Preference-based Argumentation

Transcription

1 Preference-based Argumentation Yannis Dimopoulos Department of Computer Science University of Cyprus Nicosia, Cyprus Joint work with Leila Amgoud (IRIT, Toulouse) and Pavlos Moraitis (Uni Paris 5) Y. Dimopoulos (UCY) Preference-based Argumentation 1 / 23

2 Outline 1 General Argumentation 2 Preference-based Argumentation (PBA) 3 Structural properties of PBA 4 Computational properties of PBA 5 PBA and Negotiation Y. Dimopoulos (UCY) Preference-based Argumentation 2 / 23

3 What is Argumentation? Argumentation = a reasoning model based on the construction, exchange and evaluation of arguments Y. Dimopoulos (UCY) Preference-based Argumentation 3 / 23

4 What is Argumentation? Argumentation = a reasoning model based on the construction, exchange and evaluation of arguments Argument = a reason / justification for some claim Y. Dimopoulos (UCY) Preference-based Argumentation 3 / 23

5 What is Argumentation? Argumentation = a reasoning model based on the construction, exchange and evaluation of arguments Argument = a reason / justification for some claim The core of an argument: Reasons + a claim Reason: Because Tweety is a bird and birds fly Claim: Tweety flies Y. Dimopoulos (UCY) Preference-based Argumentation 3 / 23

6 What is Argumentation? Argumentation = a reasoning model based on the construction, exchange and evaluation of arguments Argument = a reason / justification for some claim The core of an argument: Reasons + a claim Reason: Because Tweety is a bird and birds fly Claim: Tweety flies Argumentation can be used for: Internal agent s reasoning Modelling interactions between agents Y. Dimopoulos (UCY) Preference-based Argumentation 3 / 23

7 What is an Argument? A set of premises in support of a conclusion/claim Y. Dimopoulos (UCY) Preference-based Argumentation 4 / 23

8 What is an Argument? A set of premises in support of a conclusion/claim claim: Info I about John should be published because Y. Dimopoulos (UCY) Preference-based Argumentation 4 / 23

9 What is an Argument? A set of premises in support of a conclusion/claim claim: Info I about John should be published because premise/reason: John has political responsibilities and I is in the national interest and if a person has pol. resp. and info about that person is in the national interest then that info should be published Y. Dimopoulos (UCY) Preference-based Argumentation 4 / 23

10 What is Argumentation? The process of argument construction, exchange and evaluation in light of their interactions with other arguments Y. Dimopoulos (UCY) Preference-based Argumentation 5 / 23

11 What is Argumentation? The process of argument construction, exchange and evaluation in light of their interactions with other arguments A1 (publish info about John because he has responsibilities...) Y. Dimopoulos (UCY) Preference-based Argumentation 5 / 23

12 What is Argumentation? The process of argument construction, exchange and evaluation in light of their interactions with other arguments A1 (publish info about John because he has responsibilities...) A2 (John does not have pol. resp. because he resigned from parliament, and if a person resigns...) Y. Dimopoulos (UCY) Preference-based Argumentation 5 / 23

13 What is Argumentation? The process of argument construction, exchange and evaluation in light of their interactions with other arguments A1 (publish info about John because he has responsibilities...) A2 (John does not have pol. resp. because he resigned from parliament, and if a person resigns...) A3 (John does have pol. resp. because he is now middle east envoy, and if a person...) Y. Dimopoulos (UCY) Preference-based Argumentation 5 / 23

14 Arguments in Propositional Logic is a set of propositional logic formulae Args = {(H, h) H is consistent H h H is minimal} Y. Dimopoulos (UCY) Preference-based Argumentation 6 / 23

15 Arguments in Propositional Logic is a set of propositional logic formulae Args = {(H, h) H is consistent H h H is minimal} (H 1, h 1 ) and (H 2, h 2 ) rebut each other iff h 1 h 2 Y. Dimopoulos (UCY) Preference-based Argumentation 6 / 23

16 Arguments in Propositional Logic is a set of propositional logic formulae Args = {(H, h) H is consistent H h H is minimal} (H 1, h 1 ) and (H 2, h 2 ) rebut each other iff h 1 h 2 (H 1, h 1 ) undercuts (H 2, h 2 ) iff h 1 h for some h H 2 Y. Dimopoulos (UCY) Preference-based Argumentation 6 / 23

