The topic of forgetting, which loosely speaking means losing, removing, or even hiding some variables, propositions, or formulas, has been extensively studied in the field of knowledge representation and reasoning for many major formalisms. In this article, we convey this topic to the highly active field of abstract argumentation. We provide an in-depth analysis of desirable syntactical and/or semantical properties of possible forgetting operators. In doing so, we included well-known logic programming conditions, such as strong persistence or strong invariance. Further, we argue that although abstract argumentation and logic programming are closely related, it is not possible to reduce forgetting in abstract argumentation to forgetting in logic programming in a straightforward manner. The analysis of desiderata, adapted to the specifics of abstract argumentation, includes implications among them, individual and collective satisfiability, and identifying inherent limits for a set of prominent semantics. Finally, we conduct a case study on stable semantics incorporating concrete forgetting operators.

Abstract dialectical frameworks (ADFs) are one of the most powerful generalizations of classical Dung-style argumentation frameworks (AFs). The additional expressive power comes with an increase in computational complexity, namely one level up in the polynomial hierarchy in comparison to their AF counterparts. However, there is one important subclass, so-called bipolar ADFs (BADFs) which are as complex as classical AFs while offering strictly more modeling capacities. This property makes BADFs very attractive from a knowledge representation point of view and is the main reason why this class has received much attention recently. The semantics of ADFs rely on the Gamma-operator which takes as an input a three-valued interpretation and returns a new one. However, in order to obtain the output the original definition requires to consider any two-valued completion of a given three-valued interpretation. In this paper we formally prove that in case of BADFs we may bypass the computationally intensive procedure via applying Kleene's three-valued logic K. We therefore introduce the so-called bipolar disjunctive normal form which is simply a disjunctive normal form where any used atom possesses either a positive or a negative polarity. We then show that: First, this normal form is expressive enough to represent any BADF and secondly, the computation can be done via Kleene's K instead of dealing with two-valued completions. Inspired by the main correspondence result we present some first experiments showing the computational benefit of using Kleene.

The paper deals with the topic of realizability in abstract argumentation. More precisely, we consider the most basic kind of labellings, so-called conflict-free labellings. The understanding of these labellings is essential as any mature labelling-based semantics selects its output among these labellings. We show how to decide whether a given set of labellings is the conflict-free outcome of a certain Dungean framework. To this end we introduce several new criteria like L-tightness, reject-witness and reject-compositionality. These properties play a decisive role in the central characterization theorem. Moreover, we present a construction method, showing how to realize a conflict-free realizable labelling-set. Finally, we study the representational freedom in case of such labellings. This leads to an uniqueness result for a certain sub-class and a surprising connection to strong equivalence in case of extension-based semantics.

A common feature of non-monotonic logics is that the classical notion of equivalence does not preserve the intended meaning in light of additional information. Consequently, the term strong equivalence was coined in the literature and thoroughly investigated. In the present paper, the knowledge representation formalism under consideration is claimaugmented argumentation frameworks (CAFs) which provide a formal basis to analyze conclusion-oriented problems in argumentation by adapting a claim-focused perspective. CAFs extend Dung AFs by associating a claim to each argument representing its conclusion. In this paper, we investigate both ordinary and strong equivalence in CAFs. Thereby, we take the fact into account that one might either be interested in the actual arguments or their claims only. The former point of view naturally yields an extension of strong equivalence for AFs to the claim-based setting while the latter gives rise to a novel equivalence notion which is genuine for CAFs. We tailor, examine and compare these notions and obtain a comprehensive study of this matter for CAFs. We conclude by investigating the computational complexity of naturally arising decision problems.

