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Tractable Reasoning Using Logic Programs with Intensional Concepts

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Logics in Artificial Intelligence (JELIA 2021)

Abstract

Recent developments triggered by initiatives such as the Semantic Web, Linked Open Data, the Web of Things, and geographic information systems resulted in the wide and increasing availability of machine-processable data and knowledge in the form of data streams and knowledge bases. Applications building on such knowledge require reasoning with modal and intensional concepts, such as time, space, and obligations, that are defeasible. E.g., in the presence of data streams, conclusions may have to be revised due to newly arriving information. The current literature features a variety of domain-specific formalisms that allow for defeasible reasoning using specific intensional concepts. However, many of these formalisms are computationally intractable and limited to one of the mentioned application domains. In this paper, we define a general method for obtaining defeasible inferences over intensional concepts, and we study conditions under which these inferences are computable in polynomial time.

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Notes

  1. 1.

    For simplicity, we restrict ourselves to non-nested (or equivalently in view of Definition 2, composed) intensional atoms. This does not result in any loss of generality, since nested operators can straightforwardly be modelled as non-nested operators, see Remark 1.

  2. 2.

    Note that we often leave \(\mathcal {O}\) implicit as N allows to uniquely determine all elements from \(\mathcal {O}\). Also, to ease the presentation, we only consider unary intensional operators. Others can then often be represented using rules (see also [34]).

  3. 3.

    We follow the usual notation in modal logic and interpretations explicitly include the corresponding frame.

  4. 4.

    Since the intersection of an empty sequence of subsets of a set is the entire set, then, for n=0, i.e., when the body of the rule is empty, the satisfaction condition is just \(w\in \Vert A\Vert ^\dagger \) for any \(\dagger \in \{\top ,\mathsf{u}\}\).

  5. 5.

    Due to space restrictions, we are not able to provide full details and examples of this procedure.

  6. 6.

    Corresponding results for the data complexity of this problem for programs with variables can then be achieved in the usual way [20].

  7. 7.

    This also aligns well with related work, e.g., for reasoning with time, such as stream reasoning where a finite timeline is often assumed, and avoids the exponential explosion on the number of worlds for satisfiability for some epistemic structures [40].

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Acknowledgements

The authors are indebted to the anonymous reviewers of this paper for helpful feedback. The authors were partially supported by FCT project RIVER (PTDC/CCI-COM/30952/2017) and by FCT project NOVA LINCS (UIDB/04516/2020). J. Heyninck was also supported by the German National Science Foundation under the DFG-project CAR (Conditional Argumentative Reasoning) KE-1413/11-1.

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Heyninck, J., Gonçalves, R., Knorr, M., Leite, J. (2021). Tractable Reasoning Using Logic Programs with Intensional Concepts. In: Faber, W., Friedrich, G., Gebser, M., Morak, M. (eds) Logics in Artificial Intelligence. JELIA 2021. Lecture Notes in Computer Science(), vol 12678. Springer, Cham. https://doi.org/10.1007/978-3-030-75775-5_22

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