In this paper, we present a general framework for ranking sets of arguments in abstract argumentation frameworks based on their plausibility of acceptance. We present a generalisation of Dung’s extension semantics as extension-ranking semantics, which induce a preorder over the power set of all arguments, allowing us to state that one set is “closer” to being acceptable than another. To evaluate the extension-ranking semantics, we introduce a number of principles that a well-behaved extensionranking semantics should satisfy. We consider several simple base relations, each of which models a single central aspect of argumentative reasoning. The combination of these base relations provides us with a family of extension-ranking semantics.

We introduce initial models for abstract dialectical frameworks (ADFs) as a notion of minimal justifiable valuations and based on that, generalise the concept of serialisability of argumentation semantics to ADFs. In particular, we show that the characteristic operator-based semantics for ADFs can be characterised through serialisation sequences, which are, essentially, decompositions of a model into a series of initial models, representing a more fine-grained view into why a model is acceptable wrt. the semantics. We also analyse the computational complexity of tasks related to initial models.

We introduce a notion of conditional independence in (flat) assumption-based argumentation (ABA), where independence between (sets of) assumptions amounts to the presence of information about one set of assumptions not impacting the acceptability of another. We study general properties, computational complexity, and the relation to independence in abstract argumentation. In light of the high computational complexity of deciding independence, we introduce sound methods for checking independence in polynomial time via two different routes: the first utilizes the strongly connected components (SCCs) of the instantiated abstract argumentation framework; the second exploits the structure of the ABA framework directly. Along the way, we introduce the notion of SCC-recursiveness for ABA.

In the design of global logistics problems, the solution spaces are typically extremely large. To demonstrate how these challenges can be addressed in Answer Set Programming (ASP), this work investigates a representative industrial use case of a global logistics problem in the aerospace problem domain. An exploration of specific areas of the search space is done by using heuristic-driven solving for the formulation of domain heuristics that guide the solver to potentially desirable configurations. A quantitative evaluation on the Key Performance Indicators and a qualitative evaluation on the variability of the models by means of a similarity analysis shows promising results.

We consider abstract argumentation frameworks and, in particular, the problem of skeptical reasoning wrt. preferred semantics, i.e., deciding whether a given argument is contained in every preferred extension of the argumentation framework. We introduce a novel SAT-based approach, building on recent results from the literature, that searches through complete extensions to efficiently decide this problem. It also employs effective simplification procedures to shorten computation times. As our experimental evaluation shows, our algorithm significantly outperforms state-of-the-art approaches.

We consider abstract argumentation and explanations for the acceptance of arguments. Based on the notion of serialisability, we introduce sequence explanations as a procedural form of explanation for the acceptance of some argument. Intuitively, these explanations represent the process of accepting (and rejecting) arguments in order to conclude the acceptance of a certain argument. We define several variants of sequence explanations and examine them in detail. In particular, we also incorporate counterarguments into the explanations to make them dialectical. Finally, we relate our explanations to other approaches from the literature via a principle-based analysis.

We propose a framework for reasoning from inconsistent knowledge bases using minimal hitting sets, i. e., sets of interpretations such that each formula of the knowledge base is satisfied by at least one those interpretations. By additionally considering preference orders over minimal hitting sets, we can define a wide variety of non-monotonic inference relations. We consider concrete preference orders based on set inclusion, cardinality, the number of conflicting atoms within the hitting set, and using the Hamming distance between pairs of interpretations. We compare the resulting inference relations, characterize their logical properties, and position them relative to classical inference from maximal consistent subsets. Finally, we show that inference based on minimal conflicting atoms coincides with reasoning in Priest’s 3-valued logic.

We investigate the computational problem of determining the set of acceptable arguments in abstract argumentation wrt. credulous and skeptical reasoning under grounded, complete, stable, and preferred semantics. In particular, we investigate the computational complexity of that problem and its verification variant, and develop several algorithms for all problem variants, including two baseline approaches based on iterative acceptability queries and extension enumeration, and some optimised versions. We experimentally compare the runtime performance of these algorithms: our results show that our newly optimised algorithms significantly outperform the baseline algorithms in most cases.