A common feature of non-monotonic logics is that the classical notion of equivalence does not preserve the intended meaning in light of additional information. Consequently, the term strong equivalence was coined in the literature and thoroughly investigated. In the present paper, the knowledge representation formalism under consideration are claim-augmented argumentation frameworks (CAFs) which provide a formal basis to analyze conclusion-oriented problems in argumentation by adapting a claim-focused perspective. CAFs extend Dung AFs by associating a claim to each argument representing its conclusion. In this paper, we investigate both ordinary and strong equivalence in CAFs. Thereby, we take the fact into account that one might either be interested in the actual arguments or their claims only. The former point of view naturally yields an extension of strong equivalence for AFs to the claim-based setting while the latter gives rise to a novel equivalence notion which is genuine for CAFs. We tailor, examine and compare these notions and obtain a comprehensive study of this matter for CAFs. We conclude by investigating the computational complexity of naturally arising decision problems.

The topic of forgetting has been extensively studied in the field of knowledge representation and reasoning for many major formalisms. Quite recently it has been introduced to abstract argumentation. However, many already known as well as essential aspects about forgetting like strong persistence or strong invariance have been left unconsidered. We show that forgetting in abstract argumentation cannot be reduced to forgetting in logic programming. In addition, we deal with the more general problem of forgetting whole sets of arguments and show that iterative application of existing operators for single arguments does not necessarily yield a desirable result as it may not produce an informationally economic argumentation framework. As a consequence we provide a systematic and exhaustive study of forgetting desiderata and associated operations adapted to the intrinsics of abstract argumentation. We show the limits and shed light on the possibilities.

Abstract argumentation as defined by Dung in his seminal 1995 paper is by now a major research area in knowledge representation and reasoning. Dynamics of abstract argumentation frameworks (AFs) as well as syntactical consequences of semantical facts of them are the central issues of this paper. The first main part is engaged with the systematical study of the influence of attackers and supporters regarding the acceptability status of whole sets and/or single arguments. In particular, we investigate the impact of addition or removal of arguments, a line of research that has been around for more than a decade. Apart from entirely new results, we revisit, generalize and sum up similar results from the literature. To gain a comprehensive formal and intuitive understanding of the behavior of AFs we put special effort in comparing different kind of semantics. We concentrate on classical admissibility-based semantics and also give pointers to semantics based on naivity and weak admissibility, a recently introduced mediating approach. In the second main part we show how to infer syntactical information from semantical one. For instance, it is well-known that if a finite AF possesses no stable extension, then it has to contain an odd-cycle. In this paper, we even present a characterization of this issue. Moreover, we show that the change of the number of extensions if adding or removing an argument allows to conclude the existence of certain even or odd cycles in the considered AF without having further information.

We develop a notion of explanations for acceptance of arguments in an abstract argumentation framework. To this end we show that extensions returned by Dung's standard semantics can be decomposed into i) non-deterministic choices made on even cycles of the given argumentation graph and then ii) deterministic iteration of the so-called characteristic function. Naturally, the choice made in i) can be viewed as an explanation for the corresponding extension and thus the arguments it contains. We proceed to propose desirable criteria a reasonable notion of an explanation should satisfy. We present an exhaustive study of the newly introduced notion w.r.t. these criteria. Finally some interesting decision problems arise from our analysis and we examine their computational complexity, obtaining some surprising tractability results.

In his seminal 1995 paper, Dung paved the way for abstract argumentation, a by now major research area in knowledge representation. He pointed out that there is a problematic issue with self-defeating arguments underlying all traditional semantics. A self-defeat occurs if an argument attacks itself either directly or indirectly via an odd attack loop, unless the loop is broken up by some argument attacking the loop from outside. Motivated by the fact that such arguments represent self-contradictory or paradoxical arguments, he asked for reasonable semantics which overcome the problem that such arguments may indeed invalidate any argument they attack. This paper tackles this problem from scratch. More precisely, instead of continuing to use previous concepts defined by Dung we provide new foundations for abstract argumentation, so-called weak admissibility and weak defense. After showing that these key concepts are compatible as in the classical case we introduce new versions of the classical Dung-style semantics including complete, preferred and grounded semantics. We provide a rigorous study of these new concepts including interrelationships as well as the relations to their Dung-style counterparts. The newly introduced semantics overcome the issue with self-defeating arguments, and they are semantically insensitive to syntactic deletions of self-attacking arguments, a special case of self-defeat.

