Tackling Defeasible Reasoning in Bochum:

Transcription

1 Tackling Defeasible Reasoning in Bochum: the Research Group for Non-Monotonic Logic and Formal Argumentation Christian Straßer and Dunja Šešelja April 10, 2017

2 Outline The NMLFA Reasoning by Cases Unrestricted Rebut Comparative Studies Sequent-based argumentation (with Ofer Arieli, Tel Aviv) Agent-Based Models 1/43

3 The NMLFA

4 The Research Group for Non-Monotonic Logic and Formal Argumentation (NMLFA) funding: (Alexander von Humboldt-Foundation) aim: study defeasible reasoning with methods of formal argumentation location: Institute for Philosophy II, Ruhr-University Bochum online: defeasible-reasoning/index.html 2/43

5 Members AnneMarie Borg (PhD candidate) 3/43

6 Members AnneMarie Borg (PhD candidate) Jesse Heyninck (PhD candidate) 3/43

7 Members AnneMarie Borg (PhD candidate) Jesse Heyninck (PhD candidate) Pere Pardo (PostDoc researcher) 3/43

8 Members AnneMarie Borg (PhD candidate) Jesse Heyninck (PhD candidate) Pere Pardo (PostDoc researcher) Christian Straßer (Principal researcher) 3/43

9 Members AnneMarie Borg (PhD candidate) Jesse Heyninck (PhD candidate) Pere Pardo (PostDoc researcher) Christian Straßer (Principal researcher) Mathieu Beirlaen (Associated PostDoc researcher) 3/43

10 Members AnneMarie Borg (PhD candidate) Jesse Heyninck (PhD candidate) Pere Pardo (PostDoc researcher) Christian Straßer (Principal researcher) Mathieu Beirlaen (Associated PostDoc researcher) Dunja Šešelja (Associated PostDoc researcher) 3/43

11 Some research themes 1. Extending the expressive power of structured argumentation 1.1 reasoning by cases and hypothetical reasoning 1.2 expressing doubt non-greedy argumentative reasoning 1.3 unrestricted rebut 2. Comparative studies of different nonmonotonic formalisms with special attention to argumentation formalisms (ASPIC, ABA, etc.) 3. Applications of argumentation theory to deontic logic 4. Sequent-based argumentation (with Ofer Arieli, Tel Aviv) 5. Agent-based models based on techniques from abstract argumentation 4/43

12 Reasoning by Cases

13 Mathieu Beirlaen, Jesse Heyninck, and Christian Straßer, Reasoning by Cases in Structured Argumentation forthcoming in Proceedings KRR/SAC 2017, ACM Digital Library (2017) 4/43

14 Reasoning by Cases, Defeasibly strict rules ( ) vs. defeasible rules ( ) 5/43

15 Reasoning by Cases, Defeasibly strict rules ( ) vs. defeasible rules ( ) schematically: A B A C B C C 5/43

16 Reasoning by Cases, Defeasibly strict rules ( ) vs. defeasible rules ( ) schematically: or, more generally: A B A C B C C A B A C B C C 5/43

17 Reasoning by Cases, Defeasibly strict rules ( ) vs. defeasible rules ( ) schematically: or, more generally: or, more generally: A B A C B C C A B A C B C C A B A C B C Read A C: C follows defeasibly from A or There is a (defeasible) argument for C based on A. C 5/43

18 Are there good formal accounts of defeasible Reasoning by Cases? 5/43

19 The Meta-Rule Approach: OR Rules for rules: A C B C A B C [OR] 6/43

20 The Meta-Rule Approach: OR Rules for rules: A C B C A B C [OR] Illustration: 6/43

21 The Meta-Rule Approach: OR Rules for rules: A C B C A B C [OR] Illustration: 1. A C PREM 6/43

22 The Meta-Rule Approach: OR Rules for rules: A C B C A B C [OR] Illustration: 1. A C PREM 2. B C PREM 6/43

23 The Meta-Rule Approach: OR Rules for rules: A C B C A B C [OR] Illustration: 1. A C PREM 2. B C PREM 3. A B PREM 6/43

24 The Meta-Rule Approach: OR Rules for rules: A C B C A B C [OR] Illustration: 1. A C PREM 2. B C PREM 3. A B PREM 4. A B C 1,2; OR 6/43

