Reasoning and Query Answering in Description Logics

  • Magdalena Ortiz
  • Mantas Šimkus

Abstract

Description Logics (DLs) play a central role as formalisms for representing ontologies and reasoning about them. This lecture introduces the basics of DLs. We discuss the knowledge modeling capabilities of some of the most prominent DLs, including expressive ones, and present some DL reasoning services. Particular attention is devoted to the query answering problem, and to the increasingly popular framework in which data repositories are queried through DL ontologies. We give an overview of the main challenges that arise in this setting, survey some query answering techniques for both lightweight and expressive DLs, and give an overview of the computational complexity landscape.

Keywords

Description Logics Query Answering Ontology Based Data Access 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Abiteboul, S., Buneman, P., Suciu, D.: Data on the Web: from Relations to Semistructured Data and XML. Morgan Kaufmann (2000)Google Scholar
  2. 2.
    Abiteboul, S., Hull, R., Vianu, V.: Foundations of Databases. Addison Wesley Publ. Co. (1995)Google Scholar
  3. 3.
    Adali, S., Candan, K.S., Papakonstantinou, Y., Subrahmanian, V.S.: Query caching and optimization in distributed mediator systems. In: Proc. of the ACM SIGMOD Int. Conf. on Management of Data, pp. 137–148 (1996)Google Scholar
  4. 4.
    Artale, A., Calvanese, D., Kontchakov, R., Zakharyaschev, M.: The DL-Lite family and relations. J. of Artificial Intelligence Research 36, 1–69 (2009)MathSciNetMATHGoogle Scholar
  5. 5.
    Baader, F.: Augmenting concept languages by transitive closure of roles: An alternative to terminological cycles. In: Proc. of the 12th Int. Joint Conf. on Artificial Intelligence, IJCAI 1991 (1991)Google Scholar
  6. 6.
    Baader, F., Brandt, S., Lutz, C.: Pushing the \(\mathcal{EL}\) envelope. In: Proc. of the 19th Int. Joint Conf. on Artificial Intelligence, IJCAI 2005 (2005)Google Scholar
  7. 7.
    Baader, F., Calvanese, D., McGuinness, D., Nardi, D., Patel-Schneider, P. (eds.): The Description Logic Handbook: Theory, Implementation, and Applications, 2nd edn. Cambridge University Press (2007)Google Scholar
  8. 8.
    Baader, F., Hladik, J., Lutz, C., Wolter, F.: From tableaux to automata for description logics. Fundamenta Informaticae 57, 1–33 (2003)MathSciNetMATHGoogle Scholar
  9. 9.
    Bienvenu, M., Eiter, T., Lutz, C., Ortiz, M., Šimkus, M.: Query answering in the description logic \(\mathcal{S}\). In: Proc. of the 23rd International Workshop on Description Logics, DL 2010. CEUR-WS (2010)Google Scholar
  10. 10.
    Bienvenu, M., Ortiz, M., Šimkus, M.: Answering expressive path queries over lightweight DL knowledge bases. In: Description Logics (2012)Google Scholar
  11. 11.
    Bonatti, P., Lutz, C., Murano, A., Vardi, M.Y.: The complexity of enriched μ-calculi. Logical Methods in Computer Science 4(3:11), 1–27 (2008)MathSciNetMATHGoogle Scholar
  12. 12.
    Buneman, P.: Semistructured data. In: Proc. of the 16th ACM SIGACT SIGMOD SIGART Symp. on Principles of Database Systems, PODS 1997, pp. 117–121 (1997)Google Scholar
  13. 13.
    Calvanese, D., De Giacomo, G., Lembo, D., Lenzerini, M., Rosati, R.: Tractable reasoning and efficient query answering in description logics: The DL-Lite family. J. of Automated Reasoning 39(3), 385–429 (2007)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Calvanese, D., De Giacomo, G., Lenzerini, M.: On the decidability of query containment under constraints. In: Proc. of the 17th ACM SIGACT SIGMOD SIGART Symp. on Principles of Database Systems, PODS 1998, pp. 149–158 (1998)Google Scholar
  15. 15.
