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International Symposium on Logical Foundations of Computer Science

LFCS 2018: Logical Foundations of Computer Science pp 337–353Cite as

A Tableau System for Instantial Neighborhood Logic

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Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 10703))

Abstract

Extending classical propositional logic, instantial neighborhood logic (\(\mathsf {INL}\)) employs formulas like \(\Box (\alpha _1,...,\alpha _j;\alpha _0)\). The intended meaning of such a formula is: there is a neighborhood (of the current point) in which \(\alpha _0\) universally holds and none of \(\alpha _1,...,\alpha _j\) universally fails. This paper offers to \(\mathsf {INL}\) a tableau system that supports mechanical proof/counter-model search.

J. Yu—Supported by Tsinghua University Initiative Scientific Research Program 20151080426.

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Notes

  1. 1.

    By \(v_1(N-\!\ni )v_2\) we mean that \(v_2\in S\in N(v_1)\) for some \(S\subseteq W\), i.e., \(v_2\) is a point in a neighborhood of \(v_1\).

  2. 2.

    That is, if \(S_1\in N(w)\) and \(S_1\subseteq S_2\), then \(S_2\in N(w)\).

  3. 3.

    In some tableau systems, like that for intuitionistic logic [7], it is sometimes necessary to trigger rule-applications multiple times at a same node. Since all systems we consider in this paper are free of such a need, we simply exclude it at this beginning definition.

  4. 4.

    While “\(\Box \)” is an operator, “\((\Box )\)” is the name of a rule.

  5. 5.

    Since rule (MS) can only be triggered by non-empty irregular nodes, the irregular node with index I has no child. It will not be hard to see from Definition 7 and Theorem 5 (both to be presented later) that empty irregular nodes are always open and indicate empty neighborhoods of the current point.

  6. 6.

    That proof is devoted to establish soundness of \(\mathsf {G3inl}\), a sequent calculus for \(\mathsf {INL}\). Due to the length of that proof and the limit of space here, we have to omit further details and refer to [15] for a full presentation.

  7. 7.

    This vacuously covers the special case that the irregular node is empty and hence has no elements.

  8. 8.

    Suppose that \(e_d\) has children. Since \(e_d\) is an end node, all children it has are irregular. Since \(N(d)=\varnothing \), all these irregular children are closed. As \(e_d\)’s irregular children must be introduced in phases by applications of rule \((\Box )\), by Definition 7, \(e_d\) is closed, a contradiction.

  9. 9.

    Note that our proof works in a vacuous way for the special case that \(S=\varnothing \). In that case, the set \(A\cup B^I\) in Definition 6 is be empty, and hence it must be the case that \(j=0\).

  10. 10.

    Recall here that unlike \(\Box \)-prefixed formulas, \(\lnot \Box \)-prefixed formulas are not active.

References

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Acknowledgements

The tableau system introduced in this paper was invented by the author during his January 2015 visit to University of Amsterdam, where he was led by Johan van Benthem and Nick Bezhanishvili to the field of neighborhood logic. Four anonymous referees have offered helpful suggestions to the initial submission of this paper.

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Authors and Affiliations

  1. Department of Philosophy, Tsinghua University, Beijing, 100084, China

    Junhua Yu

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  1. Junhua Yu
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Correspondence to Junhua Yu .

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Editors and Affiliations

  1. City University of New York, New York, New York, USA

    Sergei Artemov

  2. Cornell University, Ithaca, New York, USA

    Anil Nerode

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Yu, J. (2018). A Tableau System for Instantial Neighborhood Logic. In: Artemov, S., Nerode, A. (eds) Logical Foundations of Computer Science. LFCS 2018. Lecture Notes in Computer Science(), vol 10703. Springer, Cham. https://doi.org/10.1007/978-3-319-72056-2_21

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  • DOI: https://doi.org/10.1007/978-3-319-72056-2_21

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