, Encoding of semantics Case of extension-based semantics

, The language is propositional with the vocabulary V = {a, b, c, d}. For the considered AF, the complete extensions are characterized by the following formula ?, See Section 7.1

, Then, given a set of arguments S, it can be checked whether S is a complete extension

, d }. For the considered AF, the complete extensions are characterized by the following formula ?: [(¬x a ? ¬x b ) ? (¬x d ? ¬x b ) ? (¬x b ? ¬x c )] ?

, The formulas can be instantiated on a given AF. For the considered AF, the complete labellings are characterized by the following formula

, For the considered AF, the complete labellings are characterized by the following formula

O. Arieli, W. A. Martin, and . Caminada, A general QBFbased formalization of abstract argumentation theory, Proceedings of the 4th Conference on Computational Models of Argument (COMMA'2012), vol.245, pp.105-116, 2012.

O. Arieli, W. A. Martin, and . Caminada, A QBF-based formalization of abstract argumentation semantics, J. Applied Logic, vol.11, issue.2, pp.229-252, 2013.

. Bbd-+-]-philippe, J. Besnard, S. Boudou, . Doutre, . Van-hieu et al., The SESAME website

C. Beierle, F. Brons, and N. Potyka, A software system using a SAT solver for reasoning under complete, stable, preferred, and grounded argumentation semantics, KI 2015: Advances in Artificial Intelligence -38th Annual German Conference on AI, pp.241-248, 2015.

P. Bisquert, C. Cayrol, F. Dupin-de-saint-cyr, and M. Bannay, Characterizing change in abstract argumentation systems, Trends in Belief Revision and Argumentation Dynamics, vol.48, pp.75-102, 2013.
URL : https://hal.archives-ouvertes.fr/hal-02875531

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

P. Besnard and S. Doutre, Checking the acceptability of a set of arguments, 10th International Workshop on Non-Monotonic Reasoning (NMR'04), pp.59-64, 2004.

P. Besnard, S. Doutre, and A. Herzig, Encoding Argument Graphs in Logic (regular paper), International Conference on Information Processing and Management of Uncertainty in Knowledge-based Systems (IPMU), vol.443, pp.345-354, 2014.

P. Besnard, S. Doutre, . Van-hieu, D. Ho, and . Longin, SESAME -A System for Specifying Semantics in Abstract Argumentation (regular paper), International Workshop on Systems and Algorithms for Formal Argumentation (SAFA), pp.40-51

G. Brewka, S. Ellmauthaler, H. Strass, J. P. Wallner, and S. Woltran, Abstract dialectical frameworks. an overview, The IfCoLog Journal of Logics and their Applications, vol.4, issue.8, pp.2263-2318, 2017.

G. Brewka, S. Ellmauthaler, H. Strass, J. P. Wallner, and S. Woltran, Abstract dialectical frameworks, Handbook of Formal Argumentation, pp.237-286, 2018.

G. Boella, D. M. Gabbay, L. Van-der-torre, and S. Villata, Support in abstract argumentation, Proceeding of the 2010 conference on Computational Models of Argument: Proceedings of COMMA 2010, pp.111-122, 2010.

H. Barringer, D. M. Gabbay, and J. Woods, Temporal dynamics of support and attack networks : From argumentation to zoology, Mechanizing Mathematical Reasoning. LNAI 2605, pp.59-98

R. Booth, S. Kaci, T. Rienstra, and L. Van-der-torre, A logical theory about dynamics in abstract argumentation, Scalable Uncertainty Management, vol.8078, pp.148-161, 2013.
URL : https://hal.archives-ouvertes.fr/lirmm-00863487

G. Brewka and S. Woltran, The ADF Project website

G. Brewka and S. Woltran, The GRAPPA Project website

F. Cerutti, P. E. Dunne, M. Giacomin, and M. Vallati, Computing preferred extensions in abstract argumentation: A sat-based approach, Theory and Applications of Formal Argumentation, TAFA 2013, vol.8306, pp.176-193

