated by S in F . T ak e an y column v ector A ∗ ,j t (1 ≤ t ≤ h ) of the s -blo ck M i 1 ,i 2 ,...,i k j 1 ,j 2 ,...,j h of S , then w e ha v e j t ∈ A \ S . By the assumption, there is some elemen t i r ∈ S (1 ≤ r ≤ k ) suc h tha t the argumen t i r attac ks the argumen t j t , i.e. , ( i r , j t ) ∈ R . It follo ws that a i r ,j t = 1 in the matrix M ( F ) and the s -blo c k M i 1 ,i 2 ,...,i k j 1 ,j 2 ,...,j h of S , and thus the column ve ctor A ∗ ,j t is non- zero. Con v ersely , supp ose that all column v ectors of the s - blo c k M i 1 ,i 2 ,...,i k j 1 ,j 2 ,...,j h = M s of S are non-zero. T ake any elemen t j t ∈ A \ S (1 ≤ t ≤ h ), then M s ∗ ,j t is a column v ector of the s -blo ck M i 1 ,i 2 ,...,i k j 1 ,j 2 ,...,j h = M s of S . By the hypothesis, w e kno w that A ∗ ,j t is non-zero. Therefore, there is some i r ∈ S (1 ≤ r ≤ k ) suc h that a i r ,j t = 1, i.e. , ( i r , j t ) ∈ R . This means t ha t the argumen t i r attac ks the argumen t j t of S in F , and th us w e claim that j t is defeated b y S in F . 6. D e termination of t he admissib l e extensio n s Example 14 Let us return to the argumentation framew ork F = ( A, R ), where A = { 1 , 2 , 3 , 4 , 5 } and R = { ( 1 , 2 ) , ( 2 , 3 ) , ( 2 , 5) , (4 , 3) , (5 , 4 ) } . Since an admissible extension is necessarily a conflict-free set, w e can lo ok f or the admissible extension from the collection {∅ , { 1 } , { 2 } , { 3 } , { 4 } , { 5 }{ 1 , 3 } , { 1 , 4 } , { 1 , 5 } , { 2 , 4 } , { 3 , 5 } , { 1 , 3 , 5 }} of conflict-free sets. By definition, it is easy to chec k that { 1 } , { 1 , 5 } and { 1 , 3 , 5 } are all the admissible extensions in F . Since { 1 , 3 , 5 } is also a stable extension and { 1 } is not typical enough as 10 an admissible extension in F , we will mainly concen trate on the admissible extension S = { 1 , 5 } which is not a stable extension in F . First, w e write out the ma t r ix of argumen tatio n f ramew or k F = ( A, R ): M ( F ) =         0 1 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0         . Secondly , we study the structure of the matrix M ( F ) of F to find out the in ternal prop erties whic h can reflect the fact that S = { 1 , 5 } is an admissible extension. In the column v ector M ( F ) ∗ , 5 of the matrix M ( F ) , a 2 , 5 = 1 means that (2 , 5) ∈ R , i.e. , the argumen t 2 atta c ks the a rgumen t 5. Under this circum- stance, the elemen t a 1 , 2 = 1 in t he row v ector M ( F ) ∗ , 2 of the matrix M ( F ) implies that (1 , 2) ∈ R , i.e. , the argumen t 1 attac ks the argument 2. This illustrates that the argumen t 5 is defende d by { 1 , 5 } in F . In the column v ector M ( F ) ∗ , 1 of the matrix M ( F ), w e hav e a i, 1 = 0 for each 1 ≤ i ≤ 5. It follo ws that the a rgumen t 1 is defended b y { 1 , 5 } in F . In the ab ov e a na lysis, the b eha vior of a 2 , 5 = 1 and a 1 , 2 = 1 in the matrix M ( F ) is intrins ic fo r the fact that the argumen t 5 is defended b y { 1 , 5 } in F . This inspires us a general idea to decide the conflict-free set S = { 1 , 5 } to b e admissible through t he structure o f the matrix M ( F ) of F . (1) In order to decide whether the arguments of { 1 , 5 } = S are defended b y S , we should firstly find the attack ers of the arg ument 1 and 5. So, w e m ust pick out the column v ectors M ( F ) ∗ , 1 and M ( F ) ∗ , 5 of the matrix M ( F ) corresp onding to the argumen