Comparative Study of Methods to Determine Arguments Acceptance in Argumentation Framework

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1 Comparative Study of Methods to Determine Arguments Acceptance in Argumentation ramework 1 1 Shuya BUNDO 1 Kazunori YAMAGUCHI The University of Tokyo Abstract: (A),, Dung,,,,, 1,, semantics ( ), Dung semantics, semantics,, semantics, ranking semantics, Dung semantics ranking semantics, Dung semantics ranking semantics,, 2 Dung semantics Dung semantics 2.1 ( [2]). A A A A = A, (Argumentation ramework, A) S, T A, s S, t T bundo@graco.c.u-tokyo.ac.jp s t S T S = {s} s T, T = {t} S t 2.2. = A, A, S A a b a, b S S conflictfree Def(S) = {a A ( b a, c S, c b)} S conflict-free S Def(S) S admissible extension S admissible extension S = Def(S) S complete extension admissible extension, complete extension adm( ), comp( ), Def( ), Def 2 ( ),..., Def n ( ),..., N Def N ( ) = Def N+1 ( ) Def N ( ) grounded extension Atk(S) = {b S b} 3 ranking semantics ranking semantics 3.1. preorder A preorder, : a b a b b a a b a b b a 3.2 (ranking semantics[3]). A A( ) preorder ( ) a b ranking semantics - 7 -

2 4 ranking semantics ranking semantics 1, 4.1., G: A G := A( ) A(G), G a, b A( ), P ab = {b = a 0 a 1 a n = a a i A} l = a 0 a n l = n. R n (a) = {b l P ab, l = n} (), R + (a) = n Z +R 2n (a), R (a) = n Z +R 2n 1 (a), a defender, a attacker R 1 (a) a direct attacker, R 2 (a) a defend 4.2. defense root (resp. attack root) nonattacked defender (resp. attacker) BR n (a) = {b R n (a) R 1 (b) = } BR + (a) = n Z +BR 2n (a) defense branch, BR (a) = n Z +BR 2n 1 (a) attack branch 4.3. = A, A : P n (a) = {x 0 = a, x 1,..., x n }, {x n x n 1, x n 1 x n 2,..., x 1, x 1 x 0 }. i 0 x i / A a A( ) defense simple, a defender, a 1 attacker, a defense distributed, a direct attacker b, b = A, = A, γ : A A (A A = ). = γ ranking semantics, ranking semantics ( 5.1 ) VP SC CP R 1 (a) =, R 1 (b) a b a a, b b a b R 1 (a) < R 1 (b) a b 1 ranking semantics axiom DP R 1 (a) = R 1 (b), R 2 (a), R 2 (b) = a b DDP R 1 (a) = R 1 (b), R 2 (a) = R 2 (b), a defense simple distributed, b defense simple distributed a b DB a A( ) γ P 2n (γ(a)) γ(a) a +DB a A( ) R 1 (a) γ P 2n (γ(a)) γ(a) a DB a, b A( ) b BR + (a), b / BR (a) γ P 2n (γ(b)) a γ(a) AB a, b A( ) b BR (a), b / BR + (a) γ P 2n (γ(b)) γ(a) a +AB a A( ) γ P 2n 1 (γ(a)) a γ(a) AvsD A acyclic BR (a) =, R 1 (b) = 1, R 2 (b) = a b 5 Dung semantics ranking semantics Dung semantics ranking semantics Dung ranking ranking semantics ranking semantics,, 2. 2 ranking semantics,, ranking semantics,, λ Λ ranking semantics { λ } λ Λ, λ ranking semantics, { λ } λ Λ,, 2 ranking semantics,