C. Cayrol, F. Dupin-de-saint-cyr, and M. Bannay, Change in abstract argumentation frameworks: Adding an argument, Journal of Artificial Intelligence Research, vol.38, pp.49-84, 2010.
URL : https://hal.archives-ouvertes.fr/hal-02875350

C. Cayrol, J. Fandinno, L. Fariñas-del-cerro, and M. , Valid attacks in argumentation frameworks with recursive attacks, Proc. of Thirteenth International Symposium on Commonsense Reasoning, volume 2052. CEUR Workshop Proceedings, 2017.
URL : https://hal.archives-ouvertes.fr/hal-01709146

C. Cayrol, J. Fandinno, L. Fariñas-del-cerro, and M. , Argumentation frameworks with recursive attacks and evidence-based support, Proc. of Tenth International Symposium on Foundations of Information and Knowledge Systems (FoIKS), volume LNCS 10833, pp.150-169, 2018.

C. Cayrol, L. Fariñas-del-cerro, and M. , Logical encodings of interactions in an argumentation graph with recursive attacks, 2017.

W. A. Martin, D. M. Caminada, and . Gabbay, A logical account of formal argumentation, Studia Logica, vol.93, issue.2-3, pp.109-145, 2009.

A. Cohen, S. Gottifredi, A. J. García, and G. R. Simari, A survey of different approaches to support in argumentation systems, The Knowledge Engineering Review, vol.29, pp.513-550, 2014.

A. Cohen, S. Gottifredi, A. J. García, and G. R. Simari, An approach to abstract argumentation with recursive attack and support, Journal of Applied Logic, vol.13, issue.4, pp.509-533, 2015.

F. Cerutti, M. Giacomin, and M. Vallati, Argsemsat: Solving argumentation problems using SAT, Computational Models of Argument -Proceedings of COMMA, pp.455-456, 2014.

C. Cayrol and M. , The Grafix website

C. Cayrol and M. , Gradual valuation for bipolar argumentation frameworks, Proc. of the eighth European Conference on Symbolic and Quantitative Approaches to Reasoning with Uncertainty (ECSQARU) -LNAI 3571, pp.366-377, 2005.
URL : https://hal.archives-ouvertes.fr/hal-02875474

C. Cayrol and M. , Bipolarity in argumentation graphs: towards a better understanding, International Journal of Approximate Reasoning, vol.54, issue.7, pp.876-899, 2013.
URL : https://hal.archives-ouvertes.fr/hal-01123535

C. Cayrol and M. , Acceptability semantics in recursive argumentation frameworks: Logical encoding and computation. Rapport de recherche IRIT/RR, 2018.

C. Cayrol and M. , Logical encoding of argumentation frameworks with higher-order attacks, Proc. of the 30th International Conference on Tools with Artificial Intelligence (ICTAI), 2018.
URL : https://hal.archives-ouvertes.fr/hal-02181912

S. Coste-marquis, C. Devred, and P. Marquis, Constrained argumentation frameworks, Proc. of KR, pp.112-122, 2006.
URL : https://hal.archives-ouvertes.fr/hal-00121160

S. Coste-marquis, S. Konieczny, J. Mailly, and P. Marquis, On the revision of argumentation systems: Minimal change of argument statuses, International Conference on Principles of Knowledge Representation and Reasoning (KR), pp.72-81, 2014.

S. Coste-marquis, S. Konieczny, J. Mailly, and P. Marquis, A translation-based approach for revision of argumentation frameworks, Logics in Artificial Intelligence, pp.397-411, 2014.

J. Luis-carballido, J. C. Nieves, and M. Osorio, Inferring preferred extensions by pstable semantics. Inteligencia Artificial, Revista Iberoamericana de Inteligencia Artificial, vol.13, issue.41, pp.38-53, 2009.