ts 1 and 5 resp ectiv ely . Since the s et S is conflict-free, there is no attack relation betw een 1 and 5, i.e. , a 1 , 1 = 0 , a 5 , 1 = 0 , a 1 , 5 = 0 , a 5 , 5 = 0. Therefore, we only need to che ck the elemen ts a 2 , 1 , a 3 , 1 , a 4 , 1 of the column v ector M ( F ) ∗ , 1 , and the elemen ts a 2 , 5 , a 3 , 5 , a 4 , 5 of the column v ector M ( F ) ∗ , 5 . Eac h non-zero elemen t of the set { a 2 , 1 , a 3 , 1 , a 4 , 1 } tells us an attac k er of the argumen t 1, and eac h non- zero elemen t of the set { a 2 , 5 , a 3 , 5 , a 4 , 5 } tells us an att ac k er of the argumen t 5. This leads to a blo c k of the matrix M ( F ) at the interse ction of column 1 , 5 and row 2 , 3 , 4, which is exactly the s - blo c k of S . (2) After ha ving determined the attack ers ( ∈ { 2 , 3 , 4 } ) of the arg umen t 1 and 5, w e should secondly to chec k whether these attack ers are defeated 11 b y S = { 1 , 5 } . F or example, a 2 , 5 = 1 means that the argumen t 2 is an attac k er o f the ar gumen t 5. So, we should c hec k the elemen t a 1 , 2 and a 5 , 2 to see whether the attac k er 2 of the argument 5 is defeated by { 1 , 5 } . Similar situation holds for any other attack ers of the argumen t 1 a nd 5. Na mely , we need also to c hec k the elemen ts a 1 , 3 , a 5 , 3 ( if the argumen t 3 is an at t a c ker of the argumen t 1 or 5 ) and elemen ts a 1 , 4 , a 5 , 4 ( if the argumen t 4 is an attack er of the argumen t 1 o r 5) . This pro cess leads to a blo ck of the matrix M ( F ) at the inters ection of columns 2 , 3 , 4 a nd ro ws 1 , 5 . In summary , w e need to c heck tw o blo c ks (related to S = { 1 , 5 } ) of the matrix M ( F ) in order to decide that the conflict-free set S = { 1 , 5 } is an admissible extension. This motiv ate us to give the follow ing definition. Definition 15 Let F = ( A, R ) b e an argumen tation fra mew ork with A = { 1 , 2 , ..., n } , and S = { i 1 , i 2 , ..., i k } ⊂ A is an admissible extension of F . The h × k blo ck M j 1 ,j 2 ,...,j h i 1 ,i 2 ,...,i k =      a j 1 ,i 1 a j 1 ,i 2 . . . a j 1 ,i k a j 2 ,i 1 a j 2 ,i 2 . . . a j 2 ,i k . . . . . . a j h ,i 1 a j h ,i 2 . . . a j h ,i k      of the matrix M ( F ) is called the a -block of S and denoted b y M a , where { j 1 , j 2 , ..., j h } = A \ S . In other w ords, the elemen ts app earing at the interse ction of row s j 1 , j 2 , ..., j h and columns i 1 , i 2 , ..., i k in the matrix M ( F ) f orm the a -blo c k M j 1 ,j 2 ,...,j h i 1 ,i 2 ,...,i k of S . Note that, there is a natural relation b et w een the a -blo c k M j 1 ,j 2 ,...,j h i 1 ,i 2 ,...,i k and the s -blo c k M i 1 ,i 2 ,...,i k j 1 ,j 2 ,...,j h in matrix theory . Namely , the a -blo c k M j 1 ,j 2 ,...,j h i 1 ,i 2 ,...,i k of S is precisely the complemen tary blo ck of the s - blo c k M i 1 ,i 2 ,...,i k j 1 ,j 2 ,...,j h of S in the matrix M ( F ). F or conv enience, in this section w e ma y assume tha t the sequence s i 1 , i 2 , ..., i k and j 1 , j 2 , ..., j h are all increasing. Theorem 16 Giv en a n argumen tation framew o r k F = ( A, R ) with A = { 1 , 2 , ..., n } , then S = { i 1 , i 2 , ..., i k } ⊂ A is an admissible extens ion in F iff the f o llo wing conditions ho ld: (1) The cf -blo ck M i 1 ,i 2 ,...,i k i 1 ,i 2 ,...,i k of S is zero, (2) The column v ector of s -blo c k M i 1 ,i 2 ,...,i k j 1 ,j 2 ,...,j h of S corresp onding to the non-zero row 