Verifiability of Argumentation Semantics

Transkript

1 Verifiability of Argumentation Semantics Ringo Baumann, Thomas Linsbichler, Stefan Woltran 6th International Conference on Computational Models of Argument Potsdam, Germany September 16, 2016

2 Introduction Abstract Argumentation Framework (AF) [Dung, 1995]: a b c d f e Baumann, Linsbichler, Woltran, September 16, 2016 Verifiability of Argumentation Semantics 1

3 Introduction Abstract Argumentation Framework (AF) [Dung, 1995]: a b c d f e Evaluation: argumentation semantics Extension: set of jointly acceptable arguments Baumann, Linsbichler, Woltran, September 16, 2016 Verifiability of Argumentation Semantics 1

4 Introduction Abstract Argumentation Framework (AF) [Dung, 1995]: a b c d f e Evaluation: argumentation semantics Extension: set of jointly acceptable arguments stb(f) = Baumann, Linsbichler, Woltran, September 16, 2016 Verifiability of Argumentation Semantics 1

5 Introduction Abstract Argumentation Framework (AF) [Dung, 1995]: a b c d f e Evaluation: argumentation semantics Extension: set of jointly acceptable arguments stb(f) = { {a, d, e}, Baumann, Linsbichler, Woltran, September 16, 2016 Verifiability of Argumentation Semantics 1

6 Introduction Abstract Argumentation Framework (AF) [Dung, 1995]: a b c d f e Evaluation: argumentation semantics Extension: set of jointly acceptable arguments stb(f) = { {a, d, e}, {b, c, e} } Baumann, Linsbichler, Woltran, September 16, 2016 Verifiability of Argumentation Semantics 1

7 Introduction Abstract Argumentation Framework (AF) [Dung, 1995]: a b c d f e Evaluation: argumentation semantics Extension: set of jointly acceptable arguments stb(f) = { {a, d, e}, {b, c, e} } Further semantics: preferred, complete, semi-stable, stage,... Baumann, Linsbichler, Woltran, September 16, 2016 Verifiability of Argumentation Semantics 1

8 Introduction Conflict-freeness: basic requirement for argumentation semantics. Baumann, Linsbichler, Woltran, September 16, 2016 Verifiability of Argumentation Semantics 2

9 Introduction Conflict-freeness: basic requirement for argumentation semantics. Example Given conflict-free sets, {a}, {b}. Baumann, Linsbichler, Woltran, September 16, 2016 Verifiability of Argumentation Semantics 2

10 Introduction Conflict-freeness: basic requirement for argumentation semantics. Example Given conflict-free sets, {a}, {b}. Can we compute semantics based on this? Baumann, Linsbichler, Woltran, September 16, 2016 Verifiability of Argumentation Semantics 2

11 Introduction Conflict-freeness: basic requirement for argumentation semantics. Example Given conflict-free sets, {a}, {b}. Can we compute semantics based on this? Yes, naive semantics (maximal conflict-free sets) Baumann, Linsbichler, Woltran, September 16, 2016 Verifiability of Argumentation Semantics 2

12 Introduction Conflict-freeness: basic requirement for argumentation semantics. Example Given conflict-free sets, {a}, {b}. Can we compute semantics based on this? Yes, naive semantics (maximal conflict-free sets) F : a b G : a b H : a b... Baumann, Linsbichler, Woltran, September 16, 2016 Verifiability of Argumentation Semantics 2

13 Introduction Conflict-freeness: basic requirement for argumentation semantics. Example Given conflict-free sets, {a}, {b}. Can we compute semantics based on this? Yes, naive semantics (maximal conflict-free sets) F : a b G : a b H : a b... na(f) = na(g) = na(h) = = {{a}, {b}}. Baumann, Linsbichler, Woltran, September 16, 2016 Verifiability of Argumentation Semantics 2

