Computational Aspects of Abstract Argumentation

Transcription

1 Computational Aspects of Abstract Argumentation PhD Defense, TU Wien (Vienna) Wolfgang Dvo ák supervised by Stefan Woltran Institute of Information Systems, Database and Articial Intelligence Group Vienna University of Technology April 11, 2012 Computational Aspects of Abstract Argumentation (PhD Defense) Slide 1

2 1. Prolog The Argumentation Process Steps Starting point: knowledge-base Form arguments Identify conicts Abstract from internal structure Resolve conicts Draw conclusions Computational Aspects of Abstract Argumentation (PhD Defense) Slide 2

3 1. Prolog The Argumentation Process Steps Starting point: knowledge-base Form arguments Identify conicts Abstract from internal structure Resolve conicts Draw conclusions Example = { x, x, x y, y, y} Computational Aspects of Abstract Argumentation (PhD Defense) Slide 2

4 1. Prolog The Argumentation Process Steps Starting point: knowledge-base Form arguments Identify conicts Abstract from internal structure Resolve conicts Draw conclusions Example = { x, x, x y, y, y} x x x y y y Computational Aspects of Abstract Argumentation (PhD Defense) Slide 2

5 1. Prolog The Argumentation Process Steps Starting point: knowledge-base Form arguments Identify conicts Abstract from internal structure Resolve conicts Draw conclusions Example = { x, x, x y, y, y} x x x y y y Computational Aspects of Abstract Argumentation (PhD Defense) Slide 2

6 1. Prolog The Argumentation Process Steps Starting point: knowledge-base Form arguments Identify conicts Abstract from internal structure Resolve conicts Draw conclusions Example = { x, x, x y, y, y} F : a e b c d Computational Aspects of Abstract Argumentation (PhD Defense) Slide 2

7 1. Prolog The Argumentation Process Steps Starting point: knowledge-base Form arguments Identify conicts Abstract from internal structure Resolve conicts Draw conclusions Example = { x, x, x y, y, y} F : a e b c d prf (F )={{ b, d}, { b, e}} Computational Aspects of Abstract Argumentation (PhD Defense) Slide 2

8 1. Prolog The Argumentation Process Steps Starting point: knowledge-base Form arguments Identify conicts Abstract from internal structure Resolve conicts Draw conclusions Example = { x, x, x y, y, y} prf (F )={{ b, d}, { b, e}} C S(F )={ x } Computational Aspects of Abstract Argumentation (PhD Defense) Slide 2

9 1. Prolog The Argumentation Process Remarks Main idea dates back to [Dung, 1995]; has then been rened by several authors (Prakken, Gordon, Caminada, etc.) Abstraction allows to compare several Knowledge Representation (KR) formalisms on a conceptual level Computational Aspects of Abstract Argumentation (PhD Defense) Slide 3

10 1. Prolog The Argumentation Process Remarks Main idea dates back to [Dung, 1995]; has then been rened by several authors (Prakken, Gordon, Caminada, etc.) Abstraction allows to compare several Knowledge Representation (KR) formalisms on a conceptual level Main Challenge All Steps in the argumentation process are, in general, intractable. This calls for: careful complexity analysis (identication of tractable fragments) re-use of established tools for implementations (reduction method) Computational Aspects of Abstract Argumentation (PhD Defense) Slide 3

11 1. Prolog Dung's Abstract Argumentation Frameworks a b c e d Main Properties Abstract from the concrete content of arguments and only consider the relation between them Semantics select subsets of arguments respecting certain criteria Simple, yet powerful, formalism Most active research area in the eld of argumentation. plethora of semantics Computational Aspects of Abstract Argumentation (PhD Defense) Slide 4

12 1. Prolog Topics of the thesis Complexity Analysis Complexity classication of standard reasoning tasks in abstract argumentation Towards Tractability Graph classes as tractable fragments Fixed-parameter tractability Intertranslatability of argumentation semantics Translations between semantics as an reduction approach within argumentation Computational Aspects of Abstract Argumentation (PhD Defense) Slide 5

13 2. Abstract Argumentation Dung's Abstract Argumentation Frameworks Denition An argumentation framework (AF) is a pair (A, R) where A is a set of arguments R A A is a relation representing the conicts (attacks) Example F=( {a,b,c,d,e}, {(a,b),(c,b),(c,d),(d,c),(d,e),(e,e)} ) a b c d e Computational Aspects of Abstract Argumentation (PhD Defense) Slide 6

14 2. Abstract Argumentation Basic Properties Conict-Free Sets Given an AF F = (A, R). A set S A is conict-free in F, if, for each a, b S, (a, b) / R. Computational Aspects of Abstract Argumentation (PhD Defense) Slide 7

15 2. Abstract Argumentation Basic Properties Conict-Free Sets Given an AF F = (A, R). A set S A is conict-free in F, if, for each a, b S, (a, b) / R. Example a b c d e cf (F ) = { {a, c}, Computational Aspects of Abstract Argumentation (PhD Defense) Slide 7

