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ExpTime Tableau Decision Procedures for Regular Grammar Logics with Converse

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Abstract

Grammar logics were introduced by Fariñas del Cerro and Penttonen in 1988 and have been widely studied. In this paper we consider regular grammar logics with converse (REG c logics) and present sound and complete tableau calculi for the general satisfiability problem of REG c logics and the problem of checking consistency of an ABox w.r.t. a TBox in a REG c logic. Using our calculi we develop ExpTime (optimal) tableau decision procedures for the mentioned problems, to which various optimization techniques can be applied. We also prove a new result that the data complexity of the instance checking problem in REG c logics is coNP-complete.

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References

  1. Baader F., Sattler U.: ‘An overview of tableau algorithms for description logics’. Studia Logica 69, 5–40 (2001)

    Article  Google Scholar 

  2. Baldoni, M., L. Giordano, and A. Martelli, ‘A tableau for multimodal logics and some (un)decidability results’, in Proceedings of TABLEAUX’1998, vol. 1397 of LNCS, Springer, 1998, pp. 44–59.

  3. De Giacomo, G., Decidability of Class-Based Knowledge Representation Formalisms, Ph.D. thesis, Universita’ di Roma “La Sapienza”, 1995.

  4. De Giacomo, G., and F. Massacci, ‘Combining deduction and model checking into tableaux and algorithms for Converse-PDL’, Information and Computation, 117-137 (2000), 87–138.

  5. Demri S.: ‘The complexity of regularity in grammar logics and related modal logics’. Journal of Logic and Computation 11(6), 933–960 (2001)

    Article  Google Scholar 

  6. Demri, S., and H. de Nivelle, ‘Deciding regular grammar logics with converse through first-order logic’, arXiv:cs.LO/0306117, 2004.

  7. Demri S., de Nivelle H.: ‘Deciding regular grammar logics with converse through first-order logic’. Journal of Logic, Language and Information 143, 289–329 (2005)

    Article  Google Scholar 

  8. Donini F., Massacci F.: ‘ExpTime tableaux for \({\mathcal{ALC}}\)’. Artificial Intelligence 124, 87–138 (2000)

    Article  Google Scholar 

  9. Fariñas del Cerro, L., and M. Penttonen, ‘Grammar logics’, Logique et Analyse, 121-122 (1988), 123–134.

  10. Fitting, M., Proof Methods for Modal and Intuitionistic Logics, volume 169 of Synthese Library. D. Reidel, Dordrecht, Holland, 1983.

  11. Goré, R., ‘Tableau methods for modal and temporal logics’, in D’Agostino et al, (ed.), Handbook of Tableau Methods, Kluwer, 1999, pp. 297–396.

  12. Goré, R., and L.A. Nguyen, ‘A tableau system with automaton-labelled formulae for regular grammar logics’, in B. Beckert, (ed.), Proceedings of TABLEAUX 2005, vol. 3702 of LNAI, Springer-Verlag, 2005, pp. 138–152.

  13. Goré, R., and L.A. Nguyen, ‘ExpTime tableaux with global caching for description logics with transitive roles, inverse roles and role hierarchies’, in N. Olivetti, (ed.), Proceedings of TABLEAUX’2007, vol. 4548 of LNAI, Springer-Verlag, 2007, pp. 133–148.

  14. Goré, R., and L.A. Nguyen, ‘Analytic cut-free tableaux for regular modal logics of agent beliefs’, in F. Sadri, and K. Satoh, (eds.), Proceedings of CLIMA VIII, vol. 5056 of LNAI, Springer-Verlag, 2008, pp. 268–287.

  15. Goré, R., and L.A. Nguyen, ‘Sound global caching for abstract modal tableaux’, in H.-D. Burkhard et al, (ed.), Proceedings of CS&P’2008, 2008, pp. 157–167.

  16. Goré, R., and F. Widmann, ‘Sound global state caching for \({\mathcal{ALC}}\) with inverse roles’, in M. Giese, and A. Waaler, (eds.), Proceedings of TABLEAUX’2009, vol. 5607 of LNCS, Springer, 2009, pp. 205–219.

  17. Goré, R., and F. Widmann, ‘Optimal and cut-free tableaux for propositional dynamic logic with converse’, in J. Giesl, and R. H¨ahnle, (eds.), Proceedings of IJCAR’ 2010, vol. 6173 of LNCS, Springer, 2010, pp. 225–239.