14 Introduction Conflict-freeness: basic requirement for argumentation semantics. Example Given conflict-free sets, {a}, {b}. Can we compute semantics based on this? Yes, naive semantics (maximal conflict-free sets) F : a b G : a b H : a b... na(f) = na(g) = na(h) = = {{a}, {b}}. not stage semantics (range-maximal conflict-free sets) stg(f) = {{a}}, stg(g) = {{b}}, stg(h) = {{a}, {b}}. Baumann, Linsbichler, Woltran, September 16, 2016 Verifiability of Argumentation Semantics 2

15 Introduction Conflict-freeness: basic requirement for argumentation semantics. Example (2) Given conflict free sets + their range: (, ), ({a}, {a, b}), ({b}, {b}) Baumann, Linsbichler, Woltran, September 16, 2016 Verifiability of Argumentation Semantics 2

16 Introduction Conflict-freeness: basic requirement for argumentation semantics. Example (2) Given conflict free sets + their range: (, ), ({a}, {a, b}), ({b}, {b}) Which semantics can we compute based on this? enough to compute stage semantics (range-maximal conflict-free sets) Baumann, Linsbichler, Woltran, September 16, 2016 Verifiability of Argumentation Semantics 2

17 Introduction Conflict-freeness: basic requirement for argumentation semantics. Example (2) Given conflict free sets + their range: (, ), ({a}, {a, b}), ({b}, {b}) Which semantics can we compute based on this? enough to compute stage semantics (range-maximal conflict-free sets) F : a b G : a b H : c a b... Baumann, Linsbichler, Woltran, September 16, 2016 Verifiability of Argumentation Semantics 2

18 Introduction Conflict-freeness: basic requirement for argumentation semantics. Example (2) Given conflict free sets + their range: (, ), ({a}, {a, b}), ({b}, {b}) Which semantics can we compute based on this? enough to compute stage semantics (range-maximal conflict-free sets) F : a b G : a b H : c a b... stg(f ) = stg(g ) = stg(h ) = = {{a}}. Baumann, Linsbichler, Woltran, September 16, 2016 Verifiability of Argumentation Semantics 2

19 Introduction Conflict-freeness: basic requirement for argumentation semantics. Example (2) Given conflict free sets + their range: (, ), ({a}, {a, b}), ({b}, {b}) Which semantics can we compute based on this? enough to compute stage semantics (range-maximal conflict-free sets) F : a b G : a b H : c a b... stg(f ) = stg(g ) = stg(h ) = = {{a}}. not preferred semantics (maximal admissible sets) pr(f ) = pr(g ) = {{a}}, pr(h ) = { }. Baumann, Linsbichler, Woltran, September 16, 2016 Verifiability of Argumentation Semantics 2

20 Introduction Conflict-freeness: basic requirement for argumentation semantics. Example (2) Given conflict free sets + their range: (, ), ({a}, {a, b}), ({b}, {b}) Which semantics can we compute based on this? enough to compute stage semantics (range-maximal conflict-free sets) F : a b G : a b H : c a b... stg(f ) = stg(g ) = stg(h ) = = {{a}}. not preferred semantics (maximal admissible sets) pr(f ) = pr(g ) = {{a}}, pr(h ) = { }. Which information on top of conflict-free sets has to be added in order to compute a certain semantics? Baumann, Linsbichler, Woltran, September 16, 2016 Verifiability of Argumentation Semantics 2

21 Introduction Systematic comparison of argumentation semantics Computational complexity [Dunne and Bench-Capon, 2002, Dvořák and Woltran, 2010] Principle-based evaluation [Baroni and Giacomin, 2007] Baumann, Linsbichler, Woltran, September 16, 2016 Verifiability of Argumentation Semantics 3

22 Introduction Systematic comparison of argumentation semantics Computational complexity [Dunne and Bench-Capon, 2002, Dvořák and Woltran, 2010] Principle-based evaluation [Baroni and Giacomin, 2007] Hierarchy of verification classes Classification of semantics into these classes Each rational semantics is exactly verifiable by one of these classes Baumann, Linsbichler, Woltran, September 16, 2016 Verifiability of Argumentation Semantics 3

