14 research outputs found

    Computational Complexity of Strong Admissibility for Abstract Dialectical Frameworks

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    Abstract dialectical frameworks (ADFs) have been introduced as a formalism for modeling and evaluating argumentation allowing general logical satisfaction conditions. Different criteria used to settle the acceptance of arguments arecalled semantics. Semantics of ADFs have so far mainly been defined based on the concept of admissibility. Recently, the notion of strong admissibility has been introduced for ADFs. In the current work we study the computational complexityof the following reasoning tasks under strong admissibility semantics. We address 1. the credulous/skeptical decision problem; 2. the verification problem; 3. the strong justification problem; and 4. the problem of finding a smallest witness of strong justification of a queried argument

    A Lightweight Defeasible Description Logic in Depth: Quantification in Rational Reasoning and Beyond

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    Description Logics (DLs) are increasingly successful knowledge representation formalisms, useful for any application requiring implicit derivation of knowledge from explicitly known facts. A prominent example domain benefiting from these formalisms since the 1990s is the biomedical field. This area contributes an intangible amount of facts and relations between low- and high-level concepts such as the constitution of cells or interactions between studied illnesses, their symptoms and remedies. DLs are well-suited for handling large formal knowledge repositories and computing inferable coherences throughout such data, relying on their well-founded first-order semantics. In particular, DLs of reduced expressivity have proven a tremendous worth for handling large ontologies due to their computational tractability. In spite of these assets and prevailing influence, classical DLs are not well-suited to adequately model some of the most intuitive forms of reasoning. The capability for abductive reasoning is imperative for any field subjected to incomplete knowledge and the motivation to complete it with typical expectations. When such default expectations receive contradicting evidence, an abductive formalism is able to retract previously drawn, conflicting conclusions. Common examples often include human reasoning or a default characterisation of properties in biology, such as the normal arrangement of organs in the human body. Treatment of such defeasible knowledge must be aware of exceptional cases - such as a human suffering from the congenital condition situs inversus - and therefore accommodate for the ability to retract defeasible conclusions in a non-monotonic fashion. Specifically tailored non-monotonic semantics have been continuously investigated for DLs in the past 30 years. A particularly promising approach, is rooted in the research by Kraus, Lehmann and Magidor for preferential (propositional) logics and Rational Closure (RC). The biggest advantages of RC are its well-behaviour in terms of formal inference postulates and the efficient computation of defeasible entailments, by relying on a tractable reduction to classical reasoning in the underlying formalism. A major contribution of this work is a reorganisation of the core of this reasoning method, into an abstract framework formalisation. This framework is then easily instantiated to provide the reduction method for RC in DLs as well as more advanced closure operators, such as Relevant or Lexicographic Closure. In spite of their practical aptitude, we discovered that all reduction approaches fail to provide any defeasible conclusions for elements that only occur in the relational neighbourhood of the inspected elements. More explicitly, a distinguishing advantage of DLs over propositional logic is the capability to model binary relations and describe aspects of a related concept in terms of existential and universal quantification. Previous approaches to RC (and more advanced closures) are not able to derive typical behaviour for the concepts that occur within such quantification. The main contribution of this work is to introduce stronger semantics for the lightweight DL EL_bot with the capability to infer the expected entailments, while maintaining a close relation to the reduction method. We achieve this by introducing a new kind of first-order interpretation that allocates defeasible information on its elements directly. This allows to compare the level of typicality of such interpretations in terms of defeasible information satisfied at elements in the relational neighbourhood. A typicality preference relation then provides the means to single out those sets of models with maximal typicality. Based on this notion, we introduce two types of nested rational semantics, a sceptical and a selective variant, each capable of deriving the missing entailments under RC for arbitrarily nested quantified concepts. As a proof of versatility for our new semantics, we also show that the stronger Relevant Closure, can be imbued with typical information in the successors of binary relations. An extensive investigation into the computational complexity of our new semantics shows that the sceptical nested variant comes at considerable additional effort, while the selective semantics reside in the complexity of classical reasoning in the underlying DL, which remains tractable in our case

