Reasoning in Argumentation Frameworks of Bounded Clique-Width

Most computational problems in the area of abstract argumentation are intractable, thus identifying tractable fragments and developing efficient algorithms for such fragments are important objectives towards practically efficient argumentation systems. One approach to tractability is to view abstract argumentation frameworks (AFs) as directed graphs and bound certain graph parameters. In particular, Dunne showed that many problems can be solved in linear time for AFs of bounded treewidth. In this paper we consider the graph-parameter clique-width, which is more general than treewidth. An additional advantage of clique-width over treewidth is that it applies well to directed graphs and takes the orientation of edges into account. We first give theoretical tractability results for AFs of bounded clique-width and then introduce dynamic-programming algorithms for credulous and skeptical reasoning.

[1]  Derek G. Corneil,et al.  Complexity of finding embeddings in a k -tree , 1987 .

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

[3]  J. Van Leeuwen,et al.  Handbook of theoretical computer science - Part A: Algorithms and complexity; Part B: Formal models and semantics , 1990 .

[4]  Paul D. Seymour,et al.  Approximating clique-width and branch-width , 2006, J. Comb. Theory, Ser. B.

[5]  Hans L. Bodlaender,et al.  On Linear Time Minor Tests with Depth-First Search , 1993, J. Algorithms.

[6]  Bruno Courcelle,et al.  Graph Rewriting: An Algebraic and Logic Approach , 1991, Handbook of Theoretical Computer Science, Volume B: Formal Models and Sematics.

[7]  Mamadou Moustapha Kanté.  The rank-width of Directed Graphs , 2007, ArXiv.

[8]  Bruno Courcelle,et al.  Handle-Rewriting Hypergraph Grammars , 1993, J. Comput. Syst. Sci..

[9]  Hans L. Bodlaender,et al.  A Tourist Guide through Treewidth , 1993, Acta Cybern..

[10]  Udi Rotics,et al.  Clique-Width is NP-Complete , 2009, SIAM J. Discret. Math..

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

[12]  Udi Rotics,et al.  On the Relationship Between Clique-Width and Treewidth , 2001, SIAM J. Comput..

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

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

[15]  Bruno Courcelle,et al.  Context-free Handle-rewriting Hypergraph Grammars , 1990, Graph-Grammars and Their Application to Computer Science.

[16]  Bruno Courcelle,et al.  Linear Time Solvable Optimization Problems on Graphs of Bounded Clique Width , 1998, WG.

[17]  Stefan Woltran,et al.  Towards Fixed-Parameter Tractable Algorithms for Argumentation , 2010, KR.

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

[19]  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..

[20]  Trevor J. M. Bench-Capon,et al.  Argumentation in artificial intelligence , 2007, Artif. Intell..