23 Introduction Systematic comparison of argumentation semantics Computational complexity [Dunne and Bench-Capon, 2002, Dvořák and Woltran, 2010] Principle-based evaluation [Baroni and Giacomin, 2007] Hierarchy of verification classes Classification of semantics into these classes Each rational semantics is exactly verifiable by one of these classes Strong equivalence Central notion in non-monotonic reasoning [Lifschitz et al., 2001, Turner, 2004, Truszczynski, 2006, Baumann and Strass, 2016] Studied for most argumentation semantics [Oikarinen and Woltran, 2011, Baumann, 2016] Baumann, Linsbichler, Woltran, September 16, 2016 Verifiability of Argumentation Semantics 3

24 Introduction Systematic comparison of argumentation semantics Computational complexity [Dunne and Bench-Capon, 2002, Dvořák and Woltran, 2010] Principle-based evaluation [Baroni and Giacomin, 2007] Hierarchy of verification classes Classification of semantics into these classes Each rational semantics is exactly verifiable by one of these classes Strong equivalence Central notion in non-monotonic reasoning [Lifschitz et al., 2001, Turner, 2004, Truszczynski, 2006, Baumann and Strass, 2016] Studied for most argumentation semantics [Oikarinen and Woltran, 2011, Baumann, 2016] Missing results for naive and strong admissible semantics Characterization theorems for intermediate semantics Baumann, Linsbichler, Woltran, September 16, 2016 Verifiability of Argumentation Semantics 3

25 Background Definition An argumentation framework (AF) is a pair (A, R) where A U is a finite set of arguments and R A A is the attack relation representing conflicts. Definition Given an AF F = (A, R) and S A, S is conflict-free (S cf(f)) if a, b S : (a, b) / R. a A is defended by S if b A : (b, a) R c S : (c, b) R S + F S F = S {a b S : (b, a) R} (the range of S) = S {a b S : (a, b) R} (the anti-range of S) Baumann, Linsbichler, Woltran, September 16, 2016 Verifiability of Argumentation Semantics 4

26 Background Semantics Given an AF F = (A, R), a set S A is admissible set if S cf(f) and each a S is defended by S, complete extension if S ad(f) and a S if a is defended by S, naive extension if S cf(f) and T cf(f) : T S, stable extension if S cf(f) and S + F = A, stage extension if S cf(f) and T cf(f) : T + F S+ F, preferred, grounded, semi-stable, ideal, eager, strongly admissible extensions Baumann, Linsbichler, Woltran, September 16, 2016 Verifiability of Argumentation Semantics 5

27 Verifiability Definition We call a function r x : 2 U 2 U ( 2 U) n which is expressible via basic set operations only a neighborhood function. a r x (A, B) is in the language X ::= A B (X X) (X X) (X \ X) Baumann, Linsbichler, Woltran, September 16, 2016 Verifiability of Argumentation Semantics 6

28 Verifiability Definition We call a function r x : 2 U 2 U ( 2 U) n which is expressible via basic set operations only a neighborhood function. The verification class induced by r x maps each AF F to F x = {( S, r x (S + F, S F )) S cf(f) }. a r x (A, B) is in the language X ::= A B (X X) (X X) (X \ X) Baumann, Linsbichler, Woltran, September 16, 2016 Verifiability of Argumentation Semantics 6

29 Verifiability Definition We call a function r x : 2 U 2 U ( 2 U) n which is expressible via basic set operations only a neighborhood function. The verification class induced by r x maps each AF F to F x = {( S, r x (S + F, S F )) S cf(f) }. a r x (A, B) is in the language X ::= A B (X X) (X X) (X \ X) Example F : a b c r + : r x (A, B) = A F + = {(, ), ({a}, {a, b}), ({c}, {b, c}), ({a, c}, {a, b, c})} Baumann, Linsbichler, Woltran, September 16, 2016 Verifiability of Argumentation Semantics 6