We present algorithms based on satisfiability problem (SAT) solving, as well as answer set programming (ASP), for solving the problem of determining inconsistency degrees in propositional knowledge bases. We consider six different inconsistency measures whose respective decision problems lie on the first level of the polynomial hierarchy. Namely, these are the contension, forgetting-based, hitting set, max-distance, sum-distance, and hit-distance inconsistency measures. In an extensive experimental analysis, we compare the SAT-based and ASP-based approaches with each other, as well as with a set of naive baseline algorithms. Our results demonstrate that, overall, both the SAT-based and the ASP-based approaches clearly outperform the naive baseline methods in terms of runtime. The results further show that the proposed ASP-based approaches perform superior to the SAT-based ones with regard to all six inconsistency measures considered in this work. Moreover, we conduct additional experiments to explain the aforementioned results in greater detail.

Abstract We analyse representatives of the class of non-admissible semantics, particularly the undisputed, strongly undisputed, weakly admissible and weakly preferred semantics, in terms of realisability and their signatures. More specifically, we determine properties of the extension sets they produce, as well as structural features of frameworks that realize extension sets under these semantics. We describe two classes of extension sets for which we show that they are not realisable under undisputed semantics. We also identify approaches that have proven to be useful in the construction of frameworks that aim to realize given extension sets under classical semantics and transfer them to the non-admissible case. While a full characterisation of the signatures of the non-admissible semantics remains elusive, we provide plausible upper bounds and discuss the challenges in establishing concrete lower bounds.

Abstract We consider problems of credulous and sceptical reasoning in abstract argumentation under a variety of semantics and present algorithms for heuristically solving these, i.e. we present algorithms that do not necessarily always give the correct answer but are more performant than correct algorithms. Our algorithms are based on using grounded semantics as a proxy for deciding acceptability w.r.t. other semantics and on bounded search for defenders. We perform a comprehensive experimental evaluation that shows competitive performance of our approaches.

We introduce the notion of serialisation equivalence, which provides a notion of equivalence that takes the underlying dialectical structure of extensions in an argumentation framework into account. Under this notion, two argumentation frameworks are considered equivalent if they possess not only the same extensions wrt. some semantics but also the same serialisation sequences. A serialisation sequence is a decomposition of an extension into a series of minimal acceptable sets and essentially offers insight into the order in which arguments need to brought forward to resolve the conflicts and to justify a particular position in the argumentation framework. We analyse serialisation equivalence in detail and show that it is generally more strict than standard equivalence and less strict than strong equivalence. Furthermore, we provide a full analysis of the computational complexity of deciding serialisation equivalence.

We consider the notion of a vacuous reduct semantics for abstract argumentation frameworks, which, given two abstract argumentation semantics σ and τ, refines σ (base condition) by accepting only those σ-extensions that have no non-empty τ-extension in their reduct (vacuity condition). We give a systematic overview on vacuous reduct semantics resulting from combining different admissibility-based and conflict-free semantics and present a principle-based analysis of vacuous reduct semantics in general. We provide criteria for the inheritance of principle satisfaction by a vacuous reduct semantics from its base and vacuity condition for established as well as recently introduced principles in the context of weak argumentation semantics. We also conduct a principle-based analysis for the special case of undisputed semantics.

This paper studies the realizability of belief revision and belief contraction operators in epistemic spaces. We observe that AGM revision and AGM contraction operators for epistemic spaces are only realizable in precisely determined epistemic spaces. We define the class of linear change operators, which are a special kind of maxichoice operators. When AGM revision, respectively, AGM contraction, is realizable, linear change operators are a canonical realization.

We propose two soft notions of the notion of admissibility in abstract argumentation. The first one weakens the defence notion by allowing, to a certain degree, undefended attacks, and the second one allows, to a certain degree, conflicts within sets of arguments. We analyse these new semantical notions based on the computational complexity of optimisation and approximation. Finally, we discuss and analyse soft notions for preferred semantics.

