Strong Normalization from Weak Normalization in Typed λ-Calculi

Information and Computation

Volumn 133, Issue 1, 1997, Pages 35-71

  Sørensen, Morten Heine ^a,b  

^a RADBOUD UNIVERSITY NIJMEGEN   (Netherlands)

^b UNIVERSITY OF COPENHAGEN   (Denmark)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0347308490     PISSN: 08905401     EISSN: None     Source Type: Journal    
DOI: 10.1006/inco.1996.2622     Document Type: Article

References (77)

1. Ariola, Z. M., Felleisen, M., Maraist, M., Odersky, J., and Wadler, P. (1995), A call-by-need lambda-calculus, in "Conference Record of the Annual ACM SIGPLAN-SIGACT Symposium on Principles of Programming Languages," pp. 233-246, Assoc. Comput. Mach., New York.
2. Barendregt, H. P. (1984), "The Lambda Calculus: Its Syntax and Semantics," 2nd (rev.) ed., North-Holland, Amsterdam.
3. Barendregt, H. P. (1992), Lambda calculi with types, in "Handbook of Logic in Computer Science" (S. Abramsky, D. M. Gabbay, and T. S. E. Maibaum, Eds.), Vol. II, pp. 117-309, Oxford Univ. Press, London.
4. Barendregt, H. P., Bergstra, J., Klop, J. W., and Volken, H. (1976), "Degrees, Reductions and Representability in the Lambda Calculus," Technical Report Preprint 22, Department of Mathematics, University of Utrecht.
5. Barendsen, E. (1995), "Types and Computations in Lambda Calculi and Graph Rewrite Systems," Ph.D. thesis, University of Nijmegen.
6. Barthe, G., and Sørensen, M. H. (1996), Domain-free pure type systems, submitted for publication.
7. Bergstra, J. A., and Klop, J. W. (1982), Strong normalization and perpetual reductions in the lambda calculus, J. Inform. Process. Cybernet. 18, 403-417.
8. Bezem, M., and Groote, J. F. Eds. (1993), "Typed Lambda Calculus and Applications," Lecture Notes in Computer Science, Vol. 664, Springer-Verlag, Berlin/New York.
9. Bloo, R., Kammareddine, F., and Nederpelt, R. (1996), The Barendregt cube with definitions and generalised reduction, Inform. Comput. 126, 123-143.
10. Consel, C., and Danvy, O. (1991), For a better support of static data flow, in "Conference on Functional Programming and Computer Architecture" (J. Hughes, Ed.), Lecture Notes in Computer Science, Vol. 523, pp. 495-519, Springer-Verlag, Berlin/New York.
11. Coquand, T., and Herbelin, H. (1994), A-translation and looping combinators in pure type systems, J. Funct. Programming 4(1), 77-88.
12. Curry, H. B., and Feys, R. (1958), "Combinatory Logic," North-Holland, Amsterdam.
13. de Bruijn, N. G. (1980), A survey of the project Automath, in [61, pp. 579-606].
14. de Groote, P. (1993), The conservation theorem revisited, in [8, pp. 163-178].
15. de Vrijer, R. C. (1985), A direct proof of the finite developments theorem, J. Symbolic Logic 50, 339-343.
16. de Vrijer, R. C. (1987), "Surjective Pairing and Strong Normalization: Two Themes in Lambda Calculus," Ph.D. thesis, University of Amsterdam.
17. Filinski, A. (1994), Representing monads, in "Conference Record of the Annual ACM SIGPLAN-SIGACT Symposium on Principles of Programming Languages," pp. 446-457, Assoc. Comput. Mach., New York.
18. Gallier, J. H. (1990), On Girard's "candidats de reductibilité," in "Logic and Computer Science" (P. Oddifreddi, Ed.), pp. 123-203, Academic Press, London.
19. Gandy, R. O. (1980), An early proof of normalization by Turing, A. M., in [61, pp. 453-455].
20. Gandy, R. O. (1980), Proofs of strong normalization, in [61, pp. 457-477].
21. Geuvers, J. H. (1993), "Logics and Type Systems," Ph.D. thesis, University of Nijmegen.
22. Girard, J.-Y. (1972), "Interprétation fonctionelle et élimination des coupres dans l'arithmétique d'ordre supérieur," Ph.D. Thesis, Université Paris VII.
23. Girard, J.-Y., Lafont, Y., and Taylor, P. (1989), "Proofs and Types," Cambridge Tracts in Theoretical Computer Science, Vol. 7, Cambridge Univ. Press, Cambridge, UK.
24. Griffin, T. G. (1990), A formulae-as-types notion of control, in "Conference Record of the Annual ACM SIGPLAN-SIGACT Symposium on Principles of Programming Languages," pp. 47-58, Assoc. Comput. Mach., New York.
25. Harper, R., and Lillibridge, M. (1993), Explicit polymorphism and CPS conversion, in "Conference Record of the Annual ACM SIGPLAN-SIGACT Symposium on Principles of Programming Languages," pp. 206-219, Assoc. Comput. Mach., New York.
