71 i n ( X , I )← i t e r a t i o n ( I ) , J = I + 1 , inA ( X , J ) . 72 o u t ( X , I )← i t e r a t i o n ( I ) , J = I + 1 , inR ( X , J ) . 73
74 okA ( I )←s e l e c t ( X , I ) , a c ( X , F ) , i s m o d e l ( F , I ) . 75 okA ( I )←A: = { s e l e c t ( _ , I ) } , i t e r a t i o n ( I ) , A= 0 . 76 inA ( X , I )←okA ( I ) , s e l e c t ( X , I ) .
77
78 i n ( X , I )←okA ( I ) , u n d e c ( X , J ) , J : = I + 1 , i t e r a t i o n ( I ) . 79 o u t ( X , I )←okA ( I ) , u n d e c ( X , J ) , J : = I + 1 , i t e r a t i o n ( I ) . 80
81
82 % c h e c k w h e t h e r t h e s e l e c t e d e l e m e n t i s i n R o r n o t . 83 okR ( I )←s e l e c t ( X , I ) , a c ( X , F ) , nomodel ( F , I ) , n o t okA ( I ) . 84 i n ( X , I )←okR ( I ) , u n d e c ( X , J ) , J : = I + 1 , i t e r a t i o n ( I ) . 85 o u t ( X , I )←okR ( I ) , u n d e c ( X , J ) , J : = I + 1 , i t e r a t i o n ( I ) . 86 inR ( X , I )←okR ( I ) , s e l e c t ( X , I ) .
87
88 ok ( I )←okA ( I ) . 89 ok ( I )←okR ( I ) . 90
91 ←n o t ok ( I ) , i t e r a t i o n ( I ) . 92
93
94 wf (X)←inA ( X , 0 ) . 95 r (X)←inR ( X , 0 ) . 96
97 # m a x i m i z e [ wf (X)@2 ] . 98 # m a x i m i z e [ r (X)@1 ] . 99
100 # h i d e . 101 # show wf / 1 . 102 # show r / 1 .
Bibliography
Amgoud, L., Belabbes, S., and Prade, H. (2005). Towards a formal framework for the search of a consensus between autonomous agents. In Dignum, F., Dignum, V., Koenig, S., Kraus, S., Singh, M. P., and Wooldridge, M., editors,4th International Joint Conference on Autonomous Agents and Multiagent Systems (AAMAS 2005), pages 537–543. ACM.
Amgoud, L. and Cayrol, C. (2002). Inferring from inconsistency in preference-based argumen-tation frameworks. J. Autom. Reasoning, 29(2):125–169.
Amgoud, L., Cayrol, C., Lagasquie-Schiex, M.-C., and Livet, P. (2008). On bipolarity in argu-mentation frameworks. Int. J. Intell. Syst., 23(10):1062–1093.
Apt, K. R. (1997). From Logic Programming to Prolog, volume 368. Prentice Hall.
Baroni, P., Caminada, M., and Giacomin, M. (2011a). An introduction to argumentation seman-tics. Knowledge Eng. Review, 26(4):365–410.
Baroni, P., Cerutti, F., Giacomin, M., and Guida, G. (2011b). AFRA: Argumentation framework with recursive attacks. Int. J. Approx. Reasoning, 52(1):19–37.
Barth, E. M. and Krabbe, E. C. (1982). From Axiom to Dialogue: A Philosophical Study of Logics and Argumentation. W. de Gruyter.
Bench-Capon, T. J. M. (2002). Representation of case law as an argumentation framework.
In Bench-Capon, T. J. M., Daskalopoulu, A., and Winkels, R., editors, Proceedings of the Fifteenth Annual Conference on Legal Knowledge and Information Systems (JURIX 2002), pages 103–112. IOS Press.
Bench-Capon, T. J. M. and Dunne, P. E. (2005). Argumentation in AI and law: Editors’ intro-duction. Artif. Intell. Law, 13(1):1–8.
Bench-Capon, T. J. M. and Dunne, P. E. (2007). Argumentation in Artificial Intelligence. Artif.
Intell., 171(10-15):619–641.
Bench-Capon, T. J. M., Prakken, H., and Sartor, G. (2009). Argumentation in legal reasoning. In Simari, G. and Rahwan, I., editors, Argumentation in Artificial Intelligence, pages 363–382.
Springer US.
Besnard, P. and Hunter, A. (2005). Practical first-order argumentation. In Veloso, M. M. and Kambhampati, S., editors,The Twentieth National Conference on Artificial Intelligence and the Seventeenth Innovative Applications of Artificial Intelligence Conference (AAAI), pages 590–595. AAAI Press / The MIT Press.
