Argumentation Theory and Modal Logic

Transcription

1 Argumentation Theory and Modal Logic Davide Grossi ILLC, University of Amsterdam

2 Preface Argumentation in a nutshell

3 Arguing

4 Arguing The Economist: Mr. Berlusconi is unfit to lead Italy because His election as prime minister would perpetuate Italy s bad old ways

5 Arguing The Economist: Mr. Berlusconi is unfit to lead Italy because His election as prime minister would perpetuate Italy s bad old ways Mr. Berlusconi: Berlusconi is the fittest to lead Italy because: Only Napoleon did more than I have done

6 Arguing e The Economist: Mr. Berlusconi is unfit to lead Italy because His election as prime minister would perpetuate Italy s bad old ways b Mr. Berlusconi: Berlusconi is the fittest to lead Italy because: Only Napoleon did more than I have done

7 Arguing e b

8 Abstract argumentation (i) P. M. Dung (1995) "On the Acceptability of Arguments and Its Fundamental Role in Argumentation, n-persons games, and Logic Programming", Artificial Intelligence e b

9 Abstract argumentation (i) P. M. Dung (1995) "On the Acceptability of Arguments and Its Fundamental Role in Argumentation, n-persons games, and Logic Programming", Artificial Intelligence e b (A, )

10 Abstract argumentation (i) P. M. Dung (1995) "On the Acceptability of Arguments and Its Fundamental Role in Argumentation, n-persons games, and Logic Programming", Artificial Intelligence e b (A, ) Abstract argumentation is about arguments (points) and attacks (relations)

11 Abstract argumentation (i) P. M. Dung (1995) "On the Acceptability of Arguments and Its Fundamental Role in Argumentation, n-persons games, and Logic Programming", Artificial Intelligence e b (A, ) Abstract argumentation is about arguments (points) and attacks (relations) Given an argumentation framework, which (sets of) arguments should be considered justified, or acceptable?

12 Abstract argumentation (i) P. M. Dung (1995) "On the Acceptability of Arguments and Its Fundamental Role in Argumentation, n-persons games, and Logic Programming", Artificial Intelligence e b j Abstract argumentation is about arguments (points) and attacks (relations) Given an argumentation framework, which (sets of) arguments should be considered justified, or acceptable?

13 Abstract argumentation (ii) P. M. Dung (1995) "On the Acceptability of Arguments and Its Fundamental Role in Argumentation, n-persons games, and Logic Programming", Artificial Intelligence e b j X conflict-free in A iff a, b X s.t. a b c A characteristic function of A iff c A : 2 A 2 A s.t. c A (X) = {a b : [b a : c X : c b]}

14 Abstract argumentation (ii) P. M. Dung (1995) "On the Acceptability of Arguments and Its Fundamental Role in Argumentation, n-persons games, and Logic Programming", Artificial Intelligence e b j

15 Abstract argumentation (ii) P. M. Dung (1995) "On the Acceptability of Arguments and Its Fundamental Role in Argumentation, n-persons games, and Logic Programming", Artificial Intelligence e b j X complete extension of A iff X is a conflict-free fixpoint of c A X stable extension of A iff X is a complete extension of A iff X = {a A b X : b a} X grounded extension of A iff X is the least fixpoint of c A X preferred extension of A iff X is a maximal complete extension of A

16 Abstract argumentation (ii) P. M. Dung (1995) "On the Acceptability of Arguments and Its Fundamental Role in Argumentation, n-persons games, and Logic Programming", Artificial Intelligence e b X complete extension of A iff X is a conflict-free fixpoint of c A X stable extension of A iff X is a complete extension of A iff X = {a A b X : b a} X grounded extension of A iff X is the least fixpoint of c A X preferred extension of A iff X is a maximal complete extension of A

17 Outline

18 Outline PART I: Dung Frameworks = Kripke Frames

19 Outline PART I: Dung Frameworks = Kripke Frames PART II: Argumentation in Modal Logic

20 Outline PART I: Dung Frameworks = Kripke Frames PART II: Argumentation in Modal Logic Axiomatizations, completeness, complexity

21 Outline PART I: Dung Frameworks = Kripke Frames PART II: Argumentation in Modal Logic Axiomatizations, completeness, complexity PART III: Dialogue Games via Semantic Games

22 Outline PART I: Dung Frameworks = Kripke Frames PART II: Argumentation in Modal Logic Axiomatizations, completeness, complexity PART III: Dialogue Games via Semantic Games Model-checking games

23 Outline PART I: Dung Frameworks = Kripke Frames PART II: Argumentation in Modal Logic Axiomatizations, completeness, complexity PART III: Dialogue Games via Semantic Games Model-checking games PART IV: When are two arguments the same?

