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Register a new userAnalytic Tableaux Calculi for KLM Logics of Nonmonotonic Reasoning
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Added by
Gian Luca Pozzato
Publication date
2006
fields
Informatics Engineering
and research's language is
English
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We present tableau calculi for some logics of nonmonotonic reasoning, as defined by Kraus, Lehmann and Magidor. We give a tableau proof procedure for all KLM logics, namely preferential, loop-cumulative, cumulative and rational logics. Our calculi are obtained by introducing suitable modalities to interpret conditional assertions. We provide a decision procedure for the logics considered, and we study their complexity.
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Data streams occur widely in various real world applications. The research on streaming data mainly focuses on the data management, query evaluation and optimization on these data, however the work on reasoning procedures for streaming knowledge bases on both the assertional and terminological levels is very limited. Typically reasoning services on large knowledge bases are very expensive, and need to be applied continuously when the data is received as a stream. Hence new techniques for optimizing this continuous process is needed for developing efficient reasoners on streaming data. In this paper, we survey the related research on reasoning on expressive logics that can be applied to this setting, and point to further research directions in this area.
The preferential conditional logic PCL, introduced by Burgess, and its extensions are studied. First, a natural semantics based on neighbourhood models, which generalise Lewis sphere models for counterfactual logics, is proposed. Soundness and completeness of PCL and its extensions with respect to this class of models are proved directly. Labelled sequent calculi for all logics of the family are then introduced. The calculi are modular and have standard proof-theoretical properties, the most important of which is admissibility of cut, that entails a syntactic proof of completeness of the calculi. By adopting a general strategy, root-first proof search terminates, thereby providing a decision procedure for PCL and its extensions. Finally, the semantic completeness of the calculi is established: from a finite branch in a failed proof attempt it is possible to extract a finite countermodel of the root sequent. The latter result gives a constructive proof of the finite model property of all the logics considered.
We present a generic framework that facilitates object level reasoning with logics that are encoded within the Higher Order Logic theorem proving environment of HOL Light. This involves proving statements in any logic using intuitive forward and backward chaining in a sequent calculus style. It is made possible by automated machinery that take care of the necessary structural reasoning and term matching automatically. Our framework can also handle type theoretic correspondences of proofs, effectively allowing the type checking and construction of computational processes via proof. We demonstrate our implementation using a simple propositional logic and its Curry-Howard correspondence to the lambda-calculus, and argue its use with linear logic and its various correspondences to session types.
We propose modal Markov logic as an extension of propositional Markov logic to reason under the principle of maximum entropy for modal logics K45, KD45, and S5. Analogous to propositional Markov logic, the knowledge base consists of weighted formulas, whose weights are learned from data. However, in contrast to Markov logic, in our framework we use the knowledge base to define a probability distribution over non-equivalent epistemic situations (pointed Kripke structures) rather than over atoms, and use this distribution to assign probabilities to modal formulas. As in all probabilistic representations, the central task in our framework is inference. Although the size of the state space grows doubly exponentially in the number of propositions in the domain, we provide an algorithm that scales only exponentially in the size of the knowledge base. Finally, we briefly discuss the case of languages with an infinite number of propositions.
The Shapes Constraint Language (SHACL) allows for formalizing constraints over RDF data graphs. A shape groups a set of constraints that may be fulfilled by nodes in the RDF graph. We investigate the problem of containment between SHACL shapes. One shape is contained in a second shape if every graph node meeting the constraints of the first shape also meets the constraints of the second. To decide shape containment, we map SHACL shape graphs into description logic axioms such that shape containment can be answered by description logic reasoning. We identify several, increasingly tight syntactic restrictions of SHACL for which this approach becomes sound and complete.
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