26. Hindley, J. R., and Seldin, J. P. (1986), "Introduction to Combinators and λ-Calculus," Cambridge Univ. Press, Cambridge, UK.
27. Howard, W. (1980), The formulae-as-types notion of construction, in [61, pp. 479-490].
28. Howard, W. (1980), Ordinal analysis of terms of finite type, J. Symbolic Logic 45(3), 493-504.
29. Kammareddine, F. (1996), "A Reduction Relation for Which Postponement of k-Contractions, Conservation, and Preservation of Strong Normalization Hold," Technical Report, Glasgow University, 1996.
30. Kammareddine, F., and Nederpelt, R. (1994), A unified approach to type theory through a refined λ-calculus, Theoret. Comput. Sci. 136, 183-216.
31. Kammareddine, F., and Nederpelt, R. (1995), Refining reduction in the lambda calculus, J. Funct. Programming 5(4), 637-651.
32. Kammareddine, F., and Nederpelt, R. (1996), A useful λ-notation, Theoret. Comput. Sci. 155, 85-109.
33. Kammareddine, F., and Rios, A. (1996), Generalized β-reduction and explicit substitution, in "Programming Languages: Implementations, Logics and Progams" (H. Kuchen and D. S. Swierstra, Eds.), Lecture Notes in Computer Science, Vol. 1140, pp. 378-392, Springer-Verlag, Berlin/ New York.
34. Karr, M. (1985), "Delayability" in proofs of strong normalizability in the typed lambda calculus, in "Mathematical Foundations of Computer Software" (H. Ehrig, C. Floyd, M. Nivat, and J. Thatcher, Eds.), Lecture Notes in Computer Science, Vol. 185, pp. 208-222, Springer-Verlag, Berlin/ New York.
35. Kfoury, A., and Tiuryn, J. (1992), Type reconstruction in finite-rank fragments of the second-order λ-calculus, J. Inform. Comput. 98(2), 228-257.
36. Kfoury, A., and Tiuryn, J., and Urzyczyn, P. (1994), An analysis of ML-typability, J. Assoc. Comput. Mach. 41(2), 368-398.
37. Kfoury, A. J., and Wells, J. (1994), A direct algorithm for type inference in the rank-2 fragment of second-order λ-calculus, in "ACM Conference on Lisp and Functional Programming."
38. Kfoury, A. J., and Wells, J. (1994), "New Notions of Reduction and Non-semantic Proofs of β-Strong Normalization in Typed λ-Calculi," Technical Report 94-014, Boston University Computer Science Department; also in "Logic in Computer Science," 1995.
39. Kfoury, A. J., and Wells, J. (1994), "New Notions of Reduction and Non-semantic Proofs of β-Strong Normalization in Typed λ-Calculi," Technical Report 94-014, Boston University Computer Science Department; also in "Logic in Computer Science," 1995.
40. Kfoury, A. J., and Wells, J. (1995), "Addendum to 'New Notions of Reduction and Non-semantic Proofs of β-Strong Normalization in Typed λ-Calculi,'" Technical Report 95-007, Boston University Computer Science Department.
41. Khasidashvili, Z. (1988), "Form Reduction Systems and Reductions of Contracted Forms and Lambda-Terms," Ph.D. thesis, Tbilisi State University, [In Russian]
42. Khasidashvili, Z. (1994), The longest perpetual reductions in orthogonal expression reduction systems, in "Symposium on Logical Foundations of Computer Science" (A. Nerode and Yu. V. Matiyasevich, Ed.), Lecture Notes in Computer Science, Vol. 813, pp. 191-203, Springer-Verlag, Berlin/New York.
43. Khasidashvili, Z. (1994), Perpetuality and strong normalization in orthogonal term rewriting systems, in "Symposium on Theoretical Aspects of Computer Science" (P. Enjalbert, E. W. Mayr, E. W., and K. W. Wagner, Eds.), Lecture Notes in Computer Science, Vol. 775, pp. 163-174, Springer-Verlag, Berlin/New York.
44. Klop, J. W. (1980), "Combinatory Reduction Systems," Ph.D. Thesis, Utrecht University; CWI Tract, Amsterdam.
45. Klop, J. W., van Oostrom, V., and van Raamsdonk, F. (1993), Combinatory reduction systems, introduction and survey, Theoret. Comput. Sci. 121(1-2), 279-308.
46. Krivine, J.-L. (1993), "Lambda-Calculus, Types and Models," English ed., Ellis Horwood Series in Computers and their Applications, Masson and Ellis Horwood.
47. Leivant, D. (1985), Syntactic translations and provably recursive functions, J. Symbolic Logic 50(3), 682-688.
48. Loader, R. (1995), Normalization by translation, presented at the BRA TYPES workshop, Turin.