24 Outline PART I: Dung Frameworks = Kripke Frames PART II: Argumentation in Modal Logic Axiomatizations, completeness, complexity PART III: Dialogue Games via Semantic Games Model-checking games PART IV: When are two arguments the same? Bisimulation, bisimulation games

25 Why a logic for argumentation?

26 Why a logic for argumentation? 1. Systematize argumentation theory

27 Why a logic for argumentation? 1. Systematize argumentation theory 2. Import techniques (e.g., calculi, logical games)

28 Why a logic for argumentation? 1. Systematize argumentation theory 2. Import techniques (e.g., calculi, logical games) 3. Import results (e.g., completeness, complexity, adequacy)

29 Why a logic for argumentation? 1. Systematize argumentation theory 2. Import techniques (e.g., calculi, logical games) 3. Import results (e.g., completeness, complexity, adequacy)... for free!

30 Part I Dung Frameworks = Kripke Frames

31 ... just a relational structure (i) A = (A, ) State = Argument Accessibility relation = Inverse of attack

32 ... just a relational structure (i) A = (A, ) M, a = ϕ iff b A : a 1 b and M, b = ϕ

33 Dung Fr. + Labellings = Kripke Models M = (A, I) States = Arguments Accessibility relation = Inverse of attack Valuation = Function from a vocabulary P to sets of arguments

34 Part II Argumentation in Modal Disguise

35 K + Global modality (i) L KU : ϕ ::= p ϕ ϕ ϕ ϕ U ϕ Definition 2 (Satisfaction for L KU in argumentation models) Let ϕ L KU. The satisfaction of ϕ by a pointed argumentation model (M, a) is inductively defined as follows (Boolean clauses are omitted): M, a = ϕ iff b A : (a, b) 1 and M, b = ϕ M, a = U ϕ iff b A : M, b = ϕ

36 K + Global modality (ii) The logic K U is axiomatized as follows: (Prop) propositional tautologies (K) [i](ϕ 1 ϕ 2 ) ([i]ϕ 1 [i]ϕ 2 ) (T) [U]ϕ ϕ (4) [U]ϕ [U][U]ϕ (5) [U]ϕ [U] [U]ϕ (Incl) [U]ϕ [i]ϕ (Dual) i ϕ [i] ϕ with i {, U}. This axiomatics is sound and strongly complete w.r.t. the class of argumentation frameworks under the given semantics

37 K + Global modality (iii) We list the following known results, which are relevant for our purposes. The complexity of deciding whether a formula of L KU is satisfiable is EXPcomplete [Hemaspaandra, 1996]. The complexity of checking whether a formula of L KU is satisfied by a pointed model M is P-complete [Graedel and Otto, 1999]. If we can express extensions as modal formulae in this logic we can import these results for free to argumentation theory.

38 Doing argumentation in Modal Logic (i) Acc(ϕ, ψ) := [U](ϕ [ ] ψ) CFree(ϕ) := [U](ϕ ϕ) Adm(ϕ) := [U](ϕ ([ ] ϕ [ ] ϕ)) Complete(ϕ) := [U]((ϕ [ ] ϕ) (ϕ [ ] ϕ)) Stable(ϕ) := [U](ϕ ϕ) Simple modal formulae express some of the main argumentation-theoretic notions

39 Doing argumentation in Modal Logic (i) Acc(ϕ, ψ) := [U](ϕ [ ] ψ) CFree(ϕ) := [U](ϕ ϕ) Adm(ϕ) := [U](ϕ ([ ] ϕ [ ] ϕ)) Complete(ϕ) := [U]((ϕ [ ] ϕ) (ϕ [ ] ϕ)) Stable(ϕ) := [U](ϕ ϕ) Simple modal formulae express some of the main argumentation-theoretic notions