The notion of forgetting, as considered in the famous paper by Lin and Reiter in 1994 has been extensively studied in classical logic and more recently, in non-monotonic formalisms like logic programming. In this paper, we convey the idea of forgetting to another major AI formalism, namely Dung-style argumentation frameworks. Our approach is axiomatic-driven and not limited to any specific semantics: we propose semantical and syntactical desiderata encoding different criteria for what forgetting an argument might mean; analyze how these criteria relate to each other; and check whether the criteria can be satisfied in general. The analysis is done for a number of widely used argumentation semantics. Our investigation shows that almost all desiderata are individually satisfiable. However, combinations of semantical and/or syntactical conditions reveal a much more interesting landscape. For instance, we found that the ad hoc approach to forgetting an argument, i.e., by the syntactical removal of the argument and all of its associated attacks, is too restrictive and only compatible with the two weakest semantical desiderata. Amongst the several interesting combinations identified, we showed that one satisfies a notion of minimal change and presented an algorithm that given an AF F and argument x, constructs a suitable AF G satisfying the conditions in the combination.

Semantics based on weak admissibility were recently introduced to overcome a problem with self-defeating arguments that has not been solved for more than 25 years. The recursive definition of weak admissibility mainly relies on the notion of a reduct regarding a set E which only contains arguments which are neither in E, nor attacked by E. At first glance the reduct seems to be tailored for the weaker versions of Dung-style semantics only. In this paper we show that standard Dung semantics can be naturally reformulated using the reduct revealing that this concept is already implicit. We further identify a new abstract principle for semantics, so-called modularization describing how to obtain further extensions given an initial one. Its importance for the study of abstract argumentation semantics is shown by its ability to alternatively characterize classical and non-classical semantics. Moreover, we tackle the notion of strong equivalence via characterizing kernels and give a complete classification of the weak versions regarding well-known properties and postulates known from the literature.

This paper continues the rather recent line of research on the dynamics of non-monotonic formalisms. In particular, we consider semantic changes in Dung’s abstract argumentation formalism. One of the most studied problems in this context is the so-called enforcing problem which is concerned with manipulating argumentation frameworks (AFs) such that a certain desired set of arguments becomes an extension. Here we study the inverse problem, namely the extension removal problem: is it possible – and if so how – to modify a given argumentation framework in such a way that certain undesired extensions are no longer generated? Analogously to the well known AGM paradigm we develop an axiomatic approach to the removal problem, i.e. a certain set of axioms will determine suitable manipulations. Although contraction (that is, the elimination of a particular belief) is conceptually quite different from extension removal, there are surprisingly deep connections between the two: it turns out that postulates for removal can be directly obtained as reformulations of the AGM contraction postulates. We prove a series of formal results including conditional and unconditional existence and semantical uniqueness of removal operators as well as various impossibility results – and show possible ways out.

Conflicting information in an agent's knowledge base may lead to a semantical defect, that is, a situation where it is impossible to draw any plausible conclusion. Finding out the reasons for the observed inconsistency (so-called diagnoses) and/or restoring consistency in a certain minimal way (so-called repairs) are frequently occurring issues in knowledge representation and reasoning. In this article we provide a series of first results for these problems in the context of abstract argumentation theory regarding the two most important reasoning modes, namely credulous as well as sceptical acceptance. Our analysis includes the following problems regarding minimal repairs/diagnoses: existence, verification, computation of one and enumeration of all solutions. The latter problem is tackled with a version of the so-called hitting set duality first introduced by Raymond Reiter in 1987. It turns out that grounded semantics plays an outstanding role not only in terms of complexity, but also as a useful tool to reduce the search space for diagnoses regarding other semantics.

Notions of equivalence that 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 non-monotonic reasoning. In this article, 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 that are located between strong equivalence (Oikarinen and Woltran, 2011, Art. Intell, 175, 1985–2009) 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.

