L. Amgoud and F. Nouioua, Undercutting in argumentation systems, 9th International Conference on Scalable Uncertainty Management, SUM'15, pp.267-281, 2015.
URL : https://hal.archives-ouvertes.fr/hal-01592025

A. Bondarenko, P. Dung, R. Kowalski, and F. Toni, An abstract, argumentationtheoretic approach to default reasoning, Artificial Intelligence, vol.93, pp.63-101, 1997.

P. Dung, On the acceptability of arguments and its fundamental role in nonmonotonic reasoning, logic programming and n-person games, Artificial Intelligence Journal, vol.77, issue.2, pp.321-357, 1995.

L. Amgoud and J. Ben-naim, Ranking-based semantics for argumentation frameworks, 7th International Conference on Scalable Uncertainty Management (SUM'13), vol.8078, pp.134-147, 2013.
URL : https://hal.archives-ouvertes.fr/hal-01239719

C. Cayrol and M. Lagasquie-schiex, Graduality in argumentation, Journal of Artificial Intelligence Research, vol.23, pp.245-297, 2005.
URL : https://hal.archives-ouvertes.fr/hal-01925532

L. Amgoud and C. Cayrol, Inferring from inconsistency in preference-based argumentation frameworks, International Journal of Automated Reasoning, vol.29, issue.2, pp.125-169, 2002.

N. Gorogiannis and A. Hunter, Instantiating abstract argumentation with classical logic arguments: Postulates and properties, Artificial Intelligence Journal, vol.175, issue.9, pp.1479-1497, 2011.

L. Amgoud and P. Besnard, Logical limits of abstract argumentation frameworks, Journal of Applied Non-Classical Logics, vol.23, issue.3, pp.229-267, 2013.
URL : https://hal.archives-ouvertes.fr/hal-01124391

L. Amgoud, M. Caminada, C. Cayrol, M. Lagasquie, and H. Prakken, Towards a consensual formal model: inference part, Deliverable of ASPIC project

H. Prakken, An abstract framework for argumentation with structured arguments, Journal of Argument and Computation, vol.1, issue.2, pp.93-124, 2010.

A. García and G. Simari, Defeasible logic programming: an argumentative approach, Theory and Practice of Logic Programming, vol.4, issue.1-2, pp.95-138, 2004.

G. Governatori, M. Maher, G. Antoniou, and D. Billington, Argumentation semantics for defeasible logic, Journal of Logic and Computation, vol.14, issue.5, pp.675-702, 2004.

A. Wyner, T. Bench-capon, and P. Dunne, On the instantiation of knowledge bases in abstract argumentation frameworks, Proceedings of the 14th International Workshop on Computational Logic in Multi-Agent Systems, CLIMA XIV, pp.34-50, 2013.

A. Wyner, T. Bench-capon, P. Dunne, and F. Cerutti, Senses of 'argument' in instantiated argumentation frameworks, Argument & Computation, vol.6, issue.1, pp.50-72, 2015.

H. Strass, Instantiating knowledge bases in abstract dialectical frameworks, Proceedings of the 14th International Workshop on Computational Logic in Multi-Agent Systems, CLIMA XIV, pp.86-101, 2013.

M. Caminada and L. Amgoud, On the evaluation of argumentation formalisms, Artificial Intelligence Journal, vol.171, issue.5-6, pp.286-310, 2007.

J. Pollock, How to reason defeasibly, Artificial Intelligence Journal, vol.57, issue.1, pp.1-42, 1992.

M. Elvang-gøransson, J. Fox, and P. Krause, Acceptability of arguments as 'logical uncertainty, 2nd European Conference on Symbolic and Quantitative Approaches to Reasoning and Uncertainty (ECSQARU'93), vol.747, pp.85-90, 1993.

L. Amgoud and P. Besnard, A formal characterization of the outcomes of rulebased argumentation systems, 7th International Conference on Scalable Uncertainty Management SUM'13, pp.78-91, 2013.
URL : https://hal.archives-ouvertes.fr/hal-01239724

S. Benferhat and F. Bannay, 6th International Conference on Information Processing and Management of Uncertainty in Knowledge-Based Systems IPMU'96, pp.1-7, 1996.

H. Lam and G. Governatori, What are the necessity rules in defeasible reasoning?, Lecture Notes in Computer Science, vol.6645, pp.187-192, 2011.

L. Amgoud, Postulates for logic-based argumentation systems, International Journal of Approximate Reasoning, vol.55, issue.9, pp.2028-2048, 2014.
URL : https://hal.archives-ouvertes.fr/hal-01123709

R. Reiter, A logic for default reasoning, Artificial Intelligence Journal, vol.13, issue.1-2, pp.81-132, 1980.

W. Lukaszewicz, Considerations on default logic : An alternative approach, vol.4, pp.1-16, 1988.

M. Gelfond and V. Lifschitz, Classical negation in logic programs and disjunctive databases, New Generation Computing, vol.9, pp.365-385, 1991.