40 Doing argumentation in Modal Logic (i) Acc(ϕ, ψ) := [U](ϕ [ ] ψ) CFree(ϕ) := [U](ϕ ϕ) Adm(ϕ) := [U](ϕ ([ ] ϕ [ ] ϕ)) Complete(ϕ) := [U]((ϕ [ ] ϕ) (ϕ [ ] ϕ)) Stable(ϕ) := [U](ϕ ϕ) Simple modal formulae express some of the main argumentation-theoretic notions

41 Doing argumentation in Modal Logic (ii) Theorem 1 (Fundamental Lemma) The following formula is a theorem of K U : Adm(ϕ) Acc(ψ ξ, ϕ) Adm(ϕ ψ) Acc(ξ, ϕ ψ) We can state theorems of argumentation as formulae...

42 Doing argumentation in Modal Logic (iii) 1. ((α γ) (β γ)) (α β γ) Prop 2. ([U](α γ) [U](β γ)) [U](α β γ) 2, N, K, MP 3. ([U](ϕ [ ] ϕ) [U](ψ [ ] ϕ)) [U](ϕ ψ [ ] ϕ) Instance of 3 4. [ ] ϕ [ ] (ϕ ψ) Prop, K, N 5. ([U](ϕ [ ] ϕ) [U](ψ [ ] ϕ)) [U](ϕ ψ [ ] ϕ ψ) 4, Prop, K, N 6. Acc(ϕ, ϕ) Acc(ψ, ϕ) Acc(ϕ ψ, ϕ ψ) 5, definition... and prove them via formal derivations!

43 What about Grounded Extension? This is known to be the least fixpoint of the characteristic function of an argumentation framework The characteristic function corresponds, in modal terms, to the operator: [ ] So the grounded extension of an argumentation framework is just the smallest proposition p which is a fixed point of the characteristic function, i.e., the smallest p s.t.:

44 What about Grounded Extension? This is known to be the least fixpoint of the characteristic function of an argumentation framework The characteristic function corresponds, in modal terms, to the operator: [ ] So the grounded extension of an argumentation framework is just the smallest proposition p which is a fixed point of the characteristic function, i.e., the smallest p s.t.: p [ ] p

45 mu-calculus (i) L Kµ : ϕ ::= p ϕ ϕ ϕ ϕ µp.ϕ(p) Definition 3 (Satisfaction for L Kµ in argumentation models) Let ϕ L Kµ. The satisfaction of ϕ by a pointed argumentation model (M, a) is inductively defined as follows: M, a = µp.ϕ(p) iff a {X 2 A ϕ M[p:=X] X} The mu operator allows us to express the least fixpoint of a formula viewed as set-transformer So the grounded extension of an argumentation framework is denoted by the formula:

46 mu-calculus (i) L Kµ : ϕ ::= p ϕ ϕ ϕ ϕ µp.ϕ(p) Definition 3 (Satisfaction for L Kµ in argumentation models) Let ϕ L Kµ. The satisfaction of ϕ by a pointed argumentation model (M, a) is inductively defined as follows: M, a = µp.ϕ(p) iff a {X 2 A ϕ M[p:=X] X} The mu operator allows us to express the least fixpoint of a formula viewed as set-transformer So the grounded extension of an argumentation framework is denoted by the formula: µp.[ ] p

47 mu-calculus (ii) (Prop) propositional schemata (K) [ ](ϕ 1 ϕ 2 ) ([ ]ϕ 1 [ ]ϕ 2 ) (Fixpoint) ϕ(µp.ϕ(p)) µp.ϕ(p) (MP) if ϕ 1 ϕ 2 and ϕ 1 then ϕ 2 (N) if ϕ then [ ]ϕ (Least) if ϕ 1 (ϕ 2 ) ϕ 2 then µp.ϕ 1 (p) ϕ 2 This axiomatics is sound and complete for argumentation frameworks [Walukiewicz, 2000]

48 mu-calculus (iii) We list some relevant known results. The satisfiability problem of K µ is decidable [Streett, 1989]. The complexity of the model-checking problem for K µ is known to be in NP co-np [Graedel, 1999], however, it is still an open question whether it is in P. The complexity of the model-checking problem for a formula of size m and alternation depth d on a system of size n is O(m n d+1 ) [Emerson, 1986].