17 Arguments in Propositional Logic = {nat, pol, nat pol pub, mid, mid pol} res, res pol, Y. Dimopoulos (UCY) Preference-based Argumentation 7 / 23

18 Arguments in Propositional Logic = {nat, pol, nat pol pub, mid, mid pol} res, res pol, A 1 = ({nat, pol, nat pol pub}, pub) Y. Dimopoulos (UCY) Preference-based Argumentation 7 / 23

19 Arguments in Propositional Logic = {nat, pol, nat pol pub, mid, mid pol} res, res pol, A 1 = ({nat, pol, nat pol pub}, pub) A 2 = ({res, res pol}, pol) A 2 A 1 Y. Dimopoulos (UCY) Preference-based Argumentation 7 / 23

20 Arguments in Propositional Logic = {nat, pol, nat pol pub, mid, mid pol} res, res pol, A 1 = ({nat, pol, nat pol pub}, pub) A 2 = ({res, res pol}, pol) A 2 A 1 A 3 = ({mid, mid pol}, pol) A 3 A 2 Y. Dimopoulos (UCY) Preference-based Argumentation 7 / 23

21 Abstract argumentation theories (Dung 1995) An argumentation theory is a pair A, R where: A = a set of arguments R A A = an attack relation between arguments Y. Dimopoulos (UCY) Preference-based Argumentation 8 / 23

22 Abstract argumentation theories (Dung 1995) An argumentation theory is a pair A, R where: A = a set of arguments R A A = an attack relation between arguments For a, b A, a attacks b if (a, b) R Y. Dimopoulos (UCY) Preference-based Argumentation 8 / 23

23 Abstract argumentation theories (Dung 1995) An argumentation theory is a pair A, R where: A = a set of arguments R A A = an attack relation between arguments For a, b A, a attacks b if (a, b) R Example Usually, Quakers are pacifists Usually, Republicans are not pacifists Nixon is both a Quaker and a Republican Y. Dimopoulos (UCY) Preference-based Argumentation 8 / 23

24 Abstract argumentation theories (Dung 1995) An argumentation theory is a pair A, R where: A = a set of arguments R A A = an attack relation between arguments For a, b A, a attacks b if (a, b) R Example Usually, Quakers are pacifists Usually, Republicans are not pacifists Nixon is both a Quaker and a Republican = two arguments: a : Nixon is a pacifist since he is a Quaker b : Nixon is not a pacifist since he is a Republican Y. Dimopoulos (UCY) Preference-based Argumentation 8 / 23

25 Abstract argumentation theories (Dung 1995) An argumentation theory is a pair A, R where: A = a set of arguments R A A = an attack relation between arguments For a, b A, a attacks b if (a, b) R Example Usually, Quakers are pacifists Usually, Republicans are not pacifists Nixon is both a Quaker and a Republican = two arguments: a : Nixon is a pacifist since he is a Quaker b : Nixon is not a pacifist since he is a Republican a b Y. Dimopoulos (UCY) Preference-based Argumentation 8 / 23

26 Abstract argumentation theories Which arguments to accept together? = acceptability semantics Y. Dimopoulos (UCY) Preference-based Argumentation 9 / 23

27 Abstract argumentation theories Which arguments to accept together? = acceptability semantics Let B A. B is conflict-free iff a, b B such that (a, b) R Y. Dimopoulos (UCY) Preference-based Argumentation 9 / 23

28 Abstract argumentation theories Which arguments to accept together? = acceptability semantics Let B A. B is conflict-free iff a, b B such that (a, b) R B defends an argument a iff b A, if (b, a) R, then c B such that (c, b) R Y. Dimopoulos (UCY) Preference-based Argumentation 9 / 23

29 Abstract argumentation theories Which arguments to accept together? = acceptability semantics Let B A. B is conflict-free iff a, b B such that (a, b) R B defends an argument a iff b A, if (b, a) R, then c B such that (c, b) R For instance: c b a Y. Dimopoulos (UCY) Preference-based Argumentation 9 / 23

30 Abstract argumentation theories Which arguments to accept together? = acceptability semantics Let B A. B is conflict-free iff a, b B such that (a, b) R B defends an argument a iff b A, if (b, a) R, then c B such that (c, b) R For instance: c b a The set {c} is conflict-free and defends a Y. Dimopoulos (UCY) Preference-based Argumentation 9 / 23