Besnard, P. and Hunter, A. (2008). Elements of Argumentation, volume 47. MIT Press Cam-bridge.
Boella, G., Gabbay, D. M., van der Torre, L. W. N., and Villata, S. (2009). Meta-argumentation modelling I: Methodology and techniques. Studia Logica, 93(2-3):297–355.
Bondarenko, A., Dung, P. M., Kowalski, R. A., and Toni, F. (1997). An abstract, argumentation-theoretic approach to default reasoning. Artif. Intell., 93(1–2):63 – 101.
Brewka, G., Dunne, P. E., and Woltran, S. (2011a). Relating the semantics of abstract dialectical frameworks and standard AFs. InProceedings of the 22nd International Joint Conference on Artificial Intelligence (IJCAI 2011), pages 780–785. AAAI Press.
Brewka, G. and Eiter, T. (2007). Equilibria in heterogeneous nonmonotonic multi-context sys-tems. InProceedings of the Twenty-Second AAAI Conference on Artificial Intelligence, pages 385–390. AAAI Press.
Brewka, G. and Eiter, T. (2009). Argumentation context systems: A framework for abstract group argumentation. In Erdem, E., Lin, F., and Schaub, T., editors,10th International Con-ference on Logic Programming and Nonmonotonic Reasoning (LPNMR), volume 5753 of Lecture Notes in Computer Science, pages 44–57. Springer.
Brewka, G., Eiter, T., Fink, M., and Weinzierl, A. (2011b). Managed multi-context systems.
In Walsh, T., editor, Proceedings of the 22nd International Joint Conference on Artificial Intelligence (IJCAI 2011), pages 786–791. AAAI Press.
Brewka, G., Eiter, T., and Truszczy´nski, M. (2011c). Answer set programming at a glance.
Commun. ACM, 54(12):92–103.
Brewka, G. and Gordon, T. F. (2010). Carneades and abstract dialectical frameworks: A recon-struction. In Baroni, P., Cerutti, F., Giacomin, M., and Simari, G. R., editors,Computational Models of Argument: Proceedings of COMMA 2010, volume 216 of Frontiers in Artificial Intelligence and Applications, pages 3–12. IOS Press.
Brewka, G. and Woltran, S. (2010). Abstract dialectical frameworks. In Lin, F., Sattler, U., and Truszczy´nski, M., editors,Principles of Knowledge Representation and Reasoning: Proceed-ings of the Twelfth International Conference, KR 2010, pages 102–111. AAAI Press.
Calimeri, F., Ianni, G., Ricca, F., Alviano, M., Bria, A., Catalano, G., Cozza, S., Faber, W., Febbraro, O., Leone, N., Manna, M., Martello, A., Panetta, C., Perri, S., Reale, K., Santoro, M., Sirianni, M., Terracina, G., and Veltri, P. (2011). The third answer set programming com-petition: Preliminary report of the system competition track. In Delgrande, J. and Faber, W.,
BIBLIOGRAPHY editors,Logic Programming and Nonmonotonic Reasoning, volume 6645 ofLecture Notes in Computer Science, pages 388–403. Springer Berlin / Heidelberg.
Caminada, M. (2006). On the issue of reinstatement in argumentation. In Fisher, M., van der Hoek, W., Konev, B., and Lisitsa, A., editors,10th European Conference on Logics in Artifi-cial Intelligence (JELIA), volume 4160 ofLecture Notes in Computer Science, pages 111–123.
Springer.
Caminada, M. and Amgoud, L. (2007). On the evaluation of argumentation formalisms. Artif.
Intell., 171(5-6):286–310.
Caminada, M. and Wu, Y. (2011). On the limitations of abstract argumentation. In Causmaecker, P. D., Maervoet, J., Messelis, T., Verbeeck, K., and Vermeulen, T., editors,Proceedings of the 23rd Benelux Conference on Artificial Intelligence (BNAIC 2011), pages 51–58.
Chesñevar, C. I., Maguitman, A. G., and Loui, R. P. (2000). Logical models of argument. ACM Comput. Surv., 32(4):337–383.
Church, A. (1996). Introduction to Mathematical Logic, pt. 1 Reprint. Princeton Univeristy Press.
Coste-Marquis, S., Devred, C., and Marquis, P. (2006). Constrained argumentation frameworks.
In Doherty, P., Mylopoulos, J., and Welty, C. A., editors, Tenth International Conference on Principles of Knowledge Representation and Reasoning (KR), pages 112–122. AAAI Press.
