Classical Natural Deduction for S4 Modal Logic

  • Daisuke Kimura
  • Yoshihiko Kakutani
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5904)


This paper proposes a natural deduction system CNDS4 for classical S4 modal logic with necessity and possibility modalities. This new system is an extension of Parigot’s Classical Natural Deduction with dual-context to formulate S4 modal logic. The modal λμ-calculus is also introduced as a computational extraction of CNDS4. It is an extension of both the λμ-calculus and the modal λ-calculus. Subject reduction, confluency, and strong normalization of the modal λμ-calculus are shown. Finally, the computational interpretation of the modal λμ-calculus, especially the computational meaning of the modal possibility operator, is discussed.


Modal Logic Classical Logic Natural Deduction Sequent Calculus Exception Handling 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. 1.
    Bierman, G.M., de Paiva, V.: On an intuitionistic modal logic. Studia Logica 65(3), 383–416 (2000)zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Curien, P.L., Herbelin, H.: The duality of computation. In: Proceedings of the 5th ACM SIGPLAN International Conference on Functional Programming, ICFP, pp. 233–243 (2000)Google Scholar
  3. 3.
    Davies, R.: A temporal-logic approach to binding-time analysis. In: Proceedings of 11 th Annual IEEE Symposium on Logic in Computer Science, pp. 184–195. IEEE Computer Society Press, Los Alamitos (1996)CrossRefGoogle Scholar
  4. 4.
    Davies, R., Pfenning, F.: A modal analysis of staged computation. Journal of the ACM 48(3), 555–604 (2001)CrossRefMathSciNetGoogle Scholar
  5. 5.
    de Groote, P.: On the relation between the lambda-mu-calculus and the syntactic theory of sequential control. In: Pfenning, F. (ed.) LPAR 1994. LNCS, vol. 822, pp. 31–43. Springer, Heidelberg (1994)Google Scholar
  6. 6.
    Ghani, N., de Paiva, V., Ritter, E.: Explicit Substitutions for Constructive Necessity. In: Larsen, K.G., Skyum, S., Winskel, G. (eds.) ICALP 1998. LNCS, vol. 1443, pp. 743–754. Springer, Heidelberg (1998)CrossRefGoogle Scholar
  7. 7.
    Griffin, T.G.: A formulae-as-types notion of control. In: Proc. of the 1990 Principles of Programming Languages Conference, pp. 47–58. IEEE Computer Society Press, Los Alamitos (1990)Google Scholar
  8. 8.
    Groote, P.D.: An environment machine for the λμ-calculus. Mathematical Structures in Computer Science 8(6), 637–669 (1998)zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Kakutani, Y.: Duality between Call-by-Name Recursion and Call-by-Value Iteration. In: Bradfield, J.C. (ed.) CSL 2002. LNCS, vol. 2471, pp. 506–521. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  10. 10.
    Kakutani, Y.: Call-by-Name and Call-by-Value in Normal Modal Logic. In: Shao, Z. (ed.) APLAS 2007. LNCS, vol. 4807, pp. 399–414. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  11. 11.
    Kimura, D.: Duality between Call-by-value Reductions and Call-by-name Reductions. IPSJ Journal 48(4), 1721–1757 (2007)MathSciNetGoogle Scholar
  12. 12.
    de Paz, M., Medeiros, N.: A new S4 classical modal logic in natural deduction. Journal of Symbolic Logic 71(3), 799–809 (2006)zbMATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Nanevski, A.: A Modal Calculus for Exception Handling. In: The 3rd intuitionistic modal logics and applications workshop (2005)Google Scholar
  14. 14.
    Newman, M.H.A.: On theories with a combinatorial definition of “equivalence”. Annals of Mathematics 43(2), 223–243 (1942)CrossRefMathSciNetGoogle Scholar
  15. 15.
    Ong, C.-H.L., Stewart, C.A.: A Curry-Howard foundation for functional computation with control. In: Proc. of the Symposium on Principles of Programming Languages, pp. 215–227 (1997)Google Scholar
  16. 16.
    Parigot, M.: λμ-calculus: an algorithmic interpretation of classical natural deduction. In: Voronkov, A. (ed.) LPAR 1992. LNCS, vol. 624, pp. 190–201. Springer, Heidelberg (1992)CrossRefGoogle Scholar
  17. 17.
    Parigot, M.: Strong normalization for second order classical natural deduction. In: Proceedings of Eighth Annual IEEE Symposium on Logic in Computer Science, pp. 39–46 (1993)Google Scholar
  18. 18.
    Pfenning, F., Davies, R.: A judgmental reconstruction of modal logic. Mathematical Structures in Computer Science 11, 511–540 (2001)zbMATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Prawitz, D.: Natural Deduction: A Proof-Theoretical Study. Almqvist and Wiksell, Stockholm (1965)Google Scholar
  20. 20.
    Selinger, P.: Control Categories and Duality: on the Categorical Semantics of the Lambda-Mu Calculus. Mathematical Structures in Computer Science, 207–260 (2001)Google Scholar
  21. 21.
    Shan, C.-C.: A Computational Interpretation of Classical S4 Modal Logic. In: The 3rd intuitionistic modal logics and applications workshop (2005)Google Scholar
  22. 22.
    Taha, W., Sheard, T.: MetaML and multi-stage programming with explicit annotations. Theoretical Computer Science 248(1-2), 211–242 (2000)zbMATHCrossRefGoogle Scholar
  23. 23.
    Wadler, P.: Call-by-Value is Dual to Call-by-Name – Reloaded. In: Giesl, J. (ed.) RTA 2005. LNCS, vol. 3467, pp. 185–203. Springer, Heidelberg (2005)Google Scholar
  24. 24.
    Yuse, Y., Igarashi, A.: A modal type system for multi-level generating extensions with persistent code. In: Proceedings of the 8th ACM SIGPLAN Symposium on Principles and Practice of Declarative Programming, pp. 201–212 (2006)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  • Daisuke Kimura
    • 1
  • Yoshihiko Kakutani
    • 1
  1. 1.Department of Computer ScienceUniversity of TokyoTokyoJapan

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