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Second-Order Quantifier Elimination on Relational Monadic Formulas – A Basic Method and Some Less Expected Applications

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Automated Reasoning with Analytic Tableaux and Related Methods (TABLEAUX 2015)

Part of the book series: Lecture Notes in Computer Science ((LNAI,volume 9323))

Abstract

For relational monadic formulas (the Löwenheim class) second-order quantifier elimination, which is closely related to computation of uniform interpolants, forgetting and projection, always succeeds. The decidability proof for this class by Behmann from 1922 explicitly proceeds by elimination with equivalence preserving formula rewriting. We reconstruct Behmann’s method, relate it to the modern DLS elimination algorithm and show some applications where the essential monadicity becomes apparent only at second sight. In particular, deciding \(\mathcal{ALCOQH}\) knowledge bases, elimination in DL-Lite knowledge bases, and the justification of the success of elimination methods for Sahlqvist formulas.

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References

  1. Ackermann, W.: Untersuchungen über das Eliminationsproblem der mathematischen Logik. Math. Ann. 110, 390–413 (1935)

    Article  MathSciNet  MATH  Google Scholar 

  2. Ackermann, W.: Zum Eliminationsproblem der mathematischen Logik. Math. Ann. 111, 61–63 (1935)

    Article  MathSciNet  MATH  Google Scholar 

  3. Artale, A., Calvanese, D., Kontchakov, R., Zakharyaschev, M.: The DL-Lite family and relations. JAIR 36, 1–69 (2009)

    MathSciNet  MATH  Google Scholar 

  4. Bachmair, L., Ganzinger, H., Waldmann, U.: Superposition with simplification as a decision procedure for the monadic class with equality. In: Mundici, D., Gottlob, G., Leitsch, A. (eds.) KGC 1993. LNCS, vol. 713, pp. 83–96. Springer, Heidelberg (1993)

    Chapter  Google Scholar 

  5. Behmann, H.: Beiträge zur Algebra der Logik, insbesondere zum Entscheidungsproblem. Math. Ann. 86(3–4), 163–229 (1922)

    Article  MathSciNet  MATH  Google Scholar 

  6. Behmann, H. (Bestandsbildner): Nachlass Heinrich Johann Behmann, Staatsbibliothek zu Berlin – Preußischer Kulturbesitz, Handschriftenabt., Nachl. 335

    Google Scholar 

  7. van Benthem, J.: Modal Logic and Classical Logic. Bibliopolis (1983)

    Google Scholar 

  8. Blackburn, P., de Rijke, M., Venema, Y.: Modal Logic. Cambr. Univ. Press (2001)

    Google Scholar 

  9. Börger, E., Grädel, E., Gurevich, Y.: The Classical Decision Problem. Springer (1997)

    Google Scholar 

  10. Calvanese, D., et al.: Tractable reasoning and efficient query answering in description logics: The DL-Lite family. JAR 39(3), 385–429 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  11. Conradie, W.: On the strength and scope of DLS. JANCL 16(3–4), 279–296 (2006)

    MathSciNet  MATH  Google Scholar 

  12. Conradie, W., Goranko, V., Vakarelov, D.: Algorithmic correspondence and completeness in modal logic. I. The core algorithm SQEMA. LMCS 2(1:5), 1–26 (2006)

    Google Scholar 

  13. Conradie, W., Goranko, V., Vakarelov, D.: Elementary canonical formulae: A survey on syntactic, algorithmic, and model-theoretic aspects. In: Advances in Modal Logic, vol. 5, pp. 17–51. College Pub. (2005)

    Google Scholar 

  14. Craig, W.: Elimination problems in logic: A brief history. Synthese 164, 321–332 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  15. Doherty, P., Łukaszewicz, W., Szałas, A.: Computing circumscription revisited: A reduction algorithm. JAR 18(3), 297–338 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  16. Egly, U.: On the value of antiprenexing. In: Pfenning, F. (ed.) LPAR 1994. LNCS, vol. 822, pp. 69–83. Springer, Heidelberg (1994)

