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We extend the expressivity of classical conditional reasoning by introducing context as a new parameter. The enriched
conditional logic generalises the defeasible conditional setting in the style of Kraus, Lehmann, and Magidor, and allows for a refined semantics that is able to distinguish, for example, between expectations and counterfactuals. In this paper we introduce the language for the enriched logic and define an appropriate semantic framework for it. We analyse which properties generally associated with conditional reasoning are still satisfied by the new semantic framework, provide a suitable representation result, and define an entailment relation based on Lehmann and Magidor’s generally-accepted notion of Rational Closure.

@{430,
  author = {Giovanni Casini, Tommie Meyer, Ivan Varzinczak},
  title = {Contextual Conditional Reasoning},
  abstract = {We extend the expressivity of classical conditional reasoning by introducing context as a new parameter. The enriched
conditional logic generalises the defeasible conditional setting in the style of Kraus, Lehmann, and Magidor, and allows for a refined semantics that is able to distinguish, for example, between expectations and counterfactuals. In this paper we introduce the language for the enriched logic and define an appropriate semantic framework for it. We analyse which properties generally associated with conditional reasoning are still satisfied by the new semantic framework, provide a suitable representation result, and define an entailment relation based on Lehmann and Magidor’s generally-accepted notion of Rational Closure.},
  year = {2021},
  journal = {35th AAAI Conference on Artificial Intelligence},
  pages = {6254-6261},
  month = {02/02/2021-09/02/2021},
  publisher = {AAAI Press},
  address = {Online},
}

The past 25 years have seen many attempts to introduce defeasible-reasoning capabilities into a description logic setting. Many, if not most, of these attempts are based on preferential extensions of description logics, with a significant number of these, in turn, following the so-called KLM approach to defeasible reasoning initially advocated for propositional logic by Kraus, Lehmann, and Magidor. Each of these attempts has its own aim of investigating particular constructions and variants of the (KLM-style) preferential approach. Here our aim is to provide a comprehensive study of the formal foundations of preferential defeasible reasoning for description logics in the KLM tradition. We start by investigating a notion of defeasible subsumption in the spirit of defeasible conditionals as studied by Kraus, Lehmann, and Magidor in the propositional case. In particular, we consider a natural and intuitive semantics for defeasible subsumption, and we investigate KLM-style syntactic properties for both preferential and rational subsumption. Our contribution includes two representation results linking our semantic
constructions to the set of preferential and rational properties considered. Besides showing that our semantics is appropriate, these results pave the way for more effective decision procedures for defeasible reasoning in description logics. Indeed, we also analyse the problem of non-monotonic reasoning in description logics at the level of entailment and present an algorithm for the computation of rational closure of a defeasible knowledge base. Importantly, our algorithm relies completely on classical entailment and shows that the computational complexity of reasoning over defeasible knowledge bases is no worse than that of reasoning in the underlying classical DL ALC.

@article{433,
  author = {Katarina Britz, Giovanni Casini, Tommie Meyer, Kody Moodley, Uli Sattler, Ivan Varzinczak},
  title = {Principles of KLM-style Defeasible Description Logics},
  abstract = {The past 25 years have seen many attempts to introduce defeasible-reasoning capabilities into a description logic setting. Many, if not most, of these attempts are based on preferential extensions of description logics, with a significant number of these, in turn, following the so-called KLM approach to defeasible reasoning initially advocated for propositional logic by Kraus, Lehmann, and Magidor. Each of these attempts has its own aim of investigating particular constructions and variants of the (KLM-style) preferential approach. Here our aim is to provide a comprehensive study of the formal foundations of preferential defeasible reasoning for description logics in the KLM tradition. We start by investigating a notion of defeasible subsumption in the spirit of defeasible conditionals as studied by Kraus, Lehmann, and Magidor in the propositional case. In particular, we consider a natural and intuitive semantics for defeasible subsumption, and we investigate KLM-style syntactic properties for both preferential and rational subsumption. Our contribution includes two representation results linking our semantic
constructions to the set of preferential and rational properties considered. Besides showing that our semantics is appropriate, these results pave the way for more effective decision procedures for defeasible reasoning in description logics. Indeed, we also analyse the problem of non-monotonic reasoning in description logics at the level of entailment and present an algorithm for the computation of rational closure of a defeasible knowledge base. Importantly, our algorithm relies completely on classical entailment and shows that the computational complexity of reasoning over defeasible knowledge bases is no worse than that of reasoning in the underlying classical DL ALC.},
  year = {2020},
  journal = {Transactions on Computational Logic},
  volume = {22 (1)},
  pages = {1-46},
  publisher = {ACM},
  url = {https://dl-acm-org.ezproxy.uct.ac.za/doi/abs/10.1145/3420258},
  doi = {10.1145/3420258},
}

We present a formal framework for modelling belief change within a non-monotonic reasoning system. Belief change and non-monotonic reasoning are two areas that are formally closely related, with recent attention being paid towards the analysis of belief change within a non-monotonic environment. In this paper we consider the classical AGM belief change operators, contraction and revision, applied to a defeasible setting in the style of Kraus, Lehmann, and Magidor. The investigation leads us to the formal characterisation of a number of classes of defeasible belief change operators. For the most interesting classes we need to consider the problem of iterated belief change, generalising the classical work of Darwiche and Pearl in the process. Our work involves belief change operators aimed at ensuring logical consistency, as well as the characterisation of analogous operators aimed at obtaining coherence—an important notion within the field of logic-based ontologies