25 The Meta-Rule Approach: OR Rules for rules: A C B C A B C [OR] Illustration: 1. A C PREM 2. B C PREM 3. A B PREM 4. A B C 1,2; OR 5. C 3,4; DefeasibleMP 6/43

26 A Problematic Example for OR Suppose we have Σ = {p q r, q s, s v, r u, u v, p}. q s v p q r v r u v 7/43

27 A Problematic Example for OR q s v p q r v r u v 8/43

28 A Problematic Example for OR q s v p q r s u v r u v by (OR): from s v and u v 8/43

29 A Problematic Example for OR q s v p q r s u v r u v by (OR): from s v and u v by (Right-Weakening), from q s and r u 8/43

30 A Problematic Example for OR q s v p q r s u v r u v by (OR): from s v and u v by (Right-Weakening), from q s and r u by (OR): from q s u and r s u 8/43

31 A Problematic Example for OR t s! q s v p q r s u v r u v Suppose now we also have t and t s. the possible defeater has no effect on the generalized path 9/43

32 A Problematic Example for OR t s! q s v p q r s u v t! r r u v Suppose now we also have t and t r. the additional possible defeater has no effect on the generalized path 10/43

33 Extension-based Approaches: Default Logic (Reiter) Input: set of defaults and a set of formulas ( facts ) Build extensions by applying Modus Ponens to defaults while maintaining consistency 11/43

34 Extension-based Approaches: Default Logic (Reiter) Input: set of defaults and a set of formulas ( facts ) Build extensions by applying Modus Ponens to defaults while maintaining consistency For instance: Nixon is a Republican Pacifist is a Quaker Pacifist 11/43

35 Extension-based Approaches: Default Logic (Reiter) Input: set of defaults and a set of formulas ( facts ) Build extensions by applying Modus Ponens to defaults while maintaining consistency For instance: Nixon is a Republican Pacifist is a Quaker Pacifist Extensions: 1. {Nixon, Republican, Quaker, Pacifist} 2. {Nixon, Republican, Quaker, Pacifist} 11/43

36 Extension-based Approaches: Default Logic (Reiter) no handling of disjunctive facts out-of-the-box for instance: Σ = {Republican Democrat, Republican political, Democrat political}.? Republican Republican Democrat? political Democrat since the default is not triggered by the fact, MP cannot be applied 12/43

37 Extension-based Approaches: Default Logic (Reiter) idea: split the factual part of the knowledge base (Gelfond, Lifschitz, Przymusinska, 1991) Republican Republican Base 1 Republican Democrat political Base 2 Democrat Democrat 13/43

38 Extension-based Approaches: Default Logic (Reiter) idea: split the factual part of the knowledge base (Gelfond, Lifschitz, Przymusinska, 1991) Republican Republican Base 1 Republican Democrat political Base 2 Democrat Democrat two extensions: 1. Republican, political 2. Democrat, political 13/43

39 Problematic Example for Disjunctive Defaults Consider the following example: 1. Either his left hand or his right hand is broken. lhb rhb 14/43

40 Problematic Example for Disjunctive Defaults Consider the following example: 1. Either his left hand or his right hand is broken. lhb rhb 2. If somebody writes legibly then usually the right hand is not broken. wl rhb 14/43

41 Problematic Example for Disjunctive Defaults Consider the following example: 1. Either his left hand or his right hand is broken. lhb rhb 2. If somebody writes legibly then usually the right hand is not broken. wl rhb 3. He writes legibly. wl 14/43

42 Problematic Example for Disjunctive Defaults Consider the following example: 1. Either his left hand or his right hand is broken. lhb rhb 2. If somebody writes legibly then usually the right hand is not broken. wl rhb 3. He writes legibly. wl With disjunctive default logic we get two extensions: 1. wl, rhb, lhb 2. wl, rhb 14/43

43 General Stratagem So far: Manipulate the database! 1. produce new defeasible rules from the given ones 2. produce new factual knowledge bases when confronted with disjunctive information 15/43