We introduce the notion of an undisputed set for abstract argumentation frameworks, which is a conflict-free set of arguments, such that its reduct contains no non-empty admissible set. We show that undisputed sets, and the stronger notion of strongly undisputed sets, provide a meaningful approach to weaken admissibility and deal with the problem of attacks from self-attacking arguments, in a similar manner as the recently introduced notion of weak admissibility. We investigate the properties of our new semantical notions and show certain relationships to classical semantics, in particular that undisputed sets are a generalisation of preferred extensions and strongly undisputed sets are a generalisation of stable extensions. We also investigate the computational complexity of standard reasoning tasks with these new notions and show that they lie on the second and third level of the polynomial hierarchy, respectively.

Inconsistency measurement aims at obtaining a quantitative assessment of the level of inconsistency in knowledge bases. While having such a quantitative assessment is beneficial in various settings, inconsistency measurement of propositional knowledge bases is under most existing measures a significantly challenging computational task. In this work, we harness Boolean satisfiability (SAT) based solving techniques for developing practical inconsistency measurement algorithms. Our algorithms—some of which constitute, to the best of our knowledge, the first practical approaches for specific inconsistency measures—are based on using natural choices of SAT-based techniques for the individual inconsistency measures, ranging from direct maximum satisfiability (MaxSAT) encodings to MaxSAT-based column generation techniques making use of incremental computations. We show through an extensive empirical evaluation that our approaches scale well in practice and significantly outperform recently-proposed answer set programming approaches to inconsistency measurement.

We consider the recently proposed notion of serialisability of semantics for abstract argumentation frameworks. This notion describes a method for the serialised non-deterministic construction of extensions through iterative addition of non-empty minimal admissible sets. Depending on the semantics, the task of enumerating all extensions for an argumentation framework can be computationally complex. Serialisability provides a natural way of parallelising the construction of extensions for most admissible-based semantics. In this work, we investigate the feasibility of using the serialisable construction scheme for a more efficient enumeration of extensions on the example of the recently introduced unchallenged semantics and provide an experimental evaluation.

We consider the recently introduced vacuous reduct semantics in abstract argumentation that allows the composition of arbitrary argumentation semantics through the notion of the reduct. We show that by recursively applying the principle of vacuous reduct semantics we are able to cover a broad range of semantical approaches. Our main result shows that we can recover the weakly preferred semantics as the unique solution of a fixed point equation involving an infinite application of the vacuous reduct semantics based only on the very simple property of conflict-freeness. We also conduct an extensive study of the computational complexity of the recursive application of vacuous reduct semantics, which shows that it completely covers each level of the polynomial hierarchy, depending on the recursion depth.

For propositional beliefs, there are well-established connections between belief revision, defeasible conditionals, and nonmonotonic inference. In argumentative contexts, such connections have not yet been investigated. On the one hand, the exact relationship between formal argumentation and nonmonotonic inference relations is a research topic that keeps on eluding researchers despite recently intensified efforts, whereas argumentative revision has been studied in numerous works during recent years. In this paper, we show that relationships between belief revision, defeasible conditionals, and nonmonotonic inference similar to those in propositional logic hold in argumentative contexts as well. We first define revision operators for abstract dialectical frameworks, and use such revision operators to define dynamic conditionals by means of the Ramsey test. We show that such conditionals can be equivalently defined using a total preorder over three-valued interpretations, and study the inferential behaviour of the resulting conditional inference relations.

Abstract dialectical frameworks (in short, ADFs) are a unifying model of formal argumentation, where argumentative relations between arguments are represented by assigning acceptance conditions to atomic arguments. This idea is generalized by letting acceptance conditions being assigned to complex formulas, resulting in conditional abstract dialectical frameworks (in short, cADFs). We define the semantics of cADFs in terms of a non-truth-functional four-valued logic, and study the semantics in-depth, by showing existence results and proving that all semantics are generalizations of the corresponding semantics for ADFs.