16 2. Abstract Argumentation Basic Properties Conict-Free Sets Given an AF F = (A, R). A set S A is conict-free in F, if, for each a, b S, (a, b) / R. Example a b c d e cf (F ) = { {a, c}, {a, d}, Computational Aspects of Abstract Argumentation (PhD Defense) Slide 7

17 2. Abstract Argumentation Basic Properties Conict-Free Sets Given an AF F = (A, R). A set S A is conict-free in F, if, for each a, b S, (a, b) / R. Example a b c d e cf (F ) = { {a, c}, {a, d}, {b, d}, Computational Aspects of Abstract Argumentation (PhD Defense) Slide 7

18 2. Abstract Argumentation Basic Properties Conict-Free Sets Given an AF F = (A, R). A set S A is conict-free in F, if, for each a, b S, (a, b) / R. Example a b c d e cf (F ) = { {a, c}, {a, d}, {b, d}, {a}, {b}, {c}, {d}, } Computational Aspects of Abstract Argumentation (PhD Defense) Slide 7

19 2. Abstract Argumentation Basic Properties Admissible Sets [Dung, 1995] Given an AF F = (A, R). A set S A is admissible in F, if S is conict-free in F each a S is defended by S in F a A is defended by S in F, if for each b A with (b, a) R, there exists a c S, such that (c, b) R. Computational Aspects of Abstract Argumentation (PhD Defense) Slide 8

20 2. Abstract Argumentation Basic Properties Admissible Sets [Dung, 1995] Given an AF F = (A, R). A set S A is admissible in F, if S is conict-free in F each a S is defended by S in F a A is defended by S in F, if for each b A with (b, a) R, there exists a c S, such that (c, b) R. Example a b c d e adm(f ) = { {a, c}, Computational Aspects of Abstract Argumentation (PhD Defense) Slide 8

21 2. Abstract Argumentation Basic Properties Admissible Sets [Dung, 1995] Given an AF F = (A, R). A set S A is admissible in F, if S is conict-free in F each a S is defended by S in F a A is defended by S in F, if for each b A with (b, a) R, there exists a c S, such that (c, b) R. Example a b c d e adm(f ) = { {a, c}, {a, d}, Computational Aspects of Abstract Argumentation (PhD Defense) Slide 8

22 2. Abstract Argumentation Basic Properties Admissible Sets [Dung, 1995] Given an AF F = (A, R). A set S A is admissible in F, if S is conict-free in F each a S is defended by S in F a A is defended by S in F, if for each b A with (b, a) R, there exists a c S, such that (c, b) R. Example a b c d e adm(f ) = { {a, c}, {a, d}, {b, d}, Computational Aspects of Abstract Argumentation (PhD Defense) Slide 8

23 2. Abstract Argumentation Basic Properties Admissible Sets [Dung, 1995] Given an AF F = (A, R). A set S A is admissible in F, if S is conict-free in F each a S is defended by S in F a A is defended by S in F, if for each b A with (b, a) R, there exists a c S, such that (c, b) R. Example a b c d e adm(f ) = { {a, c}, {a, d}, {b, d}, {a}, {b}, {c}, {d}, } Computational Aspects of Abstract Argumentation (PhD Defense) Slide 8

24 2. Abstract Argumentation 2.1. Argumentation Semantics Semantics Denition An extension-based semantics is a function σ mapping each AF F to a set of extensions σ(f) 2 A F. If for each F, σ(f) = 1 then we call σ a unique status semantics, otherwise multiple status semantics. We consider 9 semantics, namely: naive stable complete preferred stage grounded admissible resolution-based grounded semi-stable Computational Aspects of Abstract Argumentation (PhD Defense) Slide 9

25 2. Abstract Argumentation 2.1. Argumentation Semantics Semantics Grounded Extension [Dung, 1995] Given an AF F = (A, R). The unique grounded extension of F is dened as the outcome S of the following algorithm: 1 put each argument a A which is not attacked in F into S; if no such argument exists, return S; 2 remove from F all (new) arguments in S and all arguments attacked by them and continue with Step 1. Computational Aspects of Abstract Argumentation (PhD Defense) Slide 10

26 2. Abstract Argumentation 2.1. Argumentation Semantics Semantics Grounded Extension [Dung, 1995] Given an AF F = (A, R). The unique grounded extension of F is dened as the outcome S of the following algorithm: 1 put each argument a A which is not attacked in F into S; if no such argument exists, return S; 2 remove from F all (new) arguments in S and all arguments attacked by them and continue with Step 1. Example a b c d e grd(f ) = { {a}} Computational Aspects of Abstract Argumentation (PhD Defense) Slide 10

27 2. Abstract Argumentation 2.1. Argumentation Semantics Semantics Preferred Extensions [Dung, 1995] Given an AF F = (A, R). A set S A is a preferred extension of F, if S is admissible in F for each T A admissible in F, S T Computational Aspects of Abstract Argumentation (PhD Defense) Slide 11

28 2. Abstract Argumentation 2.1. Argumentation Semantics Semantics Preferred Extensions [Dung, 1995] Given an AF F = (A, R). A set S A is a preferred extension of F, if S is admissible in F for each T A admissible in F, S T Example a b c d e prf (F ) = { {a, c}, {a, d}, {a}, {c}, {d}, } Computational Aspects of Abstract Argumentation (PhD Defense) Slide 11