    Calvanese, D., De Giacomo, G., Lenzerini, M.: 2ATAs make DLs easy. In: CEUR Electronic Workshop Proceedings of Proc. of the 2002 Description Logic Workshop,DL 2002, pp. 107–118 (2002)Google Scholar
  16. 16.
    Calvanese, D., De Giacomo, G., Lenzerini, M.: Conjunctive query containment and answering under description logics constraints. ACM Trans. on Computational Logic 9(3), 22.1–22.31 (2008)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Calvanese, D., De Giacomo, G., Lenzerini, M., Rosati, R.: Logical foundations of peer-to-peer data integration. In: Proc. of the 23rd ACM SIGACT SIGMOD SIGART Symp. on Principles of Database Systems, PODS 2004, pp. 241–251 (2004)Google Scholar
  18. 18.
    Calvanese, D., De Giacomo, G., Lenzerini, M., Vardi, M.Y.: Rewriting regular expressions in semi-structured data. In: Proc. of ICDT 1999 Workshop on Query Processing for Semi-Structured Data and Non-Standard Data Formats (1999)Google Scholar
  19. 19.
    Calvanese, D., De Giacomo, G., Lenzerini, M., Vardi, M.Y.: Containment of conjunctive regular path queries with inverse. In: Proc. of the Seventh International Conference on the Principles of Knowledge Representation and Reasoning (KR 2006), pp. 176–185 (2000)Google Scholar
  20. 20.
    Calvanese, D., De Giacomo, G., Lenzerini, M., Vardi, M.Y.: Reasoning on regular path queries. SIGMOD Record 32(4), 83–92 (2003)CrossRefGoogle Scholar
  21. 21.
    Calvanese, D., De Giacomo, G., Vardi, M.Y.: Decidable containment of recursive queries. Theoretical Computer Science 336(1), 33–56 (2005)MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Calvanese, D., Eiter, T., Ortiz, M.: Answering regular path queries in expressive description logics: An automata-theoretic approach. In: Proc. of the 22nd AAAI Conference on Artificial Intelligence, AAAI 2007, pp. 391–396 (2007)Google Scholar
  23. 23.
    Calvanese, D., Eiter, T., Ortiz, M.: Regular path queries in expressive description logics with nominals. In: Boutilier, C. (ed.) Proc. of the 21st Int. Joint Conf. on Artificial Intelligence, IJCAI 2009, pp. 714–720 (2009)Google Scholar
  24. 24.
    Chandra, A.K., Merlin, P.M.: Optimal implementation of conjunctive queries in relational data bases. In: Proc. of the 9th ACM Symp. on Theory of Computing, STOC 1977, pp. 77–90 (1977)Google Scholar
  25. 25.
    Chortaras, A., Trivela, D., Stamou, G.: Optimized Query Rewriting for OWL 2 QL. In: Bjørner, N., Sofronie-Stokkermans, V. (eds.) CADE 2011. LNCS, vol. 6803, pp. 192–206. Springer, Heidelberg (2011), http://dl.acm.org/citation.cfm?id=2032266.2032282 CrossRefGoogle Scholar
  26. 26.
    Cuenca Grau, B., Horrocks, I., Motik, B., Parsia, B., Patel-Schneider, P., Sattler, U.: OWL 2: The next step for OWL. Journal of Web Semantics 6(4), 309–322 (2008)CrossRefGoogle Scholar
  27. 27.
    De Giacomo, G.: Decidability of Class-Based Knowledge Representation Formalisms. Ph.D. thesis, Dipartimento di Informatica e Sistemistica, Università di Roma “La Sapienza” (1995)Google Scholar
  28. 28.
    De Giacomo, G., Lenzerini, M.: Boosting the correspondence between description logics and propositional dynamic logics. In: Proc. of the 12th Nat. Conf. on Artificial Intelligence, AAAI 1994, pp. 205–212 (1994)Google Scholar
  29. 29.