30 Verifiability Definition We call a function r x : 2 U 2 U ( 2 U) n which is expressible via basic set operations only a neighborhood function. The verification class induced by r x maps each AF F to F x = {( S, r x (S + F, S F )) S cf(f) }. a r x (A, B) is in the language X ::= A B (X X) (X X) (X \ X) Example F : a b c r + : r x (A, B) = A F + = {(, ), ({a}, {a, b}), ({c}, {b, c}), ({a, c}, {a, b, c})} r ± : r x (A, B) = (B, A \ B) F ± = {(,, ), ({a}, {a, b}, ), ({c}, {c}, {b}), ({a, c}, {a, b, c}, )} Baumann, Linsbichler, Woltran, September 16, 2016 Verifiability of Argumentation Semantics 6

31 Verifiability Neighborhood functions for n = 1: r ɛ (A, B) = r + (A, B) = A r (A, B) = B r (A, B) = B \ A r ± (A, B) = A \ B r (A, B) = A B r (A, B) = A B r (A, B) = (A B) \ (A B) syntactically different neighborhood functions r x 1,...,x n (A, B) ::= (r x 1(A, B),..., r xn (A, B)) Baumann, Linsbichler, Woltran, September 16, 2016 Verifiability of Argumentation Semantics 7

32 Verifiability Definition r x is more informative than r y (r x r y ): there is a function δ : ( 2 U) n ( 2 U ) m such that δ (r x (A, B)) = r y (A, B) for any A, B U. In case r x r y (r x r y and r y r x ), we say that r x represents r y. Baumann, Linsbichler, Woltran, September 16, 2016 Verifiability of Argumentation Semantics 8

33 Verifiability Definition r x is more informative than r y (r x r y ): there is a function δ : ( 2 U) n ( 2 U ) m such that δ (r x (A, B)) = r y (A, B) for any A, B U. In case r x r y (r x r y and r y r x ), we say that r x represents r y. Example δ 1 (r +± (A, B)) = δ 1 (A, A \ B) = def (A, A \ (A \ B)) = (A, A B) = r + (A, B); δ 2 (r + (A, B)) = δ 2 (A, A B) = def (A \ (A B), A B) = (A \ B, A B) = r ± (A, B); δ 3 (r ± (A, B)) = δ 3 (A \ B, A B) = def ((A \ B) (A B), A \ B) = (A, A \ B) = r +± (A, B). r +± r + r ± Baumann, Linsbichler, Woltran, September 16, 2016 Verifiability of Argumentation Semantics 8

34 Verifiability Lemma All neighborhood functions are represented by the ones depicted below and the -relation represented by arcs holds. + +± + ± ± + ± ɛ Baumann, Linsbichler, Woltran, September 16, 2016 Verifiability of Argumentation Semantics 9

35 Verifiability Definition A semantics σ is verifiable by the verification class induced by the neighborhood function r x (x-verifiable) iff there is a function γ σ : ( 2 U) n 2 2 U s.t. F : γ σ ( F x ) = σ(f). Moreover, σ is exactly x-verifiable iff σ is x-verifiable and there is no r y with r y r x such that σ is y-verifiable. Baumann, Linsbichler, Woltran, September 16, 2016 Verifiability of Argumentation Semantics 10

36 Verifiability Proposition Complete semantics is exactly + -verifiable. Baumann, Linsbichler, Woltran, September 16, 2016 Verifiability of Argumentation Semantics 11

37 Verifiability Proposition Complete semantics is exactly + -verifiable. Proof Verifiability: γ co ( F + ) = {S (S, S +, S ) F +, (S \ S + ) =, ( S, S +, S ) F + : S S ( S \ S + ) )} Baumann, Linsbichler, Woltran, September 16, 2016 Verifiability of Argumentation Semantics 11

