Towards Fixed-Parameter Tractable Algorithms for Argumentation

Abstract argumentation frameworks have received a lot of interest in recent years. Most computational problems in this area are intractable but several tractable fragments have been identified. In particular, Dunne showed that many problems can be solved in linear time for argumentation frameworks of bounded tree-width. However, these tractability results, which were obtained via Courcelle's Theorem, do not directly lead to efficient algorithms. The goal of this paper is to turn the theoretical tractability results into efficient algorithms and to explore the potential of directed notions of tree-width for defining larger tractable fragments.

[1]  Yannis Dimopoulos,et al.  Graph theoretical structures in logic programs and default theories , 1996 .

[2]  Stephan Kreutzer,et al.  Digraph measures: Kelly decompositions, games, and orderings , 2007, SODA '07.

[3]  Jürgen Dix,et al.  Research challenges for argumentation , 2009, Computer Science - Research and Development.

[4]  Paolo Mancarella,et al.  Computing ideal sceptical argumentation , 2007, Artif. Intell..

[5]  Phan Minh Dung,et al.  On the Acceptability of Arguments and its Fundamental Role in Nonmonotonic Reasoning, Logic Programming and n-Person Games , 1995, Artif. Intell..

[6]  B. A. Reed,et al.  Algorithmic Aspects of Tree Width , 2003 .

[7]  Sylvie Doutre,et al.  Preferred Extensions of Argumentation Frameworks: Query Answering and Computation , 2001, IJCAR.

[8]  Pietro Baroni,et al.  Semantics of Abstract Argument Systems , 2009, Argumentation in Artificial Intelligence.

[9]  Paul D. Seymour,et al.  Graph Minors. II. Algorithmic Aspects of Tree-Width , 1986, J. Algorithms.

[10]  Ton Kloks.  Treewidth, Computations and Approximations , 1994, Lecture Notes in Computer Science.

[11]  Trevor J. M. Bench-Capon,et al.  Coherence in finite argument systems , 2002, Artif. Intell..

[12]  Bruno Courcelle,et al.  The Monadic Second-Order Logic of Graphs. I. Recognizable Sets of Finite Graphs , 1990, Inf. Comput..

[13]  Bruno Courcelle,et al.  The Monadic Second-Order Logic of Graphs X: Linear Orderings , 1996, Theor. Comput. Sci..

[14]  Stephan Kreutzer,et al.  DAG-Width and Parity Games , 2006, STACS.

[15]  L. Eggan.  Transition graphs and the star-height of regular events. , 1963 .

[16]  Bruno Courcelle,et al.  Upper bounds to the clique width of graphs , 2000, Discret. Appl. Math..

[17]  Paul E. Dunne,et al.  Semi-stable semantics , 2006, J. Log. Comput..

[18]  Sanjay Modgil,et al.  Proof Theories and Algorithms for Abstract Argumentation Frameworks , 2009, Argumentation in Artificial Intelligence.

[19]  Pierre Marquis,et al.  Symmetric Argumentation Frameworks , 2005, ECSQARU.

[20]  Robin Thomas,et al.  Directed Tree-Width , 2001, J. Comb. Theory, Ser. B.

[21]  Hans L. Bodlaender,et al.  A linear time algorithm for finding tree-decompositions of small treewidth , 1993, STOC.

[22]  Bart Verheij,et al.  Two Approaches to Dialectical Argumentation: Admissible Sets and Argumentation Stages , 1999 .

[23]  Paul D. Seymour,et al.  Graph Minors: XV. Giant Steps , 1996, J. Comb. Theory, Ser. B.

[24]  Paul E. Dunne,et al.  Computational properties of argument systems satisfying graph-theoretic constraints , 2007, Artif. Intell..

[25]  Hermann Gruber,et al.  Digraph Complexity Measures and Applications in Formal Language Theory , 2011, Discret. Math. Theor. Comput. Sci..

[26]  Dov M. Gabbay,et al.  A Logical Account of Formal Argumentation , 2009, Stud Logica.