    Eiter, T., Gottlob, G., Ortiz, M., Šimkus, M.: Query answering in the description logic horn-\(\mathcal{SHIQ}\). In: Hölldobler, S., Lutz, C., Wansing, H. (eds.) JELIA 2008. LNCS (LNAI), vol. 5293, pp. 166–179. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  30. 30.
    Eiter, T., Lutz, C., Ortiz, M., Šimkus, M.: Query Answering in Description Logics: The Knots Approach. In: Ono, H., Kanazawa, M., de Queiroz, R. (eds.) WoLLIC 2009. LNCS, vol. 5514, pp. 26–36. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  31. 31.
    Eiter, T., Lutz, C., Ortiz, M., Šimkus, M.: Query answering in description logics with transitive roles. In: Boutilier, C. (ed.) Proc. of the 21st Int. Joint Conf. on Artificial Intelligence, IJCAI 2009, pp. 759–764 (2009)Google Scholar
  32. 32.
    Eiter, T., Ortiz, M., Simkus, M.: Conjunctive query answering in the description logic SH using knots. J. Comput. Syst. Sci. 78(1), 47–85 (2012)MathSciNetCrossRefMATHGoogle Scholar
  33. 33.
    Eiter, T., Ortiz, M., Šimkus, M., Tran, T.K., Xiao, G.: Query rewriting for Horn-SHIQ plus rules. In: Proc. of the 26th AAAI Conference on Artificial Intelligence, AAAI 2012 (to appear, 2012)Google Scholar
  34. 34.
    Franconi, E.: Description logics for natural language processing. In: Working Notes of the AAAI Fall Symposium on “Knowledge Representation for Natural Language Processing in Implemented Systems”, pp. 37–44 (1994)Google Scholar
  35. 35.
    Gehrke, M., Burkert, G., Forster, P., Franconi, E.: Natural language processing and description logics. In: Peltason, C., von Luck, K., Kindermann, C. (eds.) Proc. of the Terminological Logic Users Workshop, pp. 162–164. Department of Computer Science, Technische Universität Berlin (1991)Google Scholar
  36. 36.
    Glimm, B., Horrocks, I., Lutz, C., Sattler, U.: Conjunctive query answering for the description logic \(\mathcal{SHIQ}\). Journal of Artificial Intelligence Research 31, 151–198 (2008)MathSciNetMATHGoogle Scholar
  37. 37.
    Glimm, B., Horrocks, I., Sattler, U.: Unions of conjunctive queries in SHOQ. In: Proc. of the 11th International Conference on the Principles of Knowledge Representation and Reasoning (KR 2008), pp. 252–262. AAAI Press/The MIT Press (2008)Google Scholar
  38. 38.
    Glimm, B., Kazakov, Y.: Role Conjunctions in Expressive Description Logics. In: Cervesato, I., Veith, H., Voronkov, A. (eds.) LPAR 2008. LNCS (LNAI), vol. 5330, pp. 391–405. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  39. 39.
    Glimm, B., Kazakov, Y., Lutz, C.: Status QIO: An update. In: Proc. of the 2009 Description Logic Workshop, DL 2009. CEUR Workshop Proceedings, vol. 745 (2011)Google Scholar
  40. 40.
    Gottlob, G., Orsi, G., Pieris, A.: Ontological queries: Rewriting and optimization. In: IEEE 27th International Conference on Data Engineering, ICDE, pp. 2–13 (April 2011)Google Scholar
  41. 41.
    Gottlob, G., Orsi, G., Pieris, A., Šimkus, M.: Datalog and its Extensions for the Semantic Web Databases. In: Eiter, T., Krennwallner, T. (eds.) Reasoning Web 2012, vol. 7487, pp. 54–77. Springer, Heidelberg (2012)Google Scholar
  42. 42.