31 Abstract argumentation theories Which arguments to accept together? = acceptability semantics Let B A. B is conflict-free iff a, b B such that (a, b) R B defends an argument a iff b A, if (b, a) R, then c B such that (c, b) R For instance: c b a The set {c} is conflict-free and defends a The sets {a, b}, {b, c} and {a, b, c} are not conflict-free Y. Dimopoulos (UCY) Preference-based Argumentation 9 / 23

32 Admissible extensions Let B A. B is an admissible extension iff 1 B is conflict-free 2 B defends all its elements Y. Dimopoulos (UCY) Preference-based Argumentation 10 / 23

33 Admissible extensions Let B A. B is an admissible extension iff 1 B is conflict-free 2 B defends all its elements Example (Nixon Cont.) a b Y. Dimopoulos (UCY) Preference-based Argumentation 10 / 23

34 Admissible extensions Let B A. B is an admissible extension iff 1 B is conflict-free 2 B defends all its elements Example (Nixon Cont.) a b, {a}, {b} are admissible extensions Y. Dimopoulos (UCY) Preference-based Argumentation 10 / 23

35 Admissible extensions Let B A. B is an admissible extension iff 1 B is conflict-free 2 B defends all its elements Example (Nixon Cont.) a b, {a}, {b} are admissible extensions {a, b} is not an admissible extension Y. Dimopoulos (UCY) Preference-based Argumentation 10 / 23

36 Stable extensions and graph kernels Let B A. B is a stable extension iff 1 B is conflict-free 2 B attacks any argument in A\B Y. Dimopoulos (UCY) Preference-based Argumentation 11 / 23

37 Stable extensions and graph kernels Let B A. B is a stable extension iff 1 B is conflict-free 2 B attacks any argument in A\B Example (Nixon Cont.) a b Y. Dimopoulos (UCY) Preference-based Argumentation 11 / 23

38 Stable extensions and graph kernels Let B A. B is a stable extension iff 1 B is conflict-free 2 B attacks any argument in A\B Example (Nixon Cont.) a b {a}, {b} are stable extensions, {a, b} are not stable extensions Y. Dimopoulos (UCY) Preference-based Argumentation 11 / 23

39 Stable extensions and graph kernels Let B A. B is a stable extension iff 1 B is conflict-free 2 B attacks any argument in A\B Example (Nixon Cont.) a b {a}, {b} are stable extensions, {a, b} are not stable extensions A kernel of a (di)graph G = (V, E) is a set K V such that v i, v j K it holds that (v i, v j ) E and (v j, v i ) E v i K, v j K such that (v j, v i ) E Y. Dimopoulos (UCY) Preference-based Argumentation 11 / 23

40 Stable extensions and graph kernels Let B A. B is a stable extension iff 1 B is conflict-free 2 B attacks any argument in A\B Example (Nixon Cont.) a b {a}, {b} are stable extensions, {a, b} are not stable extensions A kernel of a (di)graph G = (V, E) is a set K V such that v i, v j K it holds that (v i, v j ) E and (v j, v i ) E v i K, v j K such that (v j, v i ) E Introduced by Von Neumann and Morgenstern in 1944 Y. Dimopoulos (UCY) Preference-based Argumentation 11 / 23

41 Stable extensions and graph kernels Stable extensions of T correspond exactly to the kernels of the associated graph G T (Dimopoulos+Torres 1996 ) Y. Dimopoulos (UCY) Preference-based Argumentation 12 / 23

42 Stable extensions and graph kernels Stable extensions of T correspond exactly to the kernels of the associated graph G T (Dimopoulos+Torres 1996 ) A graph may have one or many kernels... Y. Dimopoulos (UCY) Preference-based Argumentation 12 / 23

43 Stable extensions and graph kernels Stable extensions of T correspond exactly to the kernels of the associated graph G T (Dimopoulos+Torres 1996 ) A graph may have one or many kernels......or no kernels at all Y. Dimopoulos (UCY) Preference-based Argumentation 12 / 23

44 Stable extensions and graph kernels Stable extensions of T correspond exactly to the kernels of the associated graph G T (Dimopoulos+Torres 1996 ) A graph may have one or many kernels......or no kernels at all Reasoning with stable/admissible extensions is hard Deciding the existence of stable extensions is NP-hard Deciding the existence of an non-empty admissible extension is NP-hard Y. Dimopoulos (UCY) Preference-based Argumentation 12 / 23