  18. Harel, D., D. Kozen, and J. Tiuryn, Dynamic Logic, MIT Press, 2000.

  19. Horrocks, I., O. Kutz, and U. Sattler, ‘The even more irresistible SROIQ’, in P. Doherty, J. Mylopoulos, and C.A. Welty, (eds.), Proceedings of KR’2006, AAAI Press, 2006, pp. 57–67.

  20. Horrocks I., Patel-Schneider P.F.: ‘Optimizing description logic subsumption’. Journal of Logic and Computation 9(3), 267–293 (1999)

    Article  Google Scholar 

  21. Horrocks I., Sattler U.: ‘Decidability of \({\mathcal{SHIQ}}\) with complex role inclusion axioms.’. Artificial Intelligence 160(1–2), 79–104 (2004)

    Article  Google Scholar 

  22. Hustadt, U., B. Motik, and U. Sattler, ‘Data complexity of reasoning in very expressive description logics’, in L.P. Kaelbling, and A. Saffiotti, (eds.), Proceedings of IJCAI-05, Professional Book Center, 2005, pp. 466–471.

  23. Mateescu, A., and A. Salomaa, ‘Formal languages: an introduction and a synopsis’, in Handbook of Formal Languages - Volume 1, Springer, 1997, pp. 1–40.

  24. Nguyen L.A.: ‘Analytic tableau systems and interpolation for the modal logics KB, KDB, K5, KD5’. Studia Logica 69(1), 41–57 (2001)

    Article  Google Scholar 

  25. Nguyen, L.A., ‘On the deterministic Horn fragment of test-free PDL’, in I. Hodkinson, and Y. Venema, (eds.), Advances in Modal Logic - Volume 6, King’s College Publications, 2006, pp. 373–392.

  26. Nguyen L.A.: ‘An efficient tableau prover using global caching for the description logic ALC’. Fundamenta Informaticae 93(1–3), 273–288 (2009)

    Google Scholar 

  27. Nguyen L.A.: ‘Horn knowledge bases in regular description logics with PTime data complexity’. Fundamenta Informaticae 104(4), 349–384 (2010)

    Google Scholar 

  28. Nguyen, L.A., and A. Szałas, ‘An optimal tableau decision procedure for Converse-PDL’, in N.-T. Nguyen, T.-D. Bui, E. Szczerbicki, and N.-B. Nguyen, (eds.), Proceedings of KSE’2009, IEEE Computer Society, 2009, pp. 207–214.

  29. Nguyen, L.A., and A. Szałas, ‘A tableau calculus for regular grammar logics with converse’, in R.A. Schmidt, (ed.), Proceedings of CADE-22, vol. 5663 of LNAI, Springer-Verlag, 2009, pp. 421–436.

  30. Nguyen L.A., Szałas A.: ‘Checking consistency of an ABox w.r.t. global assumptions in PDL’. Fundamenta Informaticae 102(1), 97–113 (2010)

    Google Scholar 

  31. Pratt V.R.: ‘A near-optimal method for reasoning about action’. J. Comput. Syst. Sci. 20(2), 231–254 (1980)

    Article  Google Scholar 

  32. Rautenberg W.: ‘Modal tableau calculi and interpolation’. JPL 12, 403–423 (1983)

    Article  Google Scholar 

  33. Schaerf A.: ‘Reasoning with individuals in concept languages’. Data Knowl. Eng. 13(2), 141–176 (1994)

    Article  Google Scholar 

  34. Vardi, M.Y., ‘Reasoning about the past with two-way automata’, in K.G. Larsen, S. Skyum, and G. Winskel, (eds.), Proceedings of ICALP’98, vol. 1443 of LNCS, Springer, 1998, pp. 628–641.

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Authors and Affiliations

  1. Institute of Informatics, Warsaw University, Warsaw, Poland

    Linh Anh Nguyen & Andrzej Szałas

  2. Department of Computer and Information Science, University of Linköping, Linkoping, Sweden

    Andrzej Szałas

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  1. Linh Anh Nguyen
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  2. Andrzej Szałas
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Correspondence to Linh Anh Nguyen.

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Nguyen, L.A., Szałas, A. ExpTime Tableau Decision Procedures for Regular Grammar Logics with Converse. Stud Logica 98, 387–428 (2011). https://doi.org/10.1007/s11225-011-9341-3

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