L. Amgoud and P. Besnard, A Formal Characterization of the Outcomes of Rule-Based Argumentation Systems, 3rd International Conference on Scalable Uncertainty Management (SUM'13), pp.78-91, 2013.
DOI : 10.1007/978-3-642-40381-1_7
URL : https://hal.archives-ouvertes.fr/hal-01239724

L. Amgoud, M. Caminada, C. Cayrol, M. C. Lagasquie, and H. Prakken, Towards a Consensual Formal Model: inference part, 2004.

M. Caminada and L. Amgoud, On the evaluation of argumentation formalisms, Artificial Intelligence, vol.171, issue.5-6, pp.286-310, 2007.
DOI : 10.1016/j.artint.2007.02.003

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.
DOI : 10.1007/978-3-642-39091-3_9

A. Cohen, A. J. García, and G. R. Simari, Backing and Undercutting in Defeasible Logic Programming, 11th European Conference on Symbolic and Quantitative Approaches to Reasoning with Uncertainty, pp.50-61, 2011.
DOI : 10.1007/s10503-005-4421-z

A. Cohen, A. J. García, and G. R. Simari, Backing and Undercutting in Abstract Argumentation Frameworks, 7th International Conference on Foundations of Information and Knowledge Systems, FoIKS'12, pp.107-123, 2012.
DOI : 10.1007/s10503-005-4421-z

P. M. Dung, On the acceptability of arguments and its fundamental role in nonmonotonic reasoning, logic programming and n-person games, Artificial Intelligence, vol.77, issue.2, pp.321-357, 1995.
DOI : 10.1016/0004-3702(94)00041-X

A. J. García and G. R. Simari, Defeasible logic programming: an argumentative approach, Theory and Practice of Logic Programming, pp.95-138, 2004.
DOI : 10.1017/S1471068403001674

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.
DOI : 10.1093/logcom/exn040

M. Gelfond and V. Lifschitz, Classical negation in logic programs and disjunctive databases, New Generation Computing, vol.38, issue.No. 3, pp.365-385, 1991.
DOI : 10.1145/116825.116838
URL : http://ai.ustc.edu.cn/cn/seminar/files/Classic_Negation_in_Logic_Programming_and_Disjunctive_Database.pdf

G. Governatori, M. J. Maher, G. Antoniou, and D. Billington, Argumentation Semantics for Defeasible Logic, Journal of Logic and Computation, vol.14, issue.5, pp.675-702, 2004.
DOI : 10.1093/logcom/14.5.675
URL : https://academic.oup.com/logcom/article-pdf/14/5/675/2776887/140675.pdf

W. Lukaszewicz, Considerations on default logic: an alternative approach, Computational Intelligence, vol.4, issue.1, pp.1-16, 1988.
DOI : 10.1111/j.1467-8640.1988.tb00086.x

J. L. Pollock, How to reason defeasibly, Artificial Intelligence, vol.57, issue.1, pp.1-42, 1992.
DOI : 10.1016/0004-3702(92)90103-5

H. Prakken, An abstract framework for argumentation with structured arguments, Argument & Computation, vol.4, issue.2, pp.93-124, 2010.
DOI : 10.1017/CBO9780511802034
URL : http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.149.3092

R. Reiter, A logic for default reasoning, Since E is closed under strict rules and ¬x ? Concs(E), then r n ? Concs(E). Thus, CN(Th(E)) ? Defs(Th(E)) =, pp.81-132, 1980.
DOI : 10.1016/0004-3702(80)90014-4
URL : http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.250.9224

. Sub, Since H is closed under sub-arguments, then (d i , y i ) ? E and thus y i ? Concs(E) Since H is closed under strict rules, ¬y j ? Concs(E) for all j = 1, . . . , l. The same reasoning holds for each strict rule y 1

¬. Indeed and ?. Concs, E) for all i = 1, . . . , l. By definition of derivation, there exists a strict rule r after r i such that Head(r i ) ? Body(r) thus ¬Head(r i ) ? Concs(E)

. Thus and ?. ¬head-(-r-i-)-?-r-i, Since H is closed under strict rules, r i ? Concs(E) But, r i ? Defs(E) (since r i ? Def(d)), This contradicts the fact that Th(E) is coherent

E. Let, . Ext-n-(-h-), O. Let, and . Th, It is clear that O is uniquely determined from E. Let us show that O is an option

D. It-is-obvious-that-s-?-?-s, Now, for every x ? F there is an argument ((x, ?) , x) ? Arg(T ) By definiteion of undercutting, such argument has no conflict with any other argument. Thus, all arguments of this form belong to every naive extension

?. S. Let-r, r is used in at least one argument, say a, of E. So, a has a subargument b=((x 1 , r 1 ), . . . (x n , r n ) , x n ) with r n = r and x n = Head(r) By closeness ander sub-arguments, b is also an argument of E. From the definition of derivation schema, for every x ? Body(r), x = x i for some i s

?. Suppose, D. ??, S. ?. ??-s.-t.-(-f-?, D. ?. , S. ?? et al., satisfies the previous conditions Every there is at least an argument a = (d, x) s.t. r ? Strict(d) ? Def(d) Clearly, E. But from the coherence

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

E. Let and . Ext-n, Since Option(E) = Th(E) and from the definition of functions Th and Arg it is obvious that E ? Arg(Option(E))

. Arg, This means that a = (d, x) is constructed from Option(E). So, x ? CN(Option(E)) and Def(d) ? Defs(Option(E))

. (. Arg, 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 ?

E. Let and . Ext, Since Th(E) ? POption(E) and from the definition of functions Th and Arg it is obvious that E ? Arg(POption(E))

. Arg, POption(E)). a = (d, x) is constructed from POption(E) So, Def

?. From-b and . We, But then, from bR u a, POption(E) must be incoherent. Contradiction with the fact that POption(E) is a preferred option. The second case is that E does not attack a but it does not defend it: there is b = (d ? , x ? ) / ? E such that bR u a and E does not attack b. From bR u a we have x ? ? d, Since Def(d) ? Defs(POption(E)) then x ? Defs(POption(E)

O. Let, S. , D. Popt, and . Arg, We prove that E is conflictfree , ?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 is 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

T. Tf, S. , D. ?f, S. , and D. ??, Def(d)) and x ? Def(d ? ) From the fourth conditions of the definition of a preferred option, there is r ?? ? Def(d) s.t. r ?? ? CN So, there is an argument c = (d ?? , r ?? ) with d ?? a minimal derivation of r ?? in O. Clearly, cR u b. Finally, Suppose that E is not maximal w.r.t. previous conditions. Thus, there is E ? s.t. E ? E ? and E ? is preferred, i.e., E ? is an maximal conflict-free set of arguments that defends all its elements, POption(E ? ). Clearly, D ? = D, because there every argument in E ? \ E uses at least a rule which is

. Proof, of Corollary 6. Follows immediately from the bijection between preferred options and preferred extensions

. Proof, of Corollary 7. Follows immediately from the bijection between preferred options and preferred extensions