3 2. 1, 3. 3 VP ranking semantics,, VP a b, a b a = b, ranking semantics, ranking semantics, ranking semantics λ preorder λ Λ { λ },,, a b b a, 5.2. { λ } λ Λ, λ Λ { λ } λ 2 ranking semantics, 5.3. n 0, a 0 b 0 a 1 b 1 a 2 b 2 b n 1 a n b n a 0, b n a 0 b 0 preorder a 0 b 0 a 1 b 1 a n b n a 0, a 0 b 0, a 0 b 0 a 0 b 0 b 0 a 0,, a b a b a, b. a b a b, a b,, a = a 0 b = b 0 a 1 b 1 a n b n a , n 0, a 0 b 0 a 1 b 1 a 2 b 2 b n 1 a n b n a , a b a b or a = b, preorder 1: A 1 ([2] igure 1), admissible semantics complete semantics ranking semantics, semantics, semantics semantics, σ : A 2 2U = A, A σ( ) 2 A σ( ) semantics σ adm comp Dung semantics. σ( ) semantics, a A( ) uni-accepted, exi-accepted, cleanlyaccepted, not-accepted 4 [4],,, σ( ) adm( ) E σ( ) {E, Atk(E), Undec(E)} A( ) 5.5. E A( ) a, b E preorder 1. a E,3 b iff [a] [b] [a] A( ) {E, Atk(E), Undec(E)},, E > Undec(E) > Atk(E) > 2. a E,2 b iff [a] [b] [a] A( ) {E, A( ) E} E > A( ) E v = 2, 3, a, b A( ) a σ,v b E σ( ), a E,v b preorder. preorder Dung ranking ([2] igure 1, 1 ) A 1, b grounded, E = {b, e} complete extension E, {E, Atk(E), Undec(E)} {{b, e}, {a, c, d}, }, {E, E Atk(E)} {{b, e}, {a, c, d}}, b, e comp,v a, c, d, σ,v VP ( ranking semantics) - 9 -

4 VP σ = adm (admissible), {a, b, c, d}, {b c d} A a d VP adm,v VP 5.4, a 0 VP b 0 adm,v a 1 VP b 1 adm,v VP b n 1 adm,v a n VP b n adm,v a 0 R 1 (a 1 ) = E = {a 1 } admissible b 0 adm,v a 1 b 0 E,v a 1 b 0 E, b 0 = a 1. R 1 (b 0 ) = a 0 VP b 0. σ = comp a b c a c SC σ( ) adm( ) σ, v = 2, 3 c a, b b adm( ) = {, {c}} a a, b b, b σ,v a CP σ( ) adm( ) σ, v = 2, 3 c 1 d 1, c 2 d 2, d 1 c 2, d 2 c 1, c 1 a, d 1 a, c 1 b, c 2 b, d 1 b adm( ) = {, {c 1, c 2 }, {d 1, d 2 }} R 1 (a) = 2 < R 1 (b) = 3, b σ,v a DP σ,v v = 2, v = 3 σ( ) comp( ), (, a DP b, b σ,v ) adm,3 DP 5.4, a 0 adm,3 b 0 DP a 1 adm,3 b 1 DP a 2 adm,3 b 2 DP adm,3 a n adm,3 b n DP a n+1 = a 0 x R 1 (a i ) R 2 (a i ) =, E = {x} E adm( ) a i Atk(E), a i adm,3 b i a i E,3 b i b i Atk(E), x R 1 (b i ) R 1 (a i ) R 1 (b i ) R 1 (a i ) R 1 (b i ) R 1 (b i ) = R 1 (a i+1 ), R 1 (a 0 ) R 1 (b 0 ) = R 1 (b 1 ) R 1 (a 2 ) = R 1 (b n ) = R 1 (a 0 ), R 1 (a 0 ) = R 1 (b 0 ) R 1 (a 0 ) R 1 (b 0 ) R 1 (a 0 ) = R 1 (b 0 ) R 2 (a 0 ) =, R 2 (b 0 ) DDP σ,v v = 2, v = 3 σ( ) comp( ), (, a DDP b R 1 (a) = R 1 (b), R 2 (a) = R 2 (b) a distributed, R 1 (a) R 2 (a), R 1 (b) R 2 (b) b defense simple distributed, c R 1 (b), R 1 (c) =, v = 2, v = 3 σ( ) comp( ), a DDP b, b σ,v ) adm,3 DP 5.4, a 0 adm,3 b 0 DDP a 1 adm,3 b 1 DDP DDP a n adm,3 b n DDP a n+1 = a 0 φ(a) = {b R 1 (a) R 1 (b) = } b i DDP a i+1 φ(b i ) < φ(a i+1 ) a i adm,3 b i φ(a i ) φ(b i ) b φ(a i ), R 1 (b) = E = {b} E adm( ) a i Atk(E), a i adm,3 b i b i Atk(E) b R 1 (b i ) φ(a i ) φ(b i ) φ(a i ) φ(b i ) DB +DB +DB σ,v, a a, c b γ(a) γ(a) A, γ(a) +DB a adm( ) = {, {c}} a σ,v γ(a) AB σ( ) comp( ), v = 2, 3., b a, d c γ(b) γ(a) A γ(a) AB a comp( ) = {{b, d, γ(b)}} a σ,v γ(a), σ = adm v = 2 A b adm,2 γ(b) adm,3 +AB +AB σ( ) comp( ), v = 2, 3., b a, γ(b) γ(a), c γ(a) A a +AB γ(a) comp( ) = {{b, γ(b), c}} a σ,v γ(a), σ = adm v = 2 A a adm,2 γ(a) adm,3 5.4, a 0 adm,3 b 0 +AB a 1 adm,3 b 1 +AB +AB a n adm,3 b n +AB a n+1 = a 0 b i +AB a i+1, a i 1 attack branch 1 x 0 x 1 x 2k a E = {x 0, x 2,..., x 2(k 1), x 2k }, E admissible a Atk(E), a i adm,3 b i, b i Atk(E), a i b i w i a i b i

