;. Conflict-free-iff-s-?-de and Q. ,

, Admissible iff it is conflict free and (S ? Q) ? Acc(U)

, Complete iff it is conflict free and (S ? Q) = Acc(U)

, Preferred iff it is a -maximal admissible structure

, Grounded iff it is a -minimal complete structure

, Arg-preferred iff it is a ar -maximal admissible structure

;. Stable-iff-s-=-a-\-de and . Inh,

, L(x) = out) =? (?? ? K s.t. t(?) = x, K (?) = in and A (s(?)) = in)

, L(x) = out) ?= (?? ? K s.t. t(?) = x, K (?) = in and A (s(?)) = in)

, L(x) = in) =? (?? ? K s.t. t(?) = x, K (?) = out or A (s(?)) = out)

, L(x) = in) ?= (?? ? K s.t. t(?) = x, K (?) = out or A (s(?)) = out)

, x) = out) =? (?? ? K s.t. t(?) = x, K (?) = in and A (s(?)) = in

, According to the definition of Struct2Lab(U), we have x ? (De f (U) ? Inh(U)). Following the definitions of De f (U) and Inh(U), we can state that there exists an attack ? such that ? ? Q, s(?) ? S and t(?) = x. According to the definition of Struct2Lab(U), we have so K (?) =, Let x ? (A ? K) be an argument or an attack such that L(x) = out

, notion) and also between reinstatement labellings (AF notion) and reinstatement RAF labellings (RAF notion)

. Bibliography,

P. Baroni, M. Caminada, and M. Giacomin, An introduction to argumentation semantics, Knowledge Eng. Review, vol.26, issue.4, pp.365-410, 2011.

P. Baroni, F. Cerutti, P. E. Dunne, and M. Giacomin, Computing with infinite argumentation frameworks: The case of AFRAs, Proc. of TAFA, Revised Selected Papers, pp.197-214, 2011.

P. Baroni, F. Cerutti, M. Giacomin, and G. Guida, AFRA: Argumentation framework with recursive attacks, Intl. Journal of Approximate Reasoning, vol.52, pp.19-37, 2011.

H. Barringer, D. M. Gabbay, and J. Woods, Temporal dynamics of support and attack networks : From argumentation to zoology, Mechanizing Mathematical Reasoning, Essays in Honor of Jörg H. Siekmann on the Occasion of His 60th Birthday. LNAI 2605, pp.59-98, 2005.

M. Caminada, On the issue of reinstatement in argumentation, JELIA, pp.111-123, 2006.

C. Cayrol, J. Fandinno, L. Del-cerro, and M. Lagasquie-schiex, Valid attacks in argumentation frameworks with recursive attacks, 13th International Symposium on Commonsense Reasoning (Commonsense), volume 2052. CEUR-WS : Workshop proceedings, 2017.
URL : https://hal.archives-ouvertes.fr/hal-01709146

C. Cayrol, J. Fandinno, L. Del-cerro, and M. Lagasquie-schiex, Valid attacks in Argumentation Frameworks with Recursive Attacks (long version), 2019.

G. Charwat, W. Dvo?ák, S. A. Gaggl, J. P. Wallner, and S. Woltran, Methods for solving reasoning problems in abstract argumentation -A survey, Artificial Intelligence, vol.220, pp.28-63, 2015.

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, pp.321-357, 1995.

H. Li and J. Wu, Semantics of extended argumentation frameworks defined by renovation sets, International Conference on Principles and Practice of Multi-Agent Systems (PRIMA), pp.532-540, 2019.

S. , An abstract theory of argumentation that accommodates defeasible reasoning about preferences, Proc. of ECSQARU, pp.648-659, 2007.

S. , Reasoning about preferences in argumentation frameworks, Artif. Intell, vol.173, pp.901-934, 2009.