12 v ector of the a -blo c k M j 1 ,j 2 ,...,j h i 1 ,i 2 ,...,i k of S is non-zero, where A \ S = { j 1 , j 2 , ..., j h } . Pro of Let S b e a conflict-free set and A \ S = { j 1 , j 2 , ..., j h } . W e need only to prov e t ha t ev ery i r ∈ S (1 ≤ r ≤ k ) is defended b y S in F iff the column v ector of s -blo ck M i 1 ,i 2 ,...,i k j 1 ,j 2 ,...,j h of S corresp onding to the non-zero row v ector o f the a - blo c k M j 1 ,j 2 ,...,j h i 1 ,i 2 ,...,i k of S is non-zero Assume that ev ery i r ∈ S (1 ≤ r ≤ k ) is defended by S in F . If the ro w ve ctor M a t, ∗ (1 ≤ t ≤ h ) of the a - blo c k M j 1 ,j 2 ,...,j h i 1 ,i 2 ,...,i k = M a of S is non-zero, then there is some i r (1 ≤ r ≤ k ) such that a j t ,i r = 1. Note that a j t ,i r is at the in tersection of ro w t and column r of the a -block M a of S , and at the interse ction of row j t and column i r of the matrix M ( F ). This implies that ( j t , i r ) ∈ R , i.e. , the argumen t j t attac ks the argumen t i r . By the assumption, there is some i q ∈ S (1 ≤ q ≤ k ) suc h that the argumen t i q attac ks the ar gumen t j t , i.e. , ( i q , j t ) ∈ R . It follo ws that a i q ,j t = 1 in the matrix M ( F ). But, a i q ,j t is also an elemen t of the s -blo ck M s , whic h is at the in tersection of row q and column t of M s . Namely , a i q ,j t is an elemen t of the column v ector M s ∗ ,t of M s . Therefore, w e conclude tha t the column v ector M s ∗ ,t of s - blo c k M i 1 ,i 2 ,...,i k j 1 ,j 2 ,...,j h = M s of S is non-zero. Con v ersely , supp ose that the column ve ctor of s - blo c k M i 1 ,i 2 ,...,i k j 1 ,j 2 ,...,j h of S corresp onding to the non-zero ro w v ector of the a - blo c k M j 1 ,j 2 ,...,j h i 1 ,i 2 ,...,i k of S is non-zero. F or an y fixed i r ∈ S (1 ≤ r ≤ k ) , if there is no j t ∈ A \ S (1 ≤ t ≤ h ) suc h that the argumen t j t attac ks the ar gumen t i r , then b y the fact that S is a conflict-free set w e claim that there is no i ∈ A such that the arg umen t i a ttac ks the argumen t i r . It follows that arg umen t i r ∈ S is defended b y S in F . Otherwise, there is some j t ∈ A \ S ( 1 ≤ t ≤ h ) suc h that the argumen t j t attac ks the argumen t i r . It f ollo ws that ( j t , i r ) ∈ R , i.e. , a j t ,i r = 1. Since the elemen t a j t ,i r is at the intersec tion of ro w t and column r of the a -blo c k M j 1 ,j 2 ,...,j h i 1 ,i 2 ,...,i k = M a of S , the row vec tor M a t, ∗ of the a -blo c k M a of S is non-zero. By the assumption, w e conclude that the corresp onding column ve ctor M s ∗ ,t of the s -blo c k M i 1 ,i 2 ,...,i k j 1 ,j 2 ,...,j h = M s of S is non-zero. Therefore, there is s ome i q ∈ S (1 ≤ q ≤ k ) s uc h that a i q ,j t = 1. Note that, the elemen t a i q ,j t is at the in tersection of ro w q and column t of the s - blo c k M i 1 ,i 2 ,...,i k j 1 ,j 2 ,...,j h and at the in tersection of row i q and column j t of the matrix M ( F ). Consequen tly , w e ha v e that ( i q , j t ) ∈ R , i.e. , the argument i q ∈ S at t ac ks the argumen t j t . No w, w e hav e prov ed tha t the argument i r ∈ S is also defended by S in F . 