We investigate the recently proposed notion of serialisability of semantics for abstract argumentation frameworks. This notion describes semantics where the construction of extensions can be serialised through iterative addition of minimal non-empty admissible sets. We investigate general relationships between serialisability and other principles from the literature. We also investigate the novel unchallenged semantics as a new instance of a serialisable semantics and, in particular, analyse it in terms of satisfied principles and computational complexity.

We revisit the foundations of ranking semantics for abstract argumentation frameworks by observing that most existing approaches are incompatible with classical extension-based semantics. In particular, most ranking semantics violate the principle of admissibility, meaning that admissible arguments are not necessarily better ranked than inadmissible arguments. We propose new postulates for capturing said compatibility with classical extension-based semantics and present a new ranking semantics that complies with these postulates. This ranking semantics is based on the recently proposed notion of serialisability that allows to rank arguments according to the number of conflicts needed to be solved in order to include that argument in an admissible set.

We introduce probo2, an end-to-end benchmark framework for abstract argumentation solvers. It offers evaluation capabilities and analysis features for a wide range of computational problems and is easily customizable.

We examine the impact of both training and test data selection in machine learning applications for abstract argumentation, in terms of prediction accuracy and generalizability. For that, we first review previous studies from a data-centric perspective and conduct some experiments to back up our analysis. We further present a novel algorithm to generate particularly challenging argumentation frameworks wrt. the task of deciding skeptical acceptability under preferred semantics. Moreover, we investigate graph-theoretical aspects of the existing datasets and perform some experiments which show that some simple properties (such as in-degree and out-degree of an argument) are already quite strong indicators of whether or not an argument is skeptically accepted under preferred semantics.

We interpret and formalise ordinal conditional functions (OCFs) in abstract argumentation frameworks based on ideas and concepts defined for conditional logics. There, these functions are used to rank interpretations, and we adapt them to rank extensions instead. Using conflict-freeness and admissibility as two essential principles to define the semantics of OCFs, we obtain a framework that allows to rank sets of arguments wrt. their plausibility. We analyse the properties of this framework in-depth, and in doing so we establish a formal bridge between the approaches of abstract argumentation and conditional logics.

Abstract dialectical frameworks (in short, ADFs) are one of the most general and unifying approaches to formal argumentation. As the semantics of ADFs are based on three-valued interpretations, we ask which monotonic three-valued logic allows to capture the main semantic concepts underlying ADFs. We show that possibilistic logic is the unique logic that can faithfully encode all other semantical concepts for ADFs. Based on this result, we also characterise strong equivalence and introduce possibilistic ADFs.

Extension-based semantics in abstract argumentation provide a criterion to determine whether a set of arguments is acceptable or not. In this paper, we present the notion of extension-ranking semantics, which determines a preordering over sets of arguments, where one set is deemed more plausible than another if it is somehow more acceptable. We obtain extension-based semantics as a special case of this new approach, but it also allows us to make more fine-grained distinctions, such as one set being "more complete'' or "more admissible'' than another. We define a number of general principles to classify extension-ranking semantics and develop concrete approaches. We also study the relation between extension-ranking semantics and argument-ranking based semantics, which rank individual arguments instead of sets of arguments.

We address the problem of deciding skeptical acceptance wrt. preferred semantics of an argument in abstract argumentation frameworks, i.e., the problem of deciding whether an argument is contained in all maximally admissible sets, a.k.a. preferred extensions. State-of-the-art algorithms solve this problem with iterative calls to an external SAT-solver to determine preferred extensions. We provide a new characterisation of skeptical acceptance wrt. preferred semantics that does not involve the notion of a preferred extension. We then develop a new algorithm that also relies on iterative calls to an external SAT-solver but avoids the costly part of maximising admissible sets. We present the results of an experimental evaluation that shows that this new approach significantly outperforms the state of the art. We also apply similar ideas to develop a new algorithm for computing the ideal extension.