49. Meyer, A. R. and Wand, M. (1985), Continuation semantics in typed lambda-calculi (summary), in "Logics of Programs" (R. Parikh, Ed.), Lecture Notes in Computer Science, Vol. 193, pp. 219-224, Springer-Verlag, Berlin/New York.
50. Mitchell, J. C. (1991), Type inference and simple subtypes, J. Funct. Programming 1(3), 245-285.
51. Moggi, E. (1989), Computational lambda-calculus and monads, in "Logic in Computer Science," pp. 14-23, IEEE Comput. Soc. Press.
52. Moggi, E. (1991), Notions of computation and monads, J. Inform. and Comput. 93, 55-92.
53. Nederpelt, R. (1973), "Strong Normalization for a Typed Lambda Calculus with Lambda Structured Types," Ph.D. thesis, Eindhoven.
54. Plotkin, G. (1975), Call-by-name, call-by-value and the λ-calculus, Theoret. Comput. Sci. 1, 125-159.
55. Prawitz, D. (1965), "Natural Deduction: A Proof Theoretical Study," Almquist & Wiksell, Stockholm.
56. Regnier, L. (1994), Une équivalence sur le lambda-termes, Theoret. Comput. Sci. 126, 281-292.
57. Reynolds, J. C. (1993), The discoveries of continuations, LISP Symbolic Comput. 6, 233-248.
58. Sabry, A. (1996), A reflection on call-by-value, in "International Conference on Functional Programming," pp. 13-24, Assoc. Comput. Mach., New York.
59. Sabry, A., and Felleisen, M. (1993), Reasoning about programs in continuation-passing style, Lisp Symbolic Comput. 6, 289-360.
60. Schwichtenberg, H. (1982), Complexity of normalization in the pure type lambda-calculus, in "The L.E.J. Brouwer Centenary Symposium" (A. S. Troelstra and D. van Dalen, Eds.), pp. 453-457, North-Holland, Amsterdam.
61. Schwichtenberg, H. (1991), An upper bound for reduction sequences in the typed lambda-calculus, Arch. Math. Logic 30, 405-408.
62. Seldin, J. P., and Hindley, J. R., Eds. (1980), "To H.B. Curry: Essays on Combinatory Logic, Lambda Calculus and Formalism," Academic Press Limited, London.
63. Sørensen, M. H. (1996), Effective longest and infinite reduction paths in untyped λ-calculi, in "Colloquium on Trees in Algebra and Programming" (H. Kirchner, Ed.), Lecture Notes in Computer Science, Vol. 1059, pp. 287-301, Springer-Verlag.
64. Sørensen, M. H. (1996), Properties of infinite reduction paths in untyped λ-calculus, in "Tbilisi Symposium on Language, Logic, and Computation," CLSI Lecture Notes, to appear.
65. Springintveld, J. (1993), Lower and upper bounds for reductions of types in λω and λΡ, in [8, pp. 391-405].
66. Tait, W. W. (1967), Intensional interpretations of functionals of finite type I, J. Symbolic Logic 32(2), 190-212.
67. Tait, W. W. (1975), A realizability interpretation of the theory of species, in "Logic Colloquium" (R. Parikh, Ed.), Lecture Notes in Mathematics, Vol. 453, pp. 240-251, Springer-Verlag, Berlin/ New York.
68. Troelstra, A. S. (1973), "Metamathematical Investigation of Intuitionistic Arithmetic and Analysis," Lecture Notes in Mathematics, Vol. 344, Springer-Verlag, Berlin/New York.
69. P. Urzyczyn, P. (1995), Positive recursive type assigment, in "Mathematical Foundations of Computer Science" (J. Wiedermann and P. Hájek, Eds.), Lecture Notes in Computer Science, Vol. 969, pp. 382-391, Springer-Verlag, Berlin/New York.
70. van de Pol, J. (1994), Termination proofs for higher-order rewrite systems, in "Higher Order Algebra, Logic and Term Rewriting" (J. Heering et al., Eds.), Lecture Notes in Computer Science, Vol. 816, pp. 305-325, Springer-Verlag, Berlin/New York.
71. van de Pol, J. (1995), Two different strong normalization proofs? in [69].
72. van de Pol, J., and Schwichtenberg, H. (1993), Strict functionals for determination proofs, in [8, pp. 350-364].
73. van Raamsdonk, F., and Severi, P. (1995), "On Normalization," Technical Report CS-R9545, CWI.
74. Vidal, D. (1989), "Nouvelles notions de réduction en lambda calcul," Ph.D. thesis, Université de Nancy.
75. Werner, B. (1992), Continuations, evaluation styles and types systems, manuscript.
76. Xi, H. (1996), "An Induction Measure on λ-Terms," Technical report, Department of Mathematics, Carnegie Mellon University.
77. Xi, H. (1996), On weak and strong normalization, manuscript announced on the types mailing list.