44 Enters: the Argumentative Approach Instead of manipulating the knowledge base and reasoning on top of the manipulated database, we will, in what follows, use a more direct approach to the modeling of Reasoning by Cases in the context of defeasible reasoning, following the inference scheme: or, more generally: A B A C B C C A B A C B C C This will allow us to have more control over defeating conditions and to avoid pitfalls as the ones demonstrated above. 16/43

45 A New Type of Argument: RbC-Arguments Basic idea: Given an argument a 1 Arg(T) for which Conc(a 1 ) = A B, 17/43

46 A New Type of Argument: RbC-Arguments Basic idea: Given an argument a 1 Arg(T) for which Conc(a 1 ) = A B, an argument a 2 Arg( D, K {A} ) with Conc(a 2 ) = C, and 17/43

47 A New Type of Argument: RbC-Arguments Basic idea: Given an argument a 1 Arg(T) for which Conc(a 1 ) = A B, an argument a 2 Arg( D, K {A} ) with Conc(a 2 ) = C, and an argument a 3 Arg( D, K {B} ) with Conc(a 3 ) = C, 17/43

48 A New Type of Argument: RbC-Arguments Basic idea: Given an argument a 1 Arg(T) for which Conc(a 1 ) = A B, an argument a 2 Arg( D, K {A} ) with Conc(a 2 ) = C, and an argument a 3 Arg( D, K {B} ) with Conc(a 3 ) = C, we introduce a new RbC-Argument a 1, [a 2 ], [a 3 ] C. 17/43

49 More general: RbC-Argument Definition Where a 0 Arg(T) with Conc(a 0 ) = n i=1 A i and a i Arg( D, K {A i } ) \ Arg(T) (1 i n), a 0, [a 1 ],..., [a n ] n i=1 Conc(A i) is an RbC-argument. 18/43

50 More general: RbC-Argument Definition Where a 0 Arg(T) with Conc(a 0 ) = n i=1 A i and a i Arg( D, K {A i } ) \ Arg(T) (1 i n), a 0, [a 1 ],..., [a n ] n i=1 Conc(A i) is an RbC-argument. We say that a 1,..., a n are hypothetical sub-arguments of a, in signs: a 1,..., a n HSub(a). 18/43

51 More general: RbC-Argument Definition Where a 0 Arg(T) with Conc(a 0 ) = n i=1 A i and a i Arg( D, K {A i } ) \ Arg(T) (1 i n), a 0, [a 1 ],..., [a n ] n i=1 Conc(A i) is an RbC-argument. We say that a 1,..., a n are hypothetical sub-arguments of a, in signs: a 1,..., a n HSub(a). For each a i, Hyp(a i ) = A i. 18/43

52 Let T = D, K consist of D = {p q r, q s, s v, r u, u v, t s} and K = {p, t}. 19/43

53 Let T = D, K consist of D = {p q r, q s, s v, r u, u v, t s} and K = {p, t}. We have for instance the arguments: a 1 = p q r Arg(T) a 2 = q s v Arg( D, K {q} ) a 3 = r u v Arg( D, K {r} ) a 4 = a 1, [a 2 ], [a 3 ] v Arg(T). a2: q s v a1: p q r v a3: r u v 19/43

54 What about attacks? 19/43

55 a 1 = p q r Arg(T) a 2 = q s v Arg( S, D, K {q} ) a 3 = r u v Arg( S, D, K {r} ) a 4 = a 1, [a 2 ], [a 3 ] v Arg(T) a2: q s v a1: p q r v a3: r u v 20/43

56 a 1 = p q r Arg(T) a 2 = q s v Arg( S, D, K {q} ) a 3 = r u v Arg( S, D, K {r} ) a 4 = a 1, [a 2 ], [a 3 ] v Arg(T) a 5 = t s Arg(T) a5: t s a2: q s v a1: p q r v a3: r u v 20/43

57 a 1 = p q r Arg(T) a 2 = q s v Arg( D, K {q} ) a 3 = r u v Arg( D, K {r} ) a 4 = a 1, [a 2 ], [a 3 ] v Arg(T) a2: q s v a1: p q r v a3: r u v 21/43

