{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2021,5,12]],"date-time":"2021-05-12T00:41:42Z","timestamp":1620780102922},"reference-count":29,"publisher":"Association for Computing Machinery (ACM)","issue":"4","funder":[{"DOI":"10.13039\/501100000266","name":"Engineering and Physical Sciences Research Council","doi-asserted-by":"publisher","award":["GR\/S61966GR\/S63182GR\/S63175GR\/S61973"]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["ACM Trans. Comput. Logic"],"published-print":{"date-parts":[[2008,8]]},"abstract":"We show that the unification problem \u201cis there a substitution instance of a given formula that is provable in a given logic?\u201d 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).<\/jats:p>","DOI":"10.1145\/1380572.1380574","type":"journal-article","created":{"date-parts":[[2008,9,4]],"date-time":"2008-09-04T12:51:35Z","timestamp":1220532695000},"page":"1-20","source":"Crossref","is-referenced-by-count":21,"title":["Undecidability of the unification and admissibility problems for modal and description logics"],"prefix":"10.1145","volume":"9","author":[{"given":"Frank","family":"Wolter","sequence":"first","affiliation":[{"name":"University of Liverpool, Liverpool, U.K."}]},{"given":"Michael","family":"Zakharyaschev","sequence":"additional","affiliation":[{"name":"Birkbeck College London, London, U.K."}]}],"member":"320","reference":[{"key":"e_1_2_1_1_1","doi-asserted-by":"publisher","DOI":"10.1093\/jigpal\/8.5.653"},{"key":"e_1_2_1_2_1","doi-asserted-by":"crossref","unstructured":"Areces C. and ten Cate B. 2007. Hybrid logics. See Blackburn et al. {2007} 821--867. Areces C. and ten Cate B. 2007. Hybrid logics. See Blackburn et al. {2007} 821--867.","DOI":"10.1016\/S1570-2464(07)80017-6"},{"key":"e_1_2_1_3_1","volume-title":"2003. The Description Logic Handbook: Theory, Implementation, and Applications","author":"Baader F."},{"key":"e_1_2_1_4_1","doi-asserted-by":"publisher","DOI":"10.5555\/645710.664329"},{"key":"e_1_2_1_5_1","doi-asserted-by":"publisher","DOI":"10.1006\/jsco.2000.0426"},{"key":"e_1_2_1_6_1","unstructured":"Baader F. and Siekmann J. 1994. Unification theory. In Handbook of Logic in Artificial Intelligence and Logic Programming D. Gabbay C. Hogger and J. Robinson Eds. Oxford University Press Oxford U.K. 41--125. Baader F. and Siekmann J. 1994. Unification theory. In Handbook of Logic in Artificial Intelligence and Logic Programming D. Gabbay C. Hogger and J. Robinson Eds. 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Flum J. and Thomas W. 1994. Mathematical Logic. Springer Berlin Germany. Ebbinghaus H.-D. Flum J. and Thomas W. 1994. Mathematical Logic. Springer Berlin Germany.","DOI":"10.1007\/978-1-4757-2355-7"},{"key":"e_1_2_1_13_1","doi-asserted-by":"publisher","DOI":"10.2307\/2586506"},{"key":"e_1_2_1_14_1","doi-asserted-by":"publisher","DOI":"10.1016\/S0168-0072(99)00032-9"},{"key":"e_1_2_1_15_1","doi-asserted-by":"publisher","DOI":"10.1016\/j.apal.2003.11.010"},{"key":"e_1_2_1_16_1","doi-asserted-by":"publisher","DOI":"10.2178\/jsl\/1096901773"},{"key":"e_1_2_1_17_1","doi-asserted-by":"publisher","DOI":"10.1093\/logcom\/2.1.5"},{"key":"e_1_2_1_18_1","doi-asserted-by":"crossref","unstructured":"Harel D. Kozen D. and Tiuryn J. 2000. Dynamic Logic. MIT Press Cambridge MA. Harel D. Kozen D. and Tiuryn J. 2000. Dynamic Logic. 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