    Pseudo-contractions as Gentle Repairs

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    Updating a knowledge base to remove an unwanted consequence is a challenging task. Some of the original sentences must be either deleted or weakened in such a way that the sentence to be removed is no longer entailed by the resulting set. On the other hand, it is desirable that the existing knowledge be preserved as much as possible, minimising the loss of information. Several approaches to this problem can be found in the literature. In particular, when the knowledge is represented by an ontology, two different families of frameworks have been developed in the literature in the past decades with numerous ideas in common but with little interaction between the communities: applications of AGM-like Belief Change and justification-based Ontology Repair. In this paper, we investigate the relationship between pseudo-contraction operations and gentle repairs. Both aim to avoid the complete deletion of sentences when replacing them with weaker versions is enough to prevent the entailment of the unwanted formula. We show the correspondence between concepts on both sides and investigate under which conditions they are equivalent. Furthermore, we propose a unified notation for the two approaches, which might contribute to the integration of the two areas

    Rational Defeasible Reasoning for Description Logics

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    In this paper, we extend description logics (DLs) with non-monotonic reasoning fea- tures. We start by investigating a notion of defeasible subsumption in the spirit of defeasible conditionals as studied by Kraus and colleagues in the propositional case. In particular, we consider a natural and intuitive semantics for defeasible subsumption, and we investi- gate syntactic properties (à la Gentzen) for both preferential and rational subsumptions and prove representation results for the description logic ALC. Such representation results pave the way for more effective decision procedures for defeasible reasoning in DLs. We analyse the problem of non-monotonic reasoning in DL at the level of entailment for both TBox and ABox reasoning, and present an adaptation of rational closure for the DL en- vironment. Importantly, we also show that computing it can be reduced to classical ALC entailment. One of the stumbling blocks to evaluating performance scalability of rational closure is the absence of naturally occurring DL-based ontologies with defeasible features. We overcome this barrier by devising an approach to introduce defeasible subsumption into classical real-world ontologies. Such semi-natural defeasible ontologies, together with a purely artificial set, are used to test our rational closure algorithms. We found that performance is scalable on the whole with no major bottlenecks

    Practical reasoning for defeasable description logics.

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    Doctor of Philosophy in Mathematics, Statistics and Computer Science. University of KwaZulu-Natal, Durban 2016.Description Logics (DLs) are a family of logic-based languages for formalising ontologies. They have useful computational properties allowing the development of automated reasoning engines to infer implicit knowledge from ontologies. However, classical DLs do not tolerate exceptions to speci ed knowledge. This led to the prominent research area of nonmonotonic or defeasible reasoning for DLs, where most techniques were adapted from seminal works for propositional and rst-order logic. Despite the topic's attention in the literature, there remains no consensus on what \sensible" defeasible reasoning means for DLs. Furthermore, there are solid foundations for several approaches and yet no serious implementations and practical tools. In this thesis we address the aforementioned issues in a broad sense. We identify the preferential approach, by Kraus, Lehmann and Magidor (KLM) in propositional logic, as a suitable abstract framework for de ning and studying the precepts of sensible defeasible reasoning. We give a generalisation of KLM's precepts, and their arguments motivating them, to the DL case. We also provide several preferential algorithms for defeasible entailment in DLs; evaluate these algorithms, and the main alternatives in the literature, against the agreed upon precepts; extensively test the performance of these algorithms; and ultimately consolidate our implementation in a software tool called Defeasible-Inference Platform (DIP). We found some useful entailment regimes within the preferential context that satisfy all the KLM properties, and some that have scalable performance in real world ontologies even without extensive optimisation

    Proceedings of the 11th Workshop on Nonmonotonic Reasoning

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    These are the proceedings of the 11th Nonmonotonic Reasoning Workshop. The aim of this series is to bring together active researchers in the broad area of nonmonotonic reasoning, including belief revision, reasoning about actions, planning, logic programming, argumentation, causality, probabilistic and possibilistic approaches to KR, and other related topics. As part of the program of the 11th workshop, we have assessed the status of the field and discussed issues such as: Significant recent achievements in the theory and automation of NMR; Critical short and long term goals for NMR; Emerging new research directions in NMR; Practical applications of NMR; Significance of NMR to knowledge representation and AI in general
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