We introduce a parametrized equivalence notion for abstract argumentation that subsumes standard and strong equivalence as corner cases. Under this notion, two argumentation frameworks are equivalent if they deliver the same extensions under any addition of arguments and attacks that do not affect a given set of core arguments. As we will see, this notion of equivalence nicely captures the concept of local simplifications. We provide exact characterizations and complexity results for deciding our new notion of equivalence.

Research in abstract argumentation typically per-tains to finite argumentation frameworks (AFs). Ac-tual or potential infinite AFs frequently occur if theyare used for the purpose of nonmonotonic entail-ment, so-called instantiation-based argumentation,or if they are involved as modeling tool for dia-logues, n-person-games or action sequences. Apartfrom these practical cases a profound analysis yieldsa better understanding of how the nonmonotonic the-ory of abstract argumentation works in general. Inthis paper we study a bunch of abstract propertieslike SCC-recursiveness, expressiveness or intertrans-latability for unrestricted AFs.

A Boolean function is bipolar iff it is monotone or anti-monotone in each of its arguments. We investigate the number |$b(n)$| of |$n$|-ary bipolar Boolean functions. We present an (almost) closed-form expression for |$b(n)$| that uses the number |$a(n)$| of antichain covers of an |$n$|-element set. This is closely related to Dedekind’s problem, which can be rephrased as determining the number |$d(n)$| of Boolean functions that are monotone in all arguments. Indeed, a closed-form solution of |$a(n)$| would directly yield a closed-form solution of |$d(n)$|⁠, suggesting that determining |$a(n)$| is a non-trivial problem of itself.

Dung's abstract argumentation theory is a widely used formalism to model conflicting information and to draw conclusions in such situations. Hereby, the knowledge is represented by argumentation frameworks (AFs) and the reasoning is done via semantics extracting acceptable sets. All reasonable semantics are based on the notion of conflict-freeness which means that arguments are only jointly acceptable when they are not linked within the AF. In this paper, we study the question which information on top of conflict-free sets is needed to compute extensions of a semantics at hand. We introduce a hierarchy of verification classes specifying the required amount of information and show that well-known semantics are exactly verifiable through a certain such class. This also gives a means to study semantics lying between known semantics, thus contributing to a more abstract understanding of the different features argumentation semantics offer.

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 article, 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.

Notions of equivalence which guarantee intersubstitutability 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.

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.

Reasoning about actions is a subfield of artificial intelligence that is concerned with representing and reasoning about dynamic domains. We propose to employ abstract argumentation for this purpose. Specifically, we present a translation of action domains from a specification language into Dung-style argumentation frameworks (AFs). As the key advantage of our approach, we use existing semantics for argumentation to make predictions about the domain in various manners and utilise existing results about argumentation to show that the approach can be efficiently implemented. This demonstrates the practical value not only of its theoretical results, but also abstract argumentation itself.

Time, events, changes and processes play a major role in conceptual modeling, and in information systems and computer science altogether. Accordingly, the representation of time structures and reasoning about temporal data and knowledge are important theoretical and practical research areas. We assume that a formal representation of temporal knowledge must use as a framework some toplevel ontology that describes the most general categories of temporal entities. In the current paper we discuss an ontology of time which is part of the foundational ontology GFO (General Formal Ontology). This ontology of time is inspired by ideas of Franz Brentano [1]. It is used to propose novel contributions to a number of problematic issues related to temporal representation and reasoning, among others, the Dividing Instant Problem and the problem of persistence and change. We present an axiomatization of the ontology as a theory in first-order logic. Eventually, metalogical analysis shows the consistency, completeness, and decidability of this theory.

This paper addresses the problem of revising a Dung-style argumentation framework by adding finitely many new arguments which may interact with old ones. We study the behavior of the extensions of the augmented argumentation frameworks, taking also into account possible changes of the underlying semantics (which may be interpreted as corresponding changes of proof standards). We show both possibility and impossibility results related to the problem of enforcing a desired set of arguments. Furthermore, we prove some monotonicity results for a special class of expansions with respect to the cardinality of the set of extensions and the justification state.