58 a 1 = p q r Arg(T) a 2 = q s v Arg( D, K {q} ) a 3 = r u v Arg( D, K {r} ) a 4 = a 1, [a 2 ], [a 3 ] v Arg(T) a 6 = q s Arg( D, K {q} ) a6: q s a2: q s v a1: p q r v a3: r u v 21/43

59 Attacks again (non-nested case) Let HArg(T) be the set of all arguments a for which there is a b Arg(T) for which a HSub(b). 22/43

60 Attacks again (non-nested case) Let HArg(T) be the set of all arguments a for which there is a b Arg(T) for which a HSub(b). Argumentation frameworks are now triples Arg(T), HArg(T), Attacks(T). 22/43

61 Attacks again (non-nested case) Let HArg(T) be the set of all arguments a for which there is a b Arg(T) for which a HSub(b). Argumentation frameworks are now triples Arg(T), HArg(T), Attacks(T). Altogether attacks are defined as follows: Attacks(T) (Arg(T) Arg(T)) (Arg(T) HArg(T)) (HArg(T) HArg(T)) 22/43

62 Attacks again (non-nested case) Let HArg(T) be the set of all arguments a for which there is a b Arg(T) for which a HSub(b). Argumentation frameworks are now triples Arg(T), HArg(T), Attacks(T). Altogether attacks are defined as follows: Attacks(T) (Arg(T) Arg(T)) (Arg(T) HArg(T)) (HArg(T) HArg(T)) where a rebuts b = b 1,..., b n B iff Conc(a) = B or B = Conc(a) and 1. a Arg(T) and b Arg(T) HArg(T) or 22/43

63 Attacks again (non-nested case) Let HArg(T) be the set of all arguments a for which there is a b Arg(T) for which a HSub(b). Argumentation frameworks are now triples Arg(T), HArg(T), Attacks(T). Altogether attacks are defined as follows: Attacks(T) (Arg(T) Arg(T)) (Arg(T) HArg(T)) (HArg(T) HArg(T)) where a rebuts b = b 1,..., b n B iff Conc(a) = B or B = Conc(a) and 1. a Arg(T) and b Arg(T) HArg(T) or 2. a HArg(T) and b HArg(T) and Hyp(a) = Hyp(b). 22/43

64 Unrestricted Rebut

65 joint work: Jesse Heyninck and Christian Straßer 22/43

66 Status quo in ASPIC + only restricted rebut: a rebuts b iff 1. the conclusion of a is contrary to the conclusion of b 2. and b has a defeasible top rule 23/43

67 Status quo in ASPIC + only restricted rebut: a rebuts b iff 1. the conclusion of a is contrary to the conclusion of b 2. and b has a defeasible top rule unrestricted rebut: only the first requirement pro: natural (Caminada) contra: leads to trouble for many semantics such as preferred, stable, etc 23/43

68 Enters: Caminada et al. (COMMA 2014) for grounded semantics unrestricted rebut works just fine (really?) 24/43

69 Enters: Caminada et al. (COMMA 2014) for grounded semantics unrestricted rebut works just fine (really?) Rationality postulates: Sub-argument closure: where a E and b Sub(a), b E. 24/43

70 Enters: Caminada et al. (COMMA 2014) for grounded semantics unrestricted rebut works just fine (really?) Rationality postulates: Sub-argument closure: where a E and b Sub(a), b E. Closure under strict rules: where a 1,..., a n E Arg(T) and Conc(a 1 ),..., Conc(a n ) B, also a 1,..., a n B E Arg(T) 24/43

71 Enters: Caminada et al. (COMMA 2014) for grounded semantics unrestricted rebut works just fine (really?) Rationality postulates: Sub-argument closure: where a E and b Sub(a), b E. Closure under strict rules: where a 1,..., a n E Arg(T) and Conc(a 1 ),..., Conc(a n ) B, also a 1,..., a n B E Arg(T) Consistency: {Conc(a) a E} is consistent. 24/43

72 Grounded Semantics c d a b e f h i g 25/43

73 Grounded Semantics c d a b e f h i g first select unattacked arguments 25/43

74 Grounded Semantics c d a b e f h i g first select unattacked arguments remove the arguments attacked by the selected arguments 25/43