Denecker, M., Vennekens, J., Bond, S., Gebser, M., and Truszczy´nski, M. (2009). The second answer set programming competition. In Erdem, E., Lin, F., and Schaub, T., editors, 10th International Conference on Logic Programming and Nonmonotonic Reasoning (LPNMR), volume 5753 ofLecture Notes in Computer Science, pages 637–654. Springer.
Dimopoulos, Y., Moraitis, P., and Amgoud, L. (2009). Extending argumentation to make good decisions. In Rossi, F. and Tsoukiàs, A., editors, Algorithmic Decision Theory, First In-ternational Conference (ADT), volume 5783 of Lecture Notes in Computer Science, pages 225–236. Springer.
Dimopoulos, Y. and Torres, A. (1996). Graph theoretical structures in logic programs and default theories. Theor. Comput. Sci., 170(1-2):209–244.
Dung, P. M. (1995). On the acceptability of arguments and its fundamental role in nonmonotonic reasoning, logic programming and n-person games. Artif. Intell., 77(2):321–358.
Dvoˇrák, W. (2011). On the complexity of computing the justification status of an argument.
In First International Workshop on the Theory and Applications of Formal Argumentation (TAFA-11), Barcelona, Catalonia, Spain,.
Dvoˇrák, W., Järvisalo, M., Wallner, J. P., and Woltran, S. (2012). Complexity-sensitive decision procedures for abstract argumentation. InProceedings of the 13th International Conference on Principles of Knowledge Representation and Reasoning (KR 2012), pages 54–64. AAAI Press.
Dvoˇrák, W., Morak, M., Nopp, C., and Woltran, S. (2011). dynpartix - a dynamic programming reasoner for abstract argumentation. Proceedings of the 19th International Conference on Applications of Declarative Programming and Knowledge Management (INAP 2011).
Dvoˇrák, W., Pichler, R., and Woltran, S. (2010). Towards fixed-parameter tractable algorithms for argumentation. InProceedings of the 12th International Conference on the Principles of Knowledge Representation and Reasoning (KR’10), pages 112–122.
Egly, U., Gaggl, S. A., and Woltran, S. (2008). Aspartix: Implementing argumentation frame-works using answer-set programming. In de la Banda, M. G. and Pontelli, E., editors,Logic Programming, 24th International Conference, ICLP 2008, Proceedings, volume 5366 of Lec-ture Notes in Computer Science, pages 734–738. Springer.
Egly, U., Gaggl, S. A., and Woltran, S. (2010). Answer-set programming encodings for argu-mentation frameworks. Argument and Computation, 1(2):147–177.
Eiter, T. and Gottlob, G. (1995). On the computational cost of disjunctive logic programming:
Propositional case. Ann. Math. Artif. Intell., 15(3-4):289–323.
Ellmauthaler, S. and Wallner, J. P. (2012). Evaluating Abstract Dialectical Frameworks with ASP. In Verheij, B., Szeider, S., and Woltran, S., editors,Proceedings of the 4th Conference on Computational Models of Argument (COMMA 2012), volume 245, pages 505–506. IOS Press.
Gabbay, D. M. (2009). Fibring argumentation frames.Studia Logica, 93(2-3):231–295.
Gebser, M., Kaminski, R., and Schaub, T. (2011a). Complex optimization in answer set pro-gramming. Computing Research Repository (CoRR), abs/1107.5742.
Gebser, M., Kaufmann, B., Kaminski, R., Ostrowski, M., Schaub, T., and Schneider, M. T.
(2011b). Potassco: The potsdam answer set solving collection. AI Commun., 24(2):107–124.
Gebser, M., Kaufmann, B., Neumann, A., and Schaub, T. (2007a). clasp: A Conflict-Driven Answer Set Solver. In Baral, C., Brewka, G., and Schlipf, J. S., editors, 9th International Conference on Logic Programming and Nonmonotonic Reasoning (LPNMR), volume 4483 of Lecture Notes in Computer Science, pages 260–265. Springer.
Gebser, M., Liu, L., Namasivayam, G., Neumann, A., Schaub, T., and Truszczy´nski, M. (2007b).
The first answer set programming system competition. In Baral, C., Brewka, G., and Schlipf, J. S., editors,9th International Conference on Logic Programming and Nonmonotonic Rea-soning (LPNMR), volume 4483 ofLecture Notes in Computer Science, pages 3–17. Springer.
Gelfond, M. and Lifschitz, V. (1988). The stable model semantics for logic programming. In Proceedings of the Fifth International Conference and Symposium on Logic Programming (ICLP/SLP), pages 1070–1080.
Giunchiglia, F. and Serafini, L. (1994). Multilanguage hierarchical logics or: How we can do without modal logics. Artif. Intell., 65(1):29–70.