49 mu-calculus (iii) We list some relevant known results. The satisfiability problem of K µ is decidable [Streett, 1989]. The complexity of the model-checking problem for K µ is known to be in NP co-np [Graedel, 1999], however, it is still an open question whether it is in P. The complexity of the model-checking problem for a formula of size m and alternation depth d on a system of size n is O(m n d+1 ) [Emerson, 1986]. We can tractably model-check grounded extensions!

50 What about Preferred Extensions? ST x (Compl(p)) q((st x (Compl(q)) p q)

51 What about Preferred Extensions? ST x (Compl(p)) q((st x (Compl(q)) p q) They are the maximal, conflict-free, post fixed points of the characteristic function of an argumentation framework

52 What about Preferred Extensions? ST x (Compl(p)) q((st x (Compl(q)) p q) They are the maximal, conflict-free, post fixed points of the characteristic function of an argumentation framework However this formula turns out to be invariant under total bisimulation

53 What about Preferred Extensions? ST x (Compl(p)) q((st x (Compl(q)) p q) They are the maximal, conflict-free, post fixed points of the characteristic function of an argumentation framework However this formula turns out to be invariant under total bisimulation Is it expressible in the mu-calculus + universal modality?

54 Part IV Dialogue games = Evaluation Games

55 Dialogue games in Argumentation The proof-theory of argumentation is commonly given in terms of dialogue games The semantics of modal logic offers a unified framework for systematizing games that check the membership of arguments to admissible sets, complete, grounded and stable extensions

56 Evaluation games (i) Eve (the proponent) tries to prove that an argument belongs to a given set which enjoys a specific property in the argumentation model Adam (the opponent) tries to falsify Eve s claim Positions consists of pairs (formula, argument) Who plays depends on the formula in the position A player wins iff its adversary runs out of available moves

57 Evaluation games (ii) Position Turn Available moves (ϕ 1 ϕ 2, a) {(ϕ 1, a), (ϕ 2, a)} (ϕ 1 ϕ 2, a) {(ϕ 1, a), (ϕ 2, a)} ( ϕ, a) {(ϕ, b) (a, b) 1 } ([ ]ϕ, a) {(ϕ, b) (a, b) 1 } ( U ϕ, a) {(ϕ, b) b A} ([U]ϕ, a) {(ϕ, b) b A} (, a) (, a) (p, a) & a I(p) (p, a) & a I(p) ( p, a) & a I(p) ( p, a) & a I(p)

58 Evaluation games (iii) 1 0 a b

59 Evaluation games (iii) (1 [U](1 1), a) a 1 0 b

60 Evaluation games (iii) (1 [U](1 1), a) a 1 0 b (1, a) ve wins

61 Evaluation games (iii) (1 [U](1 1), a) a 1 0 b (1, a) ve wins ([U](1 1), a)

62 Evaluation games (iii) (1 [U](1 1), a) a 1 0 b (1, a) ([U](1 1), a) ve wins (1 1, b)

63 Evaluation games (iii) (1 [U](1 1), a) a 1 0 b (1, a) ([U](1 1), a) ve wins (1 1, a) (1 1, b)

64 Evaluation games (iii) (1 [U](1 1), a) a 1 0 b (1, a) ([U](1 1), a) ve wins (1 1, a) (1 1, b) (1 1, a)

65 Evaluation games (iii) (1 [U](1 1), a) a 1 0 b (1, a) ([U](1 1), a) ve wins (1 1, a) (1 1, b) ( 1 1, a) (1 1, a)

66 Evaluation games (iii) (1 [U](1 1), a) a 1 0 b (1, a) ([U](1 1), a) ve wins (1 1, a) (1 1, b) ( 1 1, a) (1 1, a) ( 1, a) dam wins

