Strong Normalization of the Dual Classical Sequent Calculus

  • Daniel Dougherty
  • Silvia Ghilezan
  • Pierre Lescanne
  • Silvia Likavec
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3835)

Abstract

We investigate some syntactic properties of Wadler’s dual calculus, a term calculus which corresponds to classical sequent logic in the same way that Parigot’s λμ calculus corresponds to classical natural deduction. Our main result is strong normalization theorem for reduction in the dual calculus; we also prove some confluence results for the typed and untyped versions of the system.

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References

  1. 1.
    Ariola, Z.M., Herbelin, H.: Minimal classical logic and control operators. In: Baeten, J.C.M., Lenstra, J.K., Parrow, J., Woeginger, G.J. (eds.) ICALP 2003. LNCS, vol. 2719, pp. 871–885. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  2. 2.
    Bakel, S.: The language \(\mathcal{X}\): circuits, computations and classical logic. In: Coppo, M., Lodi, E., Pinna, G.M. (eds.) ICTCS 2005. LNCS, vol. 3701, pp. 81–96. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  3. 3.
    Barbanera, F., Berardi, S.: A symmetric lambda calculus for classical program extraction. Information and Computation 125(2), 103–117 (1996)MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Barendregt, H.P.: The Lambda Calculus: its Syntax and Semantics. North-Holland, Amsterdam (1984) (revised edition)MATHGoogle Scholar
  5. 5.
    Bierman, G.M.: A computational interpretation of the λμ-calculus. In: Brim, L., Gruska, J., Zlatuška, J. (eds.) MFCS 1998. LNCS, vol. 1450, pp. 336–345. Springer, Heidelberg (1998)CrossRefGoogle Scholar
  6. 6.
    Cosmo, R.D., Kesner, D.: Simulating expansions without expansions. Mathematical Structures in Computer Science 4(3), 315–362 (1994)MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Curien, P.-L.: Symmetry and interactivity in programming. Archive for Mathematical Logic (2001) (to appear)Google Scholar
  8. 8.
    Curien, P.-L.: Abstract machines, control, and sequents. In: Barthe, G., Dybjer, P., Pinto, L., Saraiva, J. (eds.) APPSEM 2000. LNCS, vol. 2395, pp. 123–136. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  9. 9.
    Curien, P.-L., Herbelin, H.: The duality of computation. In: Proc. of the 5th ACM SIGPLAN Int. Conference on Functional Programming (ICFP 2000), Montreal, Canada. ACM Press, New York (2000)Google Scholar
  10. 10.
    David, R., Nour, K.: Arithmetical proofs of strong normalization results for the symmetric λμ-calculus. In: Urzyczyn, P. (ed.) TLCA 2005. LNCS, vol. 3461, pp. 162–178. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  11. 11.
    de Groote, P.: On the relation between the λμ-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
  12. 12.
    Dougherty, D., Ghilezan, S., Lescanne, P.: Characterizing strong normalization in a language with control operators. In: Sixth ACM SIGPLAN Conference on Principles and Practice of Declarative Programming PPDP 2004, pp. 155–166. ACM Press, New York (2004)CrossRefGoogle Scholar
  13. 13.
    Dougherty, D.J.: Some lambda calculi with categorical sums and products. In: Kirchner, C. (ed.) RTA 1993. LNCS, vol. 690, pp. 137–151. Springer, Heidelberg (1993)Google Scholar
  14. 14.
    Filinski, A.: Declarative continuations and categorical duality. Master’s thesis, DIKU, Computer Science Department, University of Copenhagen, DIKU Rapport 89/11 (August 1989)Google Scholar
  15. 15.
    Gentzen, G.: Unterschungen über das logische Schliessen, Math Z., vol. 39, pp. 176–210 (1935); Szabo, M. (ed.): Collected papers of Gerhard Gentzen, pp. 68–131. North-Holland, Amsterdam (1969)MATHGoogle Scholar
  16. 16.
    Griffin, T.: A formulae-as-types notion of control. POPL 17, 47–58 (1990)Google Scholar
  17. 17.
    Herbelin, H.: Séquents qu’on calcule : de l’interprétation du calcul des séquents comme calcul de λ-termes et comme calcul de stratégies gagnantes. Thèse, U. Paris 7, Janvier (1995)Google Scholar
  18. 18.
    Howard, W.A.: The formulas-as-types notion of construction. In: Seldin, J.P., Hindley, J.R. (eds.) To H.B. Curry: Essays on Combinatory Logic, Lambda Calculus and Formalism, New York, pp. 479–490. Academic Press, London (1980)Google Scholar
  19. 19.
    Lengrand, S.: Call-by-value, call-by-name, and strong normalization for the classical sequent calculus. In: Gramlich, B., Lucas, S. (eds.) ENTCS, vol. 86. Elsevier, Amsterdam (2003)Google Scholar
  20. 20.
    Likavec, S.: Types for object oriented and functional programming languages. PhD thesis, Università di Torino, Italy, ENS Lyon, France (2005)Google Scholar
  21. 21.
    Murthy, C.R.: Classical proofs as programs: How, what, and why. In: Myers Jr., J.P., O’Donnell, M.J. (eds.) Constructivity in CS 1991. LNCS, vol. 613, pp. 71–88. Springer, Heidelberg (1992)CrossRefGoogle Scholar
  22. 22.
    Ong, C.-H.L., Stewart, C.A.: A Curry-Howard foundation for functional computation with control. In: POPL, vol. 24, pp. 215–227 (1997)Google Scholar
  23. 23.
    Parigot, M.: An algorithmic interpretation of classical natural deduction. In: Voronkov, A. (ed.) LPAR 1992. LNCS, vol. 624, pp. 190–201. Springer, Heidelberg (1992)CrossRefGoogle Scholar
  24. 24.
    Parigot, M.: Proofs of strong normalisation for second order classical natural deduction. The J. of Symbolic Logic 62(4), 1461–1479 (1997)MATHCrossRefMathSciNetGoogle Scholar
  25. 25.
    Polonovski, E.: Strong normalization of \(\lambda\mu\tilde{\mu}\)-calculus with explicit substitutions. In: Walukiewicz, I. (ed.) FOSSACS 2004. LNCS, vol. 2987, pp. 423–437. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  26. 26.
    Pym, D., Ritter, E.: On the semantics of classical disjunction. J. of Pure and Applied Algebra 159, 315–338 (2001)MATHCrossRefMathSciNetGoogle Scholar
  27. 27.
    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)MATHCrossRefMathSciNetGoogle Scholar
  28. 28.
    Takahashi, M.: Parallel reduction in λ-calculus. Information and Computation 118, 120–127 (1995)MATHCrossRefMathSciNetGoogle Scholar
  29. 29.
    Urban, C., Bierman, G.M.: Strong normalisation of cut-elimination in classical logic. In: Girard, J.-Y. (ed.) TLCA 1999. LNCS, vol. 1581, pp. 365–380. Springer, Heidelberg (1999)CrossRefGoogle Scholar
  30. 30.
    Wadler, P.: Call-by-value is dual to call-by-name. In: Proc. of the 8th Int. Conference on Functional Programming, pp. 189–201 (2003)Google Scholar
  31. 31.
    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)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  • Daniel Dougherty
    • 1
  • Silvia Ghilezan
    • 2
  • Pierre Lescanne
    • 3
  • Silvia Likavec
    • 2
    • 4
  1. 1.Worcester Polytechnic InstituteUSA
  2. 2.Faculty of EngineeringUniversity of Novi SadSerbia
  3. 3.ENS LyonFrance
  4. 4.Dipartimento di InformaticaUniversità di TorinoItaly

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