29 2. Abstract Argumentation 2.1. Argumentation Semantics Semantics Stable Extensions [Dung, 1995] Given an AF F = (A, R). A set S A is a stable extension of F, if S is conict-free in F for each a A \ S, there exists a b S, such that (b, a) R Computational Aspects of Abstract Argumentation (PhD Defense) Slide 12

30 2. Abstract Argumentation 2.1. Argumentation Semantics Semantics Stable Extensions [Dung, 1995] Given an AF F = (A, R). A set S A is a stable extension of F, if S is conict-free in F for each a A \ S, there exists a b S, such that (b, a) R Example a b c d e stb(f ) = { {a, c} Computational Aspects of Abstract Argumentation (PhD Defense) Slide 12

31 2. Abstract Argumentation 2.1. Argumentation Semantics Semantics Stable Extensions [Dung, 1995] Given an AF F = (A, R). A set S A is a stable extension of F, if S is conict-free in F for each a A \ S, there exists a b S, such that (b, a) R Example a b c d e stb(f ) = { {a, c}, {a, d}, Computational Aspects of Abstract Argumentation (PhD Defense) Slide 12

32 2. Abstract Argumentation 2.1. Argumentation Semantics Semantics Stable Extensions [Dung, 1995] Given an AF F = (A, R). A set S A is a stable extension of F, if S is conict-free in F for each a A \ S, there exists a b S, such that (b, a) R Example a b c d e stb(f ) = { {a, c}, {a, d}, {b, d}, Computational Aspects of Abstract Argumentation (PhD Defense) Slide 12

33 2. Abstract Argumentation 2.1. Argumentation Semantics Semantics Stable Extensions [Dung, 1995] Given an AF F = (A, R). A set S A is a stable extension of F, if S is conict-free in F for each a A \ S, there exists a b S, such that (b, a) R Example a b c d e stb(f ) = { {a, c}, {a, d}, {b, d}, {a}, {b}, {c}, {d},, } Computational Aspects of Abstract Argumentation (PhD Defense) Slide 12

34 2. Abstract Argumentation 2.1. Argumentation Semantics Semantics Semi-Stable Extensions [Caminada, 2006, Verheij, 1996] Given an AF F = (A, R). For a set S A, dene the range S + = S {a b S with (b, a) R}. A set S A is a semi-stable extension of F, if S is admissible in F for each T A admissible in F, S + T + Computational Aspects of Abstract Argumentation (PhD Defense) Slide 13

35 2. Abstract Argumentation 2.1. Argumentation Semantics Semantics Semi-Stable Extensions [Caminada, 2006, Verheij, 1996] Given an AF F = (A, R). For a set S A, dene the range S + = S {a b S with (b, a) R}. A set S A is a semi-stable extension of F, if S is admissible in F for each T A admissible in F, S + T + Example a b c d e sem(f ) = { {a, c}, {a, d}, {a}, {c}, {d}, } Computational Aspects of Abstract Argumentation (PhD Defense) Slide 13

36 2. Abstract Argumentation 2.1. Argumentation Semantics Semantics Some Relations For any AF F the following relations hold: 1 Each stable extension of F is admissible in F. 2 Each stable extension of F is also a preferred one. 3 Each semi-stable extension of F is also a preferred one. 4 Each stable extension of F is also a semi-stable one. stb(f ) sem(f ) prf (F ) adm(f ) cf (F ) Computational Aspects of Abstract Argumentation (PhD Defense) Slide 14

37 2. Abstract Argumentation 2.1. Argumentation Semantics Parametrised Ideal Semantics Generalising [Dung et al., 2007, Caminada, 2007] we dene: Denition Given an AF F = (A, R). A set S A is a ideal set w.r.t. base semantics σ of F, if I1. S adm(f) I2. S E σ(f) E We say that S is an ideal extension of F w.r.t. σ, if S is a -maximal ideal set (of F) w.r.t. σ. For typical base semantics there is a unique ideal extension. Computational Aspects of Abstract Argumentation (PhD Defense) Slide 15

38 3. Complexity Analysis Complexity Analysis Why doing Complexity Analysis? Complexity Theoretic View: To understand the Computational Costs that underlie a certain reasoning problem. Knowledge-Representation View: Measuring Expressivness of a formalism. Practitioners View: For applying the Reduction Approach, i.e. encoding a problem in other formalisms, the target formalism must be at least of the same complexity. Computational Aspects of Abstract Argumentation (PhD Defense) Slide 16

39 3. Complexity Analysis Decision Problems on AFs Credulous Acceptance Cred σ : Given AF F = (A, R) and a A; is a contained in at least one σ-extension of F? Skeptical Acceptance Skept σ : Given AF F = (A, R) and a A; is a contained in every σ-extension of F? If no extension exists then all arguments are skeptically accepted and no argument is credulously accepted. Computational Aspects of Abstract Argumentation (PhD Defense) Slide 17

