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Metalogical Contributions to the nonmonotonic theory of abstract argumentation
... More precisely, splitting methods try to divide a theory in subtheories such that the formal semantics of the entire theory can be obtained by constructing the semantics of the subtheories. For AFs, splitting was considered in several works (Baumann, 2011(Baumann, , 2014. We briefly recall the required notions here and then demonstrate how to infer properties of repairs and diagnoses. ...
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.
... More precisely, splitting methods try to divide a theory in subtheories such that the formal semantics of the entire theory can be obtained by constructing the semantics of the subtheories. For AFs, splitting was considered in (Baumann, 2011(Baumann, , 2014. We briefly recall the required notions here and then demonstrate how to infer properties of repairs and diagnoses. ...
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 diagnosis) and/or restoring consistency in a certain minimal way (so-called repairs) are frequently occurring issues in knowledge representation and reasoning. In this paper 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/diagnosis: 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.
... Quite surprisingly, it was shown that, in case of stable, preferred, complete and admissible semantics there are local criteria to determine the characteristic, although infinitely many possibilities to modify a given AF exist (see [9] for detailled explanations including all proofs). Let us consider again the dialouge depicted in Figure 4. Using the results in [7] one may show that the characteristic equals 1 if we allow arbitrary modifications, 2 if the deletion of former attacks is forbidden and ∞ (i.e. it is impossible to enforce {a 1 , a 2 , a 3 }) if A only can come up with weaker arguments. ...
We give a list of currently unsolved problems in abstract argumentation. For each of the problems, we motivate why it is interesting and what makes it (apparently) hard to solve.
We study the computational complexity of abstract argumentation semantics based on weak admissibility, a recently introduced concept to deal with arguments of self-defeating nature. Our results reveal that semantics based on weak admissibility are of much higher complexity (under typical assumptions) compared to all argumentation semantics which have been analysed in terms of complexity so far. In fact, we show PSPACE-completeness of all non-trivial standard decision problems for weak-admissible based semantics. We then investigate potential tractable fragments and show that restricting the frameworks under consideration to certain graph-classes significantly reduces the complexity. We also show that weak-admissibility based extensions can be computed by dividing the given graph into its strongly connected components (SCCs). This technique ensures that the bottleneck when computing extensions is the size of the largest SCC instead of the size of the graph itself and therefore contributes to the search for fixed-parameter tractable implementations for reasoning with weak admissibility.
Within argumentation dynamics, a major strand of research is concerned with how changing an argumentation framework affects the acceptability of arguments, and how to modify an argumentation framework in order to guarantee that some arguments have a given acceptance status. In this chapter, we overview the main approaches for enforcement in formal argumentation. We mainly focus on extension enforcement, i.e., on how to modify an argumenta-tion framework to ensure that a given set of arguments becomes (part of) an extension. We present different forms of extension enforcement defined in the literature, as well as several possibility and impossibility results. The question of minimal change is also considered, i.e., what is the minimal number of modifications that must be made to the argumentation framework for enforcing an extension. Computational complexity and algorithms based on a declarative approach are discussed. Finally, we briefly describe several notions that do not directly fit our definition of extension enforcement, but are closely related.
Given the large variety of existing logical formalisms it is of utmost importance to select the most adequate one for a specific purpose, e.g. for representing the knowledge relevant for a particular application or for using the formalism as a modeling tool for problem solving. Awareness of the nature of a logical formalism, in other words, of its fundamental intrinsic properties, is indispensable and provides the basis of an informed choice. In this treatise we consider the existence characterization logics as well as properties like existence and uniqueness, expressibility, replaceability and verifiability in the realm of abstract argumentation
Given the large variety of existing logical formalisms it is of utmost importance
to select the most adequate one for a specific purpose, e.g. for representing
the knowledge relevant for a particular application or for using the formalism
as a modeling tool for problem solving. Awareness of the nature of a logical
formalism, in other words, of its fundamental intrinsic properties, is indispensable
and provides the basis of an informed choice.
