Approximating operators and semantics for abstract dialectical frameworks

We provide a systematic in-depth study of the semantics of abstract dialectical frameworks (ADFs), a recent generalisation of Dung@?s abstract argumentation frameworks. This is done by associating with an ADF its characteristic one-step consequence operator and defining various semantics for ADFs as different fixpoints of this operator. We first show that several existing semantical notions are faithfully captured by our definition, then proceed to define new ADF semantics and show that they are proper generalisations of existing argumentation semantics from the literature. Most remarkably, this operator-based approach allows us to compare ADFs to related nonmonotonic formalisms like Dung argumentation frameworks and propositional logic programs. We use polynomial, faithful and modular translations to relate the formalisms, and our results show that both abstract argumentation frameworks and abstract dialectical frameworks are at most as expressive as propositional normal logic programs.

[1]  Victor W. Marek,et al.  Stable Semantics for Logic Programs and Default Theories , 1989, NACLP.

[2]  Georg Gottlob,et al.  Translating default logic into standard autoepistemic logic , 1995, JACM.

[3]  Tomi Janhunen,et al.  On the intertranslatability of non‐monotonic logics , 1999, Annals of Mathematics and Artificial Intelligence.

[4]  Davide Grossi,et al.  Fixpoints and Iterated Updates in Abstract Argumentation , 2012, KR.

[5]  Jack Minker,et al.  Logic-Based Artificial Intelligence , 2000 .

[6]  Thomas Eiter,et al.  Paracoherent Answer Set Programming , 2010, KR.

[7]  Stefan Woltran,et al.  Relating the Semantics of Abstract Dialectical Frameworks and Standard AFs , 2011, IJCAI.

[8]  K. U. LeuvenCelestijnenlaan,et al.  Approximations, Stable Operators, Well-founded Fixpoints and Applications in Nonmonotonic Reasoning , 2000 .

[9]  Gerhard Brewka,et al.  Carneades and Abstract Dialectical Frameworks: A Reconstruction , 2010, COMMA.

[10]  Martin Caminada,et al.  On the Issue of Reinstatement in Argumentation , 2006, JELIA.

[11]  Melvin Fitting,et al.  Fixpoint Semantics for Logic Programming a Survey , 2001, Theor. Comput. Sci..

[12]  Victor W. Marek,et al.  Uniform semantic treatment of default and autoepistemic logics , 2000, Artif. Intell..

[13]  Hannes Strass,et al.  Instantiating Knowledge Bases in Abstract Dialectical Frameworks , 2013, CLIMA.

[14]  Henry Prakken,et al.  A System for Defeasible Argumentation, with Defeasible Priorities , 1996, Artificial Intelligence Today.

[15]  Raymond Reiter,et al.  A Logic for Default Reasoning , 1987, Artif. Intell..

[16]  Katsumi Nitta,et al.  Computing Argumentation Semantics in Answer Set Programming , 2009, JSAI.

[17]  Stefan Ellmauthaler,et al.  Evaluating Abstract Dialectical Frameworks with ASP , 2012, COMMA.

[18]  Douglas Walton,et al.  Proof Burdens and Standards , 2009, Argumentation in Artificial Intelligence.

[19]  Michael Gelfond,et al.  An A Prolog decision support system for the Space Shuttle , 2001, Answer Set Programming.

[20]  Claudette Cayrol,et al.  Bipolar abstract argumentation systems , 2009, Argumentation in Artificial Intelligence.

[21]  Robert C. Moore.  Semantical Considerations on Nonmonotonic Logic , 1985, IJCAI.

[22]  Sanjay Modgil,et al.  Reasoning about preferences in argumentation frameworks , 2009, Artif. Intell..

[23]  Kurt Konolige,et al.  On the Relation Between Default and Autoepistemic Logic , 1987, Artif. Intell..

[24]  Henry Prakken,et al.  Relating Carneades with Abstract Argumentation , 2011, IJCAI.

[25]  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..

[26]  Dov M. Gabbay,et al.  Logical Modes of Attack in Argumentation Networks , 2009, Stud Logica.

[27]  Dov M. Gabbay,et al.  What Is Negation as Failure? , 2012, Logic Programs, Norms and Action.

[28]  Henry Prakken,et al.  An abstract framework for argumentation with structured arguments , 2010, Argument Comput..

[29]  Hannes Strass,et al.  Abstract Dialectical Frameworks Revisited , 2013, IJCAI.

[30]  Stefan Woltran,et al.  Abstract Dialectical Frameworks , 2010, KR.

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

[32]  Carlo Zaniolo,et al.  Deterministic and Non-Deterministic Stable Models , 1997, J. Log. Comput..

[33]  Francesca Toni,et al.  Argumentation and answer set programming , 2011 .

[34]  Trevor J. M. Bench-Capon,et al.  Instantiating Knowledge Bases in Abstract Argumentation Frameworks , 2009, AAAI Fall Symposium: The Uses of Computational Argumentation.

[35]  Philippe Besnard,et al.  Characterization of Semantics for Argument Systems , 2004, KR.

[36]  Dirk Vermeir,et al.  Robust Semantics for Argumentation Frameworks , 1999, J. Log. Comput..

[37]  Dov M. Gabbay,et al.  Complete Extensions in Argumentation Coincide with 3-Valued Stable Models in Logic Programming , 2009, Stud Logica.

[38]  Juan Carlos Nieves,et al.  A Schema for Generating Relevant Logic Programming Semantics and its Applications in Argumentation Theory , 2011, Fundam. Informaticae.

[39]  Stefan Woltran,et al.  Answer-set programming encodings for argumentation frameworks , 2010, Argument Comput..

[40]  Martin Caminada,et al.  An Algorithm for Stage Semantics , 2010, COMMA.

[41]  A. Tarski.  A LATTICE-THEORETICAL FIXPOINT THEOREM AND ITS APPLICATIONS , 1955 .

[42]  Martin Caminada,et al.  On the evaluation of argumentation formalisms , 2007, Artif. Intell..

[43]  Victor W. Marek,et al.  Ultimate approximation and its application in nonmonotonic knowledge representation systems , 2004, Inf. Comput..

[44]  Simon Parsons,et al.  A Generalization of Dung's Abstract Framework for Argumentation: Arguing with Sets of Attacking Arguments , 2006, ArgMAS.

[45]  Henry Prakken,et al.  The Carneades model of argument and burden of proof , 2007, Artif. Intell..