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Undecidability of the unification and admissibility problems for modal and description logics

Wolter, F. and Zakharyaschev, Michael (2008) Undecidability of the unification and admissibility problems for modal and description logics. ACM Transactions on Computational Logic 9 (4), ISSN 1529-3785.

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Official URL: http://doi.acm.org/10.1145/1380572.1380574

Abstract

We show that the unification problem “is there a substitution instance of a given formula that is provable in a given logic?” is undecidable for basic modal logics K and K4 extended with the universal modality. It follows that the admissibility problem for inference rules is undecidable for these logics as well. These are the first examples of standard decidable modal logics for which the unification and admissibility problems are undecidable. We also prove undecidability of the unification and admissibility problems for K and K4 with at least two modal operators and nominals (instead of the universal modality), thereby showing that these problems are undecidable for basic hybrid logics. Recently, unification has been introduced as an important reasoning service for description logics. The undecidability proof for K with nominals can be used to show the undecidability of unification for Boolean description logics with nominals (such as ALCO and SHIQO). The undecidability proof for K with the universal modality can be used to show that the unification problem relative to role boxes is undecidable for Boolean description logics with transitive roles, inverse roles, and role hierarchies (such as SHI and SHIQ).

Item Type: Article
School or Research Centre: Birkbeck Schools and Research Centres > School of Business, Economics & Informatics > Computer Science and Information Systems
Depositing User: Administrator
Date Deposited: 01 Feb 2011 11:49
Last Modified: 11 Jun 2013 09:29
URI: http://eprints.bbk.ac.uk/id/eprint/1918

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