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Call-by-Name and Call-by-Value in Normal Modal Logic

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Programming Languages and Systems (APLAS 2007)

Part of the book series: Lecture Notes in Computer Science ((LNPSE,volume 4807))

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Abstract

This paper provides a call-by-name and a call-by-value calculus, both of which have a Curry-Howard correspondence to the minimal normal logic K. The calculi are extensions of the λμ-calculi, and their semantics are given by CPS transformations into a calculus corresponding to the intuitionistic fragment of K. The duality between call-by-name and call-by-value with modalities is investigated in our calculi.

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References

  1. Abe, T.: Completeness of modal proofs in first-order predicate logic. Computer Software, JSSST Journal (to appear)

    Google Scholar 

  2. Barber, A.: Dual intuitionistic linear logic. Technical report, LFCS, University of Edinburgh (1996)

    Google Scholar 

  3. Barendregt, H.P.: Lambda calculi with types. In: Abramski, S., Gabbay, D.M., Maibaum, T.S.E. (eds.) Handbook of Logic in Computer Science, vol. 2, pp. 117–309. Oxford University Press, Oxford (1992)

    Google Scholar 

  4. Bellin, G., de Paiva, V.C.V., Ritter, E.: Extended Curry-Howard correspondence for a basic constructive modal logic. In: Proceedings of Methods for Modalities (2001)

    Google Scholar 

  5. Bierman, G.M.: What is a categorical model of intuitionistic linear logic. In: Dezani-Ciancaglini, M., Plotkin, G. (eds.) TLCA 1995. LNCS, vol. 902, pp. 78–93. Springer, Heidelberg (1995)

    Chapter  Google Scholar 

  6. Bierman, G.M., de Paiva, V.C.V.: On an intuitionistic modal logic. Studia Logica 65(3), 383–416 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  7. Blackburn, P., de Rijke, M., Venema, Y.: Modal Logic. Cambridge University Press, Cambridge (2001)

    MATH  Google Scholar 

  8. Davies, R., Pfenning, F.: A modal analysis of staged computation. Journal of the ACM 48(3), 555–604 (2001)

    Article  MathSciNet  Google Scholar 

  9. de Groote, P.: A cps-translation of the λμ-calculus. In: Tison, S. (ed.) CAAP 1994. LNCS, vol. 787, pp. 85–99. Springer, Heidelberg (1994)

    Chapter  Google Scholar 

  10. Filinski, A.: Declarative continuations and categorical duality. Master’s thesis, Computer Science Department, University of Copenhagen (1989)

    Google Scholar 

  11. Fischer, M.: Lambda calculus schemata. In: Proving Assertions about Programs, pp. 104–109. ACM Press, New York (1972)

    Chapter  Google Scholar 

  12. Girard, J.-Y.: Linear logic. Theoretical Computer Science 50(1), 1–102 (1987)

    Article  MATH  MathSciNet  Google Scholar 

  13. Griffin, T.G.: A formulae-as-types notion of control. In: Principles of Programming Languages, pp. 47–58. ACM Press, New York (1990)

    Google Scholar 

  14. Hofmann, M., Streicher, T.: Continuation models are universal for λμ-calculus. In: Logic in Computer Science, pp. 387–397. IEEE Computer Society Press, Los Alamitos (1997)

    Google Scholar 

  15. Howard, W.A.: The formulae-as-types notion of construction. In: Essays on Combinatory Logic, Lambda Calculus and Formalism, pp. 479–490. Academic Press, London (1980)

    Google Scholar 

  16. Kakutani, Y.: Duality between call-by-name recursion and call-by-value iteration. In: Bradfield, J.C. (ed.) CSL 2002 and EACSL 2002. LNCS, vol. 2471, pp. 506–521. Springer, Heidelberg (2002)

    Chapter  Google Scholar 

  17. Kakutani, Y.: Calculi for intuitionistic normal modal logic. In: Proceedings of Programming and Programming Languages (2007)

    Google Scholar 

  18. Kripke, S.: Semantic analysis of modal logic I, normal propositional logic. Zeitschrift für Mathemathische Logik und Grundlagen der Mathematik 9, 67–96 (1963)

    Article  MATH  MathSciNet  Google Scholar 

  19. Lambek, J., Scott, P.J.: Introduction to Higher-Order Categorical Logic. Cambridge University Press, Cambridge (1986)

    MATH  Google Scholar 

  20. Mac Lane, S.: Categories for the Working Mathematician, 2nd edn. Springer, Heidelberg (1997)

    Google Scholar 

  21. Maietti, M.E., Maneggia, P., de Paiva, V.C.V., Ritter, E.: Relating categorical semantics for intuitionistic linear logic. Applied Categorical Structures 13(1), 1–36 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  22. Martini, S., Masini, A.: A computational interpretation of modal proofs. In: Proof Theory of Modal Logics, pp. 213–241. Kluwer Academic Publishers, Dordrecht (1996)

    Google Scholar 

  23. Miyamoto, K., Igarashi, A.: A modal foundation for secure information flow. In: Proceedings of Foundations of Computer Security (2004)

    Google Scholar 

  24. Moggi, E.: Computational lambda-calculus and monads. In: Logic in Computer Science, pp. 14–23. IEEE Computer Society Press, Los Alamitos (1989)

    Google Scholar 

  25. Ong, C.-H.L., Stewart, C.A.: A Curry-Howard foundation for functional computation with control. In: Principle of Programming Languages, pp. 215–227. ACM Press, New York (1997)

    Google Scholar 

  26. 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)

    Chapter  Google Scholar 

  27. Plotkin, G.D.: Call-by-name, call-by-value and the lambda calculus. Theoretical Computer Science 1(2), 125–159 (1975)

    Article  MATH  MathSciNet  Google Scholar 

  28. Pym, D., Ritter, E.: On the semantics of classical disjunction. Journal of Pure and Applied Algebra 159(2,3), 315–338 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  29. Seely, R.A.G.: Linear logic, ∗-autonomous categories and cofree coalgebras. In: Categories in Computer Science and Logic. Contemporary Mathematics, vol. 92, pp. 371–389. AMS (1989)

    Google Scholar 

  30. Selinger, P.: Control categories and duality: on the categorical semantics of the lambda-mu calculus. Mathematical Structures in Computer Science 11(2), 207–260 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  31. Selinger, P.: Some remarks on control categories. Manuscript (2003)

    Google Scholar 

  32. Shan, C.-C.: A computastional interpretation of classical S4 modality. In: Proceedings of Intuitionistic Modal Logics and Applications (2005)

    Google Scholar 

  33. Simpson, A.K.: The Proof Theory and Semantics of Intuitionistic Modal Logics. PhD thesis, University of Edinburgh (1993)

    Google Scholar 

  34. Wadler, P.: Call-by-value is dual to call-by-name. In: International Conference on Functional Programming, pp. 189–201. ACM Press, New York (2003)

    Google Scholar 

  35. Wijesekera, D.: Constructive modal logic I. Annals of Pure and Applied Logic 50, 271–301 (1990)

    Article  MATH  MathSciNet  Google Scholar 

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Zhong Shao

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Kakutani, Y. (2007). Call-by-Name and Call-by-Value in Normal Modal Logic. In: Shao, Z. (eds) Programming Languages and Systems. APLAS 2007. Lecture Notes in Computer Science, vol 4807. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-76637-7_27

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  • DOI: https://doi.org/10.1007/978-3-540-76637-7_27

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-76636-0

  • Online ISBN: 978-3-540-76637-7

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