67 Evaluation games (iii) (1 [U](1 1), a) a 1 0 b (1, a) ([U](1 1), a) ve wins (1 1, a) (1 1, b) ( 1 1, a) (1 1, a) ( 1, a) ( 1, a) dam wins

68 Evaluation games (iii) (1 [U](1 1), a) a 1 0 b (1, a) ([U](1 1), a) ve wins (1 1, a) (1 1, b) ( 1 1, a) (1 1, a) ( 1, a) (1, b) ve wins ( 1, a) dam wins

69 Evaluation games (iv) Theorem 2 (Adequacy of the evaluation game for K U ) Let ϕ L KU, and let M = (A, I) be an argumentation model. Then, for any argument a A, it holds that: (ϕ, a) Win (E(ϕ, M)) M, a = ϕ. So, in the previous game, Adam could not possibly force Eve to loose!

70 Evaluation games for argumentation Adm : E(ϕ [U](ϕ ([ ] ϕ [ ] ϕ)), M)@(ϕ [U](ϕ ([ ] ϕ [ ] ϕ), a) Complete : E(ϕ [U](ϕ [ ] ϕ)), M)@(ϕ [U](ϕ [ ] ϕ), a) Stable : E(ϕ [U](ϕ ϕ)), M)@(ϕ [U](ϕ ϕ), a) Grounded : E(µp.[ ] p, M)@(µp.[ ] p, a) Evaluation games provide a systematization of the gametheoretical proof-theory of argumentation The game is the same, what changes is the formula to be checked.

71 Evaluation games for argumentation Adm : E(ϕ [U](ϕ ([ ] ϕ [ ] ϕ)), M)@(ϕ [U](ϕ ([ ] ϕ [ ] ϕ), a) Complete : E(ϕ [U](ϕ [ ] ϕ)), M)@(ϕ [U](ϕ [ ] ϕ), a) Stable : E(ϕ [U](ϕ ϕ)), M)@(ϕ [U](ϕ ϕ), a) Grounded : E(µp.[ ] p, M)@(µp.[ ] p, a) Evaluation games provide a systematization of the gametheoretical proof-theory of argumentation The game is the same, what changes is the formula to be checked.

72 Evaluation games vs. Dialogue games Given: (A, I), a, ϕ (A, I), a = ϕ? Given: A, a, ϕ I : (A, I), a = ϕ? A = p 1... p n ST a ( ϕ(p 1... p n ))? Evaluation games are algorithms for modal model-checking Dialogue games as defined in argumentation theory are inherently more complex (checking on pointed frames)!

73 Part V When are two arguments the same?

74 Sameness = Behavioral equivalence guilty guilty a b innocent c

75 Sameness = Behavioral equivalence guilty guilty a b x guilty innocent c innocent y

76 Sameness = Behavioral equivalence guilty guilty a b x guilty innocent c innocent Are c and y different from the point of view of abstract argumentation? What about the rest? E.g., if guilty denotes the grounded extension on the left, so should guilty do on the right y

77 Sameness = Behavioral equivalence guilty guilty a b x guilty innocent c innocent Are c and y different from the point of view of abstract argumentation? What about the rest? E.g., if guilty denotes the grounded extension on the left, so should guilty do on the right y

78 Sameness = Behavioral equivalence guilty guilty a b x guilty innocent c innocent Are c and y different from the point of view of abstract argumentation? What about the rest? E.g., if guilty denotes the grounded extension on the left, so should guilty do on the right y

79 Sameness = Behavioral equivalence guilty guilty a b x guilty innocent c innocent Are c and y different from the point of view of abstract argumentation? What about the rest? E.g., if guilty denotes the grounded extension on the left, so should guilty do on the right y

80 Bisimulation (i) Definition 3 (Bisimulation) Let M = (A,, I) and M = (A,, I ) be two argumentation models. A bisimulation between M and M is a non-empty relation Z A A such that for any aza : Atom: a and a are propositionally equivalent; Zig: if a b for some b A, then a b for some b A and bzb ; Zag: if a b for some b A then a b for some b A and aza. A total bisimulation is a bisimulation Z A A such that its left projection covers A and its right projection covers A. Two arguments are the same iff they are labelled in the same way, they are attacked by arguments with same labels (bisimulation) and this holds for all arguments in the framework (total bisimulation)