40 3. Complexity Analysis Decision Problems on AFs Credulous Acceptance Cred σ : Given AF F = (A, R) and a A; is a contained in at least one σ-extension of F? Skeptical Acceptance Skept σ : Given AF F = (A, R) and a A; is a contained in every σ-extension of F? If no extension exists then all arguments are skeptically accepted and no argument is credulously accepted. Ideal Acceptance Ideal σ : Given AF F = (A, R) and a A; is a contained in the ideal extension (w.r.t. base-semantics σ) of F? Computational Aspects of Abstract Argumentation (PhD Defense) Slide 17

41 3. Complexity Analysis Further Decision Problems Verifying an extension Ver σ : Given AF F = (A, R) and S A; is S a σ-extension of F? Computational Aspects of Abstract Argumentation (PhD Defense) Slide 18

42 3. Complexity Analysis Further Decision Problems Verifying an extension Ver σ : Given AF F = (A, R) and S A; is S a σ-extension of F? Does there exist an extension? Exists σ : Given AF F = (A, R); Does there exist a σ-extension for F? Does there exist a nonempty extension? Exists σ : Does there exist a non-empty σ-extension for F? Computational Aspects of Abstract Argumentation (PhD Defense) Slide 18

43 3. Complexity Analysis Complexity Landscape (State-of-the-Art) σ Cred σ Skept σ Ideal σ Ver σ Exists σ Exists σ cf in P trivial? in P trivial in P naive in P in P? in P trivial in P grd in P in P? in P trivial in P stb NP-c conp-c? in P NP-c NP-c adm NP-c trivial? in P trivial NP-c com NP-c in P? in P trivial NP-c resgr NP-c conp-c? in P trivial in P prf NP-c Π P 2 -c in ΘP 2 conp-c trivial NP-c sem in Σ P 2 in Π P 2? conp-c trivial NP-c stg?????? Table: State-of-the art complexity landscape for abstract argumentation. Computational Aspects of Abstract Argumentation (PhD Defense) Slide 19

44 3. Complexity Analysis Complexity Analysis - Contributions We contribute in three directions: Exact complexity classications for semi-stable and stage semantics Complexity analysis for ideal reasoning Generic complexity results referring to the complexity of other reasoning tasks (membership and hardness results) Exact complexity classications for concrete base semantics P-completeness classication for tractable problems Computational Aspects of Abstract Argumentation (PhD Defense) Slide 20

45 3. Complexity Analysis 3.1. Complexity of semi-stable and stage semantics Complexity of semi-stable and stage semantics Theorem Cred sem is Σ P 2 -complete and Skept sem is ΠP 2 -complete. Computational Aspects of Abstract Argumentation (PhD Defense) Slide 21

46 3. Complexity Analysis 3.1. Complexity of semi-stable and stage semantics Complexity of semi-stable and stage semantics Theorem Cred sem is Σ P 2 -complete and Skept sem is ΠP 2 -complete. Hardness is via the following reduction: Given a QBF 2 formula Φ = Y ZC, we dene F Φ = (A, R), where A = {ϕ, ϕ, b} C Y Ȳ Y Ȳ Z Z R = {(c, ϕ) c C} {(ϕ, ϕ), ( ϕ, ϕ), (ϕ, b), (b, b)} {(x, x), ( x, x) x Y Z} {(y, y ), (ȳ, ȳ ), (y, y ), (ȳ, ȳ ) y Y } {(l, c) l C, c C}. One can show that Φ is valid i ϕ is skeptically accepted w.r.t. sem, i ϕ is not credulously accepted w.r.t. sem. Computational Aspects of Abstract Argumentation (PhD Defense) Slide 21

47 3. Complexity Analysis 3.1. Complexity of semi-stable and stage semantics Complexity of semi-stable and stage semantics Φ = y 1, y 2 z 3, z 4 (y 1 y 2 z 3 ) (ȳ 2 z 3 z 4 ) (ȳ 1 ȳ 2 z 4 ). ϕ ϕ b c 1 c 2 c 3 y 1 ȳ 1 y 2 ȳ 2 z 3 z 3 z 4 z 4 y 1 ȳ 1 y 2 ȳ 2 Computational Aspects of Abstract Argumentation (PhD Defense) Slide 22

48 3. Complexity Analysis 3.1. Complexity of semi-stable and stage semantics Complexity of semi-stable and stage semantics Φ = y 1, y 2 z 3, z 4 (y 1 y 2 z 3 ) (ȳ 2 z 3 z 4 ) (ȳ 1 ȳ 2 z 4 ). ϕ ϕ b c 1 c 2 c 3 y 1 ȳ 1 y 2 ȳ 2 z 3 z 3 z 4 z 4 y 1 ȳ 1 y 2 ȳ 2 true assignment τ: τ(y 1 ) = f, τ(y 2 ) = f, τ(z 3 ) = f, τ(z 4 ) = f Computational Aspects of Abstract Argumentation (PhD Defense) Slide 22

