K. Let, R. , N. , R. , and N. {?x, ? b (X ) ? c (X ) ? ?), ?X (e (X ) ? d (X ) ? ?)}. The corresponding argumentation framework is composed of 161 attacks and the following 20 arguments: ? a 0 : ({a(m)}, {a(m)})

, ? a 1 : ({b (m)}, {b (m)})

, ? a 2 : ({a(m), b (m)}, {a(m), b (m)})

, ? a 3 : ({c (m)}, {c (m)})

, ? a 4 : ({a(m), c (m)}, {a(m), c (m)})

, ? a 5 : ({b (m), c (m)}, {b (m), c (m)})

, ? a 6 : ({d (m)}, {d (m)})

, ? a 7 : ({a(m), d (m)}, {a(m), d (m)})

, ? a 8 : ({b (m), d (m)}, {b (m), d (m)})), d (m)})

, ? a 9 : ({a(m), b (m), d (m)}, {a(m), b (m), d (m)})

, ? a 10 : ({c (m), d (m)}, {c (m), d (m)})

, ? a 11 : ({a(m), c (m), d (m)}, {a(m), c (m), d (m)})

, ? a 12 : ({b (m), c (m), d (m)}, {b (m), c (m), d (m)})

, ? a 13 : ({e (m)}, {e (m)})

, ? a 14 : ({a(m), e (m)}, {a(m), e (m)})

, ? a 15 : ({b (m), e (m)}, {b (m), e (m)})

, ? a 16 : ({a(m), b (m), e (m)}, {a(m), b (m), e (m)})

, ? a 17 : ({c (m), e (m)}, {c (m), e (m)})

, ? a 18 : ({a(m), c (m), e (m)}, {a(m), c (m), e (m)})

, The set of cf2 extensions is the set {? 1, ? a 19 : ({b (m), c (m), e (m)}, {b (m), c (m), e (m)}) The preferred extensions will be composed of the following sets: ? ? 1 = {a 0, vol.3

, 11 (Characterisation of dummy arguments) Let KB = (F, R, N) be knowledge base such that R = ? and |F| = n, vol.3

, Att + (a) = ?}, the set of dummy arguments. 1. Let us prove that |U nn| ? 2 k ? 1. The set Free (KB) corresponds to the set of facts that are not in any conflict. Since k = |Free (KB)|, we conclude that there are at least 2 k ?1 arguments of the form (X , X ) that have a non empty subset X of Free (KB) as support, Let us consider U nn = {a ? A | Att ? (a) =

, Let us prove that |U nn| ? 2 k ? 1. By means of contradiction, we suppose that |U nn| > 2 k ? 1. It means that that there is a dummy argument a ? U nn such that Supp(a) Free (KB). Thus, there exists a minimal inconsistent set X ? MI (KB)

. Now, Y. Us-consider, and . Supp, We know that Y is not empty otherwise there is a contradiction with the consistency of the support of a. Furthermore, Y is R-consistent since Y ? X . Thus, there is an argument b = (Y , Y ) such that (b, a) ? R, contradiction. Proposition 3.12 (Number of arguments) Let KB = (F, R, N) be a knowledge base with R = ?. If |Free (KB)| = k then there is a subgraph of F KB = (A , R) that is a (2 k )-cop graph of

, If Free (KB) = ?, then it is obvious that F KB is a 1-cop graph of itself

, Hence, |A | = |{X | X is a non empty R-consistent subset of F \ Free (KB)}|. Now, let us consider F KB = (A , R), the argumentation framework corresponding to the knowledge base KB. We show that the subgraph F KB = (A , R ) of F KB where A = {a ? A | Supp(a) Free (KB)} and R = R |A is a (2 |F r ee (KB) | )-cop graph of F KB : ? We know that for any set X that is an R-consistent subset of F \ Free (KB), X ?X , where X is a subset of Free (KB), is an R-consistent set. Thus |A | = |{X ? X | X ? Free (KB) and X is a non empty R-consistent subset of F \ Free (KB)}|, Moreover, since R = ?, the arguments can only be of the form (X , X ) where X is an R-consistent subset of F \ Free (KB)