13 Remark: The fact that any stable extension m ust b e admissible is clearly expresse d b y the prop erties of s -blo c ks in the matrix. In other w o rds, the condition every column v ector of the s -blo ck M i 1 ,i 2 ,...,i k j 1 ,j 2 ,...,j h of S are non-zero is stronger than that the column v ector of the s -blo ck M i 1 ,i 2 ,...,i k j 1 ,j 2 ,...,j h of S corre- sp onding to the non-zero ro w vector of the a -blo c k M j 1 ,j 2 ,...,j h i 1 ,i 2 ,...,i k of S is non-zero. 7. D e termination of t he complet e e xt e nsions Example 17 Consider the ar gumen ta tion framew o rk F = ( A, R ), where A = { 1 , 2 , 3 , 4 , 5 } and R = { (1 , 2) , (2 , 3) , { 2 , 4 } , (2 , 5) , (4 , 3 ) , (5 , 4) } . Since the admissible extension is necessarily a conflict- f ree set, w e can find out t he admissible extension from the collection of conflict-free sets {∅ , { 1 } , { 2 } , { 3 } , { 4 } , { 5 }{ 1 , 3 } , { 1 , 4 } , { 1 , 5 } , { 3 , 5 } , { 1 , 3 , 5 }} . By the directed graph of F , it is easy t o che ck that { 1 , 5 } and { 1 , 3 , 5 } are all the admissible extensions in F . F urthermore, one can ve rify that S 1 = { 1 , 3 , 5 } is the only complete extension in F , while S 2 = { 1 , 5 } is not. Next, we will ana lysis the differen t expressions in the matrix M ( F ) of F b et w een { 1 , 3 , 5 } (as a complete extension but not an a dmissible extension) and { 1 , 5 } (as an admissible extension). By comparing them, w e extract the matrix metho d to decide that an admissible extension is complete. Let us firstly write out the matrix of the arg umen tat io n fra mew ork F : M ( F ) =         0 1 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0         . In the column v ector M ( F ) ∗ , 2 of the matrix M ( F ) , a 1 , 2 = 1 means that (1 , 2) ∈ R , i.e. , the argument 1 a ttac ks the argumen t 2 . Since S 1 = { 1 , 3 , 5 } is a conflict-free set, there is no elemen t of S 1 whic h attacks the argumen t 1. It follows that the arguments 2 is not defended by S 1 in F . In the column v ector M ( F ) ∗ , 4 of the matrix M ( F ), a 5 , 4 = 1 means that (5 , 4) ∈ R , i.e. , the argumen t 5 attac ks the argumen t 4. Also b ecause that S 1 = { 1 , 3 , 5 } is a conflict-free set, there is no elemen t of S 1 whic h attac ks the ar g umen t 5. Th us, w e ha v e that the argumen ts 4 is not defended b y S 1 in F . These are e xactly the re asons for the admissible extension S 1 = { 1 , 3 , 5 } to b e a complete extension. 14 Next, we will mainly fo cus our atten tion on the argument 3 with resp ect to S 2 = { 1 , 5 } . In the column v ector M ( F ) ∗ , 3 of the matrix M ( F ) , a 2 , 3 = 1 means that (2 , 3) ∈ R , and a 4 , 3 = 1 means tha t (4 , 3) ∈ R . Therefore, b oth argumen ts 2 and 4 attac k the argumen t 3. On the other hand, in the column v ector M ( F ) ∗ , 2 of the matrix M ( F ), a 1 , 2 = 1 means that (1 , 2) ∈ R , i.e. , the argumen t 1 attacks the argumen t 2. In the column v ector M ( F ) ∗ , 4 of the matrix M ( F ), a 5 , 4 = 1 means that (5 , 4) ∈ R , i.e. , the argumen t 5 attac ks the argumen t 4. Consequen tly , w e hav e that the argument 3 is defended b y S 2 = { 1 , 5 } in F . It is precisely that the argumen t 3 is not included in S 2 whic h leads to the fact that S 2 = { 1 , 5 } is not a complete extension. F rom the ab ov e analysis, w e find a simple f act: In an argumen tation framew ork F = ( A, R ) w ith A = { 1 , 2 , ..., n } , an admissible e xtension S = { i 1 , i 2 , ..., i k } is complete iff eac h a rgumen t of A \ S = { j 1 , j 2 , ..., j h } is not defended b y S in F . And, w e can summarize the pro cess to decide an ad- missible extension S to b e complete b y the blo ck s