81 Bisimulation (ii) Theorem 3 (Bisimilar arguments) Let (M, a) and (M, a ) be two pointed argumentation models, and let Z be a total bisimulation between M and M. It holds that a belongs to an admissible set (complete extension, stable extension, grounded extension) if and only if a belongs to an admissible set (complete extension, stable extension, grounded extension) both denoted by a same label ϕ. Follows directly from the fact that the logics expressing those concepts are invariant under (total) bisimulation

82 Bisimulation games (i) The game is played by a Spoiler who tries to show that two given pointed models are not bisimilar, and a Duplicator who tries to show the contrary Position consist of pairs: (pointed model, pointed model) Spoiler starts, Duplicator responds Spoiler wins iff a position is reached where the pointed models do not satisfy the same labels, or when Duplicator is out of moves

83 Sameness = Behavioral equivalence guilty guilty a b x guilty innocent c innocent y

84 Sameness = Behavioral equivalence guilty guilty a b x guilty innocent (M, c)(m, y) c y innocent

85 Sameness = Behavioral equivalence guilty guilty a b x guilty innocent (M, c)(m, y) c y innocent

86 Sameness = Behavioral equivalence guilty guilty a b x guilty innocent (M, c)(m, y) c y innocent (M, a)(m, y) (M, b)(m, y) (M, c)(m, x)

87 Sameness = Behavioral equivalence guilty guilty a b x guilty innocent (M, c)(m, y) c y innocent (M, a)(m, y) (M, b)(m, y) (M, c)(m, x)

88 Sameness = Behavioral equivalence guilty guilty a b x guilty innocent (M, c)(m, y) c y innocent (M, a)(m, y) (M, b)(m, y) (M, c)(m, x) (M, b)(m, x)

89 Sameness = Behavioral equivalence guilty guilty a b x guilty innocent (M, c)(m, y) c y innocent (M, a)(m, y) (M, b)(m, y) (M, c)(m, x) (M, b)(m, x) Duplicator wins!

90 Bisimulation games (ii) Theorem 4 (Adequacy of bisimulation games) Let (M, a) and (M, a ) be two argumentation models. Duplicator has a winning strategy in the (total) bisimulation game B(M, M )@(a, a ) if and only if M, a and M, a are (totally) bisimilar. Bisimulation games are an adequate proof procedure for checking whether two labelled argumentation frameworks behave in the same way from the point of view of argumentation theory

91 Part VI Conclusions

92 The Uses of Argument (1958) What things about the form and merits of our arguments are field-invariant and what things about them are field-dependent? [...] The force of the conclusion [...] is the same regardless of fields: the criteria or sorts of grounds required to justify such a conclusion vary from field to field [Toulmin,1958]

93 The Uses of Argument (1958) What things about the form and merits of our arguments are field-invariant and what things about them are field-dependent? [...] The force of the conclusion [...] is the same regardless of fields: the criteria or sorts of grounds required to justify such a conclusion vary from field to field [Toulmin,1958] Argumentation theory studies the form(s) of argumentation

94 The Uses of Argument (1958) What things about the form and merits of our arguments are field-invariant and what things about them are field-dependent? [...] The force of the conclusion [...] is the same regardless of fields: the criteria or sorts of grounds required to justify such a conclusion vary from field to field [Toulmin,1958] Argumentation theory studies the form(s) of argumentation Such forms can naturally ground logical semantics

95 The Uses of Argument (1958) What things about the form and merits of our arguments are field-invariant and what things about them are field-dependent? [...] The force of the conclusion [...] is the same regardless of fields: the criteria or sorts of grounds required to justify such a conclusion vary from field to field [Toulmin,1958] Argumentation theory studies the form(s) of argumentation Such forms can naturally ground logical semantics... logic was thrown out through the door and comes back through the window...

96 Future work Dialogue games = MSO model-checking games Lorentzen games = Argumentation games Dominance structures in games = Dung s frameworks