45 Preference-based Argumentation An extension of classical argumentation Basic Idea: We often have preferences over arguments Y. Dimopoulos (UCY) Preference-based Argumentation 13 / 23

46 Preference-based Argumentation An extension of classical argumentation Basic Idea: We often have preferences over arguments Example Small cars have low running cost Y. Dimopoulos (UCY) Preference-based Argumentation 13 / 23

47 Preference-based Argumentation An extension of classical argumentation Basic Idea: We often have preferences over arguments Example Small cars have low running cost Big cars are safe Y. Dimopoulos (UCY) Preference-based Argumentation 13 / 23

48 Preference-based Argumentation An extension of classical argumentation Basic Idea: We often have preferences over arguments Example Small cars have low running cost Big cars are safe Safety is more important than running cost Y. Dimopoulos (UCY) Preference-based Argumentation 13 / 23

49 Preference-based Argumentation An extension of classical argumentation Basic Idea: We often have preferences over arguments Example Small cars have low running cost Big cars are safe Safety is more important than running cost Preferences present in previous works on argumentation But no systematic study Y. Dimopoulos (UCY) Preference-based Argumentation 13 / 23

50 Preference-based Argumentation An extension of classical argumentation Basic Idea: We often have preferences over arguments Example Small cars have low running cost Big cars are safe Safety is more important than running cost Preferences present in previous works on argumentation But no systematic study This work: Study the properties of a specific Preference-based Argumentation Framework Y. Dimopoulos (UCY) Preference-based Argumentation 13 / 23

51 Abstract Preference-based Argumentation The attacking relation R is the combination of A conflict relation, C, capturing incompatibility between arguments A preference relation,, capturing the relative strength of arguments Y. Dimopoulos (UCY) Preference-based Argumentation 14 / 23

52 Abstract Preference-based Argumentation The attacking relation R is the combination of A conflict relation, C, capturing incompatibility between arguments A preference relation,, capturing the relative strength of arguments a b means a b and b a Y. Dimopoulos (UCY) Preference-based Argumentation 14 / 23

53 Abstract Preference-based Argumentation The attacking relation R is the combination of A conflict relation, C, capturing incompatibility between arguments A preference relation,, capturing the relative strength of arguments a b means a b and b a C is assumed irreflexive and symmetric Y. Dimopoulos (UCY) Preference-based Argumentation 14 / 23

54 Abstract Preference-based Argumentation The attacking relation R is the combination of A conflict relation, C, capturing incompatibility between arguments A preference relation,, capturing the relative strength of arguments a b means a b and b a C is assumed irreflexive and symmetric is assumed reflexive and transitive, i.e. a pre-order Y. Dimopoulos (UCY) Preference-based Argumentation 14 / 23

55 Abstract Preference-based Argumentation The attacking relation R is the combination of A conflict relation, C, capturing incompatibility between arguments A preference relation,, capturing the relative strength of arguments a b means a b and b a C is assumed irreflexive and symmetric is assumed reflexive and transitive, i.e. a pre-order A Preference-based Argumentation Theory (PBAT) is a pair A, R : A = a set of arguments (a, b) R iff (a, b) C and b a Y. Dimopoulos (UCY) Preference-based Argumentation 14 / 23

56 Preference-based Argumentation - Example A = {a, b, c} Y. Dimopoulos (UCY) Preference-based Argumentation 15 / 23

57 Preference-based Argumentation - Example A = {a, b, c} C = {(a, b),(b, a)(a, c),(c, a)} Y. Dimopoulos (UCY) Preference-based Argumentation 15 / 23

58 Preference-based Argumentation - Example A = {a, b, c} C = {(a, b),(b, a)(a, c),(c, a)} a b, a c b c, c b Y. Dimopoulos (UCY) Preference-based Argumentation 15 / 23

59 Preference-based Argumentation - Example A = {a, b, c} C = {(a, b),(b, a)(a, c),(c, a)} a b, a c b c, c b b a c Y. Dimopoulos (UCY) Preference-based Argumentation 15 / 23

60 The graph of PBATs The (di)graph G T of a PBAT T has some useful properties Y. Dimopoulos (UCY) Preference-based Argumentation 16 / 23