    Gottlob, G., Schwentick, T.: Rewriting ontological queries into small nonrecursive datalog programs. In: Rosati, R., Rudolph, S., Zakharyaschev, M. (eds.) Description Logics. CEUR Workshop Proceedings, vol. 745, CEUR-WS.org (2011)Google Scholar
  43. 43.
    Grädel, E.: Why are modal logics so robustly decidable? Bulletin of the European Association for Theoretical Computer Science 68, 90–103 (1999)MathSciNetMATHGoogle Scholar
  44. 44.
    Gupta, A., Ullman, J.D.: Generalizing conjunctive query containment for view maintenance and integrity constraint verification (abstract). In: Workshop on Deductive Databases (In conjunction with JICSLP), Washington D.C. (USA), p. 195 (1992)Google Scholar
  45. 45.
    Hladik, J.: A tableau system for the description logic SHIO. In: Sattler, U. (ed.) IJCAR Doctoral Programme. CEUR Workshop Proceedings, vol. 106, CEUR-WS.org (2004)Google Scholar
  46. 46.
    Horrocks, I., Kutz, O., Sattler, U.: The irresistible \(\mathcal{SRIQ}\). In: Proc. of the 1st Int. Workshop on OWL: Experiences and Directions, OWLED 2005 (2005)Google Scholar
  47. 47.
    Horrocks, I., Kutz, O., Sattler, U.: The even more irresistible \(\mathcal{SROIQ}\). In: Doherty, P., Mylopoulos, J., Welty, C.A. (eds.) Proc. of the 10th Int. Conf. on the Principles of Knowledge Representation and Reasoning (KR 2006), pp. 57–67. AAAI Press (2006)Google Scholar
  48. 48.
    Horrocks, I., Rector, A., Goble, C.: A description logic based schema for the classification of medical data. In: CEUR Electronic Workshop Proceedings, Proc. of the 3rd Int. Workshop on Knowledge Representation Meets Databases, KRDB 1996, pp. 24–28 (1996), http://ceur-ws.org/Vol-4/
  49. 49.
    Horrocks, I., Sattler, U., Tobies, S.: Practical reasoning for very expressive description logics. J. of the Interest Group in Pure and Applied Logic 8(3), 239–264 (2000)MathSciNetMATHGoogle Scholar
  50. 50.
    Horrocks, I., Sattler, U., Tobies, S.: Reasoning with Individuals for the Description Logic \(\mathcal{SHIQ}\). In: McAllester, D. (ed.) CADE 2000. LNCS, vol. 1831, pp. 482–496. Springer, Heidelberg (2000)CrossRefGoogle Scholar
  51. 51.
    Horrocks, I., Tessaris, S.: A conjunctive query language for description logic ABoxes. In: Proc. of the 17th Nat. Conf. on Artificial Intelligence, AAAI 2000, pp. 399–404 (2000)Google Scholar
  52. 52.
    Hustadt, U., Motik, B., Sattler, U.: A Decomposition Rule for Decision Procedures by Resolution-Based Calculi. In: Baader, F., Voronkov, A. (eds.) LPAR 2004. LNCS (LNAI), vol. 3452, pp. 21–35. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  53. 53.
    Hustadt, U., Motik, B., Sattler, U.: Data complexity of reasoning in very expressive description logics. In: Proc. of the 19th Int. Joint Conf. on Artificial Intelligence, IJCAI 2005, pp. 466–471 (2005)Google Scholar
  54. 54.
    Immerman, N.: Expressibility and parallel complexity. SIAM J. Comput. 18(3), 625–638 (1989)MathSciNetCrossRefMATHGoogle Scholar
  55. 55.
    Kazakov, Y.: \(\mathcal{RIQ}\) and \(\mathcal{SROIQ}\) are harder than \(\mathcal{SHOIQ}\). In: Proc. of the Eleventh International Conference on the Principles of Knowledge Representation and Reasoning, KR 2008, pp. 274–284 (2008)Google Scholar
  56. 56.