BIBLIOGRAPHY Gordon, T. F., Prakken, H., and Walton, D. N. (2007). The Carneades model of argument and
burden of proof. Artif. Intell., 171:875–896.
Gordon, T. F. and Walton, D. (2009). Proof burdens and standards. In Simari, G. and Rahwan, I., editors,Argumentation in Artificial Intelligence, pages 239–258. Springer US.
Ierusalimschy, R., de Figueiredo, L. H., and Celes, W. (2007). The evolution of lua. In Pro-ceedings of the third Association for Computing Machinery (ACM) Special Interest Group on Programming Languages (SIGPLAN) conference on History of Programming Languages (HOPL), pages (2–1)–(2–26). ACM.
Johnson, D. S. (1992).Handbook of Theoretical Computer Science: Algorithms and Complexity, chapter 2: A Catalog of Complexity Classes, pages 68–162. MIT Press Cambridge.
Kakas, A. C. and Moraitis, P. (2006). Adaptive agent negotiation via argumentation. In Nakashima, H., Wellman, M. P., Weiss, G., and Stone, P., editors, 5th International Joint Conference on Autonomous Agents and Multiagent Systems (AAMAS 2006), pages 384–391.
ACM.
Leitsch, A. (1997). The Resolution Calculus. Springer-Verlag.
Lundström, J. E. (2009).On the Formal Modeling of Games of Language and Adversarial Argu-mentation: A Logic-Based Artificial Intelligence Approach. Universitetsbiblioteket, Uppsala University. Ph.D. Thesis.
Lundström, J. E., Aceto, G., and Hamfelt, A. (2011a). A dynamic metalogic argumenta-tion framework implementaargumenta-tion. In Bassiliades, N., Governatori, G., and Paschke, A., ed-itors,Rule-Based Reasoning, Programming, and Applications - 5th International Symposium (RuleML Europe), volume 6826 ofLecture Notes in Computer Science, pages 83–98. Springer.
Lundström, J. E., Aceto, G., and Hamfelt, A. (2011b). Towards a dynamic metalogic imple-mentation of legal arguimple-mentation. In Ashley, K. D. and van Engers, T. M., editors,The 13th International Conference on Artificial Intelligence and Law (ICAIL), pages 91–95. ACM.
Modgil, S. (2009). Reasoning about preferences in argumentation frameworks. Artif. Intell., 173(9–10):901 – 934.
Modgil, S. and Bench-Capon, T. J. M. (2011). Metalevel argumentation. J. Log. Comput., 21(6):959–1003.
Papadimitriou, C. H. (1994). Computational Complexity. Addison-Wesley.
Prakken, H. and Sartor, G. (1997). Argument-based extended logic programming with defeasible priorities. Journal of Applied Non-Classical Logics, 7(1).
Prakken, H. and Vreeswijk, G. (2002). Logics for defeasible argumentation. Handbook of Philosophical Logic, 4:219–318.
Rahwan, I. and Simari, G., editors (2009). Argumentation in Artificial Intelligence. Springer US.
Rothmaler, P. (2000).Introduction to Model Theory. Algebra, Logic, and Applications. Gordon
& Breach.
South, M., Vreeswijk, G., and Fox, J. (2008). Dungine: A java dung reasoner. In Besnard, P., Doutre, S., and Hunter, A., editors,Proceedings of the 2nd Conference on Computational Models of Argument (COMMA 208), volume 172 of Frontiers in Artificial Intelligence and Applications, pages 360–368. IOS Press.
Toulmin, S. E. (2003). The uses of Argument. Press Syndicate of the University of Cambridge, 2nd edition.
Verheij, B. (2007). A labeling approach to the computation of credulous acceptance in argumen-tation. In Veloso, M. M., editor,Proceedings of the 20th International Joint Conference on Artificial Intelligence (IJCAI 2007), pages 623–628.
Villata, S., Boella, G., and van der Torre, L. W. N. (2011). Attack semantics for abstract argu-mentation. In Walsh, T., editor, Proceedings of the 22nd International Joint Conference on Artificial Intelligence (IJCAI 2011), pages 406–413. IJCAI/AAAI.
Walton, D. N. (1996). Argumentation Schemes for Presumptive Reasoning. Lawrence Erlbaum.
Weydert, E. (2011). Semi-stable extensions for infinite frameworks. In Causmaecker, P. D., Maervoet, J., Messelis, T., Verbeeck, K., and Vermeulen, T., editors,Proceedings of the 23rd Benelux Conference on Artificial Intelligence, pages 336–343.