    Chapter  Google Scholar 

  17. Fermüller, C., Leitsch, A., Hustadt, U., Tammet, T.: Resolution decision procedures. In: Robinson, A., Voronkov, A. (eds.) Handb. of Autom. Reasoning, vol. 2, pp. 1793–1849. Elsevier (2001)

    Google Scholar 

  18. Gabbay, D.M., Schmidt, R.A., Szałas, A.: Second-Order Quantifier Elimination: Foundations, Computational Aspects and Applications. College Pub. (2008)

    Google Scholar 

  19. Gabbay, D., Ohlbach, H.J.: Quantifier elimination in second-order predicate logic. In: KR 1992, pp. 425–435. Morgan Kaufmann (1992)

    Google Scholar 

  20. Ghilardi, S., Lutz, C., Wolter, F.: Did I damage my ontology? A case for conservative extensions in description logics. In: KR 2006, pp. 187–197. AAAI Press (2006)

    Google Scholar 

  21. Goranko, V., Hustadt, U., Schmidt, R.A., Vakarelov, D.: SCAN is complete for all Sahlqvist formulae. In: Berghammer, R., Möller, B., Struth, G. (eds.) RelMiCS 2003. LNCS, vol. 3051, pp. 149–162. Springer, Heidelberg (2004)

    Chapter  Google Scholar 

  22. Gustafsson, J.: An implementation and optimization of an algorithm for reducing formulae in second-order logic. Tech. Rep. LiTH-MAT-R-96-04, Univ. Linköping (1996)

    Google Scholar 

  23. Hustadt, U., Schmidt, R.A., Georgieva, L.: A survey of decidable first-order fragments and description logics. JoRMiCS 1, 251–276 (2004)

    Google Scholar 

  24. Konev, B., Walther, D., Wolter, F.: Forgetting and uniform interpolation in large-scale description logic terminologies. In: IJCAI 2009, pp. 830–835 (2009)

    Google Scholar 

  25. Kontchakov, R., Wolter, F., Zakharyaschev, M.: Logic-based ontology comparison and module extraction, with an application to DL-Lite. AI 174(15), 1093–1141 (2010)

    MathSciNet  MATH  Google Scholar 

  26. Koopmann, P., Schmidt, R.A.: Uniform interpolation of \(\mathcal{ALC}\)-ontologies using fixpoints. In: Fontaine, P., Ringeissen, C., Schmidt, R.A. (eds.) FroCoS 2013. LNCS, vol. 8152, pp. 87–102. Springer, Heidelberg (2013)

    Chapter  Google Scholar 

  27. Koopmann, P., Schmidt, R.A.: Count and forget: Uniform interpolation of \(\mathcal{SHQ}\)-ontologies. In: Demri, S., Kapur, D., Weidenbach, C. (eds.) IJCAR 2014. LNCS (LNAI), vol. 8562, pp. 434–448. Springer, Heidelberg (2014)

    Google Scholar 

  28. Koopmann, P., Schmidt, R.A.: Uniform interpolation and forgetting for \(\mathcal{ALC}\) ontologies with ABoxes. In: AAAI 2015, pp. 175–181. AAAI Press (2015)

    Google Scholar 

  29. Lewis, H.R.: Complexity results for classes of quantificational formulas. Journal of Computer and System Sciences 21, 317–353 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  30. Ludwig, M., Konev, B.: Practical uniform interpolation and forgetting for \(\mathcal{ALC}\) TBoxes with applications to logical difference. In: KR 2014. AAAI Press (2014)

    Google Scholar 

  31. Lutz, C., Wolter, F.: Foundations for uniform interpolation and forgetting in expressive description logics. In: IJCAI 2011, pp. 989–995. AAAI Press (2011)

    Google Scholar 

  32. Löwenheim, L.: Über Möglichkeiten im Relativkalkül. Math. Ann. 76, 447–470 (1915)

    Article  MathSciNet  MATH  Google Scholar 

  33. Mancosu, P., Zach, R.: Heinrich Behmann’s 1921 lecture on the algebra of logic and the decision problem. The Bulletin of Symbolic Logic 21, 164–187 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  34. Murray, N.V., Rosenthal, E.: Tableaux, path dissolution and decomposable negation normal form for knowledge compilation. In: Cialdea Mayer, M., Pirri, F. (eds.) TABLEAUX 2003. LNCS (LNAI), vol. 2796, pp. 165–180. Springer, Heidelberg (2003)

