Abstract
We extend Pratt’s worst-case optimal decision procedure for PDL to a richer logic with nominals, difference modalities, and inverse actions. We prove correctness and worst-case optimality. Our correctness proof is based on syntactic models called demos. The main theorem states that a formula is satisfiable if and only if it is contained in a demo. From this theorem the correctness of the decision procedure is easily obtained. Our development is modular and we extend it stepwise from modal logic with eventualities to the full logic.
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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)
Baader, F., Sattler, U.: Tableau algorithms for description logics. In: Dyckhoff, R. (ed.) TABLEAUX 2000. LNCS, vol. 1847, pp. 1–18. Springer, Heidelberg (2000)
Blackburn, P., de Rijke, M., Venema, Y.: Modal Logic. CUP (2001)
Bolander, T., Blackburn, P.: Termination for hybrid tableaus. J. Log. Comput. 17(3), 517–554 (2007)
De Giacomo, G.: Decidability of class-based knowledge representation formalisms. Ph.D. thesis, Università degli Studi di Roma “La Sapienza” (1995)
De Giacomo, G., Massacci, F.: Combining deduction and model checking into tableaux and algorithms for converse-PDL. Inf. Comput. 162(1-2), 117–137 (2000)
Donini, F.M., Massacci, F.: EXPtime tableaux for \(\mathcal{ALC}\). AI 124(1), 87–138 (2000)
Emerson, E.A., Halpern, J.Y.: Decision procedures and expressiveness in the temporal logic of branching time. J. Comput. System Sci. 30(1), 1–24 (1985)
Fischer, M.J., Ladner, R.E.: Propositional modal logic of programs. In: Proc. STOC, pp. 286–294. ACM, New York (1977)
Fischer, M.J., Ladner, R.E.: Propositional dynamic logic of regular programs. J. Comput. System Sci., 194–211 (1979)
Gargov, G., Goranko, V.: Modal logic with names. J. Philos. L. 22, 607–636 (1993)
Goré, R., Widmann, F.: An optimal on-the-fly tableau-based decision procedure for PDL-satisfiability. In: Schmidt, R.A. (ed.) CADE-22. LNCS, vol. 5663, pp. 437–452. Springer, Heidelberg (2009)
Goré, R.P., 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., Widmann, F.: Optimal and cut-free tableaux for propositional dynamic logic with converse. In: Giesl, J., Hähnle, R. (eds.) IJCAR 2010. LNCS, vol. 6173, pp. 225–239. Springer, Heidelberg (2010)
Haarslev, V., Möller, R.: RACER system description. In: Goré, R.P., Leitsch, A., Nipkow, T. (eds.) IJCAR 2001. LNCS (LNAI), vol. 2083, pp. 701–705. Springer, Heidelberg (2001)
Harel, D.: Dynamic logic. In: Gabbay, D., Guenthner, F. (eds.) Handbook of Philosophical Logic, vol. II, pp. 497–604. Reidel, Dordrechtz (1984)
Harel, D., Kozen, D., Tiuryn, J.: Dynamic Logic. The MIT Press, Cambridge (2000)
Hollunder, B., Nutt, W., Schmidt-Schauß, M.: Subsumption algorithms for concept description languages. In: Proc. ECAI, pp. 348–353 (1990)
Kaminski, M., Schneider, T., Smolka, G.: Correctness and worst-case optimality of Pratt-style decision procedures for modal and hybrid logics. Technical report, Saarland University (2011), http://tinyurl.com/hpdldc
Kaminski, M., Smolka, G.: Clausal graph tableaux for hybrid logic with eventualities and difference. In: Fermüller, C.G., Voronkov, A. (eds.) LPAR-17. LNCS, vol. 6397, pp. 417–431. Springer, Heidelberg (2010)
Kaminski, M., Smolka, G.: Terminating tableaux for hybrid logic with eventualities. In: Giesl, J., Hähnle, R. (eds.) IJCAR 2010. LNCS, vol. 6173, pp. 240–254. Springer, Heidelberg (2010)
Kozen, D., Tiuryn, J.: Logics of programs. In: Handbook of Theoretical Computer Science. Formal Models and Sematics, vol. B, pp. 789–840. Elsevier, Amsterdam (1990)
Passy, S., Tinchev, T.: An essay in combinatory dynamic logic. Inf. Comput. 93(2), 263–332 (1991)
Pratt, V.R.: Models of program logics. In: Proc. FOCS, pp. 115–122 (1979)
Pratt, V.R.: A near-optimal method for reasoning about action. J. Comput. System Sci. 20(2), 231–254 (1980)
Prior, A.: Past, Present and Future. OUP, England (1967)
Schmidt-Schauß, M., Smolka, G.: Attributive concept descriptions with compliments. AI 48(1), 1–26 (1991)
Segerberg, K.: A note on the logic of elsewhere. Theoria 46(2-3), 183–187 (1980)
Streett, R.S., Emerson, E.A.: An automata theoretic decision procedure for the propositional μ-calculus. Inform. and Control 81, 249–264 (1989)
Tsarkov, D., Horrocks, I., Patel-Schneider, P.F.: Optimizing terminological reasoning for expressive description logics. J. Autom. Reas. 39(3), 277–316 (2007)
Vardi, M.Y., Wolper, P.: Automata-theoretic techniques for modal logics of programs. J. Comput. System Sci. 32, 183–221 (1986)
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Kaminski, M., Schneider, T., Smolka, G. (2011). Correctness and Worst-Case Optimality of Pratt-Style Decision Procedures for Modal and Hybrid Logics. In: Brünnler, K., Metcalfe, G. (eds) Automated Reasoning with Analytic Tableaux and Related Methods. TABLEAUX 2011. Lecture Notes in Computer Science(), vol 6793. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-22119-4_16
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DOI: https://doi.org/10.1007/978-3-642-22119-4_16
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