49 3. Complexity Analysis 3.1. Complexity of semi-stable and stage semantics Complexity of semi-stable and stage semantics Φ = y 1, y 2 z 3, z 4 (y 1 y 2 z 3 ) (ȳ 2 z 3 z 4 ) (ȳ 1 ȳ 2 z 4 ). ϕ ϕ b c 1 c 2 c 3 y 1 ȳ 1 y 2 ȳ 2 z 3 z 3 z 4 z 4 y 1 ȳ 1 y 2 ȳ 2 true assignment τ: τ(y 1 ) = f, τ(y 2 ) = f, τ(z 3 ) = f, τ(z 4 ) = f Computational Aspects of Abstract Argumentation (PhD Defense) Slide 22

50 3. Complexity Analysis 3.1. Complexity of semi-stable and stage semantics Complexity of semi-stable and stage semantics Φ = y 1, y 2 z 3, z 4 (y 1 y 2 z 3 ) (ȳ 2 z 3 z 4 ) (ȳ 1 ȳ 2 z 4 ). ϕ ϕ b c 1 c 2 c 3 y 1 ȳ 1 y 2 ȳ 2 z 3 z 3 z 4 z 4 y 1 ȳ 1 y 2 ȳ 2 true assignment τ: τ(y 1 ) = f, τ(y 2 ) = f, τ(z 3 ) = t, τ(z 4 ) = f Computational Aspects of Abstract Argumentation (PhD Defense) Slide 22

51 3. Complexity Analysis 3.1. Complexity of semi-stable and stage semantics Complexity of semi-stable and stage semantics Φ = y 1, y 2 z 3, z 4 (y 1 y 2 z 3 ) (ȳ 2 z 3 z 4 ) (ȳ 1 ȳ 2 z 4 ). ϕ ϕ b c 1 c 2 c 3 y 1 ȳ 1 y 2 ȳ 2 z 3 z 3 z 4 z 4 y 1 ȳ 1 y 2 ȳ 2 true assignment τ: τ(y 1 ) = t, τ(y 2 ) = f, τ(z 3 ) = t, τ(z 4 ) = f Computational Aspects of Abstract Argumentation (PhD Defense) Slide 22

52 3. Complexity Analysis 3.1. Complexity of semi-stable and stage semantics Complexity of semi-stable and stage semantics Φ = y 1, y 2 z 3, z 4 (y 1 y 2 z 3 ) (ȳ 2 z 3 z 4 ) (ȳ 1 ȳ 2 z 4 ). ϕ ϕ b c 1 c 2 c 3 y 1 ȳ 1 y 2 ȳ 2 z 3 z 3 z 4 z 4 y 1 ȳ 1 y 2 ȳ 2 true assignment τ: τ(y 1 ) = f, τ(y 2 ) = f, τ(z 3 ) = f, τ(z 4 ) = f Computational Aspects of Abstract Argumentation (PhD Defense) Slide 22

53 3. Complexity Analysis 3.1. Complexity of semi-stable and stage semantics Complexity of semi-stable and stage semantics Φ = y 1, y 2 z 3, z 4 (y 1 y 2 z 3 ) (ȳ 2 z 3 z 4 ) (ȳ 1 ȳ 2 z 4 ). ϕ ϕ b c 1 c 2 c 3 y 1 ȳ 1 y 2 ȳ 2 z 3 z 3 z 4 z 4 y 1 ȳ 1 y 2 ȳ 2 true assignment τ: τ(y 1 ) = f, τ(y 2 ) = f, τ(z 3 ) = f, τ(z 4 ) = f Computational Aspects of Abstract Argumentation (PhD Defense) Slide 22

54 3. Complexity Analysis 3.1. Complexity of semi-stable and stage semantics Complexity of semi-stable and stage semantics Φ = y 1, y 2 z 3, z 4 (y 1 y 2 z 3 ) (ȳ 2 z 3 z 4 ) (ȳ 1 ȳ 2 z 4 ). ϕ ϕ b c 1 c 2 c 3 y 1 ȳ 1 y 2 ȳ 2 z 3 z 3 z 4 z 4 y 1 ȳ 1 y 2 ȳ 2 true assignment τ: τ(y 1 ) = f, τ(y 2 ) = f, τ(z 3 ) = t, τ(z 4 ) = f Computational Aspects of Abstract Argumentation (PhD Defense) Slide 22

55 3. Complexity Analysis 3.1. Complexity of semi-stable and stage semantics Complexity of semi-stable and stage semantics Φ = y 1, y 2 z 3, z 4 (y 1 y 2 z 3 ) (ȳ 2 z 3 z 4 ) (ȳ 1 ȳ 2 z 4 ). ϕ ϕ b c 1 c 2 c 3 y 1 ȳ 1 y 2 ȳ 2 z 3 z 3 z 4 z 4 y 1 ȳ 1 y 2 ȳ 2 true assignment τ: τ(y 1 ) = f, τ(y 2 ) = f, τ(z 3 ) = t, τ(z 4 ) = f Computational Aspects of Abstract Argumentation (PhD Defense) Slide 22

