Studia Logica

, 98:387

ExpTime Tableau Decision Procedures for Regular Grammar Logics with Converse

Article
  • 64 Downloads

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 (REGc logics) and present sound and complete tableau calculi for the general satisfiability problem of REGc logics and the problem of checking consistency of an ABox w.r.t. a TBox in a REGc 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 REGc logics is coNP-complete.

Keywords

Modal logic regular grammar logics with converse automated reasoning tableaux global caching 

References

  1. 1.
    Baader F., Sattler U.: ‘An overview of tableau algorithms for description logics’. Studia Logica 69, 5–40 (2001)CrossRefGoogle Scholar
  2. 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.Google Scholar
  3. 3.
    De Giacomo, G., Decidability of Class-Based Knowledge Representation Formalisms, Ph.D. thesis, Universita’ di Roma “La Sapienza”, 1995.Google Scholar
  4. 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.Google Scholar
  5. 5.
    Demri S.: ‘The complexity of regularity in grammar logics and related modal logics’. Journal of Logic and Computation 11(6), 933–960 (2001)CrossRefGoogle Scholar
  6. 6.
    Demri, S., and H. de Nivelle, ‘Deciding regular grammar logics with converse through first-order logic’, arXiv:cs.LO/0306117, 2004.Google Scholar
  7. 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)CrossRefGoogle Scholar
  8. 8.
    Donini F., Massacci F.: ‘ExpTime tableaux for \({\mathcal{ALC}}\)’. Artificial Intelligence 124, 87–138 (2000)CrossRefGoogle Scholar
  9. 9.
    Fariñas del Cerro, L., and M. Penttonen, ‘Grammar logics’, Logique et Analyse, 121-122 (1988), 123–134.Google Scholar
  10. 10.
    Fitting, M., Proof Methods for Modal and Intuitionistic Logics, volume 169 of Synthese Library. D. Reidel, Dordrecht, Holland, 1983.Google Scholar
  11. 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.Google Scholar
  12. 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.Google Scholar
  13. 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.Google Scholar
  14. 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.Google Scholar
  15. 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.Google Scholar
  16. 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.Google Scholar
  17. 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.Google Scholar
  18. 18.
    Harel, D., D. Kozen, and J. Tiuryn, Dynamic Logic, MIT Press, 2000.Google Scholar
  19. 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.Google Scholar
  20. 20.
    Horrocks I., Patel-Schneider P.F.: ‘Optimizing description logic subsumption’. Journal of Logic and Computation 9(3), 267–293 (1999)CrossRefGoogle Scholar
  21. 21.
    Horrocks I., Sattler U.: ‘Decidability of \({\mathcal{SHIQ}}\) with complex role inclusion axioms.’. Artificial Intelligence 160(1–2), 79–104 (2004)CrossRefGoogle Scholar
  22. 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.Google Scholar
  23. 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.Google Scholar
  24. 24.
    Nguyen L.A.: ‘Analytic tableau systems and interpolation for the modal logics KB, KDB, K5, KD5’. Studia Logica 69(1), 41–57 (2001)CrossRefGoogle Scholar
  25. 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.Google Scholar
  26. 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. 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. 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.Google Scholar
  29. 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.Google Scholar
  30. 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. 31.
    Pratt V.R.: ‘A near-optimal method for reasoning about action’. J. Comput. Syst. Sci. 20(2), 231–254 (1980)CrossRefGoogle Scholar
  32. 32.
    Rautenberg W.: ‘Modal tableau calculi and interpolation’. JPL 12, 403–423 (1983)CrossRefGoogle Scholar
  33. 33.
    Schaerf A.: ‘Reasoning with individuals in concept languages’. Data Knowl. Eng. 13(2), 141–176 (1994)CrossRefGoogle Scholar
  34. 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.Google Scholar

Copyright information

© Springer Science+Business Media B.V. 2011

Authors and Affiliations

  1. 1.Institute of InformaticsWarsaw UniversityWarsawPoland
  2. 2.Department of Computer and Information ScienceUniversity of LinköpingLinkopingSweden

Personalised recommendations