Complexity of Abstract Argumentation

  • Paul E. Dunne
  • Michael Wooldridge

The semantic models discussed in Chapter 11 provide an important element of the formal computational theory of abstract argumentation. Such models offer a variety of interpretations for “collection of acceptable arguments” but are unconcerned with issues relating to their implementation. In other words, the extension-based semantics described earlier distinguish different views of what it means for a set, S, of arguments to be acceptable, but do not consider the procedures by which such a set might be identified.


Polynomial Time Decision Problem Abstract Argumentation Argumentation Framework Satisfying Assignment 
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  1. 1.
    D. Angluin and L. Valiant. Fast probabilistic algorithms for hamiltonian circuits and matchings. Jnl. of Comp. and System Sci., 18:82–93, 1979.MathSciNetGoogle Scholar
  2. 2.
    S. Arnborg, J. Lagergren, and D. Seese. Easy problems for tree-decomposable graphs. Jnl. of Algorithms, 12:308–340, 1991.MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    T. J. M. Bench-Capon, S. Doutre, and P. E. Dunne. Audiences in argumentation frameworks. Artificial Intelligence, 171:42–71, 2007.CrossRefMathSciNetGoogle Scholar
  4. 4.
    H. L. Bodlaender. A partial k-arboretum of graphs with bounded treewidth. Theoretical Computer Science, 209:1–45, 1998.MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    A. Bondarenko, P. Dung, R. Kowalski, and F. Toni. An abstract, argumentation-theoretic approach to default reasoning. Artificial Intelligence, 93:63–101, 1997.MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    M. Caminada. Semi-stable semantics. In P. E. Dunne and T. J. M. Bench-Capon, editors, Proc. 1st Int. Conf. on Computational Models of Argument, volume 144 of FAIA, pages 121–130. IOS Press, 2006.Google Scholar
  7. 7.
    M. Caminada. An algorithm for computing semi-stable semantics. In Proc. of ECSQARU 2007, 9th uropean Conference on Symbolic and Quantitative Approaches to Reasoning with Uncertainty, pages 222–234, Hammamet, Tunisia, 2007.CrossRefGoogle Scholar
  8. 8.
    R. Chang and J. Kadin. On computing Boolean connectives of characteristic functions. Math. Syst. Theory, 28:173–198, 1995.MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    R. Chang, J. Kadin, and P. Rohatgi. On unique satisfiability and the threshold behavior of randomised reductions. Jnl. of Comp. and Syst. Sci., pages 359–373, 1995.Google Scholar
  10. 10.
    S. A. Cook. The complexity of theorem-proving procedures. In STOC ’71: Proc. of the 3rd Annual ACM Symposium on Theory of Computing, pages 151–158, New York, NY, USA, 1971. ACM.CrossRefGoogle Scholar
  11. 11.
    S. A. Cook and R. A. Reckhow. The relative complexity of propositional proof systems. Journal of Symbolic Logic, 44(1):36–50, 1979.MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    S. Coste-Marquis, C. Devred, and P. Marquis. Prudent semantics for argumentation frameworks. In Proc. 17th IEEE Intnl.Conf. on Tools with AI (ICTAI 2005), pages 568–572. IEEE Computer Society, 2005.Google Scholar
  13. 13.
    S. Coste-Marquis, C. Devred, and P. Marquis. Symmetric argumentation frameworks. In L. Godo, editor, Proc. 8th European Conf. on Symbolic and Quantitative Approaches to Reasoning With Uncertainty (ECSQARU), volume 3571 of LNAI, pages 317–328. Springer-Verlag, 2005.Google Scholar
  14. 14.
    B. Courcelle. The monadic second-order logic of graphs. I. recognizable sets of finite graphs. Information and Computation, 85(1):12–75, 1990.MATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    B. Courcelle. The monadic second-order logic of graphs III: tree-decompositions, minor and complexity issues. Informatique Théorique et Applications, 26:257–286, 1992.MATHMathSciNetGoogle Scholar
  16. 16.
    N. Creignou. The class of problems that are linearly equivalent to satisfiability or a uniform method for proving np-completeness. Theoretical Computer Science, 145(1-2):111–145, 1995.MATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    W. F. de la Vega. Kernels in random graphs. Discrete Math., 82(2):213–217, 1990.CrossRefMathSciNetGoogle Scholar
  18. 18.
    Y. Dimopoulos, B. Nebel, and F. Toni. Preferred arguments are harder to compute than stable extensions. In D. Thomas, editor, Proc. of the 16th International Joint Conference on Artificial Intelligence (IJCAI-99-Vol1), pages 36–43, San Francisco, 1999. Morgan Kaufmann Publishers.Google Scholar
  19. 19.
    Y. Dimopoulos, B. Nebel, and F. Toni. Finding admissible and preferred arguments can be very hard. In A. G. Cohn, F. Giunchiglia, and B. Selman, editors, KR2000: Principles of Knowledge Representation and Reasoning, pages 53–61, San Francisco, 2000. Morgan Kaufmann.Google Scholar
  20. 20.
    Y. Dimopoulos, B. Nebel, and F. Toni. On the computational complexity of assumption-based argumentation for default reasoning. Artificial Intelligence, 141:55–78, 2002.CrossRefMathSciNetGoogle Scholar