, We denote by f the function from A to A such that f (a ) = a iff Supp(a ) = Supp(a ) ? (F \ Free (KB)). We now show that this function CHAPTER

, Let ? ? Sat R (Output x (F KB )). Then there exist a minimal set {? 1 , . . . , ? k } ? Output x (F KB ) and a sequence of rule applications such that ? is produced from {? 1 , . . . , ? k }. Since ? 1 , . . . , ? k ? Output x (F KB ) then for every E ? Ext x (F KB ), we have ? 1

, From the first part of the proof, Sat R (Concs (E)) = Concs (E). Thus, for every E ? Ext x (F KB ), ? ? Concs (E). Thus, ? ? Output x (F KB )

, Given a logical argumentation framework F K = (A , R) with A being the set of arguments defined by Definition 3.1 and R the set of attacks defined according to Definition 3.2, the set R enjoys the following properties: 1. C1b : for every a, b, c ? A , if Conc (a) Conc (b) then

, C2b : for every a, b, c ? A , if Supp(a) Supp(b) then ((c, a) ? R if and only if (c, b) ? R)

, ) be a knowledge base expressed using existential rules and F KB = (A , R) the corresponding argumentation framework. Now

, Suppose that Conc (a)

, Conc (b). If (a, c) ? R, it means that there exists ? ? Supp(c) such that Sat R?N (Conc (a) ?{?}) |= ?. However, since Conc (a) Conc (b), we can infer that Sat R (Conc (a)) = Sat R (Conc (b)), thus Sat R?N (Conc (b) ? {?}) |= ? and (b, c) ? R. Likewise

, Conc (c) ?{?}) |= ?. However, since Supp(a) Supp(b), by definition, we have that there exists ? ? Supp(b) s.t. ? ?, i.e. Sat R (? ) = Sat R (?). Therefore, we can infer that Sat R (Conc (c) ? {? }) |= ? and (c, b) ? R. Likewise, (c, b) ? R implies (c, a) ? R which ends the proof, Suppose now that Supp(a) Supp(b). If (c, a) ? R, it means that there exists ? ? Supp(a) such that Sat R?N, vol.50, p.93

, Bipolar argumentation framework 114

.. .. Body,

, Boolean conjunctive query, vol.17, p.30

. .. Bottom-facts, , p.149

. .. Breadth-first-derivation,

, Burden-based ranking-based semantics

. .. Cf2-extension, , vol.36, p.63

.. .. Chase,

. .. Chase-finiteness-order,

.. .. Closure,

. .. Combined-complexity, , vol.26

. .. Complete-extension, , vol.35, p.37

.. .. Complete,

. .. Component-defeated, , p.36

. .. Conclusion, , vol.49, p.50

. .. Concrete-classes, , p.24

, Conflict-based argument, p.55

, Conflict-based complete graph, p.55

. .. Conflict-freeness, , vol.34, p.94

. .. Conflicting-facts, , vol.27, p.28

. .. Conjunction, , vol.14, p.15

. .. Conjunctive-query, , p.17

.. .. Constant,

.. .. Coquiaas,

. .. Core, , vol.116, p.118

. .. Counter-transitivity, , p.136

. .. Cp, , vol.126, p.133

. .. Credulously-accepted,

. .. Cycle, , vol.55, p.135

.. .. Datalog,

, Deductive argumentation, vol.7, p.49

, Defeasible logic programming

.. .. Defeasible,

. .. Delp, , vol.7, p.48

. .. Density, , vol.82, p.105

. .. Derivation, , vol.18, p.19

, Derivation for a fact, vol.19, p.20

.. .. Direct, , vol.33, p.132

, Discussion-based ranking-based semantics

. .. Distributed-defense, , p.133

, Distributed-defense precedence, p.133

. .. Dlgp, , vol.72, p.106

. .. Dot,

, Drastic Shapley inconsistency value 143, p.146

. .. Dummy-argument, , vol.56

. .. Ex-falso-quodlibet, 3 Exhaustive breadth-first derivation 20 Existence property