56 3. Complexity Analysis 3.1. Complexity of semi-stable and stage semantics Complexity of semi-stable and stage semantics Φ = y 1, y 2 z 3, z 4 (y 1 y 2 z 3 ) (ȳ 2 z 3 z 4 ) (ȳ 1 ȳ 2 z 4 ). ϕ ϕ b c 1 c 2 c 3 y 1 ȳ 1 y 2 ȳ 2 z 3 z 3 z 4 z 4 y 1 ȳ 1 y 2 ȳ 2 true assignment τ: τ(y 1 ) = f, τ(y 2 ) = f, τ(z 3 ) = t, τ(z 4 ) = f Computational Aspects of Abstract Argumentation (PhD Defense) Slide 22

57 3. Complexity Analysis 3.2. Overview Complexity Landscape σ Cred σ Skept σ Ideal σ Ver σ Exists σ Exists σ cf in L trivial trivial in L trivial in L naive in L in L P-c in L trivial in L grd P-c P-c P-c P-c trivial in L stb NP-c conp-c D P -c in L NP-c NP-c adm NP-c trivial trivial in L trivial NP-c com NP-c P-c P-c in L trivial NP-c resgr NP-c conp-c conp-c P-c trivial in P prf NP-c Π P 2 -c in ΘP 2 conp-c trivial NP-c sem Σ P-c 2 ΠP-c 2 ΠP 2 -c conp-c trivial NP-c stg Σ P-c 2 ΠP-c 2 ΠP 2 -c conp-c trivial in L Table: Complexity of abstract argumentation. Computational Aspects of Abstract Argumentation (PhD Defense) Slide 23

58 4. Towards Tractability Towards Tractability Tractability for Abstract Argumentation Increasing interest for reasoning in argumentation frameworks (AFs). Many reasoning tasks are computationally intractable. As AFs can be considered as graphs, there are several graph classes where some in general hard problems have been shown to be tractable (Tractable Fragments) there is broad range of graph parameters we can consider to identify tractable fragments (Fixed-Parameter Tractability) Computational Aspects of Abstract Argumentation (PhD Defense) Slide 24

59 4. Towards Tractability 4.1. Tractable Fragments Tractable Fragments We study four tractable fragments proposed by the literature: acyclic AFs [Dung, 1995] AFs without even length cycles (noeven) [Dunne and Bench-Capon, 2001] symmetric AFs [Coste-Marquis et al., 2005] bipartite AFs [Dunne, 2007] We complement existing results by generalising them to all semantics under our considerations, classifying them w.r.t. P-completeness, solving an open problem concerning resolution-based grounded semantics and bipartite AFs. Computational Aspects of Abstract Argumentation (PhD Defense) Slide 25

60 4. Towards Tractability 4.1. Tractable Fragments Tractable Fragments The P-hardness for acyclic, noeven, and bipartite is by the following: Theorem Cred grd is P-complete even for acyclic bipartite AFs. Computational Aspects of Abstract Argumentation (PhD Defense) Slide 26

61 4. Towards Tractability 4.1. Tractable Fragments Tractable Fragments The P-hardness for acyclic, noeven, and bipartite is by the following: Theorem Cred grd is P-complete even for acyclic bipartite AFs. Hardness is by a reduction from the Mon. Circuit Value Problem (β, a) x y a x x ȳ y 1 2 Monotone Boolean Circuit β AF F β,a, with a(x) = 0, a(y) = 1 Computational Aspects of Abstract Argumentation (PhD Defense) Slide 26

62 4. Towards Tractability 4.2. Fixed-Parameter Tractability Fixed-Parameter Tractability Often computational costs primarily depend on some problem parameters rather than on the mere size of the instances. Many hard problems become tractable if some problem parameter is xed or bounded by a xed constant. In the arena of graphs important parameters are tree-width and clique-width. They have served as the key to many xed-parameter tractability (FPT) results. We are looking for algorithms with a worst case runtime that might be exponential in the parameter but is polynomial in the size of the instance. Computational Aspects of Abstract Argumentation (PhD Defense) Slide 27

63 4. Towards Tractability 4.2. Fixed-Parameter Tractability Fixed-Parameter Tractability Positive Results: We show FPT results for the parameters tree-width and clique-width via meta-theorems by Courcelle (1987) and Courcelle, Makowsky & Rotics (2000), and MSO encodings of the argumentation semantics. Negative Results: We show that typical reasoning tasks remain intractable if we bound the parameter cycle-rank. We extend this result to the parameters directed path-width, DAG-width, Kelly-width, and directed tree-width. Computational Aspects of Abstract Argumentation (PhD Defense) Slide 28

64 4. Towards Tractability 4.2. Fixed-Parameter Tractability Negative Results Denition An AF F = (A, R), has cycle rank 0 (cr(f ) = 0) i F is acyclic, and cr(f ) 1 i each strongly connected component of F can be made acyclic by removing one argument. Computational Aspects of Abstract Argumentation (PhD Defense) Slide 29

65 4. Towards Tractability 4.2. Fixed-Parameter Tractability Negative Results Denition An AF F = (A, R), has cycle rank 0 (cr(f ) = 0) i F is acyclic, and cr(f ) 1 i each strongly connected component of F can be made acyclic by removing one argument. Theorem When restricted to AFs which have a cycle-rank of 1 1 Cred sem remains Σ P 2 -hard, and 2 Skept sem remains Π P 2 -hard. Computational Aspects of Abstract Argumentation (PhD Defense) Slide 29