  21. 21.
    Y. Dimopoulos and A. Torres. Graph theoretical structures in logic programs and default theories. Theoretical Computer Science, 170:209–244, 1996.MATHMathSciNetGoogle Scholar
  22. 22.
    P. M. Dung. On the acceptability of arguments and its fundamental role in nonmonotonic reasoning, logic programming, and N-person games. Artificial Intelligence, 77:321–357, 1995.MATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    P. M. Dung, P. Mancarella, and F. Toni. A dialectical procedure for sceptical assumption-based argumentation. In P. E. Dunne and T. J. M. Bench-Capon, editors, Proc. 1st Int. Conf. on Computational Models of Argument, volume 144 of FAIA, pages 145–156. IOS Press, 2006.Google Scholar
  24. 24.
    P. M. Dung, P. Mancarella, and F. Toni. Computing ideal sceptical argumentation. Artificial Intelligence, 171:642–674, 2007.CrossRefMathSciNetGoogle Scholar
  25. 25.
    P. E. Dunne. Computational properties of argument systems satisfying graph-theoretic constraints. Artificial Intelligence, 171:701–729, 2007.CrossRefMathSciNetGoogle Scholar
  26. 26.
    P. E. Dunne. The computational complexity of ideal semantics. Technical Report ULCS-08-015, Dept. of Comp. Sci., Univ. of Liverpool, August 2008.Google Scholar
  27. 27.
    P. E. Dunne. The computational complexity of ideal semantics I: abstract argumentation frameworks. In Proc. 2nd Int. Conf. on Computational Models of Argument, volume 172 of FAIA, pages 147–158. IOS Press, 2008.Google Scholar
  28. 28.
    P. E. Dunne and T. J. M. Bench-Capon. Coherence in finite argument systems. Artificial Intelligence, 141:187–203, 2002.MATHCrossRefMathSciNetGoogle Scholar
  29. 29.
    P. E. Dunne and T. J. M. Bench-Capon. Two party immediate response disputes: properties and efficiency. Artificial Intelligence, 149:221–250, 2003.MATHCrossRefMathSciNetGoogle Scholar
  30. 30.
    P. E. Dunne and T. J. M. Bench-Capon. Complexity in value-based argument systems. In Proc. 9th JELIA, volume 3229 of LNAI, pages 360–371. Springer-Verlag, 2004.Google Scholar
  31. 31.
    P. E. Dunne and M. Caminada. Computational complexity of semi-stable semantics in abstract argumentation frameworks. In Proc. 11th JELIA, volume 5293 of LNAI, pages 153–165. Springer-Verlag, 2008.Google Scholar
  32. 32.
    A. Fraenkel. Planar kernel and grundy with d≤3, d out≤2, d in≤2 are NP–complete. Discrete Appl. Math., 3(4):257–262, 1981.MATHCrossRefMathSciNetGoogle Scholar
  33. 33.
    M. R. Garey and D. S. Johnson. Computers and Intractability: A Guide to the Theory of NP-Completeness. W. H. Freeman: New York, 1979.MATHGoogle Scholar
  34. 34.
    G. Gentzen. Investigations into logical deductions, 1935. In M. E. Szabo, editor, The Collected Papers of Gerhard Gentzen, pages 68–131. North-Holland Publishing Co., Amsterdam, 1969.Google Scholar
  35. 35.
    A. S. Lapaugh and C. H. Papadimitriou. The even path problem for graphs and digraphs. Networks, 14(4):597–614, 1984.CrossRefMathSciNetGoogle Scholar
  36. 36.
    L. Levin. Average case complete problems. SIAM J. Comput., 15:285–286, 1986.MATHCrossRefMathSciNetGoogle Scholar
  37. 37.
    D. Mitchell, B. Selman, and H. Levesque. Hard and easy distributions of sat problems. In Proc. AAAI-92, pages 459–465. AAAI/MIT Press, 1992.Google Scholar
  38. 38.
    R. C. Moore. Semantical considerations on nonmonotonic logic. Artificial Intelligence, 25:75–94, 1985.MATHCrossRefMathSciNetGoogle Scholar
  39. 39.
    R. Reiter. A logic for default reasoning. Artificial Intelligence, 13:81–132, 1980.MATHCrossRefMathSciNetGoogle Scholar
  40. 40.
    B. Selman, H. Levesque, and D. Mitchell. A new method for solving hard satisfiability problems. In Proc. 10th National Conf. on Art. Intellig., pages 440–446, 1992.Google Scholar
  41. 41.
    I. Tomescu. Almost all digraphs have a kernel. Discrete Math., 84(2):181–192, 1990.MATHCrossRefMathSciNetGoogle Scholar
  42. 42.
    A. Urquhart. The complexity of Gentzen systems for propositional logic. Theoretical Computer Science, 66(1):87–97, 1989.MATHCrossRefMathSciNetGoogle Scholar
  43. 43.
    L. G. Valiant and V. V. Vazirani. NP is as easy as detecting unique solutions. Theoretical Computer Science, 47:85–93, 1986.MATHCrossRefMathSciNetGoogle Scholar
  44. 44.
    G. Vreeswijk and H. Prakken. Credulous and sceptical argument games for preferred semantics. In Proc. of JELIA’2000, The 7th European Workshop on Logic for Artificial Intelligence., pages 224–238, Berlin, 2000. Springer LNAI 1919, Springer Verlag.Google Scholar
  45. 45.
    C. Wrathall. Complete sets and the polynomial-time hierarchy. Theoretical Computer Science, 3:23–33, 1976.CrossRefMathSciNetGoogle Scholar
  46. 46.
    L. Wu and C. Tang. Solving the satisfiability problem by using randomized approach. Inf. Proc. Letters, 41:187–190, 1992.MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag US 2009

Authors and Affiliations

  1. 1.Dept. of Computer ScienceUniversity of LiverpoolLiverpoolUK

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