We investigate inconsistency and culpability measures for multisets of business rule bases. As companies might encounter thousands of rule bases daily, studying not only individual rule bases separately, but rather also their interrelations, becomes necessary. As current works on inconsistency measurement focus on assessing individual rule bases, we therefore present an extension of those works in the domain of business rules management. We show how arbitrary culpability measures (for single rule bases) can be automatically transformed for multisets, propose new rationality postulates for this setting, and investigate the complexity of central aspects regarding multi-rule base inconsistency measurement.

For propositional beliefs, there are well-established connections between belief revision, defeasible conditionals and nonmonotonic inference. In argumentative contexts, such connections have not yet been investigated. On the one hand, the exact relationship between formal argumentation and nonmonotonic inference relations is a research topic that keeps on eluding researchers despite recently intensified efforts, whereas argumentative revision has been studied in numerous works during recent years. In this paper, we show that similar relationships between belief revision, defeasible conditionals and nonmonotonic inference hold in argumentative contexts as well. We first define revision operators for abstract dialectical frameworks, and use such revision operators to define dynamic conditionals by means of the Ramsey test. We show that such conditionals can be equivalently defined using a total preorder over three-valued interpretations, and study the inferential behaviour of the resulting conditional inference relations.

In abstract argumentation, the admissible semantics can be said to distinguish the preferred semantics in the sense that argumentation frameworks with the same admissible extensions also have the same preferred extensions. In this paper we present an exhaustive study of such distinguishability relationships, including those between sets of semantics. We further examine restricted classes of argumentation frameworks, such as self-attack-free and acyclic frameworks. We discuss the relevance of our results in the context of the argumentation framework elicitation problem.

Restoring consistency of a knowledge base, known as consolidation, should preserve as much information as possible of the original knowledge base. On the one hand, the field of belief change captures this principle of minimal change via rationality postulates. On the other hand, within the field of inconsistency measurement, culpability measures have been developed to assess how much a formula participates in making a knowledge base inconsistent. We look at culpability measures as a tool to disclose epistemic preference relations and build rational consolidation functions. We introduce tacit culpability measures that consider semantic counterparts between conflicting formulae, and we define a special class of these culpability measures based on a fixed-point characterisation: the stable tacit culpability measures. We show that the stable tacit culpability measures yield rational consolidation functions and that these are also the only culpability measures that yield rational consolidation functions.

We investigate the computational problem of determining the set of acceptable arguments in abstract argumentation wrt. credulous and skeptical reasoning under grounded, complete, stable, and preferred semantics. In particular, we investigate the computational complexity of that problem and its verification variant, and develop four SAT-based algorithms for the case of credulous reasoning under complete semantics, two baseline approaches based on iterative acceptability queries and extension enumeration and two optimised algorithms.

We investigate the application of inconsistency measures to the problem of analysing business rule bases. Due to some intricacies of the domain of business rule bases, a straightforward application is not feasible. We therefore develop some new rationality postulates for this setting as well as adapt and modify existing inconsistency measures. We further adapt the notion of inconsistency values (or culpability measures) for this setting and give a comprehensive feasibility study.

We investigate the notion of independence in abstract argumentation, i.e., the question of whether the evaluation of one set of arguments is independent of the evaluation of another set of arguments, given that we already know the status of a third set of arguments. We provide a semantic definition of this notion and develop a method to discover independencies based on transforming an argumentation framework into a DAG on which we then apply the well-known d-separation criterion. We also introduce the SCC Markov property for argumentation semantics, which generalises the Markov property from the classical acyclic case and guarantees the soundness of our approach.

We address the issue of quantitatively assessing the severity of inconsistencies in nonmonotonic frameworks. While measuring inconsistency in classical logics has been investigated for some time now, taking the nonmonotonicity into account poses new challenges. In order to tackle them, we focus on the structure of minimal strongly kb-inconsistent subsets of a knowledge base kb---a generalization of minimal inconsistency to arbitrary, possibly nonmonotonic, frameworks. We propose measures based on this notion and investigate their behavior in a nonmonotonic setting by revisiting existing rationality postulates, analyzing the compliance of the proposed measures with these postulates, and by investigating their computational complexity.