of matrix M ( F ) of F as follo ws: (1) First, we pic k out the column ve ctors M ( F ) ∗ ,j 1 , M ( F ) ∗ ,j 2 , ..., M ( F ) ∗ ,j h of the matrix M ( F ) corresp onding to the argumen t s of A \ S = { j 1 , j 2 , ..., j h } . F or each argumen t j t ∈ A \ S (1 ≤ t ≤ h ), we c hec k the elemen ts a 1 ,j t , a 2 ,j t , ..., a n,j t in t he column v ector M ( F ) ∗ ,j t of the matrix M ( F ) to find all the attack ers of the arg umen t j t . (2) F or each arg umen t j t (1 ≤ t ≤ h ), w e consider t w o cases with resp ect to its atta c kers. (a) There is some j p ∈ A \ S ( 1 ≤ p ≤ h ) suc h that a j p ,j t = 1 in the column v ector M ( F ) ∗ ,j t of the matrix M ( F ), i.e. , ( j p , j t ) ∈ R , then the ar g umen t j p attac ks the argument j t in F . In order that the argument j t is not defended b y S , an y argumen t i r ∈ S (1 ≤ r ≤ k ) should not attac k the argumen t j p . Th us, w e hav e ( i r , j p ) / ∈ R , i.e. , a i r ,j p = 0 for all 1 ≤ r ≤ k . (b) There is no j p ∈ A \ S (1 ≤ p ≤ h ) such t ha t a j p ,j t = 1 in the column vec tor M ( F ) ∗ ,j t of the matrix M ( F ), then there m ust b e some i r ∈ S (1 ≤ r ≤ k ) suc h that a i r ,j t = 1 in the column v ector M ( F ) ∗ ,j t . Otherwise , there is no i ∈ A suc h that a i,j t = 1, i.e. , there is no i ∈ A suc h that ( i, j t ) ∈ R . It follo ws that there is no argumen t i ∈ A whic h attac ks the argumen t j t in F . This implies t hat the argumen t j t is defended by S in F , and th us S is not a complete extension. In case ( a ), the elemen ts ” a j p ,j t ”(1 ≤ p ≤ h, 1 ≤ t ≤ h ) for m a blo c k of the matr ix M ( F ) at the in tersection of ro w j 1 , j 2 , ..., j h and the same n um b er 15 of columns. The elemen ts ” a i r ,j t ”(1 ≤ r ≤ k , 1 ≤ t ≤ h ) form anther blo c k of the matrix M ( F ) at the in tersection of ro w i 1 , i 2 , ..., i k and the column j 1 , j 2 , ..., j h , whic h is exactly the s - blo c k of S . In case ( b ), one can find that the elemen ts considered fo rm the same blo c ks as in case ( a ). This mot iv ation mak es us t o giv e t he fo llo wing definition. Definition 18 Let F = ( A, R ) b e an argumen tation fra mew ork with A = { 1 , 2 , ..., n } , and S = { i 1 , i 2 , ..., i k } ⊂ A is a complete extension of F . The blo c k M j 1 ,j 2 ,...,j h j 1 ,j 2 ,...,j h =      a j 1 ,i 1 a j 1 ,i 2 . . . a j 1 ,i k a j 2 ,i 1 a j 2 ,i 2 . . . a j 2 ,i k . . . . . . a j h ,i 1 a j h ,i 2 . . . a j h ,i k      of order h in the matrix of M ( F ) is called the c -blo c k of S and denoted by M c , where { j 1 , j 2 , ..., j h } = A \ S . In other w ords, the elemen ts app earing at the interse ction of row s j 1 , j 2 , ..., j h and the same num b er o f columns in the matrix M ( F ) form the c -blo c k M j 1 ,j 2 ,...,j h j 1 ,j 2 ,...,j h of S . Note that, the c -blo ck M c = M j 1 ,j 2 ,...,j h j 1 ,j 2 ,...,j h of S is exactly the complemen tary blo c k of the s - blo c k M s = M i 1 ,i 2 ,...,i k i 1 ,i 2 ,...,i k of S , in the matrix M ( F ) o f F . No w, the fact that S 1 = { 1 , 3 , 5 } is a complete extension in the ab o v e example can b e v erified by the follow ing conditions: (1) The column v ector of s - blo c k M 1 , 3 , 5 2 , 4 of S 1 corresp onding t o the non- zero r ow v ector of c - blo c k M 2 , 4 2 , 4 of S 1 is zero; (2) The column v ector of s -blo c k M 1 , 3 , 5 2 , 4 of S 1 corresp onding to the zero column