Λμ-calculus was introduced as a Böhm-complete extension of Parigot’s λμ-calculus. Λμ-calculus, contrarily to Parigot’s calculus, is a calculus of CBN delimited control as evidenced by Herbelin and Ghilezan. In their seminal paper on (CBV) delimited control, Danvy and Filinski introduced the CPS Hierarchy of control operators (shift i /reset i ) i ∈ ω .

In a similar way, we introduce in the present paper the Stream Hierarchy, a hierarchy of calculi extending and generalizing Λμ-calculus. The \((\Lambda^n)_{n\in\omega}\)-calculi have Church-Rosser and Böhm theorems. We then present sound and complete CPS translations for the hierarchy. Next, we investigate the operational content of the hierarchy through its abstract machines, the \((\Lambda^n)_{n\in\omega}\)-KAM. Finally, we establish that the Stream hierarchy is indeed a CBN analogue to the CPS hierarchy.


Λμ-calculus delimited control CPS hierarchy Böhm theorem CPS translation Abstract machine Streams 


  1. 1.
    Ariola, Z., 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. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  2. 2.
    Ariola, Z., Herbelin, H.: Control Reduction Theories: the Benefit of Structural Substitution. In: JFP. Includes a Historical Note by Matthias Felleisen (2007)Google Scholar
  3. 3.
    Berarducci, A.: Infinite λ-calculus and non-sensible models. In: Logic and Algebra 1994. Lect. Notes in Pure and App. Math. Series, vol. 180. Marcel Dekker, New York (1996)Google Scholar
  4. 4.
    Berarducci, A., Dezani, M.: Infinite λ-calculus and types. TCS, 212 (1999)Google Scholar
  5. 5.
    Biernacka, M., Biernacki, D., Danvy, O.: An operational foundation for delimited continuations in the CPS hierarchy. Logical Meth. in Comp. Science 1(2) (2005)Google Scholar
  6. 6.
    Böhm, C.: Alcune proprietà delle forme βη-normali nel λK-calcolo. Publicazioni dell’Istituto per le Applicazioni del Calcolo 696 (1968)Google Scholar
  7. 7.
    Danvy, O., Filinski, A.: Abstracting control. In: LISP and Funct. Prog. (1990)Google Scholar
  8. 8.
    Danvy, O., Yang, Z.: An operational investigation of the CPS hierarchy. In: Swierstra, S.D. (ed.) ESOP 1999. LNCS, vol. 1576, pp. 224–242. Springer, Heidelberg (1999)CrossRefGoogle Scholar
  9. 9.
    David, R., Py, W.: λμ-calculus and Böhm’s theorem. J. of Symb. Logic (2001)Google Scholar
  10. 10.
    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)CrossRefGoogle Scholar
  11. 11.
    de Groote, P.: An environment machine for the λ-calculus. MSCS 8 (1998)Google Scholar
  12. 12.
    Felleisen, M., Friedman, D.P., Kohlbecker, E.E., Duba, B.F.: A syntactic theory of sequential control. TCS 52, 205–237 (1987)zbMATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Filinski, A.: Representing monads. In: POPL 1994, pp. 446–457. ACM, New York (1994)CrossRefGoogle Scholar
  14. 14.
    Fujita, K.: A sound and complete cps-translation for λ-calculus. In: TLCA (2003)Google Scholar
  15. 15.
    Griffin, T.: A formulae-as-types notion of control. In: POPL. IEEE, Los Alamitos (1990)Google Scholar
  16. 16.
    Herbelin, H., Ghilezan, S.: An approach to CBN delimited continuations. In: POPL. ACM Sigplan, New York (2008)Google Scholar
  17. 17.
    Howard, W.A.: The formulae-as-type notion of construction, 1969. In: Essays in Comb. Logic, λ-Calculus, and Formalism, pp. 479–490. Academic Press, London (1980)Google Scholar
  18. 18.
    Kameyama, Y., Hasegawa, M.: A Sound and Complete Axiomatization of Delimited Continuations. In: ICFP 2003, pp. 177–188. SIGPLAN Notices (2003)Google Scholar
  19. 19.
    Kameyama, Y.: Axioms for control operators in the cps hierarchy. In: HOSC (2007)Google Scholar
  20. 20.
    Kennaway, R., Klop, J.W., Sleep, M.R., de Vries, F.-J.: Infinitary lambda calculus. TCS 175(1), 93–125 (1997)zbMATHCrossRefGoogle Scholar
  21. 21.
    Kiselyov, O.: Call-by-name linguistic side effects. In: ESSLLI 2008 Workshop on Symmetric calculi and Ludics for the semantic interpretation (2008)Google Scholar
  22. 22.
    Krivine, J.-L.: Lambda-calculus, Types and Models. Ellis Horwood (1993)Google Scholar
  23. 23.
    Krivine, J.-L.: A call-by-name lambda-calculus machine. In: HOSC (2005)Google Scholar
  24. 24.
    Lafont, Y., Reus, B., Streicher, T.: Continuations semantics or expressing implication by negation. Tech. Rep. 9321, Ludwig-Maximilians-Universität (1993)Google Scholar
  25. 25.
    Lassen, S.: Head normal form bisimulation for pairs and the λ-calculus. In: Logic In Computer Science. IEEE Computer Society Press, Los Alamitos (2006)Google Scholar
  26. 26.
    Parigot, M.: Free deduction: An analysis of ”computations” in classical logic. In: Voronkov, A. (ed.) RCLP 1990 and RCLP 1991. LNCS (LNAI), vol. 592, pp. 361–380. Springer, Heidelberg (1992)Google Scholar
  27. 27.
    Parigot, M.: λμ-calculus: an algorithmic interpretation of classical natural deduction. In: Voronkov, A. (ed.) LPAR 1992. LNCS, vol. 624. Springer, Heidelberg (1992)CrossRefGoogle Scholar
  28. 28.
    Parigot, M.: Proofs of strong normalisation for second order classical natural deduction. Journal of Symbolic Logic 62(4), 1461–1479 (1997)zbMATHCrossRefMathSciNetGoogle Scholar
  29. 29.
    Py, W.: Confluence en λμ-calcul. PhD thesis, Université de Savoie (1998)Google Scholar
  30. 30.
    Saurin, A.: A hierarchy for delimited continuations in call-by-name. long version at,
  31. 31.
    Saurin, A.: Separation with streams in the Λμ-calculus. In: LICS. IEEE, Los Alamitos (2005)Google Scholar
  32. 32.
    Saurin, A.: On the relations between the syntactic theories of λμ-calculi. In: Kaminski, M., Martini, S. (eds.) CSL 2008. LNCS, vol. 5213, pp. 154–168. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  33. 33.
    Saurin, A.: Une étude logique du contrôle, appliquée à la programmation fonctionnelle et logique. PhD thesis, École Polytechnique (September 2008)Google Scholar
  34. 34.
    Saurin, A.: Typing streams in the Λμ-calculus. ACM ToCL (to appear)Google Scholar
  35. 35.
    Saurin, A.: Standardization and Böhm trees for Λμ-calculus. In: FLOPS 2010. LNCS, vol. 6009. Springer, Heidelberg (to appear, 2010)Google Scholar
  36. 36.
    Streicher, T., Reus, B.: Classical logic, continuation semantics and abstract machines. Journal of Functional Programming 8(6), 543–572 (1998)zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Alexis Saurin
    • 1
  1. 1.PPS & INRIA πr2 

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