66 4. Towards Tractability 4.2. Fixed-Parameter Tractability Negative Results Proof. Recall the reduction from the hardness proof: ϕ ϕ b c 1 c 2 c 3 y 1 ȳ 1 y 2 ȳ 2 z 3 z 3 z 4 z 4 y 1 ȳ 1 y 2 ȳ 2 every framework of the form F Φ has cycle-rank 1. Computational Aspects of Abstract Argumentation (PhD Defense) Slide 30

67 4. Towards Tractability 4.3. Overview Tractability Results stb adm com resgr prf sem stg acyclic noeven bipartite symmetric bounded tree-width bounded clique-width bounded cycle-rank bounded directed path-width bounded Kelly-width bounded DAG-width bounded directed tree-width Computational Aspects of Abstract Argumentation (PhD Defense) Slide 31

68 5. Intertranslatability of Argumentation Semantics Intertranslatability of Argumentation Semantics Why consider translations between Argumentation Semantics? Plethora of Argumentation Semantics Reduction approach within argumentation: Given a translation for semantics σ to semantics σ we can reuse sophisticated solver for σ for semantics σ. AF F Tr Translation (F) Solver σ (Tr (F)) σ(f) for σ σ for σ Filter Figure: Generalising Argumentation Systems via Translations Computational Aspects of Abstract Argumentation (PhD Defense) Slide 32

69 5. Intertranslatability of Argumentation Semantics Translations Denition A Translation Tr is a function mapping (nite) AFs to (nite) AFs. Computational Aspects of Abstract Argumentation (PhD Defense) Slide 33

70 5. Intertranslatability of Argumentation Semantics Translations Denition A Translation Tr is a function mapping (nite) AFs to (nite) AFs. We want translations to satisfy certain properties: Basic Properties of a Translation Tr ecient: for every AF F, Tr (F ) can be computed using logarithmic space wrt. to F Computational Aspects of Abstract Argumentation (PhD Defense) Slide 33

71 5. Intertranslatability of Argumentation Semantics Translations Next we connect translations with semantics. Levels of Faithfulness (for semantics σ, σ ) exact: for every AF F, σ(f ) = σ (Tr (F )) faithful: for every AF F, σ(f ) = {E A F E σ (Tr (F ))} and σ(f ) = σ (Tr (F )). Computational Aspects of Abstract Argumentation (PhD Defense) Slide 34

72 5. Intertranslatability of Argumentation Semantics Translations Next we connect translations with semantics. Levels of Faithfulness (for semantics σ, σ ) exact: for every AF F, σ(f ) = σ (Tr (F )) faithful: for every AF F, σ(f ) = {E A F E σ (Tr (F ))} and σ(f ) = σ (Tr (F )). AF F Translation Tr (F) Solver σ (Tr (F)) = σ(f) for σ σ for σ Computational Aspects of Abstract Argumentation (PhD Defense) Slide 34

73 5. Intertranslatability of Argumentation Semantics Translations Next we connect translations with semantics. Levels of Faithfulness (for semantics σ, σ ) exact: for every AF F, σ(f ) = σ (Tr (F )) faithful: for every AF F, σ(f ) = {E A F E σ (Tr (F ))} and σ(f ) = σ (Tr (F )). AF F Tr Translation (F) Solver σ (Tr (F)) σ(f) for σ σ for σ E A F Computational Aspects of Abstract Argumentation (PhD Defense) Slide 34

74 5. Intertranslatability of Argumentation Semantics 5.1. Translations for Argumentations Semantics Example Translation 1 Denition For AF F, let Tr 1 (F ) = (A, R ) where A = A F A F and R = R F {(a, a ), (a, a), (a, a ) a A F }, with A = F {a a A F }. Example a b c d e a b c d e Result: Tr 1 is an exact translation for prf sem. Computational Aspects of Abstract Argumentation (PhD Defense) Slide 35

75 5. Intertranslatability of Argumentation Semantics 5.1. Translations for Argumentations Semantics Example Translation 2 Denition For AF F, Tr 6 (F ) = (A, R ) where A = A F Ā F R F and R = R F {(a, ā), (ā, a) a A F } {(r, r) r R F } {(ā, r) r = (y, a) R F } {(a, r) r = (z, y) R F, (a, z) R F }. Example ā b c d ē a b c d e (a, b) (c, b) (d, c) (c, d) (d, e) (e, e) Result: Tr 6 is a faithful translation for adm stb. Computational Aspects of Abstract Argumentation (PhD Defense) Slide 36

76 5. Intertranslatability of Argumentation Semantics 5.2. Impossibility Results Impossibility Results Proposition There is no exact translation for adm σ with σ {stb, prf, sem} com adm Computational Aspects of Abstract Argumentation (PhD Defense) Slide 37

77 5. Intertranslatability of Argumentation Semantics 5.2. Impossibility Results Impossibility Results Proposition There is no exact translation for adm σ with σ {stb, prf, sem} com adm Proposition There is no ecient faithful translation for sem σ, σ {adm, stb}, unless Σ P 2 = NP. Follows from complexity results. Computational Aspects of Abstract Argumentation (PhD Defense) Slide 37