5 , b i +AB a i+1, w i < w i+1, w 0 < w 1 < < w n < w n+1 = w 0 DB σ,v, c b a a, e d γ(c) γ(b) γ(a) γ(a) A, a DB γ(a) a σ,v γ(a) AvsD σ( ) grounded extension v = 2, 3 E adm( ) a / Atk(E), b / E a E,v b ground extension a b a σ,v b 1 adm,v comp,v VP SC CP DP DDP DB +DB DB AB (v = 3), (v = 2) +AB (v = 3), (v = 2) AvsD 1:,, 1, 5 Dung ranking 4., Dung semantics,,, DP DDP,, AB +AB branch v = 3, AvsD, Dung ranking 6 ranking semantics, ranking semantics, [2] ranking semantics Cat, Dbs, Bds, M&T, SA ( Cat ) 5 σ,v ( ranking semantics M&T SA ), σ,v (σ = adm, comp, v = 2, 3), ranking semantics, a, b (a b) a b a adm,v b a comp,v b, ranking semantics σ,v 6.1. a σ,v b a σ,v b a = b. σ,v σ,v ranking semantics Cat, Dbs, Bds Cat, Dbs, Bds adm,v comp,v, 1 1, d δ e (δ = Cat, Dbs, Bds), e σ,v d (σ = adm, comp) M&T M&T = A,, X, Y A, Y X {(x, y) X Y, x y}, f(n) = n/(n+1), P, O A ϕ(p, O) = 1/2 [1+f( O P ) f( P O )], 0 (P conflict-free ) r(p, O) = 1 (P conflict-free P O = ) ϕ(p, O) (otherwise), a A 1, 2 : (proponent) (opponent) ( ) A P A, O A, P a P, r(p, O),, v(a) a M&T b v(a) v(b) 6.2. M&T σ,v (σ {adm, comp}, v {2, 3}), A = {a 0, a 1, a 2, b 0, b 1, b 2, c}, = {a 0 a 1, a 1 a 2, a 2 a 0 } {a i b j 0 i, j 2} {b j c 0 j 2} A = A, ( 2), v(c) 1/2, 1/3 {c, a 0 }, {c, a 1 }, {c, a 2 }, O A 1/3 [ r({c, a i}, O)] 1/2, i {c, a i } conflict-free, P conflict-free P O =