75 Grounded Semantics c d a b e f h i g first select unattacked arguments remove the arguments attacked by the selected arguments select unattacked arguments 25/43

76 Grounded Semantics c d a b e f h i g first select unattacked arguments remove the arguments attacked by the selected arguments select unattacked arguments remove the arguments attacked by the selected arguments 25/43

77 Grounded Semantics c d a b e f h i g first select unattacked arguments remove the arguments attacked by the selected arguments select unattacked arguments remove the arguments attacked by the selected arguments and so on until fixed point is reached 25/43

78 Non-Interference For a set of formulas F let Atoms(F) be the set of all propositional atoms in F. 1 Caminada, Carnielli, Dunne (JLC, 2012). Avron (2016) calls this the basic relevance criterion. 26/43

79 Non-Interference For a set of formulas F let Atoms(F) be the set of all propositional atoms in F. Non-interference 1 Where T = D, K and T = D, K are argumentation theories and A is a formula such that Atoms(D K {A}) Atoms(D K ) = then: T A iff D D, K K A. 1 Caminada, Carnielli, Dunne (JLC, 2012). Avron (2016) calls this the basic relevance criterion. 26/43

80 The problem with unrestricted rebut in ASPIC Take the knowledge base: { p}. 27/43

81 The problem with unrestricted rebut in ASPIC Take the knowledge base: { p}. Clearly: a = p is in the grounded extension. 27/43

82 The problem with unrestricted rebut in ASPIC Take the knowledge base: { p}. Clearly: a = p is in the grounded extension. Now, take the knowledge base: { p, s, s}. 27/43

83 The problem with unrestricted rebut in ASPIC Take the knowledge base: { p}. Clearly: a = p is in the grounded extension. Now, take the knowledge base: { p, s, s}. (Let the strict rules be closed under classical logic.) 27/43

84 The problem with unrestricted rebut in ASPIC Take the knowledge base: { p}. Clearly: a = p is in the grounded extension. Now, take the knowledge base: { p, s, s}. (Let the strict rules be closed under classical logic.) Now, a is attacked by s, s p. 27/43

85 The problem with unrestricted rebut in ASPIC Take the knowledge base: { p}. Clearly: a = p is in the grounded extension. Now, take the knowledge base: { p, s, s}. (Let the strict rules be closed under classical logic.) Now, a is attacked by s, s p. As a consequence, a is not in the grounded extension. 27/43

86 The problem with unrestricted rebut in ASPIC Take the knowledge base: { p}. Clearly: a = p is in the grounded extension. Now, take the knowledge base: { p, s, s}. (Let the strict rules be closed under classical logic.) Now, a is attacked by s, s p. As a consequence, a is not in the grounded extension. Thus, Non-Interference doesn t hold for unrestricted rebut. 27/43

87 Prima Facie solution sort out inconsistent arguments (Wu, 2012: this works in ASPIC + ) however, this doesn t work with unrestricted rebut 28/43

88 Prima Facie solution ii Let { 1 p, p 1 q, 2 (p q)} be our knowledge base. We have, e.g., the following arguments: a = 1 p b = a 1 q a b = a, b p q c = 2 (p q) a c = a, c q b c = b, c p a b c b c a c a b 29/43

89 Prima Facie solution ii Let { 1 p, p 1 q, 2 (p q)} be our knowledge base. We have, e.g., the following arguments: a = 1 p b = a 1 q a b = a, b p q c = 2 (p q) a c = a, c q b c = b, c p a b c b c a c a b Problem: b c is inconsistent and thus filtered out this leaves a and c in but a c out of the grounded extension. Failure of closure! 29/43

90 Enters: ASPIC : generalized unrestricted rebut lifting of the contrariness operator to (finite) sets of formulas, e.g., 30/43

91 Enters: ASPIC : generalized unrestricted rebut lifting of the contrariness operator to (finite) sets of formulas, e.g., {A 1,..., A n } = df n i=1 A i, or 30/43

92 Enters: ASPIC : generalized unrestricted rebut lifting of the contrariness operator to (finite) sets of formulas, e.g., {A 1,..., A n } = df n i=1 A i, or {A 1,..., A n } = df n i=1 A i. 30/43