We study rankings over labelings as a generalization of traditional labeling-based semantics in abstract argumentation. Our approach is an alternative to recent developments on rankings over arguments. The formal basis is a qualitative abstraction of probability theory called ranking theory. We propose a fundamental property, called SCC stratification, that a ranking-theoretic semantics can be expected to satisfy, present a general scheme to define a ranking-theoretic semantics, and determine conditions under which this scheme satisfies SCC stratification.

We present a family of stochastic local search algorithms for finding a single stable extension in an abstract argumentation framework. These incomplete algorithms work on random labellings for arguments and iteratively select a random mislabeled argument and flip its label. We present a general version of this approach and an optimisation that allows for greedy selections of arguments. We conduct an empirical evaluation with benchmark graphs from the previous two ICCMA competitions and further random instances. Our results show that our approach is competitive in general and significantly outperforms previous direct approaches and reduction-based approaches for the Barabási-Albert graph model.

We propose a new graded semantics for abstract argumentation frameworks that is based on the constellations approach to probabilistic argumentation. Given an abstract argumentation framework, our approach assigns uniform probability to all arguments and then ranks arguments according to the probability of acceptance wrt. some classical semantics. Albeit relying on a simple idea this approach (1) is based on the solid theoretical foundations of probability theory, and (2) complies with many rationality postulates proposed for graded semantics. We also investigate an application of our approach for inconsistency measurement in argumentation frameworks and show that the measure induced by the probabilistic graded semantics also complies with the basic rationality postulates from that area.

We present a probabilistic interpretation of the plausibility of attacks in abstract argumentation frameworks by extending the epistemic approach to probabilistic argumentation with probabilities on attacks. By doing so we also generalise the previously proposed attack semantics by Villata et al. to the probabilistic setting and provide a fine-grained assessment of the plausibility of attacks. We also consider the setting where partial probabilistic information on arguments and/or attacks is given and missing probabilities have to be derived.

Minimal inconsistent subsets of knowledge bases play an important role in classical logics, most notably for repair and inconsistency measurement. It turns out that for nonmonotonic reasoning a stronger notion is needed. In this paper we develop such a notion, called strong inconsistency. We show that—in an arbitrary logic, monotonic or not—minimal strongly inconsistent subsets play the same role as minimal inconsistent subsets in classical reasoning. In particular, we show that the well-known classical duality between hitting sets of minimal inconsistent subsets and maximal consistent subsets generalizes to arbitrary logics if the strong notion of inconsistency is used. We investigate the complexity of various related reasoning problems and present a generic algorithm for computing minimal strongly inconsistent subsets of a knowledge base. We also demonstrate the potential of our new notion for applications, focusing on repair and inconsistency measurement.

We survey recent approaches to inconsistency measurement in propositional logic and provide a comparative analysis in terms of their expressivity. For that, we introduce four different expressivity characteristics that quantitatively assess the number of different knowledge bases that a measure can distinguish. Our approach aims at complementing ongoing discussions on rationality postulates for inconsistency measures by considering expressivity as a desirable property. We evaluate a large selection of measures on the proposed characteristics and conclude that a distance-based measure from [Grant and Hunter, 2013] has maximal expressivity along all considered characteristics.

Abstract argumentation offers an appealing way of representing and evaluating arguments and counterarguments. This approach can be enhanced by considering probability assignments on arguments, allowing for a quantitative treatment of formal argumentation. In this paper, we regard the assignment as denoting the degree of belief that an agent has in an argument being acceptable. While there are various interpretations of this, an example is how it could be applied to a deductive argument. Here, the degree of belief that an agent has in an argument being acceptable is a combination of the degree to which it believes the premises, the claim, and the derivation of the claim from the premises. We consider constraints on these probability assignments, inspired by crisp notions from classical abstract argumentation frameworks and discuss the issue of probabilistic reasoning with abstract argumentation frameworks. Moreover, we consider the scenario when assessments on the probabilities of a subset of the arguments are given and the probabilities of the remaining arguments have to be derived, taking both the topology of the argumentation framework and principles of probabilistic reasoning into account. We generalise this scenario by also considering inconsistent assessments, i.e., assessments that contradict the topology of the argumentation framework. Building on approaches to inconsistency measurement, we present a general framework to measure the amount of conflict of these assessments and provide a method for inconsistency-tolerant reasoning.