vec tor of c - blo c k M 2 , 4 2 , 4 of S 1 is non- zero. F or conv enience, in this section w e also a ssume that the sequences i 1 , i 2 , ..., i k and j 1 , j 2 , ..., j h are all increasing. Lemma 19 Let F = ( A, R ) b e an argumen tation framew ork with A = { 1 , 2 , ..., n } , then S = { i 1 , i 2 , ..., i k } ⊂ A is a complete extension o f F iff S is an admissible extens ion and eac h argumen t j t ∈ S (1 ≤ t ≤ h ) is not defended by S in F . Theorem 20 Giv en a n argumen tation framew o r k F = ( A, R ) with A = { 1 , 2 , ..., n } , t hen the admissible extension S = { i 1 , i 2 , ..., i k } ⊂ A is a com- plete extension in F iff the follow ing conditions hold: 16 (1) the column vec tor o f s -blo c k M i 1 ,i 2 ,...,i k j 1 ,j 2 ,...,j h of S corresp onding to the non-zero row v ector of the c - blo c k M j 1 ,j 2 ,...,j h j 1 ,j 2 ,...,j h of S is zero, (2) the column vec tor of s -blo c k M i 1 ,i 2 ,...,i k j 1 ,j 2 ,...,j h of S corresp onding to the zero column vec tor of the c - blo c k M j 1 ,j 2 ,...,j h j 1 ,j 2 ,...,j h of S is non-zero, where A \ S = { j 1 , j 2 , ..., j h } . Pro of Let S be an admissible extension and A \ S = { j 1 , j 2 , ..., j h } , w e need only t o pro v e tha t ev ery j t ∈ S (1 ≤ t ≤ h ) is not defended by S in F iff the condition (1) a nd (2) are hold. Assume that ev ery j t ∈ A \ S (1 ≤ t ≤ h ) is not defended b y S in F . If the row v ector M c r, ∗ (1 ≤ r ≤ h ) of the c -blo c k M j 1 ,j 2 ,...,j h j 1 ,j 2 ,...,j h of S is non-zero, then there is some 1 ≤ t ≤ h suc h that a j r ,j t = 1, i.e. , ( j r , j t ) ∈ R . It follows that the argumen t a j r attac ks the argumen t a j t . By the assumption, there is no argumen t in S whic h atta c ks the argumen t a j r . Therefore, f o r eac h i q ∈ S (1 ≤ q ≤ k ) w e hav e ( i q , j r ) / ∈ R , i.e. , a i q ,j r = 0. This means that the column vec tor M s ∗ ,r of the s - blo c k M i 1 ,i 2 ,...,i k j 1 ,j 2 ,...,j h of S is zero. If the column v ector M c ∗ ,t (1 ≤ t ≤ h ) of the c -blo ck M j 1 ,j 2 ,...,j h j 1 ,j 2 ,...,j h of S is zero, then for eac h 1 ≤ p ≤ h we ha v e t ha t a j p ,j t = 0, i.e. , ( j p , j t ) / ∈ R . Therefore, there is no argumen t in A \ S whic h attac ks the argument j t . If there is no argument in S whic h attac ks the argumen t j t , t hen there is no argumen t in A whic h attac ks the argumen t j t . It follows that the argumen t j t is defended by S in F , a con tradiction with the assumption. Thus , there is some a r g umen t i r ∈ S (1 ≤ r ≤ k ) whic h attac ks the argumen t j t , i.e. , ( i r , j t ) ∈ R . This implies that a i r ,j t = 1, and th us the column ve ctor M s ∗ ,t of the s - blo c k M i 1 ,i 2 ,...,i k j 1 ,j 2 ,...,j h of S is non-zero. Con v ersely , supp ose t ha t the conditions (1) and (2) are hold. Let j t ∈ A \ S (1 ≤ t ≤ h ), w e consider the column ve ctor M c ∗ ,t of the c - blo c k M j 1 ,j 2 ,...,j h j 1 ,j 2 ,...,j h of S . If the column vec tor M c ∗ ,t is zero, then by condition (2) w e hav e that the column ve ctor M s ∗ ,t of the s -blo c k M i 1 ,i 2 ,...,i k j 1 ,j 2 ,...,j h of S is non-zero. It follows that there is some i q ∈ S (1 ≤ q ≤ k ) suc h that a i q ,j t = 1, i.e. , ( i q , j t ) ∈ R . This means that the argument i q attac ks the argument j t in F . Considering that S is a conflict-free set, there is no ar g umen t i r ∈ S (1 ≤ r ≤ k ) whic h attac ks