    Kikot, S., Kontchakov, R., Zakharyaschev, M.: On (in)tractability of OBDA with OWL 2 QL. In: Rosati, R., Rudolph, S., Zakharyaschev, M. (eds.) Description Logics. CEUR Workshop Proceedings, vol. 745, CEUR-WS.org (2011), http://dblp.uni-trier.de/db/conf/dlog/dlog2011.html#KikotKZ11
  57. 57.
    Krisnadhi, A., Lutz, C.: Data Complexity in the \(\mathcal{EL}\) Family of Description Logics. In: Dershowitz, N., Voronkov, A. (eds.) LPAR 2007. LNCS (LNAI), vol. 4790, pp. 333–347. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  58. 58.
    Krötzsch, M.: OWL 2 profiles: An Introduction to Lightweight Ontology Languages. In: Eiter, T., Krennwallner, T. (eds.) Reasoning Web 2012. LNCS, vol. 7487, Springer, Heidelberg (2012)Google Scholar
  59. 59.
    Krötzsch, M., Rudolph, S.: Conjunctive queries for \(\mathcal{EL}\) with composition of roles. In: Proc. of the 2007 Description Logic Workshop, DL 2007. CEUR Electronic Workshop Proceedings, vol. 250 (2007), http://ceur-ws.org/Vol-250/
  60. 60.
    Krötzsch, M., Rudolph, S., Hitzler, P.: Complexity boundaries for Horn description logics. In: Proceedings of the 22nd AAAI Conference on Artificial Intelligence, AAAI 2007, pp. 452–457. AAAI Press (2007)Google Scholar
  61. 61.
    Lenzerini, M.: Data integration: A theoretical perspective. In: Proc. of the 21st ACM SIGACT SIGMOD SIGART Symp. on Principles of Database Systems, PODS 2002, pp. 233–246 (2002)Google Scholar
  62. 62.
    Levy, A.Y., Rousset, M.C.: Combining Horn rules and description logics in CARIN. Artificial Intelligence 104(1-2), 165–209 (1998)MathSciNetCrossRefMATHGoogle Scholar
  63. 63.
    Lutz, C., Toman, D., Wolter, F.: Conjunctive query answering in the description logic \(\mathcal{EL}\) using a relational database system. In: Proc. of the 21st Int. Joint Conf. on Artificial Intelligence, IJCAI 2009, pp. 2070–2075. AAAI Press (2009)Google Scholar
  64. 64.
    Lutz, C.: Complexity of Terminological Reasoning Revisited. In: Ganzinger, H., McAllester, D., Voronkov, A. (eds.) LPAR 1999. LNCS, vol. 1705, pp. 181–200. Springer, Heidelberg (1999)CrossRefGoogle Scholar
  65. 65.
    Lutz, C.: Inverse roles make conjunctive queries hard. In: Proc. of the 2007 Description Logic Workshop, DL 2007. CEUR Electronic Workshop Proceedings, vol. 250, pp. 100–111 (2007), http://ceur-ws.org/Vol-250/
  66. 66.
    Lutz, C.: The Complexity of Conjunctive Query Answering in Expressive Description Logics. In: Armando, A., Baumgartner, P., Dowek, G. (eds.) IJCAR 2008. LNCS (LNAI), vol. 5195, pp. 179–193. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  67. 67.
    Lutz, C.: Two upper bounds for conjunctive query answering in SHIQ. In: Baader, F., Lutz, C., Motik, B. (eds.) Description Logics. CEUR Workshop Proceedings, vol. 353, CEUR-WS.org (2008)Google Scholar
  68. 68.
    McGuinness, D.L.: Ontology-enhanced search for primary care medical literature. In: Proc. of the Int. Medical Informatics Association Working Group 6 – Conference on Natural Language Processing and Medical Concept Representation, IMIA 1999 (1999)Google Scholar
  69. 69.
    Motik, B.: Reasoning in Description Logics using Resolution and Deductive Databases. Ph.D. thesis, Univesität Karlsruhe (TH), Karlsruhe, Germany (January 2006)Google Scholar
  70. 70.
