Ringo Baumann

What does it take to enforce an argument? minimal change in abstract argumentation

Argumentation is a dynamic process. The enforcing problem in argumentation, i.e. the question whether it is possible to modify a given argumentation framework (AF) in such a way that a desired set of arguments becomes an extension or a subset of an extension, was first studied in [3] and positively answered under certain conditions. In this paper, we take up this research and study the more general problem of minimal change. That is, in brief, i) is it possible to enforce a desired set of arguments, and if so, ii) what is the minimal number of modifications (additions or removals of attacks) to reach such an enforcement, the so-called characteristic. We show for several Dung semantics that this problem can be decided by local criteria encoded by the so-called value functions. Furthermore, we introduce the corresponding equivalence notions between two AFs which guarantee equal minimal efforts needed to enforce certain subsets, namely minimal-E-equivalence and the more general minimal change equivalence. We present characterization theorems for several Dung semantics and finally, we show the relations to standard and the recently proposed strong equivalence [9] for a whole range of semantics.

Splitting an argumentation framework

Splitting results in non-mononotonic formalisms have a long tradition. On the one hand, these results can be used to improve existing computational procedures, and on the other hand they yield deeper theoretical insights into how a non-monotonic approach works. In the 90s) Lifschitz and Turner [1,2] proved splitting results for logic programs and default theory. In this paper we establish similar results for Dung style argumentation frameworks (AFs) under the most important semantics, namely stable, preferred, complete and grounded semantics. Furthermore we show how to use these results in dynamical argumentation.

Normal and strong expansion equivalence for argumentation frameworks

Given a semantics @s, two argumentation frameworks (AFs) F and G are said to be standard equivalent if they possess the same extensions and strongly equivalent if, for any AF H, F conjoined with H and G conjoined with H are standard equivalent. Argumentation is a dynamic process and, in general, new arguments occur in response to a former argument or, more precisely, attack a former argument. For this reason, rather than considering arbitrary expansions we focus here on expansions where new arguments and attacks may be added but the attacks among the old arguments remain unchanged. We define and characterize two new notions of equivalence between AFs (which lie in-between standard and strong equivalence), namely normal and strong expansion equivalence. Furthermore, using the characterization theorems proved in this paper, we draw the connections between all mentioned notions of equivalence including further equivalence relations, so-called weak and local expansion equivalence.

Splitting argumentation frameworks: an empirical evaluation

In a recent paper Baumann [1] has shown that splitting results, similar to those known for logic programs under answer set semantics and default logic, can also be obtained for Dung argumentation frameworks (AFs). Under certain conditions a given AF A can be split into subparts A1 and A2 such that extensions of A can be computed by (1) computing an extension E1 of A1, (2) modifying A2 based on E1, and (3) combining E1 and an extension E2 of the modified variant of A2. In this paper we perform a systematic empirical evaluation of the effects of splitting on the computation of extensions. Our study shows that the performance of algorithms may drastically improve when splitting is applied.

Compact argumentation frameworks

Abstract argumentation frameworks (AFs) are one of the most studied formalisms in AI. In this work, we introduce a certain subclass of AFs which we call compact. Given an extension-based semantics, the corresponding compact AFs are characterized by the feature that each argument of the AF occurs in at least one extension. This not only guarantees a certain notion of fairness; compact AFs are thus also minimal in the sense that no argument can be removed without changing the outcome. We address the following questions in the paper: (1) How are the classes of compact AFs related for different semantics? (2) Under which circumstances can AFs be transformed into equivalent compact ones? (3) Finally, we show that compact AFs are indeed a non-trivial subclass, since the verification problem remains coNP-hard for certain semantics.

Context-free and context-sensitive kernels: update and deletion equivalence in abstract argumentation

Notions of equivalence which guarantee inter-substitutability w.r.t. further modifications have received considerable interest in nonmonotonic reasoning. This paper is within the context of abstract argumentation and we focus on the most general form of a dynamic scenarios, so-called updates as well as certain sub-classes, namely local, normal and arbitrary deletions. We provide characterization theorems for the corresponding equivalence notions and draw the relations to the recently proposed kinds of expansion equivalence [15, 3]. Many of the results rely on abstract concepts like context-free kernels or semantics satisfying isolate-inclusion. Therefore, the results may apply to future semantics as well as further equivalence notions.

On the Maximal and Average Numbers of Stable Extensions

We present an analytical and empirical study of the maximal and average numbers of stable extensions in abstract argumentation frameworks. As one of the analytical main results, we prove a tight upper bound on the maximal number of stable extensions that depends only on the number of arguments in the framework. More interestingly, our empirical results indicate that the distribution of stable extensions as a function of the number of attacks in the framework seems to follow a universal pattern that is independent of the number of arguments.

The equivalence zoo for Dung-style semantics

Notions of equivalence which are stronger than standard equivalence in the sense that they also take potential modifications of the available information into account have received considerable interest in nonmonotonic reasoning. In this paper we focus on equivalence notions in argumentation. More specifically, we establish a number of new results about the relationships among various equivalence notions for Dung argumentation frameworks which are located between strong equivalence [16] and standard equivalence. We provide the complete picture for this variety of equivalence relations (which we call the equivalence zoo) for stable, preferred, admissible and complete semantics. keywords: argumentation, expansion equivalence

The role of self-attacking arguments in characterizations of equivalence notions

A special case of loops in argumentation are self-attacking arguments. While their role with respect to the ontological nature of argumentation is controversially discussed, their presence (or absence) in the abstract setting of Dung-style argumentation frameworks seems to be less crucial for semantics or fundamental properties. There are, however, a few exceptions where self-attacking arguments have essential influence. One such exception concerns characterizations of (strong) equivalence notions between argumentation frameworks. Different notions of equivalence have recently been proposed in the literature and several characterization results for different semantics have been obtained. In this paper, we will survey the current state of this research direction with a particular emphasis on the effect of (dis)allowing self-conflicting arguments. We also provide some novel results for stage, eager, and naive semantics in order to present a full classification of ten prominent semantics and four equivalence notions. keywords: abstract argumentation, equivalence relations.

Analyzing the Equivalence Zoo in Abstract Argumentation

Notions of equivalence which are stronger than standard equivalence in the sense that they also take potential modifications of the available information into account have received considerable interest in nonmonotonic reasoning. In this paper we focus on equivalence notions in argumentation. More specifically, we establish a number of new results about the relationships among various equivalence notions for Dung argumentation frameworks which are located between strong equivalence [1] and standard equivalence. We provide the complete picture for this variety of equivalence relations which we call the equivalence zoo for the most important semantics.