Hostname: page-component-745bb68f8f-grxwn Total loading time: 0 Render date: 2025-02-05T13:36:40.708Z Has data issue: false hasContentIssue false
  • English
  • Français

Atomic polymorphism

Published online by Cambridge University Press:  12 March 2014

Fernando Ferreira  and
Gilda Ferreira
Fernando Ferreira
Affiliation:
Faculdade de Ciências da Universidade de Lisboa, Departamento de Matemática, Campo Grande, ED. C6, 1749-016 Lisboa, Portugal, E-mail: fjferreira@fc.ul.pt
Gilda Ferreira
Affiliation:
Universidade Lusófona de Humanidades e Tecnologias, Departamento de Matemática, AV DO Campo Grande, 376, 1749-024 Lisboa, Portugal, E-mail: gildafer@cii.fc.ul.pt
Get access Rights & Permissions [Opens in a new window]

Abstract

It has been known for six years that the restriction of Girard's polymorphic system F to atomic universal instantiations interprets the full fragment of the intuitionistic propositional calculus. We firstly observe that Tait's method of “convertibility” applies quite naturally to the proof of strong normalization of the restricted Girard system. We then show that each β-reduction step of the full intuitionistic propositional calculus translates into one or more βη-reduction steps in the restricted Girard system. As a consequence, we obtain a novel and perspicuous proof of the strong normalization property for the full intuitionistic propositional calculus. It is noticed that this novel proof bestows a crucial role to η-conversions.

Keywords

Predicative polymorphismstrong normalizationnatural deductionsecond-order λ-calculusη-conversions
Type
Research Article
Information
The Journal of Symbolic Logic , Volume 78 , Issue 1 , March 2013 , pp. 260 - 274
Copyright
Copyright © Association for Symbolic Logic 2013

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1]Aehlig, K., Parameter-free polymorphic types, Annals of Pure and Applied Logic, vol. 156 (2008), pp. 3–12.CrossRefGoogle Scholar
[2]Altenkirch, T. and Coquand, T., A finitary subsystem of the polymorphic lambda-calculus. Typed Lambda Calculi and Applications (TLCA 2001) (Abramsky, S., editor), Lecture Notes in Computer Science, vol. 2044, Springer, 2001, pp. 22–28.CrossRefGoogle Scholar
[3]Ferreira, F., Comments on predicative logic, Journal of Philosophical Logic, vol. 35 (2006), pp. 1–8.CrossRefGoogle Scholar
[4]Ferreira, F. and Ferreira, G., Commuting conversions vs. the standard conversions of the “good” connectives, Studia Logica, vol. 92 (2009), pp. 63–84.CrossRefGoogle Scholar
[5]Girard, J.-Y., Une extension de l'interprétation de Gödel à l'analyse, et son application à lélimination des coupures dans l'analyse et la théorie des types, Proceedings ofthe second Scandinavian logic symposium (Fenstad, J. E., editor), North Holland, Amsterdam, 1971, pp. 63–92.CrossRefGoogle Scholar
[6]Girard, J.-Y., Lafont, Y., and Taylor, P., Proofs and types, Cambridge University Press, 1989.Google Scholar
[7]Joachimski, F. and Matthes, R., Short proofs of normalization for the simply-typed lambda-calculus, permutative conversions and Gödel's T, Archive for Mathematical Logic, vol. 42 (2003), pp. 59–87.CrossRefGoogle Scholar
[8]Kretz, M., On the treatment of predicative polymorphism in theories of explicit mathematics, Ph.D. thesis, Universität Bern, 2002.Google Scholar
[9]Mitchell, J. C., Type systems for programming languages, Handbook of theoretical computer science (van Leeuwen, J., editor), vol. B, Elsevier, 1990, pp. 365–458.Google Scholar
[10]Mitchell, J. C. and Harper, R., The essence of ML, Proceedings of 15th ACM symposium on principles of programming languages, 1988, pp. 28–46.Google Scholar
[11]Prawitz, D., Natural deduction, Almkvist & Wiksell, Stockholm, 1965, reprinted, with a new preface, Dover Publications, 2006.Google Scholar
[12]Prawitz, D., Ideas and results in proof theory, Proceedings of the second Scandinavian logic symposium (Fenstad, J. E., editor), North-Holland, 1971, pp. 235–307.CrossRefGoogle Scholar
[13]Reynolds, J. C., Towards a theory of type structure, Collogue sur la programmation (Robinet, B., editor), Lecture Notes in Computer Science, vol. 19, Springer, 1974.Google Scholar
[14]Russell, B. and Whitehead, A. N., Principia mathematica, 2nd ed., Cambridge University Press, 1927.Google Scholar
[15]Sandqvist, T., A note on definability of logical operators in second-order logic, unpublished manuscript, 2008.Google Scholar
[16]Tait, W., Intentional interpretations of functionals offinite type I, this Journal, vol. 32 (1967), pp. 198–212.Google Scholar
[17]Troelstra, A. S. and Schwichtenberg, H., Basic proof theory, Cambridge University Press, Cambridge, 1996.Google Scholar