    Nebel, B.: Terminological reasoning is inherently intractable. Artificial Intelligence 43, 235–249 (1990)MathSciNetCrossRefMATHGoogle Scholar
  71. 71.
    Németi, I.: Free algebras and decidability in algebraic logic. DSc. thesis, Mathematical Institute of The Hungarian Academy of Sciences, Budapest (1986)Google Scholar
  72. 72.
    Noy, N.F.: Semantic integration: A survey of ontology-based approaches. SIGMOD Record 33(4), 65–70 (2004)CrossRefGoogle Scholar
  73. 73.
    Ortiz, M.: Query Answering in Expressive Description Logics: Techniques and Complexity Results. Ph.D. thesis, Vienna University of Technology (2010)Google Scholar
  74. 74.
    Ortiz, M., Calvanese, D., Eiter, T.: Characterizing data complexity for conjunctive query answering in expressive description logics. In: Proc. of the 21st Nat. Conf. on Artificial Intelligence, AAAI 2006. AAAI Press (July 2006)Google Scholar
  75. 75.
    Ortiz, M., Calvanese, D., Eiter, T.: Data complexity of query answering in expressive description logics via tableaux. J. of Automated Reasoning 41(1), 61–98 (2008)MathSciNetCrossRefMATHGoogle Scholar
  76. 76.
    Ortiz, M., Rudolph, S., Šimkus, M.: Query answering is undecidable in DLs with regular expressions, inverses, nominals, and counting. Tech. Rep. INFSYS RR-1843-10-03, Institut für Informationssysteme, Technische Universität Wien, A-1040 Vienna, Austria (April 2010)Google Scholar
  77. 77.
    Ortiz, M., Rudolph, S., Simkus, M.: Worst-case optimal reasoning for the Horn-DL fragments of OWL 1 and 2. In: Lin, F., Sattler, U., Truszczynski, M. (eds.) KR 2010, AAAI Press (2010)Google Scholar
  78. 78.
    Ortiz, M., Rudolph, S., Simkus, M.: Query answering in the Horn fragments of the description logics SHOIQ and SROIQ. In: Walsh, T. (ed.) IJCAI, pp. 1039–1044. IJCAI/AAAI (2011)Google Scholar
  79. 79.
    Ortiz, M., Šimkus, M., Eiter, T.: Worst-case optimal conjunctive query answering for an expressive description logic without inverses. In: Fox, D., Gomes, C.P. (eds.) AAAI 2008, pp. 504–510. AAAI Press (2008)Google Scholar
  80. 80.
    Patel-Schneider, P., Hayes, P., Horrocks, I.: OWL Web Ontology Language semantics and abstract syntax – W3C recommendation. Tech. rep., World Wide Web Consortium (February 2004), http://www.w3.org/TR/owl-semantics/
  81. 81.
    Pérez-Urbina, H., Motik, B., Horrocks, I.: A comparison of query rewriting techniques for DL-Lite. In: Grau, B.C., Horrocks, I., Motik, B., Sattler, U. (eds.) Description Logics. CEUR Workshop Proceedings, vol. 477, CEUR-WS.org (2009)Google Scholar
  82. 82.
    Pérez-Urbina, H., Motik, B., Horrocks, I.: Tractable query answering and rewriting under description logic constraints. J. Applied Logic 8(2), 186–209 (2010)MathSciNetCrossRefMATHGoogle Scholar
  83. 83.
    Poggi, A., Lembo, D., Calvanese, D., De Giacomo, G., Lenzerini, M., Rosati, R.: Linking Data to Ontologies. In: Spaccapietra, S. (ed.) Journal on Data Semantics X. LNCS, vol. 4900, pp. 133–173. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  84. 84.
    Pratt, V.R.: Models of program logic. In: Proc. of the 20th Annual Symp. on the Foundations of Computer Science (FOCS 1979), pp. 115–122 (1979)Google Scholar
  85. 85.
