Abstract. The study of extension-based semantics within the seminal abstract argumentation model of Dung has largely focused on definitional, algorithmic and complexity issues. In contrast, matters relating to comparisons of representational limits, in particular, the extent to which given collections of extensions are expressible within the formalism, have been under-developed. As such, little is known concerning conditions under which a candidate set of subsets of arguments are "realistic" in the sense that they correspond to the extensions of some argumentation framework AF for a semantics of interest. In this paper we present a formal basis for examining extension-based semantics in terms of the sets of extensions that these may express within a single AF. We provide a number of characterization theorems which guarantee the existence of AFs whose set of extensions satisfy specific conditions and derive preliminary complexity results for decision problems that require such characterizations.

Answer-set programming (ASP) has emerged as a declarative programming paradigm where problems are encoded as logic programs, such that the so-called answer sets of theses programs represent the solutions of the encoded problem. The efficiency of the latest ASP solvers reached a state that makes them applicable for problems of practical importance. Consequently, problems from many different areas, including diagnosis, data integration, and graph theory, have been successfully tackled via ASP. In this work, we present such ASP-encodings for problems associated to abstract argumentation frameworks (AFs) and generalisations thereof. Our encodings are formulated as fixed queries, such that the input is the only part depending on the actual AF to process. We illustrate the functioning of this approach, which is underlying a new argumentation system called ASPARTIX in detail and show its adequacy in terms of computational complexity.Keywords: abstract argumentation frameworks; answer-set programming; implementation MotivationIn Artificial Intelligence (AI), the area of argumentation (the survey by Bench-Capon and Dunne (2007) gives an excellent overview) has become one of the central issues during the last decade. Argumentation provides a formal treatment for reasoning problems arising in a number of applications fields, including Multi-Agent Systems and Law Research. In a nutshell, the so-called abstract argumentation frameworks (AFs) formalise statements together with a relation denoting rebuttals between them, such that the semantics gives an abstract handle to solve the inherent conflicts between statements by selecting admissible subsets of them. The reasoning underlying such AFs turned out to be a very general principle capturing many other important formalisms from the areas of AI and knowledge representation.The increasing interest in argumentation led to numerous proposals for formalisations of argumentation. These approaches differ in many aspects. First, there are several ways as to how "admissibility" of a subset of statements can be defined; second, the notion of rebuttal has different meanings (or even additional relationships between statements are taken into account); finally, statements are augmented with priorities, such that the semantics yields those admissible sets which contain statements of higher priority. Thus, in order to compare these different proposals, it is desirable to have a system at hand, which is capable of dealing with a large number of argumentation semantics.Argumentation problems are, in general, intractable; for instance, deciding if an argument is contained in some preferred extension is known to be NP-complete. Therefore, developing dedicated algorithms for the different reasoning problems is non-trivial. A promising way to implement such systems is to use a reduction method, where the given problem is translated into another language, for which sophisticated systems already exist.

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Abstract. In recent research on non-monotonic logic programming, repeatedly strong equivalence of logic programs P and Q has been considered, which holds if the programs P ∪ R and Q ∪ R have the same answer sets for any other program R. This property strengthens equivalence of P and Q with respect to answer sets (which is the particular case for R = ∅), and has its applications in program optimization, verification, and modular logic programming. In this paper, we consider more liberal notions of strong equivalence, in which the actual form of R may be syntactically restricted. On the one hand, we consider uniform equivalence, where R is a set of facts rather than a set of rules. This notion, which is well known in the area of deductive databases, is particularly useful for assessing whether programs P and Q are equivalent as components of a logic program which is modularly structured. On the other hand, we consider relativized notions of equivalence, where R ranges over rules over a fixed alphabet, and thus generalize our results to relativized notions of strong and uniform equivalence. For all these notions, we consider disjunctive logic programs in the propositional (ground) case, as well as some restricted classes, provide semantical characterizations and analyze the computational complexity. Our results, which naturally extend to answer set semantics for programs with strong negation, complement the results on strong equivalence of logic programs and pave the way for optimizations in answer set solvers as a tool for input-based problem solving.

Dialectical Frameworks (ADFs) generalize Dung's argumentation frameworks allowing various relationships among arguments to be expressed in a systematic way. We further generalize ADFs so as to accommodate arbitrary acceptance degrees for the arguments. This makes ADFs applicable in domains where both the initial status of arguments and their relationship are only insufficiently specified by Boolean functions. We define all standard ADF semantics for the weighted case, including grounded, preferred and stable semantics. We illustrate our approach using acceptance degrees from the unit interval and show how other valuation structures can be integrated. In each case it is sufficient to specify how the generalized acceptance conditions are represented by formulas, and to specify the information ordering underlying the characteristic ADF operator. We also present complexity results for problems related to weighted ADFs.