SCAN Is Complete for All Sahlqvist Formulae

  • V. Goranko
  • U. Hustadt
  • R. A. Schmidt
  • D. Vakarelov
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3051)


scan is an algorithm for reducing existential second-order logic formulae to equivalent simpler formulae, often first-order logic formulae. It is provably impossible for such a reduction to first-order logic to be successful for every second-order logic formula which has an equivalent first-order formula. In this paper we show that scan successfully computes the first-order equivalents of all Sahlqvist formulae in the classical (multi-)modal language.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Bibel, W., Schmitt, P.H. (eds.): Automated Deduction – A Basis for Applications, vol. I-III. Kluwer, Dordrecht (1998)Google Scholar
  2. 2.
    Blackburn, P., de Rijke, M., Venema, V.: Modal Logic. Cambridge Univ. Press, Cambridge (2001)MATHGoogle Scholar
  3. 3.
    Chagrov, A., Zakharyaschev, M.: Modal Logic. Oxford Logic Guides, vol. 35. Clarendon Press, Oxford (1997)MATHGoogle Scholar
  4. 4.
    de Rijke, M., Venema, Y.: Sahlqvist’s Theorem For Boolean Algebras with Operators with an Application to Cylindric Algebras. Studia Logica 54, 61–78 (1995)MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Doherty, P., Lukaszewics, W., Szalas, A.: Computing circumscription revisited: A reduction algorithm. Journal of Automated Reasoning 18(3), 297–336 (1997)MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Engel, T.: Quantifier Elimination in Second-Order Predicate Logic. MSc thesis, Saarland University, Saarbrücken, Germany (1996)Google Scholar
  7. 7.
    Fermüller, C.G., Leitsch, A., Hustadt, U., Tammet, T.: Resolution decision procedures. In: Handbook of Automated Reasoning, pp. 1791–1849. Elsevier, Amsterdam (2001)CrossRefGoogle Scholar
  8. 8.
    Gabbay, D.M., Ohlbach, H.J.: Quantifier elimination in second-order predicate logic. South African Computer Journal 7, 35–43 (1992)Google Scholar
  9. 9.
    Goranko, V., Vakarelov, D.: Sahlqvist formulas unleashed in polyadic modal languages. In: Advances in Modal Logic, vol. 3, pp. 221–240. World Scientific, Singapore (2002)CrossRefGoogle Scholar
  10. 10.
    Jónsson, B.: On the canonicity of Sahlqvist identities. Studia Logica 53, 473–491 (1994)MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Kracht, M.: How completeness and correspondence theory got married. In: Diamonds and Defaults, pp. 175–214. Kluwer, Dordrecht (1993)Google Scholar
  12. 12.
    Nonnengart, A.: Strong skolemization. Research Report MPI-I-96-2-010, Max- Planck-Institut für Informatik, Saarbrücken (1996)Google Scholar
  13. 13.
    Nonnengart, A., Ohlbach, H.J., Szalas, A.: Quantifier elimination for secondorder predicate logic. In: Logic, Language and Reasoning: Essays in honour of Dov Gabbay, Kluwer, Dordrecht (1999)Google Scholar
  14. 14.
    Nonnengart, A., Weidenbach, C.: Computing small clause normal forms. In: Handbook of Automated Reasoning, pp. 335–367. Elsevier, Amsterdam (2001)CrossRefGoogle Scholar
  15. 15.
    Robinson, A., Voronkov, A. (eds.): Handbook of Automated Reasoning. Elsevier Science, Amsterdam (2001)Google Scholar
  16. 16.
    Sahlqvist, H.: Completeness and correspondence in the first and second order semantics for modal logics. In: Proc. of the 3rd Scandinavian Logic Symposium, 1973, pp. 110–143. North-Holland, Amsterdam (1975)CrossRefGoogle Scholar
  17. 17.
    Sambin, G., Vaccaro, V.: A new proof of Sahlqvist’s theorem on modal definability and completeness. Journal of Symbolic Logic 54(3), 992–999 (1989)MATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Szałas, A.: On the correspondence between modal and classical logic: An automated approach. Journal of Logic and Computation 3(6), 605–620 (1993)MATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    van Benthem, J.: Modal Logic and Classical Logic. Bibliopolis (1983)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2004

Authors and Affiliations

  • V. Goranko
    • 1
  • U. Hustadt
    • 2
  • R. A. Schmidt
    • 3
  • D. Vakarelov
    • 4
  1. 1.Rand Afrikaans UniversitySouth Africa
  2. 2.University of LiverpoolUK
  3. 3.University of ManchesterUK
  4. 4.Sofia UniversityBulgaria

Personalised recommendations