78 5. Intertranslatability of Argumentation Semantics 5.3. Overview Hierarchies of intertranslatability semi-stable semi-stable preferred stage preferred stage stable admissible complete admissible, complete, stable grounded Exact Intertranslatability grounded Faithful Intertranslatability Computational Aspects of Abstract Argumentation (PhD Defense) Slide 38

79 6. Summary Summary We complemented existing Complexity Analysis by exact classications for semi-stable and stage semantics our studies on ideal reasoning P-completeness classications Towards tractable instances we studied Tractable Fragments as well as Fixed-Parameter Tractability. We complemented studies of Tractable Fragments Fixed-Parameter Tractability results for tree-width and clique-width. By the Intertranslatability of semantics we applied the reduction approach within abstract argumentation presenting translations between argumentation semantics negative results showing that certain translations are impossible Computational Aspects of Abstract Argumentation (PhD Defense) Slide 39

80 6. Summary Publications Complexity Analysis: Dvo ák, W. and Woltran, S. (2010). Complexity of semi-stable and stage semantics in argumentation frameworks. Inf. Process. Lett., 110(11): Dvo ák, W., Dunne, P. E., and Woltran, S. (2011). Parametric properties of ideal semantics. IJCAI 2011 Towards Tractability: Dvo ák, W., Pichler, R., and Woltran, S. (2012). Towards xed-parameter tractable algorithms for abstract argumentation. Articial Intelligence, 186(0):1 37. Dvo ák, W., Ordyniak, S., and Szeider, S. (2012). Augmenting tractable fragments of abstract argumentation. Articial Intelligence, in press. Intertranslatability: Dvo ák, W. and Woltran, S. (2011). On the intertranslatability of argumentation semantics. J. Artif. Intell. Res. (JAIR), 41: Computational Aspects of Abstract Argumentation (PhD Defense) Slide 40

81 7. Bibliography Bibliography I Caminada, M. (2006). Semi-stable semantics. In Dunne, P. E. and Bench-Capon, T. J. M., editors, Proceedings of the 1st Conference on Computational Models of Argument (COMMA 2006), volume 144 of Frontiers in Articial Intelligence and Applications, pages IOS Press. Caminada, M. (2007). Comparing two unique extension semantics for formal argumentation: ideal and eager. In Proceedings of the 19th Belgian-Dutch Conference on Articial Intelligence (BNAIC 2007), pages Computational Aspects of Abstract Argumentation (PhD Defense) Slide 41

82 7. Bibliography Bibliography II Coste-Marquis, S., Devred, C., and Marquis, P. (2005). Symmetric argumentation frameworks. In Godo, L., editor, Proceedings of the 8th European Conference on Symbolic and Quantitative Approaches to Reasoning with Uncertainty (ECSQARU 2005), volume 3571 of Lecture Notes in Computer Science, pages Springer. 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): Dung, P. M., Mancarella, P., and Toni, F. (2007). Computing ideal sceptical argumentation. Artif. Intell., 171(10-15): Computational Aspects of Abstract Argumentation (PhD Defense) Slide 42

83 7. Bibliography Bibliography III Dunne, P. E. (2007). Computational properties of argument systems satisfying graph-theoretic constraints. Artif. Intell., 171(10-15): Dunne, P. E. and Bench-Capon, T. J. M. (2001). Complexity and combinatorial properties of argument systems. Technical report, Dept. of Computer Science, University of Liverpool. Dvo ák, W., Dunne, P. E., and Woltran, S. (2011). Parametric properties of ideal semantics. In Walsh, T., editor, IJCAI 2011, Proceedings of the 22nd International Joint Conference on Articial Intelligence, Barcelona, Catalonia, Spain, July 16-22, 2011, pages IJCAI/AAAI. Dvo ák, W., Ordyniak, S., and Szeider, S. (2012a). Augmenting tractable fragments of abstract argumentation. Articial Intelligence, (0):. Computational Aspects of Abstract Argumentation (PhD Defense) Slide 43

84 7. Bibliography Bibliography IV Dvo ák, W., Pichler, R., and Woltran, S. (2012b). Towards xed-parameter tractable algorithms for abstract argumentation. Articial Intelligence, 186(0):1 37. Dvo ák, W. and Woltran, S. (2010). Complexity of semi-stable and stage semantics in argumentation frameworks. Inf. Process. Lett., 110(11): Dvo ák, W. and Woltran, S. (2011). On the intertranslatability of argumentation semantics. J. Artif. Intell. Res. (JAIR), 41: Computational Aspects of Abstract Argumentation (PhD Defense) Slide 44

85 7. Bibliography Bibliography V Verheij, B. (1996). Two approaches to dialectical argumentation: admissible sets and argumentation stages. In Meyer, J. and van der Gaag, L., editors, Proceedings of the 8th Dutch Conference on Articial Intelligence (NAIC'96), pages Computational Aspects of Abstract Argumentation (PhD Defense) Slide 45