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Non-clausal Connection Calculi for Non-classical Logics

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Automated Reasoning with Analytic Tableaux and Related Methods (TABLEAUX 2017)

Part of the book series: Lecture Notes in Computer Science ((LNAI,volume 10501))

Abstract

The paper introduces non-clausal connection calculi for first-order intuitionistic and several first-order modal logics. The notion of a non-clausal matrix together with the non-clausal connection calculus for classical logic are extended to intuitionistic and modal logics by adding prefixes that encode the Kripke semantics of these logics. Details of the required prefix unification and some optimization techniques are described. Furthermore, compact Prolog implementations of the introduced non-classical calculi are presented. An experimental evaluation shows that non-clausal connection calculi are a solid basis for proof search in these logics, in terms of time complexity and proof size.

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Notes

  1. 1.

    A relation \(R \subseteq W \times W\) is serial iff for all \(w_1\in W\) there is some \(w_2\in W\) with \((w_1,w_2)\in R\).

  2. 2.

    \(\hbox {pre}(x)\) for a variable x is the prefix \(\hbox {pre}(Q x G)\) of the corresponding subformula QxG, \(Q{\in }\{\forall ,\exists \}\).

  3. 3.

    \(u \preceq w\) holds iff u is an initial substring of w or \(u \,{=}\, w\). This condition, as well as the fact that there is no accessibility condition for S5, are slightly corrected conditions of [23].

  4. 4.

    The original characterization [23] uses a “tableau-like” definition and not non-clausal matrices.

  5. 5.

    This is also called the monoid problem; it is the equation theory in which there is a neutral element \(\varepsilon \) and the associativity of the string concatenation operator \(\circ \) holds.

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Acknowledgements

The author would like to thank Arild Waaler for his support through the Sirius Center at the University of Oslo funded by the Research Council of Norway. Furthermore, he would like to thank Wolfgang Bibel for his comments.

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Authors and Affiliations

  1. Department of Informatics, University of Oslo, PO Box 1080, Blindern, 0316, Oslo, Norway

    Jens Otten

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  1. Jens Otten
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Correspondence to Jens Otten .

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Editors and Affiliations

  1. University of Manchester, Manchester, United Kingdom

    Renate A. Schmidt

  2. University of Brasília, Brasília D.F., Brazil

    Cláudia Nalon

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Otten, J. (2017). Non-clausal Connection Calculi for Non-classical Logics. In: Schmidt, R., Nalon, C. (eds) Automated Reasoning with Analytic Tableaux and Related Methods. TABLEAUX 2017. Lecture Notes in Computer Science(), vol 10501. Springer, Cham. https://doi.org/10.1007/978-3-319-66902-1_13

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  • DOI: https://doi.org/10.1007/978-3-319-66902-1_13

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