93 Enters: ASPIC : generalized unrestricted rebut lifting of the contrariness operator to (finite) sets of formulas, e.g., {A 1,..., A n } = df n i=1 A i, or {A 1,..., A n } = df n i=1 A i. Concs(a) = df {Conc(b) b Sub(a)} 30/43

94 Enters: ASPIC : generalized unrestricted rebut lifting of the contrariness operator to (finite) sets of formulas, e.g., {A 1,..., A n } = df n i=1 A i, or {A 1,..., A n } = df n i=1 A i. Concs(a) = df {Conc(b) b Sub(a)} 30/43

95 Enters: ASPIC : generalized unrestricted rebut lifting of the contrariness operator to (finite) sets of formulas, e.g., {A 1,..., A n } = df n i=1 A i, or {A 1,..., A n } = df n i=1 A i. Concs(a) = df {Conc(b) b Sub(a)} Definition a gen-rebuts b iff b is defeasible and Conc(a) = for some Concs(b). Definition a gen-defeats b iff a gen-rebuts c for some c Sub(b) and c a. 30/43

96 Back to the example a b c b c a c a b 31/43

97 Rationality Where the strict rules are obtained from classical logic, for weakest link we get sub-argument closure closure under strict rules consistency non-interference 32/43

98 Comparative Studies

99 Jesse Heyninck, Christian Straßer: Relations between assumption-based approaches in nonmonotonic logic and formal argumentation (NMR 2016, Cape Town, also available on Arxiv) 32/43

100 The landscape ABA: assumption-based argumentation (Dung, Kowalski, Toni) ALs: adaptive logics (Batens) DACR: default assumptions (Makinson) KLM: preferential semantics (Shoham, Kraus/Lehman/Magidor) 33/43

101 On the argumentative side ASPIC + : defeasible and strict rules various attack types: rebuts, undercuts, undermine arguments as proof trees 34/43

102 On the argumentative side ASPIC + : defeasible and strict rules various attack types: rebuts, undercuts, undermine arguments as proof trees ABA (Assumption-based argumentation) only strict rules higher level of abstraction: arguments as sets of defeasible assumptions only assumption-attacks ( undermine) our translation: without priorities 34/43

103 Currently extending (Heyninck, Straßer, NMR 2016) with priorities relations between adaptive logics and parametrized logic programming (Jesse Heyninck, Pere Pardo, Christian Straßer) 35/43

104 Sequent-based argumentation (with Ofer Arieli, Tel Aviv)

105 Sequent-based Argumentation 36/43

106 Sequent-based Argumentation arguments are C-provable sequents, where 36/43

107 Sequent-based Argumentation arguments are C-provable sequents, where C is a sound and complete sequent-calculus 36/43

108 Sequent-based Argumentation arguments are C-provable sequents, where C is a sound and complete sequent-calculus of a (Tarskian) core logic L 36/43

109 Sequent-based attacks: elimination rules Attacker Sequent Conditions Attacked Sequent Eliminated Sequent 37/43

110 Sequent-based attacks: elimination rules Examples Attacker Sequent Conditions Attacked Sequent Eliminated Sequent Undercut: Γ 1 ψ 1 ψ 1 Γ 2 Γ 2, Γ 2 ψ 2 Γ 2, Γ 2 ψ 2 37/43

111 Sequent-based attacks: elimination rules Examples Attacker Sequent Conditions Attacked Sequent Eliminated Sequent Γ 1 ψ 1 ψ 1 Γ 2 Γ 2, Γ 2 Undercut: ψ 2 Γ 2, Γ 2 ψ 2 Γ 1 Γ 2 Γ 2, Γ 2 Compact Undercut: ψ Γ 2, Γ 2 ψ 37/43

112 Sequent-based attacks: elimination rules Examples Attacker Sequent Conditions Attacked Sequent Eliminated Sequent Γ 1 ψ 1 ψ 1 Γ 2 Γ 2, Γ 2 Undercut: ψ 2 Γ 2, Γ 2 ψ 2 Γ 1 Γ 2 Γ 2, Γ 2 Compact Undercut: ψ Γ 2, Γ 2 ψ Rebuttal: Γ 1 ψ 1 ψ 1 ψ 2 Γ 2 ψ 2 Γ 2 ψ 2 37/43