M. Gebser, M. Gharib, R. Mercer, and T. Schaub, Monotonic answer set programming, Journal of Logic and Computation, vol.19, issue.4, pp.539-564, 2009.

M. Caminada, S. Sá, and J. Alcântara, On the equivalence between logic programming semantics and argumentation semantics, 12th European Conference on Symbolic and Quantitative Approaches to Reasoning with Uncertainty ECSQARU'13, pp.97-108, 2013.

M. Caminada, S. Sá, J. Alcântara, and W. Dvorák, On the equivalence between logic programming semantics and argumentation semantics, International Journal of Approximate Reasoning, vol.58, pp.87-111, 2015.

D. Gaertner and F. Toni, Computing arguments and attacks in assumption-based argumentation, IEEE Intelligent Systems, vol.22, issue.6, pp.24-33, 2007.

S. Modgil and H. Prakken, A general account of argumentation with preferences, Artificial Intelligence, vol.195, pp.361-397, 2013.

P. M. Dung and T. C. Son, An argument-based approach to reasoning with specificity, Artif. Intell, vol.133, issue.1-2, pp.35-85, 2001.

A. Cohen, A. García, and G. Simari, Backing and undercutting in defeasible logic programming, 11th European Conference on Symbolic and Quantitative Approaches to Reasoning with Uncertainty, ECSQARU'11, pp.50-61, 2011.

A. Cohen, A. García, and G. Simari, Backing and undercutting in abstract argumentation frameworks, 7th International Conference on Foundations of Information and Knowledge Systems, FoIKS'12, pp.107-123, 2012.

A. Cohen, A. García, and G. Simari, A structured argumentation system with backing and undercutting, vol.49, pp.149-166, 2016.

L. Amgoud and J. Ben-naim, Proceedings of the 2015 International Conference on Autonomous Agents and Multiagent Systems, pp.1511-1519, 2015.

N. Rescher and R. Manor, On inference from inconsistent premises, Journal of Theory and Decision, vol.1, pp.179-219, 1970.

?. Suppose-that, D. ??, S. ??-s.t.-(f-?, D. , S. ?? et al., D ?? ) satisfies the previous conditions. For every rule r ? (S ?? ? D ?? ) \ (S ? ? D ? ), there is at least an argument a = (d, x) s.t. r ? Strict(d) ? Def(d)

E. Let and E. Ext-n, Let us show that E ? E ? . Let a = (d, x) ? E. Then, d is a derivation for x in NOption(E)

, Contradiction, since NOption(E) = NOption(E ? ). We show similarly that E ? ? E

, Th(E) and from the definition of functions Th and Arg it is obvious that E ? Arg(NOption(E)). Now let a = (d, x) ? Arg(NOption(E)). This means that a = (d, x) is constructed from NOption(E). So, x ? CN(NOption(E)) and Def(d) ? Defs(NOption(E))? ? CN(NOption(E)) and Def(d ? ) ? Defs(NOption(E)). But then

F. , S. , and D. Nopt, Let H = (Arg(T ), R u ) be a system built over a theory T . Let O =

?. , ?. , S. ??, and D. ??, But since d and d ? are derivation schemas for x and x ? respectively in O we have: x ? CN(O) and Def(d ? ) ? D ? , so x ? D ? . Contradiction with the coherence of naive option O. E = Arg(O) is conflictfree. Now, suppose that E is not maximal, NOption(E ? ) = Th(E ? ) = O ? is a naive option of T . Let O ? =, vol.3

F. , S. , D. ?-)), and . Body, Let Body(r) = {x 1 , . . . , x k } and Head(r) = y. Since for 1 ? i ? k, x i ? CN((F, S, D ? )), then there is an argument a i = (d i , x i ) ? E (1 ? i ? k) for each x i . Thus, we can construct an argument a for y using r as last rule, i.e., a = ( d, (y, r) , y) where Facts(d) = i Facts(d i ), Strict(d) = i Strict(d i ) and Def(d) = i Def, x, r) , x) is an argument outside E but since E is a stable extension, there is b ? E s.t. bR u a. So

F. , S. , and D. ?. ,

, Let us show that E ? E ? . Let a = (d, x) ? E. Then, d is a derivation for x in SOption(E)

, Contradiction, since SOption(E) = SOption(E ? ). We show similarly that E ? ? E

, SOption(E) and from the definition of functions Th and Arg it is obvious that E ? Arg(SOption(E)). Now let a = (d, x) ? Arg(SOption(E)). a = (d, x) is constructed from SOption(E). So, Def(d) ? Defs(SOption(E))

, Since E is a stable extension then there is b = (d ? , x ? ) ? E such that bR u a. From b ? E we easily deduce that x ? ? CN(SOption(E)). But then, from bR u a, SOption(E) must be incoherent. Contradiction with the fact that SOption(E) is a stable option. Proof of Theorem 6, Let H =

O. Let, S. , D. ?-)-?-sopt, and ;. Arg, We prove that E is conflict-free and ?b ? Arg(T ) \ E, ?a ? E s.t. aR u b. H = (Arg(T ), R u ) s.t. Ext p (H) = ?. Let us show that for all E ? Ext p (H)