6 f adm,v c adm,v M&T comp( ) = {{d, f}}, comp,v M&T 2: 6.2 A ϕ(p, O) r(p, O) = 1, 1/3 [ ϕ({c, a i}, O)] 1/2 a 3 = a 0 O A 0 i 2 O {c,ai} = {c, a i+1 } O i = 0 x A, x = a 0, a 2, c {x} {c,a0} = {c, a 1 } {x} = 0,, x = a 1, b 0, b 1, b 2 {x} {c,a0} = {c, a 1 } {x} = 1, x A {x} {c,a0} = {c, a 1 } {x}, O {c,a0} = x O {x} {c,a0} = x O {c, a 1} {x} = {c, a 1 } O i = 1, 2 O {c,ai} = {c, a i+1 } O, [f( O {c,ai+1} ) f( {c, a i } O )] = 0 f( O {c,ai} ) = 1 3 ϕ({c, a i }, O) = 1/2 f( {c, a i } O ) v(c) 1/2, = A, A = {d, e, f}, = {d e, e d, e e, e f}, f P conflict-free {f}, {d, f} 2, r({f}, {e}) = 1/2 [ /2] = 1/4, r({d, f}, {e}) = 1/2 [1 + 1/2 2/3] = 5/12 v(f) 5/12, = A A,, [5], A v(c), v(f) A c M&T f, adm( ) = {, {d}, {d, f}} E adm( ) c Undec(E), f / Atk(E),f {d, f}, v {2, 3} SA, SA SA, = A,, ϵ > 0 1, v : A [0, 1], v(a) = 1 (1 v(a i )) 1 + ϵ a i R 1(a), a SA b v(a) v(b) v v, SA,, well-founded A (, acyclic) v, 6.3. = A, well-founded A ( acyclic), σ,v (σ {grounded, comp}, v = 2, 3), ϵ > 0 σ,v SA [1], well-founded grounded extension G, A = G Atk(G), attack G 0 = Def( ), grounded extension G, N G = Def N (G 0 ) M = max a A R 1 (a) ϵ > 0 (M + 1) N ϵ < 1/2 a Def n (G 0 ) v(a) 1 (M + 1) n ϵ n = 0 1 ϵ < 1/(1 + ϵ). n 1, a Def n (G 0 ), v(a) = 1/(1+ϵ) a (1 v(a i R 1(a) i)) a i R 1 (a), b Def n 1 (G 0 ) b R 1 (a i ) v(b) 1 (M +1) n 1 ϵ, v(a i ) = 1/(1 + ϵ) b (1 v(b j R 1(a i) j)) 1 v(b) (M + 1) n 1 ϵ, R 1 (a) M v(a) = ϵ (1 v(a i )) a i R 1(a) (1 ϵ)(1 (M + 1) n 1 ϵ) M (1 (M + 1) n 1 ϵ) M+1 1 (M + 1) n ϵ, (1 x) n 1 nx, a G = Def N (G 0 ), v(a) 1 (M +1) N ϵ > 1/2, b Atk(G)

7 , a G a R 1 (b), v(b) = 1/(1 + ϵ) a (1 v(a i R 1(b) i)) 1 v(a) < 1/2. a σ,v b (σ {grounded, comp}, v = 2, 3) v(a) > v(b), a SA b 2 adm,v comp,v ( (grounded wellfounded )) Cat Dbs Bds M&T SA (wellfounded ), ( ) 2: ranking semantics Dung ranking (,, ) 2 SA, SA, SA Dung ranking SA ϵ, Dung ranking,, ϵ = 0, complete extension v 1, v 0, v 7 Dung semantics ranking Dung ranking Dung ranking, ranking semantics,, Dung ranking ranking semantics ranking semantics,, Dung ranking ranking semantics, SA, well-founded A grounded complete SA Dung ranking,, well-founded, A [1] Phan Minh Dung. On the acceptability of arguments and its fundamental role in nonmonotonic reasoning, logic programming and n-person games. Artificial Intelligence, 77(2), pages , [2] Elise Bonzon, Jérôme Delobelle, Sébastien Konieczny, and Nicolas Maudet. A comparative study of ranking-based semantics for abstract argumentation. In Proceedings of the Thirtieth AAAI Conference on Artificial Intelligence, AAAI 16, pages , AAAI Press, [3] L. Amgoud and J. Ben-Naim. Ranking-based semantics for argumentation frameworks. In Scalable Uncertainty Management, pages , Springer, [4] C. Cayrol and M.-Ch. Lagasquie-Schiex. Graduality in argumentation. Journal of Artificial Intelligence Research, 23, pages , [5] P.-A. Matt and. Toni. A game-theoretic measure of argument strength for abstract argumentation. Procs. of JELIA 08, LNCS 5293, pages Springer, [6] J. Leite and J. Martins. Social abstract argumentation. In Proc. 22nd Int l Joint Conf. on Artificial Intelligence, pages ,