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Register a new userSituated Conditional Reasoning
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Giovanni Casini
Publication date
2021
fields
Informatics Engineering
and research's language is
English
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Conditionals are useful for modelling, but are not always sufficiently expressive for capturing information accurately. In this paper we make the case for a form of conditional that is situation-based. These conditionals are more expressive than classical conditionals, are general enough to be used in several application domains, and are able to distinguish, for example, between expectations and counterfactuals. Formally, they are shown to generalise the conditional setting in the style of Kraus, Lehmann, and Magidor. We show that situation-based conditionals can be described in terms of a set of rationality postulates. We then propose an intuitive semantics for these conditionals, and present a representation result which shows that our semantic construction corresponds exactly to the description in terms of postulates. With the semantics in place, we proceed to define a form of entailment for situated conditional knowledge bases, which we refer to as minimal closure. It is reminiscent of and, indeed, inspired by, the version of entailment for propositional conditional knowledge bases known as rational closure. Finally, we proceed to show that it is possible to reduce the computation of minimal closure to a series of propositional entailment and satisfiability checks. While this is also the case for rational closure, it is somewhat surprising that the result carries over to minimal closure.
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This paper is an appendix to the paper Reasoning with Justifiable Exceptions in Contextual Hierarchies by Bozzato, Serafini and Eiter, 2018. It provides further details on the language, the complexity results and the datalog translation introduced in the main paper.
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A lattice-theoretic framework is introduced that permits the study of the conditional independence (CI) implication problem relative to the class of discrete probability measures. Semi-lattices are associated with CI statements and a finite, sound and complete inference system relative to semi-lattice inclusions is presented. This system is shown to be (1) sound and complete for saturated CI statements, (2) complete for general CI statements, and (3) sound and complete for stable CI statements. These results yield a criterion that can be used to falsify instances of the implication problem and several heuristics are derived that approximate this lattice-exclusion criterion in polynomial time. Finally, we provide experimental results that relate our work to results obtained from other existing inference algorithms.
Reasoning on defeasible knowledge is a topic of interest in the area of description logics, as it is related to the need of representing exceptional instances in knowledge bases. In this direction, in our previous works we presented a framework for representing (contextualized) OWL RL knowledge bases with a notion of justified exceptions on defeasible axioms: reasoning in such framework is realized by a translation into ASP programs. The resulting reasoning process for OWL RL, however, introduces a complex encoding in order to capture reasoning on the negative information needed for reasoning on exceptions. In this paper, we apply the justified exception approach to knowledge bases in $textit{DL-Lite}_{cal R}$, i.e., the language underlying OWL QL. We provide a definition for $textit{DL-Lite}_{cal R}$ knowledge bases with defeasible axioms and study their semantic and computational properties. In particular, we study the effects of exceptions over unnamed individuals. The limited form of $textit{DL-Lite}_{cal R}$ axioms allows us to formulate a simpler ASP encoding, where reasoning on negative information is managed by direct rules. The resulting materialization method gives rise to a complete reasoning procedure for instance checking in $textit{DL-Lite}_{cal R}$ with defeasible axioms. Under consideration in Theory and Practice of Logic Programming (TPLP).
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