113 Sequent-based attacks: elimination rules Examples Attacker Sequent Conditions Attacked Sequent Eliminated Sequent Γ 1 ψ 1 ψ 1 Γ 2 Γ 2, Γ 2 Undercut: ψ 2 Γ 2, Γ 2 ψ 2 Γ 1 Γ 2 Γ 2, Γ 2 Compact Undercut: ψ Γ 2, Γ 2 ψ Rebuttal: Specificity, etc. Γ 1 ψ 1 ψ 1 ψ 2 Γ 2 ψ 2 Γ 2 ψ 2 37/43

114 Dynamic proof theories 1. p p Axiom 2. p, p [ ], 1 3. p p [ ], 2 4. p p (p p) (p p) p p q q Axiom 7. p p Axiom 8. p p Ucut, 7, 7, 7, 1 p p 9. p p 10. p p 11. p p Ucut, 1, 9, 10, 7 p p 38/43

115 Dynamic proof theories (Retraction, Basic Idea) function Evaluate(D) /* D Attack := ; Elim := ; Derived := ; while (D is not empty) do { if (Top(D) = i, s, J, ) then Derived := Derived {s}; A derivation must be coherent: Attack(D) Elim(D) = if (Top(D) = i, s, J, r ) then if (r Elim) then Elim := Elim {s} and Attack := Attack {r}; D := Tail(D); } Accept := Derived Elim; return (Attack, Elim, Accept) 39/43

116 Dynamic proof theories (Retraction, Basic Idea) function Evaluate(D) /* D Attack := ; Elim := ; Derived := ; while (D is not empty) do { if (Top(D) = i, s, J, ) then Derived := Derived {s}; if (Top(D) = i, s, J, r ) then if (r Elim) then Elim := Elim {s} and Attack := Attack {r}; D := Tail(D); } Accept := Derived Elim; return (Attack, Elim, Accept) A derivation must be coherent: Attack(D) Elim(D) = A sequent A is finally derived in a dynamic derivation D if A Accept(D) and D cannot be extended to a dynamic derivation D such that A Elim(D ). 39/43

117 Sequent-based Argumentation: some publications Ofer Arieli, Christian Straßer, Sequent-Based Logical Argumentation, in Argument and Computation, Vol. 6, Issue 1, pp , 2015 Ofer Arieli, Christian Straßer, Dynamic Derivations for Sequent-Based Deductive Argumentation, Proceedings of Computational Models of Argument (Editors: S. Parsons, N. Oren, C. Reed, and F. Cerutti) in the series Frontiers in Artificial Intelligence and Applications, Volume 266, IOS Press, pp , 2014 Christian Straßer and Ofer Ariel, Normative Reasoning by Sequent-Based Argumentation, Journal of Logic and Computation, doi.org/ /logcom/exv050 (2015) Ofer Arieli and Christian Straßer, Deductive argumentation by enhanced sequent calculi and dynamic derivations, Electronic Notes in Theoretical Computer Science, 323, (2016). Ofer Arieli, Annemarie Borg, and Christian Straßer, Argumentative Approaches to Reasoning with Consistent Subsets of Premises in proceedings of IEA/AIE 2017 (full paper), Lecture Notes in Artificial Intelligence series, Springer (2017) 40/43

118 Agent-Based Models

119 joint work: AnneMarie Borg, Daniel Frey, Dunja Šešelja and Christian Straßer Borg A., Frey D., Šešelja D. and Straßer C. (accepted) An Argumentative Agent-Based Model of Scientific Inquiry, forthcoming in the Proceedings of IEA/AIE, Springer-Verlag (extended version at: Borg A., Frey D., Šešelja D. and Straßer C. (under revision) Epistemic Effects of Scientific Interaction: approaching the question with an argumentative agent-based model, special issue of Historical Social Research: Agent Based Modelling across Social Science, Economics, and Philosophy 41/43

120 Explanatory Argumentation Frameworks Šešelja and Straßer, Synthese, 2013, 190: /43

121 Abstract argumentation in our ABM Theory 1 We represent in an abstract way: arguments discovery relation attack relation Theory 2 43/43