the argumen t i q in F . If the column v ector M c ∗ ,t is non-zero, then the row v ector M c t, ∗ is also no n- zero. By condition (1), the column v ector M s ∗ ,t of the s -blo ck M i 1 ,i 2 ,...,i k j 1 ,j 2 ,...,j h = M s 17 of S is zero. It follo ws that a i r ,j t = 0, i.e. , ( i r , j t ) / ∈ R fo r each 1 ≤ r ≤ k . This implies that there is no argument i r ∈ S (1 ≤ r ≤ k ) whic h a ttac ks the argumen t j t in F . T o sum up, w e conclude that the argument j t ∈ A \ S (1 ≤ t ≤ h ) is not defended by S . 8. C onclusio ns and p ersp ecti v es In this pa p er, we introduced the matrix M ( F ) of a n a r gumen ta tion frame- w ork F = ( A, R ) , and the cf - blo c k M cf , s -blo ck M s , a -blo ck M a and c -blo c k M c of a set S ⊂ A , presen ted sev eral theorems to decide v arious extensions (stable, admissible, complete) of the AF, b y blo cks of the matrix M ( F ) of F and relations b et w een these blo ck s. In terestingly , the s - blo c k M s ( a -blo c k M a , c -blo c k M c ) of S corresp onds to the de termination for S to b e a stable extension (admissible extension, complete extension resp ectiv ely). And, the c -blo c k of S is exactly the com- plemen ta r y blo ck of the cf -blo c k of S , the a -blo ck of S is exactly the com- plemen ta r y blo c k of the s -blo c k of S . F urthermore, w e can decide basic extensions of an argumen tation framew ork b y the s p ecial feature of blo cks and relatio ns b etw een these blo c ks. These fa cts indicate that there is indeed a corresp onding relation b et w een the argumentation framew ork and its ma- trix. So, w e can inv estigate the structure and prop erties of an argumen tation framew ork b y using the theory and metho d of matrix. F or the o t her common ex tension seman tics (preferred, g rounded, ideal, semi-stable and eager) of Dung’s argumen tation f ramew or k not discussed in the ab o ve sections, w e can also pro vide the matrix metho d to describ e them, b y com bining the obtained results. F or example, if we wan t to decide tha t a complete extension S ⊂ A is grounded in F = ( A, R ) , w e could first find out all the complete extensions by theorem 20 . Then, w e compare the cf - blo c ks of these complete extensions. If the cf -blo c k of S is the minimal one in the collection of cf -blo c ks of all complete extensions, then w e claim that S is a grounded extension. The prospectiv es are that, w e can find out the in ternal pattern of AFs and the relations b et w een differen t ob jects whic h w e concerned in AFs, b y studying blo cks of the matrix of AFs. Our future goal is to dev elop the ma- trix method in the related areas, suc h as argumen t acceptabilit y , dialog ue games, alg orithmic and complexity and so on [7, 11, 8, 13 , 16 , 12]. 18 References 1. P . Baroni, M. Giacomin, On principle-based ev aluatio n of extens ion- based a r g umen tat ion seman tics, Artificial In telligence 17 1 (200 7), 675 - 700. 2. T. J. M. Benc h-Cap on, P aul E. Dunne, Arg umentation in artificial in telligence, Artificial intelligenc e 1 71(2007)619 -641 3. M.Caminada, Semi-stable seman tics, in: F r o n tiers in Artificial In telli- gence and its Applications, vol. 144, IOS Press, 2006, pp. 12 1 -130. 4. C.Ca yrol, M.C.Lagasquie-Sc hiex, G radualit y in argumentation, J. AI Res. 23 (200 5 )245-297. 5. 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