Abstract
We introduce the method of clausal graph tableaux at the example of hybrid logic with difference and star modalities. Clausal graph tableaux are prefix-free and terminate by construction. They provide an abstract method of establishing the small model property of modal logics. In contrast to the filtration method, clausal graph tableaux result in goal-directed decision procedures. Until now no goal-directed decision procedure for the logic considered in this paper was known. There is the promise that clausal graph tableaux lead to a new class of effective decision procedures.
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References
Areces, C., Blackburn, P., Marx, M.: The computational complexity of hybrid temporal logics. L. J. IGPL 8(5), 653–679 (2000)
Areces, C., ten Cate, B.: Hybrid logics. In: Blackburn, et al. (eds.) [4], pp. 821–868
Blackburn, P.: Internalizing labelled deduction. J. Log. Comput. 10(1), 137–168 (2000)
Blackburn, P., van Benthem, J., Wolter, F. (eds.): Handbook of Modal Logic, Studies in Logic and Practical Reasoning, vol. 3. Elsevier, Amsterdam (2007)
Blackburn, P., de Rijke, M., Venema, Y.: Modal Logic. Cambridge University Press, Cambridge (2001)
Emerson, E.A., Clarke, E.M.: Using branching time temporal logic to synthesize synchronization skeletons. Sci. Comput. Programming 2(3), 241–266 (1982)
Emerson, E.A., Halpern, J.Y.: “Sometimes” and “not never” revisited: On branching versus linear time temporal logic. J. ACM 33(1), 151–178 (1986)
Fischer, M.J., Ladner, R.E.: Propositional dynamic logic of regular programs. J. Comput. System Sci. 194–211 (1979)
Fitting, M.: Proof Methods for Modal and Intuitionistic Logics. Reidel, Dordrecht (1983)
Gabbay, D.M.: Selective filtration in modal logic, Part A. Semantic tableaux method. Theoria 36(3), 323–330 (1970)
Giesl, J., Hähnle, R. (eds.): IJCAR 2010. LNCS (LNAI), vol. 6173. Springer, Heidelberg (2010)
Goré, R., Nguyen, L.A.: EXPTIME tableaux with global caching for description logics with transitive roles, inverse roles and role hierarchies. In: Olivetti, N. (ed.) TABLEAUX 2007. LNCS (LNAI), vol. 4548, pp. 133–148. Springer, Heidelberg (2007)
Goré, R., Nguyen, L.A.: Clausal tableaux for multimodal logics of belief. Fund. Inform. 94(1), 21–40 (2009)
Goré, R., Widmann, F.: Optimal tableaux for propositional dynamic logic with converse. In: Giesl, Hähnle (eds.) [11], pp. 225–239
Harel, D., Kozen, D., Tiuryn, J.: Dynamic Logic. The MIT Press, Cambridge (2000)
Horrocks, I., Hustadt, U., Sattler, U., Schmidt, R.: Computational modal logic. In: Blackburn, et al. (eds.) [4], pp. 181–245
Kaminski, M., Smolka, G.: Clausal graph tableaux for hybrid logic with eventualities and difference. Technical report, Saarland University (2010), http://www.ps.uni-saarland.de/Publications/details/KaminskiSmolka:2010:ClausalGraph.html
Kaminski, M., Smolka, G.: Clausal tableaux for hybrid PDL. Technical report, Saarland University (2010), http://www.ps.uni-saarland.de/Publications/details/KaminskiSmolka:2010:HPDL.html
Kaminski, M., Smolka, G.: Terminating tableaux for hybrid logic with eventualities. In: Giesl, Hähnle (eds.) [11], pp. 240–254
Kesten, Y., Manna, Z., McGuire, H., Pnueli, A.: A decision algorithm for full propositional temporal logic. In: Courcoubetis, C. (ed.) CAV 1993. LNCS, vol. 697, pp. 97–109. Springer, Heidelberg (1993)
Kripke, S.A.: Semantical analysis of modal logic I: Normal modal propositional calculi. Z. Math. Logik Grundlagen Math. 9, 67–96 (1963)
Lemmon, E.J., Scott, D.: The ‘Lemmon Notes’: An Introduction to Modal Logic. Blackwell, Malden (1977)
Manna, Z., Wolper, P.: Synthesis of communicating processes from temporal logic specifications. ACM TOPLAS 6(1), 68–93 (1984)
Massacci, F.: Single step tableaux for modal logics. J. Autom. Reasoning 24(3), 319–364 (2000)
Nguyen, L.A.: A new space bound for the modal logics K4, KD4 and S4. In: Kutyłowski, M., Wierzbicki, T., Pacholski, L. (eds.) MFCS 1999. LNCS, vol. 1672, pp. 321–331. Springer, Heidelberg (1999)
Pnueli, A.: The temporal logic of programs. In: Proc. 18th Annual Symp. on Foundations of Computer Science (FOCS 1977), pp. 46–57. IEEE Computer Society Press, Los Alamitos (1977)
Pratt, V.R.: A near-optimal method for reasoning about action. J. Comput. System Sci. 20(2), 231–254 (1980)
Sattler, U., Vardi, M.Y.: The hybrid μ-calculus. In: Goré, R.P., Leitsch, A., Nipkow, T. (eds.) IJCAR 2001. LNCS (LNAI), vol. 2083, pp. 76–91. Springer, Heidelberg (2001)
Segerberg, K.: An Essay in Classical Modal Logic. No. 13 in Filosofiska Studier. University of Uppsala (1971)
Tanabe, Y., Takahashi, K., Hagiya, M.: A decision procedure for alternation-free modal μ-calculi. In: Areces, C., Goldblatt, R. (eds.) Advances in Modal Logic, vol. 7, pp. 341–362. College Publications (2008)
Tsarkov, D., Horrocks, I., Patel-Schneider, P.F.: Optimizing terminological reasoning for expressive description logics. J. Autom. Reasoning 39(3), 277–316 (2007)
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Kaminski, M., Smolka, G. (2010). Clausal Graph Tableaux for Hybrid Logic with Eventualities and Difference. In: Fermüller, C.G., Voronkov, A. (eds) Logic for Programming, Artificial Intelligence, and Reasoning. LPAR 2010. Lecture Notes in Computer Science, vol 6397. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-16242-8_30
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DOI: https://doi.org/10.1007/978-3-642-16242-8_30
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