E. , S. , D. , S. , D. et al., Def(d) ? D ? and Strict(d) ? S ? . Thus, a = ( d, (x, r) , x) is an argument outside E. a does not attack any argument of E. Indeed, if we suppose the contrary then, since E is a preferred extension, there is b ? E s.t. bR u a. So, there is a sub-argument of a: a ? = ( d ? , (x ? , r ? ) , x ? ) with r ? ? D ? and b = (d ?? , r ? ). However since a ? ? E (because it uses only rules from S ? ?D ? ), this means that E is not conflict-free which contradicts the fact that E is a preferred extension. So E ? {a} is conflict free. Moreover, for every c ? Arg(T ) \ (E ? {a}), if cR u a then there is a sub-argument of a: a ? = ( d ? , (x ? , r ? ) , x ? ) with x ? ? D ? and c = (d ?? , x ? ). However since a ? ? E (because it uses only rules from S ? ? D ? ) and E is a preferred extension, we complete S ? by the remaining strict rules). Clearly, O is uniquely determined from E. We have that Concs(E) = CN(Th(E)). Let us show that: CN

S. ?d-??-?-d,-if-?r-?-?-d-?-s.t.-r-?-?-cn(f, D. ??-;-f, S. , D. ??-),-i.e.-;-f, S. et al., Since E is a preferred extension, there is an argument c = (d ?? , x ?? ) ? E s.t. cR u b, i.e., there is a derivation d ?? for r ?? in (F, S, D ? ) s.t. d ?? ? Def(d ? ), then there is a minimal derivation d ? for r ? in

E. Thus, ?. Cn(f, and S. , Def(d)) and x ? Def(d ? ), i.e. x ? D ?? . But, from the fourth condition of preferred options, there is r ?? ? Def(d) such that r ?? ? CN(O). So, there is an argument a ? ? E ? such that a ?

E. and E. , We show by a similar way as in the second point of Theorem 5 that: for all E, E ? ? Ext p (H) if POption(E) = POption

, Suppose that a / ? E. Since E is a preferred extension then we have two cases. The first case is that there is b = (d ? , x ? ) ? E such that bR u a. From b ? E we easily deduce that x ? ? CN(POption(E)). But then, from bR u a, POption(E) must be incoherent. Contradiction with the fact that POption(E) is a preferred option, Since Th(E) ? POption(E) and from the definition of functions Th and Arg it is obvious that E ? Arg(POption(E))

, Since Def(d) ? Defs(POption(E)) then x ? Defs(POption(E)). So, x is used in at least an argument c = (d ?? , x ?? ) of E i.e., x ? d ?? . Thus, c is attacked by b. But since E is a preferred extension, then it must contain an argument which attacks b. This contradict the hypothesis that E does not attack b. Proof of Theorem 8, ? E such that bR u a and E does not attack b. From bR u a we have x ? ? d

O. Let, S. , D. ?-)-?-popt, and ;. Arg, We prove that E is conflict-free, ?b ? Arg(T ) \ E, if ?a ? E s.t. bR u a then ?c ? E s.t. cR u b and E is a maximal subset of Arg(T ) satisfying the previous two conditions. Suppose that there are two argument a = (d, x) and b = (d ? , x ? ) in E s.t. aR u b, i.e., x ? Def(d ? ). But since d and d ? are derivation schemas for x and x ? respectively in O we have: x ? CN(O) and Def(d ? ) ? D ?

, )) and x ? Def(d ? ). From the fourth conditions of the definition of a preferred option, Now, let us show that: ?b ? Arg(T ) \ E, if ?a ? E s.t. bR u a then ?c ? E s.t. cR u b

F. , S. , D. ??-)-=-poption-;-e-?-).-clearly, and D. ?-=-d, E ? is an maximal conflict-free set of arguments that defends all its elements, is an argument c = (d ?? , r ?? ) with d ?? a minimal derivation of r ?? in O. Clearly, cR u b. Finally

E. Let, . Ext-c-(h), S. Let-th-;-e)-=-(f-?, D. , S. et al., ) , x ? ) with r ? ? D ? and b = (d ?? , r ? ). However since a ? ? E (because it uses only rules from S ? ?D ? ), this means that E is not conflict-free which contradicts the fact that E is a complete extension, we complete S ? by the remaining strict rules). Clearly, O is uniquely determined from E. We have that Concs(E) = CN(Th(E)). Let us show that: CN

F. , S. , D. ??-))-=-?-;-f, S. , and D. ?. , It follows that there is an arguments b = (d ? , x ? ) s.t. Defs(d ? ) ? D ?? and x ? ? D 1 , i.e., b R u a. In this case, there is an argument c = (d 1 , x 1 ) s.t. c R u b, i.e., there exists r ?? ? Defs(d ? ) hence r ?? ? D ?? s.t. x 1 = r ??, Since clearly x 1 ? CN

, Follows immediately from the bijection between complete options and complete extensions, pp.9-10