61 The graph of PBATs The (di)graph G T of a PBAT T has some useful properties Every cycle of G T has at least two symmetric edges Y. Dimopoulos (UCY) Preference-based Argumentation 16 / 23

62 The graph of PBATs The (di)graph G T of a PBAT T has some useful properties Every cycle of G T has at least two symmetric edges G T has no elementary cycle of length greater than 2 Y. Dimopoulos (UCY) Preference-based Argumentation 16 / 23

63 The graph of PBATs The (di)graph G T of a PBAT T has some useful properties Every cycle of G T has at least two symmetric edges G T has no elementary cycle of length greater than 2 Duchet, 1979: kernels always exist for certain classes of graphs Y. Dimopoulos (UCY) Preference-based Argumentation 16 / 23

64 The graph of PBATs The (di)graph G T of a PBAT T has some useful properties Every cycle of G T has at least two symmetric edges G T has no elementary cycle of length greater than 2 Duchet, 1979: kernels always exist for certain classes of graphs From those (and other) results we obtain the following properties Every PBAT has at least one stable extension Y. Dimopoulos (UCY) Preference-based Argumentation 16 / 23

65 The graph of PBATs The (di)graph G T of a PBAT T has some useful properties Every cycle of G T has at least two symmetric edges G T has no elementary cycle of length greater than 2 Duchet, 1979: kernels always exist for certain classes of graphs From those (and other) results we obtain the following properties Every PBAT has at least one stable extension Every PBAT is coherent i.e. stable and maximal admissible extensions coincide Y. Dimopoulos (UCY) Preference-based Argumentation 16 / 23

66 The graph of PBATs The (di)graph G T of a PBAT T has some useful properties Every cycle of G T has at least two symmetric edges G T has no elementary cycle of length greater than 2 Duchet, 1979: kernels always exist for certain classes of graphs From those (and other) results we obtain the following properties Every PBAT has at least one stable extension Every PBAT is coherent i.e. stable and maximal admissible extensions coincide All results are based on transitivity Y. Dimopoulos (UCY) Preference-based Argumentation 16 / 23

67 Preferences on sets of arguments From a preference relation on arguments ( ) to a preference relation on sets of arguments: Y. Dimopoulos (UCY) Preference-based Argumentation 17 / 23

68 Preferences on sets of arguments From a preference relation on arguments ( ) to a preference relation on sets of arguments: For A 1, A 2 set of arguments, A 1 A 2 iff A 1 A 2, or a, b with a A 1 \ A 2 and b A 2 \ A 1, it holds that a b Y. Dimopoulos (UCY) Preference-based Argumentation 17 / 23

69 Preferences on sets of arguments From a preference relation on arguments ( ) to a preference relation on sets of arguments: For A 1, A 2 set of arguments, A 1 A 2 iff A 1 A 2, or a, b with a A 1 \ A 2 and b A 2 \ A 1, it holds that a b stable extensions = most preferred sets wrt permitted by C Y. Dimopoulos (UCY) Preference-based Argumentation 17 / 23

70 Preferences on sets on arguments - Example A = {a, b, c} C = {(a, b),(b, a)(a, c),(c, a)} a b, a c b c, c b Y. Dimopoulos (UCY) Preference-based Argumentation 18 / 23

71 Preferences on sets on arguments - Example abc ab ac A = {a, b, c} C = {(a, b),(b, a)(a, c),(c, a)} a b, a c b c, c b a bc b {} c Y. Dimopoulos (UCY) Preference-based Argumentation 18 / 23

72 Preferences on sets on arguments - Example ab ac a A = {a, b, c} C = {(a, b),(b, a)(a, c),(c, a)} a b, a c b c, c b b bc {} c Y. Dimopoulos (UCY) Preference-based Argumentation 18 / 23

73 Preferences on sets on arguments - Example a bc A = {a, b, c} C = {(a, b),(b, a)(a, c),(c, a)} a b, a c b c, c b b {} c Y. Dimopoulos (UCY) Preference-based Argumentation 18 / 23

74 Preferences on sets on arguments - Example a bc {a} is the unique stable extension b {} c Y. Dimopoulos (UCY) Preference-based Argumentation 18 / 23

