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    Topological logics with connectedness over Euclidean spaces

    Kontchakov, Roman and Nenov, Y. and Pratt-Hartmann, I. and Zakharyaschev, Michael (2013) Topological logics with connectedness over Euclidean spaces. ACM Transactions on Computational Logic 14 (2), p. 13. ISSN 1529-3785.

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    Abstract

    We consider the quantifier-free languages, Bc and Bc°, obtained by augmenting the signature of Boolean algebras with a unary predicate representing, respectively, the property of being connected, and the property of having a connected interior. These languages are interpreted over the regular closed sets of Rn (n ≥ 2) and, additionally, over the regular closed semilinear sets of Rn. The resulting logics are examples of formalisms that have recently been proposed in the Artificial Intelligence literature under the rubric Qualitative Spatial Reasoning. We prove that the satisfiability problem for Bc is undecidable over the regular closed semilinear sets in all dimensions greater than 1, and that the satisfiability problem for Bc and Bc° is undecidable over both the regular closed sets and the regular closed semilinear sets in the Euclidean plane. However, we also prove that the satisfiability problem for Bc° is NP-complete over the regular closed sets in all dimensions greater than 2, while the corresponding problem for the regular closed semilinear sets is ExpTime-complete. Our results show, in particular, that spatial reasoning is much harder over Euclidean spaces than over arbitrary topological spaces.

    Metadata

    Item Type: Article
    School: School of Business, Economics & Informatics > Computer Science and Information Systems
    Depositing User: Administrator
    Date Deposited: 11 Jun 2013 09:31
    Last Modified: 11 Oct 2016 12:02
    URI: https://eprints.bbk.ac.uk/id/eprint/7432

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