38 Verifiability Proposition Complete semantics is exactly + -verifiable. Proof Verifiability: γ co ( F + ) = {S (S, S +, S ) F +, (S \ S + ) =, ( S, S +, S ) F + : S S ( S \ S + ) )} Exactness: +± : F 1 : a b F 1 : a b +± +± F 1 = {(,, ), ({a}, {a}, )} = F 1 co(f 1 ) = { } {{a}} = co(f 1 ) co is not +±-verifiable Baumann, Linsbichler, Woltran, September 16, 2016 Verifiability of Argumentation Semantics 11

39 Verifiability Proposition Complete semantics is exactly + -verifiable. Proof (ctd.) : F 2 : a b c F 2 : a b c ± : F 3 : a b F 3 : a ± : F 4 : a b F 4 : a + : F 5 : a b F 5 : a b b b : F 6 : a b F 6 : a b Baumann, Linsbichler, Woltran, September 16, 2016 Verifiability of Argumentation Semantics 12

40 Verifiability + : co + : ss, eg ±: gr, sad +: stb, stg : ad, pr, id ɛ: na Baumann, Linsbichler, Woltran, September 16, 2016 Verifiability of Argumentation Semantics 13

41 Verifiability Definition We call a semantics σ rational if self-loop-chains are irrelevant. That is, for every AF F it holds that σ(f) = σ(f l ), where F l = (A F, R F \ {(a, b) R F (a, a), (b, b) R F, a b}). Baumann, Linsbichler, Woltran, September 16, 2016 Verifiability of Argumentation Semantics 14

42 Verifiability Definition We call a semantics σ rational if self-loop-chains are irrelevant. That is, for every AF F it holds that σ(f) = σ(f l ), where F l = (A F, R F \ {(a, b) R F (a, a), (b, b) R F, a b}). Theorem Every semantics which is rational is exactly verifiable by a verification class induced by one of the neighborhood functions below. + +± + ± ± + ± ɛ Baumann, Linsbichler, Woltran, September 16, 2016 Verifiability of Argumentation Semantics 14

43 Strong Equivalence Definition Two AFs F and G are strongly equivalent w.r.t. semanticsσ (F σ E G) iff for all AFs H: σ(f H) = σ(g H) Baumann, Linsbichler, Woltran, September 16, 2016 Verifiability of Argumentation Semantics 15

44 Strong Equivalence Definition Two AFs F and G are strongly equivalent w.r.t. semanticsσ (F σ E G) iff for all AFs H: σ(f H) = σ(g H) syntactical criteria exist Example (stable semantics) stb-kernel: F k(stb) = (A, R \ {(a, b) a b, (a, a) R}). Theorem: F k(stb) = G k(stb) F and G are strongly equivalent. F : a b G : a b We have F k(stb) = G k(stb) = G. Thus, F and G are strong equivalent. Baumann, Linsbichler, Woltran, September 16, 2016 Verifiability of Argumentation Semantics 15

45 Strong Equivalence Definition (σ-kernel) Let F = (A, R). We define σ-kernels F k(σ) = ( A, R k(σ)) whereby 1 R k(stb) = R \ {(a, b) a b, (a, a) R}, 2 R k(ad) = R \ {(a, b) a b, (a, a) R, {(b, a), (b, b)} R }, 3 R k(gr) = R \ {(a, b) a b, (b, b) R, {(a, a), (b, a)} R }, 4 R k(co) = R \ {(a, b) a b, (a, a), (b, b) R}. 5 R k(na) = R {(a, b) a b, {(a, a), (b, a), (b, b)} R }. Baumann, Linsbichler, Woltran, September 16, 2016 Verifiability of Argumentation Semantics 16

46 Strong Equivalence Theorem Strong equivalence is characterizable through kernels (see below). F σ E G F k = G k dstgp dstbp dssp degp dadp dprp didp dgrp dsadp dcop dnap k(stb) k(stb) k(ad) k(ad) k(ad) k(ad) k(ad) k(gr) k(gr) k(co) k(na) Baumann, Linsbichler, Woltran, September 16, 2016 Verifiability of Argumentation Semantics 17