    Pratt-Hartmann, I.: Data-complexity of the two-variable fragment with counting quantifiers. Information and Computation 207(8), 867–888 (2009)MathSciNetCrossRefMATHGoogle Scholar
  86. 86.
    Rector, A., Bechhofer, S., Goble, C.A., Horrocks, I., Nowlan, W.A., Solomon, W.D.: The grail concept modelling language for medical terminology. Artificial Intelligence in Medicine 9, 139–171 (1997)CrossRefGoogle Scholar
  87. 87.
    Rosati, R.: On conjunctive query answering in \(\mathcal{EL}\). In: Proc. of the 2007 Description Logic Workshop, DL 2007. CEUR Electronic Workshop Proceedings, vol. 250 (2007), http://ceur-ws.org/Vol-250/
  88. 88.
    Rosati, R., Almatelli, A.: Improving query answering over DL-Lite ontologies. In: Proceedings of the Twelfth International Conference on Principles of Knowledge Representation and Reasoning, KR 2010 (2010)Google Scholar
  89. 89.
    Rudolph, S., Glimm, B.: Nominals, inverses, counting, and conjunctive queries or: Why infinity is your friend! J. of Artificial Intelligence Research 39, 429–481 (2010)MathSciNetMATHGoogle Scholar
  90. 90.
    Sattler, U., Vardi, M.Y.: The hybrid μ-calculus. In: Proc. of the Int. Joint Conf. on Automated Reasoning, IJCAR 2001, pp. 76–91 (2001)Google Scholar
  91. 91.
    Schaerf, A.: On the complexity of the instance checking problem in concept languages with existential quantification. J. of Intelligent Information Systems 2, 265–278 (1993)CrossRefGoogle Scholar
  92. 92.
    Schild, K.: A correspondence theory for terminological logics: Preliminary report. In: Proc. of the 12th Int. Joint Conf. on Artificial Intelligence, IJCAI 1991, pp. 466–471 (1991)Google Scholar
  93. 93.
    Tessaris, S.: Questions and Answers: Reasoning and Querying in Description Logic. Ph.D. thesis, University of Manchester, Department of Computer Science (April 2001)Google Scholar
  94. 94.
    Tobies, S.: The complexity of reasoning with cardinality restrictions and nominals in expressive description logics. J. of Artificial Intelligence Research 12, 199–217 (2000)MathSciNetMATHGoogle Scholar
  95. 95.
    Tobies, S.: Complexity Results and Practical Algorithms for Logics in Knowledge Representation. Ph.D. thesis, LuFG Theoretical Computer Science, RWTH-Aachen, Germany (2001)Google Scholar
  96. 96.
    Ullman, J.D.: Information integration using logical views. Theoretical Computer Science 239(2), 189–210 (2000)MathSciNetCrossRefMATHGoogle Scholar
  97. 97.
    Vardi, M.Y.: The complexity of relational query languages. In: Proc. of the 14th ACM SIGACT Symp. on Theory of Computing, STOC 1982, pp. 137–146 (1982)Google Scholar
  98. 98.
    Vardi, M.Y.: Reasoning about the Past with Two-Way Automata. In: Larsen, K.G., Skyum, S., Winskel, G. (eds.) ICALP 1998. LNCS, vol. 1443, pp. 628–641. Springer, Heidelberg (1998)CrossRefGoogle Scholar
  99. 99.
    Vardi, M.Y., Wolper, P.: Automata-theoretic techniques for modal logics of programs. Journal of Computer and System Sciences 32, 183–221 (1986)MathSciNetCrossRefMATHGoogle Scholar
  100. 100.
    Vollmer, H.: Introduction to circuit complexity - a uniform approach. In: Texts in Theoretical Computer Science. Springer (1999)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Magdalena Ortiz
    • 1
  • Mantas Šimkus
    • 1
  1. 1.Institute of Information SystemsVienna University of TechnologyAustria

Personalised recommendations