. .. Existential-quantifier,

. .. Falsum, , vol.23, p.51

, Filtrated argumentation framework 68

, Filtrated set of arguments, p.67

. .. Finite, see FES Finite unification set . . . . . . see FUS First-order logic, vol.1, p.14

. .. Flatness-property, , p.40

.. .. Flawed,

.. .. Formula,

. .. Free-fact, , vol.31, p.57

.. .. Free-property,

. .. Frontier, , vol.18, p.21

. .. Frontier-chase, , vol.21, p.22

. .. Frontier-derivation-reducer,

. .. Fus,

. .. Graal, , vol.72, p.80

. .. Graph, 24 GRD see Graph of rule dependency Greedy bounded tree-width set . see GBTS Ground atom, vol.35, p.97

. .. Grounded-labelling, , p.39

. .. Group-comparison, 135 H H-categoriser function, p.41

, H-categoriser ranking-based semantics

.. .. Hard,

.. .. Head,

. .. Homomorphism, , vol.16, p.17

.. .. Homomorphism-restriction, , p.17

, Hypergraph argumentation framework 92

. .. Hypothesis,

. .. Icr, , vol.35, p.37

.. .. In,

. .. Incoherence, , vol.3, p.29

. .. Inconsistency, 3, 28, 29 Inconsistency measure, vol.143, p.144

. .. Independence,

. .. Interpretation,

.. .. Killing, , vol.33, p.38

. .. Labsatsolver, , p.83

. .. Language, , vol.7, p.14

. .. Lexicographical-order, , p.41

. .. Leximax, 143, 144 xx INDEX Logic-based argumentation, vol.143, p.146

, Minimal inconsistent set, p.31

. .. Minimality-property, , p.149

. .. Model, 16 Model faithful acyclicity . . see MFA More restrictive equivalence relation 121

. .. Nae, , vol.126, p.28

. .. Nh-categoriser, , p.136

. .. Nh-categoriser-function, , p.136

, Nh-categoriser ranking-based semantics

, Non entailment ejection, p.151

, Non-attacked equivalence . see NaE Non-trivial

. .. Null, , vol.30, p.142

. .. Oblivious-fes, , p.23

.. .. ,

. .. Out, , vol.33, p.131

. .. Productivity, , vol.13, p.21

.. .. R-r-append,

. .. Range-restricted, , p.24

, Ranking-based semantics, vol.39, p.114

. .. Rationality-postulates, , p.99

. .. Reducer-order-relation, , vol.52, p.54

. .. Repair, 4, 31, 95 Repair semantics

. .. Restricted-fes, , p.23

. .. Rif, , vol.143, p.144

. .. Rule,

, Rule-based logics

, Saturated set of facts, vol.40, p.41

. .. Sct, , vol.126, p.136

, Self-attacking argument, vol.54, p.98

. .. Self-contradiction, , p.133

, Semi-stable extension, vol.35, p.54

.. .. Sentinel, , vol.68, p.94

. .. Set-defense, , vol.33, p.94

, Set of attacking argument, vol.67, p.91

. .. Shapley-inconsistency-value,

.. .. Skolem-chase,

. .. Skolem-term, , vol.16, p.21

. .. Skolem-fes, , pp.23-25

. .. Stage-semnatics, 36 Strict addition of defense branch 134 Strict counter-transitivity . see, vol.36, p.83

.. .. Subgraph,

. .. Support, , vol.49, p.50

, Support argumentation framework 114

. .. Supremacy, , p.151

.. .. Symmetry,

.. .. Term,

. .. Universal-quantifier,

.. .. Vocabulary,

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