47 Intermediate Semantics stb and stg are both characterizable through k(stb). Does this also hold for arbitrary semantics σ with stb(f) σ(f) stg(f) for each AF F? (e.g. when obtained from SESAME [Besnard et al., 2016]) Baumann, Linsbichler, Woltran, September 16, 2016 Verifiability of Argumentation Semantics 18

48 Intermediate Semantics stb and stg are both characterizable through k(stb). Does this also hold for arbitrary semantics σ with stb(f) σ(f) stg(f) for each AF F? (e.g. when obtained from SESAME [Besnard et al., 2016]) Example Stagle semantics : S sta(f) S cf(f), S + F S F = A F and T cf (F) : S + F T+ F Baumann, Linsbichler, Woltran, September 16, 2016 Verifiability of Argumentation Semantics 18

49 Intermediate Semantics stb and stg are both characterizable through k(stb). Does this also hold for arbitrary semantics σ with stb(f) σ(f) stg(f) for each AF F? (e.g. when obtained from SESAME [Besnard et al., 2016]) Example Stagle semantics : S sta(f) S cf(f), S + F S F = A F and T cf (F) : S + F T+ F F : a b c stb(f) = sta(f) = {{b}} stg(f) = {{b}, {c}}. Baumann, Linsbichler, Woltran, September 16, 2016 Verifiability of Argumentation Semantics 18

50 Stagle semantics is not compatible with the stable kernel. Baumann, Linsbichler, Woltran, September 16, 2016 Verifiability of Argumentation Semantics 18 Intermediate Semantics stb and stg are both characterizable through k(stb). Does this also hold for arbitrary semantics σ with stb(f) σ(f) stg(f) for each AF F? (e.g. when obtained from SESAME [Besnard et al., 2016]) Example Stagle semantics : S sta(f) S cf(f), S + F S F = A F and T cf (F) : S + F T+ F F : a b c stb(f) = sta(f) = {{b}} stg(f) = {{b}, {c}}. F k(stb) : a b c sta ( F k(stb)) = {{b}, {c}} F sta E F k(stb), F k(stb) = ( F k(stb)) k(stb)

51 Intermediate Semantics Theorem For each semantics σ which is +-verifiable and stb-stg-intermediate, it holds that F k(stb) = G k(stb) F σ E G. Baumann, Linsbichler, Woltran, September 16, 2016 Verifiability of Argumentation Semantics 19

52 Intermediate Semantics Theorem For each semantics σ which is +-verifiable and stb-stg-intermediate, it holds that F k(stb) = G k(stb) F σ E G. Theorem For each semantics σ which is + -verifiable and ρ-ad-intermediate with ρ {ss, id, eg}, it holds that F k(ad) = G k(ad) F σ E G. Theorem For each semantics σ which is ±-verifiable and gr-sad-intermediate, it holds that F k(gr) = G k(gr) F σ E G. Baumann, Linsbichler, Woltran, September 16, 2016 Verifiability of Argumentation Semantics 19

53 Conclusion Summary: Hierarchy of verification classes Each rational semantics is exactly verifiable by a certain class Characterization of strong equivalence for intermediate semantics Future work: Semantics not captured by the approach, e.g. cf2 semantics [Baroni et al., 2005] Investigating labelling-based semantics [Caminada and Gabbay, 2009] Use classification as distance measure [Doutre and Mailly, 2016] Baumann, Linsbichler, Woltran, September 16, 2016 Verifiability of Argumentation Semantics 20

