Computational Complexity of Semi-stable Semantics in Abstract Argumentation Frameworks

  • Paul E. Dunne
  • Martin Caminada
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5293)

Abstract

Semi-stable semantics offer a further extension based formalism by which the concept of “collection of justified arguments” in abstract argumentation frameworks may be described. In contrast to the better known stable semantics, one advantage of semi-stability is that any finite argumentation framework always has at least one semi-stable extension. Although there has been some development of the formal logical theory of semi-stable semantics so that several computational properties of these extensions have been identified, with the exception of some algorithmic studies, more detailed investigation of computational complexity issues has been neglected. Our purpose in this article is to present a number of results on the complexity of some natural decision questions for semi-stable semantics.

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References

  1. 1.
    Verheij, B.: Two approaches to dialectical argumentation: admissible sets and argumentation stages. In: Meyer, J.J., van der Gaag, L. (eds.) Proceedings of the Eighth Dutch Conference on Artificial Intelligence (NAIC 1996), pp. 357–368. Utrecht University, Utrecht (1996)Google Scholar
  2. 2.
    Verheij, B.: Deflog: on the logical interpretation of prima facie justified assumptions. Journal of Logic and Computation 13, 319–346 (2003)MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Caminada, M.: Semi-stable semantics. In: Dunne, P.E., Bench-Capon, T.J.M. (eds.) Proc. 1st Int. Conf. on Computational Models of Argument. FAIA, vol. 144, pp. 121–130. IOS Press, Amsterdam (2006)Google Scholar
  4. 4.
    Caminada, M., Ben-Naim, J.: Postulates for paraconsistent reasoning and fault tolerant logic programming. Technical Report Technical Report UU-CS-2007-004, Utrecht University (2007)Google Scholar
  5. 5.
    Dung, P.M.: On the acceptability of arguments and its fundamental role in nonmonotonic reason, logic programming, and N-person games. Artificial Intelligence 77, 321–357 (1995)MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Gelfond, M., Lifschitz, V.: The stable model semantics for logic programming. In: Proceedings of the 5th International Conference/Symposium on Logic Programming, pp. 1070–1080. MIT Press, Cambridge (1988)Google Scholar
  7. 7.
    Gelfond, M., Lifschitz, V.: Classical negation in logic programs and disjunctive databases. New Generation Computing 9(3/4), 365–385 (1991)CrossRefGoogle Scholar
  8. 8.
    Arieli, O., Avron, A.: The value of four values. Artificial Intellligence 102, 97–141 (1998)MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Reiter, R.: A logic for default reasoning. Artificial Intelligence 13, 81–132 (1980)MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Baroni, P., Giacomin, M.: On principle-based evaluation of extension-based argumentation semantics. Artificial Intelligence 171, 675–700 (2007); special issue on argumentation in artificial intelligenceCrossRefMathSciNetMATHGoogle Scholar
  11. 11.
    Baroni, P., Giacomin, M.: Comparing argumentation semantics with respect to skepticism. In: Mellouli, K. (ed.) ECSQARU 2007. LNCS (LNAI), vol. 4724, pp. 210–221. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  12. 12.
    Caminada, M.: An algorithm for computing semi-stable semantics. In: Mellouli, K. (ed.) ECSQARU 2007. LNCS (LNAI), vol. 4724, pp. 222–234. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  13. 13.
    Dimopoulos, Y., Torres, A.: Graph theoretical structures in logic programs and default theories. Theoretical Computer Science 170, 209–244 (1996)MATHMathSciNetGoogle Scholar
  14. 14.
    Dunne, P.E., Bench-Capon, T.J.M.: Coherence in finite argument systems. Artificial Intelligence 141, 187–203 (2002)MATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Dunne, P.E.: The computational complexity of ideal semantics I: abstract argumentation frameworks. In: Proc. 2nd Int. Conf. on Computational Models of Argument. FAIA, vol. 172, pp. 147–158. IOS Press, Amsterdam (2008)Google Scholar
  16. 16.
    Dung, P.M., Mancarella, P., Toni, F.: A dialectical procedure for sceptical assumption-based argumentation. In: Dunne, P.E., Bench-Capon, T.J.M. (eds.) Proc. 1st Int. Conf. on Computational Models of Argument. FAIA, vol. 144, pp. 145–156. IOS Press, Amsterdam (2006)Google Scholar
  17. 17.
    Dung, P.M., Mancarella, P., Toni, F.: Computing ideal sceptical argumentation. Artificial Intelligence 171, 642–674 (2007)CrossRefMathSciNetMATHGoogle Scholar
  18. 18.
    Papadimitriou, C.H.: Computational Complexity. Addison-Wesley, Reading (1994)MATHGoogle Scholar
  19. 19.
    Wagner, K.: Bounded query computations. In: Proc. 3rd Conf. on Structure in Complexity Theory, pp. 260–277 (1988)Google Scholar
  20. 20.
    Wagner, K.: Bounded query classes. SIAM Jnl. Comput. 19, 833–846 (1990)MATHCrossRefGoogle Scholar
  21. 21.
    Jenner, B., Toran, J.: Computing functions with parallel queries to NP. Theoretical Computer Science 141, 175–193 (1995)MATHCrossRefMathSciNetGoogle Scholar
  22. 22.
    Valiant, L.G., Vazirani, V.V.: NP is as easy as detecting unique solutions. Theoretical Computer Science 47, 85–93 (1986)MATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    Chang, R., Kadin, J.: On computing Boolean connectives of characteristic functions. Math. Syst. Theory 28, 173–198 (1995)MATHCrossRefMathSciNetGoogle Scholar
  24. 24.
    Dunne, P.E.: Computational properties of argument systems satisfying graph-theoretic constraints. Artificial Intelligence 171, 701–729 (2007)CrossRefMathSciNetMATHGoogle Scholar
  25. 25.
    Verheij, B.: A labelling approach to the computation of credulous acceptance in argumentation. In: Proc. IJCAI 2007, pp. 623–628 (2007)Google Scholar
  26. 26.
    Bondarenko, A., Dung, P., Kowalski, R., Toni, F.: An abstract, argumentation-theoretic approach to default reasoning. Artificial Intelligence 93, 63–101 (1997)MATHCrossRefMathSciNetGoogle Scholar
  27. 27.
    Dimopoulos, Y., Nebel, B., Toni, F.: Preferred arguments are harder to compute than stable extensions. In: Thomas, D. (ed.) Proceedings of the 16th International Joint Conference on Artificial Intelligence (IJCAI 1999, pp. 36–43. Morgan Kaufmann, San Francisco (1999)Google Scholar
  28. 28.
    Dimopoulos, Y., Nebel, B., Toni, F.: Finding admissible and preferred arguments can be very hard. In: Cohn, A.G., Giunchiglia, F., Selman, B. (eds.) KR2000: Principles of Knowledge Representation and Reasoning, pp. 53–61. Morgan Kaufmann, San Francisco (2000)Google Scholar
  29. 29.
    Dimopoulos, Y., Nebel, B., Toni, F.: On the computational complexity of assumption-based argumentation for default reasoning. Artificial Intelligence 141, 55–78 (2002)CrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2008

Authors and Affiliations

  • Paul E. Dunne
    • 1
  • Martin Caminada
    • 2
  1. 1.Department of Computer ScienceThe University of LiverpoolUK
  2. 2.Computer Science and Communications Research UnitUniversity of Luxembourg 

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