One such intrinsic property of logic-based knowledge representation languages
is the context-dependency of pieces of knowledge. In classical propositional
logic, for example, there is no such context-dependence: whenever two
sets of formulas are equivalent in the sense of having the same models (ordinary
equivalence), then they are mutually replaceable in arbitrary contexts (strong
equivalence). However, a large number of commonly used formalisms are not
like classical logic which leads to a series of interesting developments. It turned
out that sometimes, to characterize strong equivalence in formalism L, we can
use ordinary equivalence in formalism L': for example, strong equivalence in
normal logic programs under stable models can be characterized by the standard
semantics of the logic of here-and-there. Such results about the existence of
characterizing logics has rightly been recognized as important for the study of
concrete knowledge representation formalisms and raise a fundamental question:
Does every formalism have one? In this thesis, we answer this question
with a qualified “yes”. More precisely, we show that the important case of
considering only finite knowledge bases guarantees the existence of a canonical
characterizing formalism. Furthermore, we argue that those characterizing
formalisms can be seen as classical, monotonic logics which are uniquely determined
(up to isomorphism) regarding their model theory.
The other main part of this thesis is devoted to argumentation semantics
which play the flagship role in Dung’s abstract argumentation theory. Almost
all of them are motivated by an easily understandable intuition of what should
be acceptable in the light of conflicts. However, although these intuitions equip
us with short and comprehensible formal definitions it turned out that their
intrinsic properties such as existence and uniqueness, expressibility, replaceability
and verifiability are not that easily accessible. We review the mentioned
properties for almost all semantics available in the literature. In doing so we
include two main axes: namely first, the distinction between extension-based
and labelling-based versions and secondly, the distinction of different kind of
argumentation frameworks such as finite or unrestricted ones.
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 and restoring consistency in a certain minimal way are frequently occurring issues in the research area of knowledge representation and reasoning. In a seminal paper Raymond Reiter proves a duality between maximal consistent subsets of a propositional knowledge base and minimal hitting sets of each minimal conflict -- the famous hitting set duality. We extend Reiter's result to arbitrary non-monotonic logics. To this end, we develop a refined notion of inconsistency, called \emph{strong inconsistency}. We show that minimal strongly inconsistent subsets play a similar role as minimal inconsistent subsets in propositional logic. In particular, the duality between hitting sets of minimal inconsistent subsets and maximal consistent subsets generalizes to arbitrary logics if the stronger notion of inconsistency is used. We cover various notions of repairs and characterize them using analogous hitting set dualities. Our analysis also includes an investigation of structural properties of knowledge bases with respect to our notions.
Minimal inconsistent subsets of knowledge bases in monotonic logics play an important role when investigating the reasons for conflicts and trying to handle them, but also for inconsistency measurement. Our notion of strong inconsistency thus allows us to extend existing results to non-monotonic logics. While measuring inconsistency in propositional logic has been investigated for some time now, taking the non-monotony into account poses new challenges. In order to tackle them, we focus on the structure of minimal strongly inconsistent subsets of a knowledge base. We propose measures based on this notion and investigate their behavior in a non-monotonic setting by revisiting existing rationality postulates, and analyzing the compliance of the proposed measures with these postulates.
We provide a series of first results in the context of inconsistency in abstract argumentation theory regarding the two most important reasoning modes, namely credulous as well as skeptical acceptance. Our analysis includes the following problems regarding minimal repairs: existence, verification, computation of one and characterization of all solutions. The latter will be tackled with our previously obtained duality results.
Finally, we investigate the complexity of various related reasoning problems and compare our results to existing ones for monotonic logics.
Abstract properties satisfied for finite structures do not necessarily carry over to infinite structures. Two of the most basic properties are existence and uniqueness of something. In this work we study these properties for acceptable sets of arguments, so-called extensions, in the field of abstract argumentation. We review already known results, present new proofs or explain sketchy old ones in more detail. We also contribute new results and introduce as well as study the question of existence-(in)dependence between argumentation 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.
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