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

P. Florence-dupin-de-saint-cyr, C. Bisquert, and M. Cayrol, Argumentation update in YALLA (yet another logic language for argumentation), Int. J. Approx. Reasoning, vol.75, pp.57-92, 2016.

C. Devred and S. Doutre, Claire Lefèvre, and Pascal Nicolas. Dialectical proofs for constrained argumentation, Computational Models of Argument -Proceedings of COMMA, pp.159-170, 2010.

W. Dvo?ák, S. A. Gaggl, T. Linsbichler, and J. P. Wallner, Reduction-based approaches to implement Modgil's extended argumentation frameworks, Advances in Knowledge Representation, Logic Programming, and Abstract Argumentation -Essays Dedicated to Gerhard Brewka on the Occasion of his 60th Birthday, vol.9060, pp.249-264, 2015.

W. Dvo?ák, S. A. Gaggl, and S. Woltran, The ASPARTIX website

M. Diller, A. Haret, T. Linsbichler, S. Rümmele, and S. Woltran, An extension-based approach to belief revision in abstract argumentation, Proceedings of the Twenty-Fourth International Joint Conference on Artificial Intelligence (IJCAI 2015), pp.2926-2932, 2015.

S. Doutre, A. Herzig, and L. Perrussel, A dynamic logic framework for abstract argumentation, Proceedings of the 14th International Conference on Principles of Knowledge Representation and Reasoning (KR'2014), 2014.
URL : https://hal.archives-ouvertes.fr/hal-01787378

W. Dvo?ák, M. Järvisalo, J. P. Wallner, and S. Woltran, The CEGARTIX website

W. Dvo?ák, M. Järvisalo, J. P. Wallner, and S. Woltran, CEGARTIX: a SAT-based argumentation system, Proc. of Pragmatics of SAT, workshop of SAT conference, 2012.

D. Phan-minh, On the acceptability of arguments and its fundamental role in nonmonotonic reasoning, logic programming and n-person games, Artif. Intell, vol.77, issue.2, pp.321-358, 1995.

M. Diller, J. P. Wallner, and S. Woltran, Reasoning in abstract dialectical frameworks using quantified boolean formulas, Proceedings of the 5th Conference on Computational Models of Argument (COMMA'2014), pp.241-252, 2014.

M. Diller, J. P. Wallner, and S. Woltran, Reasoning in abstract dialectical frameworks using quantified boolean formulas, Argument & Computation, vol.6, issue.2, pp.149-177, 2015.

U. Egly, S. A. Gaggl, and S. Woltran, Answer-set programming encodings for argumentation frameworks, Argument & Computation, vol.1, issue.2, pp.147-177, 2010.

J. Fandinno and L. Cerro, Constructive logic covers argumentation and logic programming, Proc. of Sixteenth International Conference on Principles of Knowledge Representation and Reasoning (KR 2018), pp.128-137, 2018.

D. M. Gabbay, Fibring argumentation frames, Studia Logica, vol.93, pp.231-295, 2009.

D. M. Gabbay, Semantics for higher level attacks in extended argumentation frames part 1: Overview. Studia Logica, vol.93, pp.357-381, 2009.

D. M. Gabbay, Meta-logical Investigations in Argumentation Networks. Number 44 in Studies in logic, 2013.

S. A. Gaggl, Towards a general argumentation system based on answer-set programming, Technical Communications of the 26th International Conference on Logic Programming, vol.7, pp.265-269, 2010.

D. M. Gabbay and M. Gabbay, The attack as strong negation, part I, Logic Journal of the IGPL, vol.23, issue.6, pp.881-941, 2015.

D. M. Gabbay and M. Gabbay, The attack as intuitionistic negation, Logic Journal of the IGPL, vol.24, issue.5, pp.807-837, 2016.

D. Grossi, On the logic of argumentation theory, Proc. of AAMAS, pp.409-416, 2010.

N. Karacapilidis and D. Papadias, Computer supported argumentation and collaborative decision making: the hermes system, Information systems, vol.26, issue.4, pp.259-277, 2001.

