Abstract
In this paper we provide the first (as far as we know) direct calculus deciding satisfiability of formulae in negation normal form in the fragment of FHL (full hybrid logic with the binder, including the global and converse modalities), where no occurrence of a universal operator is in the scope of a binder. By means of a satisfiability preserving translation of formulae, the calculus can be turned into a satisfiability decision procedure for the fragment \(\textsf{FHL}\setminus\Box \mathord\downarrow\Box\), i.e. formulae in negation normal form where no occurrence of the binder is both in the scope of and contains in its scope a universal operator. The calculus is based on tableaux and termination is achieved by means of a form of anywhere blocking with indirect blocking. Direct blocking is a relation between nodes in a tableau branch, holding whenever the respective labels (formulae) are equal up to (a proper form of) nominal renaming. Indirect blocking is based on a partial order on the nodes of a tableau branch, which arranges them into a tree-like structure.
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Areces, C., Blackburn, P., Marx, M.: A road-map on complexity for hybrid logics. In: Flum, J., Rodríguez-Artalejo, M. (eds.) Computer Science Logic of LNCS, vol. 1683, pp. 307–321. Springer (1999)
Areces, C., ten Cate, B.: Hybrid logics. In: Blackburn, P., Wolter, F., van Benthem, J. (eds.) Handbook of Modal Logics, pp. 821–868. Elsevier (2007)
Blackburn, P., Seligman, J.: Hybrid languages. J. Logic, Lang. Inf. 4, 251–272 (1995)
Bolander, T., Blackburn, P.: Termination for hybrid tableaus. J. Log. Comput. 17(3), 517–554 (2007)
Cerrito, S., Cialdea Mayer, M.: Nominal substitution at work with the global and converse modalities. In: Beklemishev, L., Goranko, V., Shehtman, V. (eds.) Advances in Modal Logic, vol. 8, pp. 57–74. College Publications (2010)
Cerrito, S., Cialdea Mayer, M.: A tableaux based decision procedure for a broad class of hybrid formulae with binders. In: Brünnler, K., Metcalfe, G. (eds.) Automated Resoning with Analytic Tableaux and Related Methods (TABLEAUX 2011) of LNAI, vol. 6793, pp. 104–118. Springer (2011)
Ganzinger, H., De Nivelle, H.: A superposition decision procedure for the guarded fragment with equality. In: Proc. 14th Symposium on Logic in Computer Science, pp. 295–305. IEEE Computer Society Press (1999)
Grädel, E.: On the restraining power of guards. J. Symb. Log. 64, 1719–1742 (1998)
Hirsch, C., Tobies, S.: A tableau algorithm for the clique guarded fragment. In: Wolter, F., Wansing, H., de Rijke, M., Zakharyaschev, M. (eds.) Advances in Modal Logic, vol. 3, pp. 257–277. CSLI Publications (2001)
Hladik, J.: Implementation and evaluation of a tableau algorithm for the guarded fragment. In: Egly, U., Fermüller, C.G. (eds.) Automated Reasoning with Analytic Tableaux and Related Methods (TABLEAUX 2002) of LNAI, vol. 2381, pp. 145–159. Springer (2002)
Horrocks, I., Glimm, B., Sattler, U.: Hybrid logics and ontology languages. Electron. Notes Theor. Comput. Sci. 174, 3–14 (2007)
Kaminski, M., Smolka, G.: Terminating tableau systems for hybrid logic with difference and converse. J. Logic, Lang. Inf. 18(4), 437–464 (2009)
Marx, M.: Narcissists, stepmothers and spies. In: Proceedings of International Workshop on Description Logics (DL 2002), vol. 53. CEUR (2002)
ten Cate, B., Franceschet, M.: Guarded fragments with constants. J. Logic, Lang. Inf. 14, 281–288 (2005)
ten Cate, B., Franceschet, M.: On the complexity of hybrid logics with binders. In: Ong, L. (ed.) Proceedings of Computer Science Logic 2005 of LNCS, vol. 3634, pp. 339–354. Springer (2005)
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Cerrito, S., Cialdea Mayer, M. A Tableau Based Decision Procedure for an Expressive Fragment of Hybrid Logic with Binders, Converse and Global Modalities. J Autom Reasoning 51, 197–239 (2013). https://doi.org/10.1007/s10817-012-9257-2
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DOI: https://doi.org/10.1007/s10817-012-9257-2