54 References I Baroni, P. and Giacomin, M. (2007). On principle-based evaluation of extension-based argumentation semantics. Artif. Intell., 171(10-15): Baroni, P., Giacomin, M., and Guida, G. (2005). SCC-Recursiveness: A general schema for argumentation semantics. Artif. Intell., 168(1-2): Baumann, R. (2016). Characterizing equivalence notions for labelling-based semantics. In Principles of Knowledge Representation and Reasoning: Proceedings of the 15th International Conference, pages Baumann, R. and Strass, H. (2016). An abstract logical approach to characterizing strong equivalence in logic-based knowledge representation formalisms. In Principles of Knowledge Representation and Reasoning: Proceedings of the 15th International Conference, pages Besnard, P., Doutre, S., Ho, V. H., and Longin, D. (2016). SESAME - A system for specifying semantics in abstract argumentation. In Proc. SAFA@COMMA, pages Baumann, Linsbichler, Woltran, September 16, 2016 Verifiability of Argumentation Semantics 21

55 References II Caminada, M. and Amgoud, L. (2007). On the evaluation of argumentation formalisms. Artif. Intell., pages Caminada, M. and Gabbay, D. M. (2009). A logical account of formal argumentation. Studia Logica, 93(2): Doutre, S. and Mailly, J.-G. (2016). Quantifying the difference between argumentation semantics. In Computational Models of Argument - Proceedings of COMMA 2016, volume 287 of Frontiers in Artificial Intelligence and Applications, pages IOS Press. Dung, P. M. (1995). On the acceptability of arguments and its fundamental role in nonmonotonic reasoning, logic programming and n-person games. Artif. Intell., 77(2): Dunne, P. E. and Bench-Capon, T. J. M. (2002). Coherence in finite argument systems. Artif. Intell., 141(1/2): Dvořák, W. and Woltran, S. (2010). Complexity of semi-stable and stage semantics in argumentation frameworks. Inf. Process. Lett., 110(11): Baumann, Linsbichler, Woltran, September 16, 2016 Verifiability of Argumentation Semantics 22

56 References III Lifschitz, V., Pearce, D., and Valverde, A. (2001). Strongly equivalent logic programs. ACM Transactions on Computational Logic, 2(4): Oikarinen, E. and Woltran, S. (2011). Characterizing strong equivalence for argumentation frameworks. Artif. Intell., 175(14-15): Truszczynski, M. (2006). Strong and uniform equivalence of nonmonotonic theories - an algebraic approach. Annals of Mathematics and Artificial Intelligence, 48(3-4): Turner, H. (2004). Strong equivalence for causal theories. In 7th International Conference on Logic Programming and Nonmonotonic Reasoning, Proceedings, volume 2923 of Lecture Notes in Computer Science, pages Springer. Baumann, Linsbichler, Woltran, September 16, 2016 Verifiability of Argumentation Semantics 23

57 Verifiability γ na( F ɛ A ) = {S S F, S is -maximal in F}; γ stg ( F + A ) = {S (S, S+ ) F +, S + is -maximal in {C + (C, C + ) F + }}; γ stb ( F + A ) = {S (S, S+ ) F +, S + = A}; γ ad ( F A ) = {S (S, S ) F, S = }; γ pr( F A ) = {S S γ ad( F A ), S is -maximal in γ ad( F A )}; + γ ss( F A ) = {S S γ ad ( F A ), S+ is -maximal in {C + (C, C +, C ) F +, C γ ad ( F A )}}; γ id ( F A ) = {S S is -maximal in {C C γ ad( F A ), C γ pr( F A )}}; + γ eg( F A ) = {S S is -maximal in {C C γ ad ( F A ), C + γ ss( F A )}}; ± γ sad ( F A ) = {S (S, S, S ± ) F ±, (S 0, S 0, S± 0 ),..., (Sn, S n, S n ± ) F ± : ± γ gr( F A ) = {S S γ sad ( ( = S 0 S n = S i {1,..., n} : S i S ± i 1 )}; ± F A ), ( S, S, S ± ) F ± : S S ( S \S ± ) )}. Baumann, Linsbichler, Woltran, September 16, 2016 Verifiability of Argumentation Semantics 24