J. Lagniez, E. Lonca, and J. Mailly, Coquiaas: A constraint-based quick abstract argumentation solver, 27th IEEE International Conference on Tools with Artificial Intelligence, ICTAI, pp.928-935, 2015.
URL : https://hal.archives-ouvertes.fr/hal-02380767

C. Lefèvre and P. Nicolas, The ASPeRiX website

T. Lehtonen, A. Niskanen, and M. Järvisalo, Satbased approaches to adjusting, repairing, and computing largest extensions of argumentation frameworks, Computational Models of Argument -Proceedings of COMMA, pp.193-204, 2018.

S. Modgil, Reasoning about preferences in argumentation frameworks, Artificial Intelligence, vol.173, pp.901-934, 2009.

F. Nouioua, AFs with necessities: further semantics and labelling characterization, Proc. of the International Conference on Scalable Uncertainty Management (SUM), pp.120-133, 2013.

F. Nouioua and V. Risch, Bipolar argumentation frameworks with specialized supports, Proc. of the IEEE International Conference on Tools with Artificial Intelligence (ICTAI), pp.215-218, 2010.

F. Nouioua and V. Risch, Argumentation frameworks with necessities, Proc. of the International Conference on Scalable Uncertainty Management (SUM), pp.163-176, 2011.

N. Oren and T. J. Norman, Semantics for evidence-based argumentation, Proc of the 2 nd international conference on Computational Models of Argument (COMMA), pp.276-284, 2008.

M. Osorio and J. C. Nieves, Range-based argumentation semantics as two-valued models, TPLP, vol.17, issue.1, pp.75-90, 2017.

M. Osorio, J. C. Nieves, and A. Santoyo, Complete extensions as clark's completion semantics, Mexican International Conference on Computer Science, ENC 2013, pp.81-88, 2013.

N. Oren, C. Reed, and M. Luck, Moving between argumentation frameworks, Proceeding of the conference on Computational Models of Argument (COMMA), pp.379-390, 2010.

F. Pu, G. Luo, and Z. Jiang, Encoding argumentation semantics by boolean algebra, IEICE Transactions, issue.4, pp.838-848, 2017.

S. Polberg and N. Oren, Revisiting support in abstract argumentation systems, 2014.

S. Polberg, Intertranslatability of abstract argumentation frameworks, 2017.

C. Sakama and T. Rienstra, Representing argumentation frameworks in answer set programming, Fundamenta Informaticae, vol.155, issue.3, pp.261-292, 2017.

. +-12]-serena, G. Villata, D. M. Boella, L. Gabbay, J. Van-der-torre et al., A logic of argumentation for specification and verification of abstract argumentation frameworks, Annals of Mathematics and Artificial Intelligence, vol.66, issue.1-4, pp.199-230, 2012.

S. Villata, G. Boella, D. M. Gabbay, and L. Van-der-torre, Modelling defeasible and prioritized support in bipolar argumentation, Annals of Mathematics and Artificial Intelligence, vol.66, issue.1-4, pp.163-197, 2012.
URL : https://hal.archives-ouvertes.fr/hal-01076568

B. Verheij, Deflog: on the logical interpretation of prima facie justified assumptions, Journal of Logic in Computation, vol.13, pp.319-346, 2003.

Y. Wu, M. Caminada, and D. M. Gabbay, Complete extensions in argumentation coincide with 3-valued stable models in logic programming, Studia Logica: An International Journal for Symbolic Logic, vol.93, pp.383-403, 2009.

M. Walicki and S. Dyrkolbotn, Finding kernels or solving SAT, Discrete Algorithms, vol.10, pp.146-164, 2012.

J. Peter-wallner, G. Weissenbacher, and S. Woltran, Advanced SAT techniques for abstract argumentation, Computational Logic in Multi-Agent Systems -14th International Workshop, pp.138-154, 2013.