Handbook of logic in artificial intelligence and logic programming
Unification of concept terms in description logics
Journal of Symbolic Computation
Dynamic Logic
The description logic handbook: theory, implementation, and applications
The description logic handbook: theory, implementation, and applications
Admissible Rules of Modal Logics
Journal of Logic and Computation
A tableaux decision procedure for SHOIQ
IJCAI'05 Proceedings of the 19th international joint conference on Artificial intelligence
Unification in the Description Logic $\mathcal{EL}$
RTA '09 Proceedings of the 20th International Conference on Rewriting Techniques and Applications
A Tableau Method for Checking Rule Admissibility in S4
Electronic Notes in Theoretical Computer Science (ENTCS)
A Unified Framework for Non-standard Reasoning Services in Description Logics
Proceedings of the 2010 conference on ECAI 2010: 19th European Conference on Artificial Intelligence
Rules admissible in transitive temporal logic TS4, sufficient condition
Theoretical Computer Science
Unification in the description logic EL without the top concept
CADE'11 Proceedings of the 23rd international conference on Automated deduction
A goal-oriented algorithm for unification in ELHR+ w.r.t. cycle-restricted ontologies
AI'12 Proceedings of the 25th Australasian joint conference on Advances in Artificial Intelligence
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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).