75 Computing a Stable Extension is Easy A stable extension of a PBAT can be computed in polynomial time Y. Dimopoulos (UCY) Preference-based Argumentation 19 / 23

76 Computing a Stable Extension is Easy A stable extension of a PBAT can be computed in polynomial time General Idea of the algorithm: Start from a top component Find an argument that defends itself against all its attackers Add the argument to the stable extension and simplify Repeat on the remaining theory Y. Dimopoulos (UCY) Preference-based Argumentation 19 / 23

77 Computing a Stable Extension is Easy A stable extension of a PBAT can be computed in polynomial time General Idea of the algorithm: Start from a top component Find an argument that defends itself against all its attackers Add the argument to the stable extension and simplify Repeat on the remaining theory Key property: There always exists a self-defending argument Y. Dimopoulos (UCY) Preference-based Argumentation 19 / 23

78 Goal Reasoning is Hard Deciding whether there is a stable extension that contains a is NP-hard Reduction from 3SAT Y. Dimopoulos (UCY) Preference-based Argumentation 20 / 23

79 Goal Reasoning is Hard Deciding whether there is a stable extension that contains a is NP-hard Reduction from 3SAT Why Complex interaction between arguments Must find the right combination of other arguments Y. Dimopoulos (UCY) Preference-based Argumentation 20 / 23

80 Goal Reasoning is Hard Deciding whether there is a stable extension that contains a is NP-hard Reduction from 3SAT Why Complex interaction between arguments Must find the right combination of other arguments Deciding whether a is included in every stable extension is co-np-hard Y. Dimopoulos (UCY) Preference-based Argumentation 20 / 23

81 Theories without incomparability Reasoning becomes a bit easier if there is no incomparability Y. Dimopoulos (UCY) Preference-based Argumentation 21 / 23

82 Theories without incomparability Reasoning becomes a bit easier if there is no incomparability i.e. there are no a, b A s.t. a b and b a Y. Dimopoulos (UCY) Preference-based Argumentation 21 / 23

83 Theories without incomparability Reasoning becomes a bit easier if there is no incomparability i.e. there are no a, b A s.t. a b and b a Key Properties Correspondence between the stable extensions of T and Maximal Independent Sets of G T Maximal Independent Sets can be computed with Polynomial Delay Y. Dimopoulos (UCY) Preference-based Argumentation 21 / 23

84 Theories without incomparability Reasoning becomes a bit easier if there is no incomparability i.e. there are no a, b A s.t. a b and b a Key Properties Correspondence between the stable extensions of T and Maximal Independent Sets of G T Maximal Independent Sets can be computed with Polynomial Delay The stable extensions can be computed with Polynomial Delay Y. Dimopoulos (UCY) Preference-based Argumentation 21 / 23

85 Theories without incomparability Reasoning becomes a bit easier if there is no incomparability i.e. there are no a, b A s.t. a b and b a Key Properties Correspondence between the stable extensions of T and Maximal Independent Sets of G T Maximal Independent Sets can be computed with Polynomial Delay The stable extensions can be computed with Polynomial Delay Exponential worst case behavior A theory with n arguments can have n n/3 stable extensions Y. Dimopoulos (UCY) Preference-based Argumentation 21 / 23

86 Negotiation Negotiation: search for a mutually acceptable agreement between two (or several agents) on one or more issues Offers ranked by their utility Reservation value Alternate Offers Protocol Y. Dimopoulos (UCY) Preference-based Argumentation 22 / 23

87 Negotiation Negotiation: search for a mutually acceptable agreement between two (or several agents) on one or more issues Offers ranked by their utility Reservation value Alternate Offers Protocol Characteristics of Negotiation Deadline? Y. Dimopoulos (UCY) Preference-based Argumentation 22 / 23

88 Negotiation Negotiation: search for a mutually acceptable agreement between two (or several agents) on one or more issues Offers ranked by their utility Reservation value Alternate Offers Protocol Characteristics of Negotiation Deadline? Can I accept an offer that I have previously rejected? Issue by issue? Y. Dimopoulos (UCY) Preference-based Argumentation 22 / 23

89 PBA and Negotiation Offers supported by arguments Argument preference determines offer preference Best offer is supported by the most preferred argument Performatives: Propose, Argue, Reject, Agree, Nothing, Withdraw... Y. Dimopoulos (UCY) Preference-based Argumentation 23 / 23