@{382,
  author = {Giovanni Casini, Tommie Meyer, Ivan Varzinczak},
  title = {Rational Defeasible Belief Change},
  abstract = {We present a formal framework for modelling belief change within a non-monotonic reasoning system. Belief change and non-monotonic reasoning are two areas that are formally closely related, with recent attention being paid towards the analysis of belief change within a non-monotonic environment. In this paper we consider the classical AGM belief change operators, contraction and revision, applied to a defeasible setting in the style of Kraus, Lehmann, and Magidor. The investigation leads us to the formal characterisation of a number of classes of defeasible belief change operators. For the most interesting classes we need to consider the problem of iterated belief change, generalising the classical work of Darwiche and Pearl in the process. Our work involves belief change operators aimed at ensuring logical consistency, as well as the characterisation of analogous operators aimed at obtaining coherence—an important notion within the field of logic-based ontologies},
  year = {2020},
  journal = {17th International Conference on Principles of Knowledge Representation and Reasoning (KR 2020)},
  pages = {213-222},
  month = {12/09/2020},
  publisher = {IJCAI},
  address = {Virtual},
  url = {https://library.confdna.com/kr/2020/},
  doi = {10.24963/kr.2020/22},
}

In recent work, we addressed an important limitation in previous ex- tensions of description logics to represent defeasible knowledge, namely the re- striction in the semantics of defeasible concept inclusion to a single preference or- der on objects of the domain. Syntactically, this limitation translates to a context- agnostic notion of defeasible subsumption, which is quite restrictive when it comes to modelling different nuances of defeasibility. Our point of departure in our recent proposal allows for different orderings on the interpretation of roles. This yields a notion of contextual defeasible subsumption, where the context is informed by a role. In the present paper, we extend this work to also provide a proof-theoretic counterpart and associated results. We define a (naïve) tableau- based algorithm for checking preferential consistency of contextual defeasible knowledge bases, a central piece in the definition of other forms of contextual defeasible reasoning over ontologies, notably contextual rational closure.

@{247,
  author = {Katarina Britz, Ivan Varzinczak},
  title = {Preferential tableaux for contextual defeasible ALC},
  abstract = {In recent work, we addressed an important limitation in previous ex- tensions of description logics to represent defeasible knowledge, namely the re- striction in the semantics of defeasible concept inclusion to a single preference or- der on objects of the domain. Syntactically, this limitation translates to a context- agnostic notion of defeasible subsumption, which is quite restrictive when it comes to modelling different nuances of defeasibility. Our point of departure in our recent proposal allows for different orderings on the interpretation of roles. This yields a notion of contextual defeasible subsumption, where the context is informed by a role. In the present paper, we extend this work to also provide a proof-theoretic counterpart and associated results. We define a (naïve) tableau- based algorithm for checking preferential consistency of contextual defeasible knowledge bases, a central piece in the definition of other forms of contextual defeasible reasoning over ontologies, notably contextual rational closure.},
  year = {2019},
  journal = {28th International Conference on Automated Reasoning with Analytic Tableaux and Related Methods (TABLEAUX)},
  pages = {39-57},
  month = {03/09-05/09},
  publisher = {Springer LNAI no. 11714},
  isbn = {ISBN 978-3-030-29026-9},
  url = {https://www.springer.com/gp/book/9783030290252},
}

Description logics have been extended in a number of ways to support defeasible reason- ing in the KLM tradition. Such features include preferential or rational defeasible concept inclusion, and defeasible roles in complex concept descriptions. Semantically, defeasible subsumption is obtained by means of a preference order on objects, while defeasible roles are obtained by adding a preference order to role interpretations. In this paper, we address an important limitation in defeasible extensions of description logics, namely the restriction in the semantics of defeasible concept inclusion to a single preference order on objects. We do this by inducing a modular preference order on objects from each modular preference order on roles, and using these to relativise defeasible subsumption. This yields a notion of contextualised rational defeasible subsumption, with contexts described by roles. We also provide a semantic construction for rational closure and a method for its computation, and present a correspondence result between the two.

@article{246,
  author = {Katarina Britz, Ivan Varzinczak},
  title = {Contextual rational closure for defeasible ALC},
  abstract = {Description logics have been extended in a number of ways to support defeasible reason- ing in the KLM tradition. Such features include preferential or rational defeasible concept inclusion, and defeasible roles in complex concept descriptions. Semantically, defeasible subsumption is obtained by means of a preference order on objects, while defeasible roles are obtained by adding a preference order to role interpretations. In this paper, we address an important limitation in defeasible extensions of description logics, namely the restriction in the semantics of defeasible concept inclusion to a single preference order on objects. We do this by inducing a modular preference order on objects from each modular preference order on roles, and using these to relativise defeasible subsumption. This yields a notion of contextualised rational defeasible subsumption, with contexts described by roles. We also provide a semantic construction for rational closure and a method for its computation, and present a correspondence result between the two.},
  year = {2019},
  journal = {Annals of Mathematics and Artificial Intelligence},
  volume = {87},
  pages = {83-108},
  issue = {1-2},
  isbn = {ISSN: 1012-2443},
  url = {https://link.springer.com/article/10.1007/s10472-019-09658-2},
  doi = {10.1007/s10472-019-09658-2},
}