    Chapter  Google Scholar 

  35. Nonnengart, A., Szałas, A.: A fixpoint approach to second-order quantifier elimination with applications to correspondence theory. In: Orlowska, E. (ed.) Logic at Work. Essays Ded. to the Mem. of Helena Rasiowa, pp. 89–108. Springer (1998)

    Google Scholar 

  36. Nonnengart, A., Weidenbach, C.: Computing small clause normal forms. In: Robinson, A., Voronkov, A. (eds.) Handb. of Autom. Reasoning, vol. 1, pp. 335–367. Elsevier (2001)

    Google Scholar 

  37. Quine, W.V.: On the logic of quantification. JSL 10(1), 1–12 (1945)

    MathSciNet  MATH  Google Scholar 

  38. Sattler, U., Calvanese, D., Molitor, R.: Relationships with other formalisms. In: Baader, F., et al. (eds.) The Description Logic Handb, pp. 137–177. Cambr. Univ. Press (2003)

    Google Scholar 

  39. Schmidt, R.A.: The Ackermann approach for modal logic, correspondence theory and second-order reduction. JAL 10(1), 52–74 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  40. Schneider, K.: Verification of Reactive Systems. Springer (2003)

    Google Scholar 

  41. Skolem, T.: Untersuchungen über die Axiome des Klassenkalküls und über Produktations- und Summationsprobleme welche gewisse Klassen von Aussagen betreffen. Videnskapsselskapets Skrifter I. Mat.-Nat. Klasse(3) (1919)

    Google Scholar 

  42. Veanes, M., Bjørner, N., Nachmanson, L., Bereg, S.: Monadic decomposition. In: Biere, A., Bloem, R. (eds.) CAV 2014. LNCS, vol. 8559, pp. 628–645. Springer, Heidelberg (2014)

    Google Scholar 

  43. Wang, Z., Wang, K., Topor, R.W., Pan, J.Z.: Forgetting for knowledge bases in DL-Lite. Ann. Math. Artif. Intell. 58, 117–151 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  44. Wernhard, C.: Tableaux for projection computation and knowledge compilation. In: Giese, M., Waaler, A. (eds.) TABLEAUX 2009. LNCS (LNAI), vol. 5607, pp. 325–340. Springer, Heidelberg (2009)

    Chapter  Google Scholar 

  45. Wernhard, C.: Abduction in logic programming as second-order quantifier elimination. In: Fontaine, P., Ringeissen, C., Schmidt, R.A. (eds.) FroCoS 2013. LNCS (LNAI), vol. 8152, pp. 103–119. Springer, Heidelberg (2013)

    Chapter  Google Scholar 

  46. Wernhard, C.: Expressing view-based query processing and related approaches with second-order operators. Tech. Rep. KRR 14–02, TU Dresden (2014)

    Google Scholar 

  47. Wernhard, C.: Heinrich Behmann’s contributions to second-order quantifier elimination. Tech. Rep. KRR 15–05, TU Dresden (2015)

    Google Scholar 

  48. Wernhard, C.: Second-order quantifier elimination on relational monadic formulas – A basic method and some less expected applications. Tech. Rep. KRR 15–04, TU Dresden (2015)

    Google Scholar 

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  1. Technische Universität Dresden, Dresden, Germany

    Christoph Wernhard

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  1. Christoph Wernhard
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  1. Uniwersytet Wrocławski, Wrocław, Poland

    Hans De Nivelle

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Wernhard, C. (2015). Second-Order Quantifier Elimination on Relational Monadic Formulas – A Basic Method and Some Less Expected Applications. In: De Nivelle, H. (eds) Automated Reasoning with Analytic Tableaux and Related Methods. TABLEAUX 2015. Lecture Notes in Computer Science(), vol 9323. Springer, Cham. https://doi.org/10.1007/978-3-319-24312-2_18

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  • DOI: https://doi.org/10.1007/978-3-319-24312-2_18

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