We introduce probo, a general benchmark framework for comparing abstract argumentation solvers. probo is intended to act as the core of an argumentation competition intended to run in 2015.

Uncertainty about which arguments or attacks should appear in an argument graph means that there is uncertainty as to the structure of the argument graph. When informal arguments are presented, there may be imprecision in the language used, and so the audience may be uncertain as to the structure of the argument graph as intended by the presenter of the arguments. For a presenter of arguments, it is useful to know the audience's argument graph, but the presenter may be uncertain as to the structure of it. To model each of these situations, we can use probabilistic argument graphs. The set of subgraphs of an argument graph is a sample space. A probability value is assigned to each subgraph such that the sum is 1, thereby reflecting the uncertainty over which is the actual subgraph. We can then determine the probability that a particular set of arguments is included or excluded from an extension according to a particular Dung semantics. We harness this to define the notion of an argument lottery, which can be used by the audience to determine the expected utility of a debate, and can be used by the presenter to decide which arguments to present by choosing those that maximize expected utility.

We investigate the space of ordinal semantics, where the status of an argument is interpreted by a natural number. In doing so we do not only consider the usual acceptability-based approach for generalizing classical semantics to multi-valued semantics, i.e., positioning “undecided” arguments to be in between “in” and “out” arguments, but also a controversiality-based approach where we interpret the value “undecided” as being the most controversial status of an argument. We introduce stratified labelings as a novel semantical approach that follows the idea of a controversiality-based order of truth-values. We investigate general properties for ordinal semantics and of our approach of stratified labelings in particular.

We consider augmenting abstract argumentation frame-works with probabilistic information and discuss different constraints to obtain meaningful probabilistic information. Moreover, we investigate the problem of incomplete probability assignments and propose a solution for completing these assignments by applying the principle of maximum entropy.

Consolidation describes the operation of restoring consistency in an inconsistent knowledge base. Here we consider this problem in the context of probabilistic conditional logic, a language that focuses on probabilistic conditionals (if-then rules). If a knowledge base, i. e., a set of probabilistic conditionals, is inconsistent traditional model-based inference techniques are not applicable. In this paper, we develop an approach to repair such knowledge bases that relies on a generalized notion of a model of a knowledge base that extends to classically inconsistent knowledge bases. We define a generalized approach to reasoning under maximum entropy on these generalized models and use it to repair the knowledge base. This approach is founded on previous work on inconsistency measures and we show that it is well-defined, provides a unique solution, and satisfies other desirable properties.

Markov logic is a robust approach for probabilistic relational knowledge representation that uses a log-linear model of weighted first-order formulas for probabilistic reasoning. This log-linear model always exists but may not represent the knowledge engineer's intentions adequately. In this paper, we develop a general framework for measuring this coherence of Markov logic networks by comparing the resulting probabilities in the model with the weights given to the formulas. Our measure takes the interdependence of different formulas into account and analyzes the degree of impact they have on the probabilities of other formulas. This approach can be used by the knowledge engineer in constructing a well-formed Markov logic network if data for learning is not available. We also apply our approach to the problem of assessing the compatibility of multiple Markov Logic networks, i. e., to measure to what extent the merging of these networks results in a change of probabilities.

Legal reasoning is one of the most obvious application areas for computational models of argumentation as the exchange of arguments and counterarguments is the established means for making decisions in law. In this paper we employ Defeasible Logic Programming (DeLP) for representing legal cases and for giving decision-support, exemplary for private law. We give a formalization of legal provisions that can be used easily by judges for supporting their decision process and present a working system that resembles the decision-making in legal reasoning, in particular, with respect to the burden of proof.