Hostname: page-component-77c89778f8-m42fx Total loading time: 0 Render date: 2024-07-16T10:35:09.803Z Has data issue: false hasContentIssue false

Introduction to generalized type systems

Published online by Cambridge University Press:  10 August 2016

Henk Barendregt*
Catholic University Nijmegen, The Netherlands
Rights & Permissions [Opens in a new window]


Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Programming languages often come with type systems. Some of these are simple, others are sophisticated. As a stylistic representation of types in programming languages several versions of typed lambda calculus are studied. During the last 20 years many of these systems have appeared, so there is some need of classification. Working towards a taxonomy, Barendregt (1991) gives a fine-structure of the theory of constructions (Coquand and Huet 1988) in the form of a canonical cube of eight type systems ordered by inclusion. Berardi (1988) and Terlouw (1988) have independently generalized the method of constructing systems in the λ-cube. Moreover, Berardi (1988, 1990) showed that the generalized type systems are flexible enough to describe many logical systems. In that way the well-known propositions-as-types interpretation obtains a nice canonical form.

Research Article
Copyright © Cambridge University Press 1991


Barendregt, H. P. 1984. The Lambda Calculus; Its Syntax and Semantics (2nd Edn). North-Holland.Google Scholar
Barendregt, H. P. 1991. Lambda calculi with types. In Abramsky, S., Gabbai, D. and Maibaum, T. (editors), Handbook of Logic in Computer Science. Oxford University Press.Google Scholar
Barendregt, H. P. and van Leeuwen, M. 1985. Functional programming and the language TALE. In Lecture Notes in Computer Science, 224, pp. 122208. Springer-Verlag.Google Scholar
Barendregt, H. P. and Hemerik, K. 1990. Types in lambda calculi and programming languages. In Proc. European Symposium on Programming, pp. 135, Copenhagen, Denmark (May).Google Scholar
Barendregt, H. P. and Dekkers, W. 1990. Typed lambda calculi.Google Scholar
Barendsen, E. 1989. Representation of Logic, Data Types and Recursive Functions in Typed Lambda Calculi. Masters thesis, University of Nijmegen, Netherlands.Google Scholar
Barendsen, E. and Geuvers, H. 1989. Conservativity of λP over PRED . Manuscript. University of Nijmegen, Netherlands.Google Scholar
Berardi, S. 1988. Personal communication.Google Scholar
Berardi, S. 1990. Type Dependence and Constructive Mathematics. PhD thesis, Mathematical Institute, Torino, Italy.Google Scholar
de Bruijn, N. G. 1970. The mathematical language AUTOMATH, its usage and some of its extensions. In Lecture Notes in Mathematics, 125, pp. 2961. Springer-Verlag.Google Scholar
de Bruijn, N. G. 1980. A survey of the AUTOMATH project. In Hindley, J. R. and Seldin, J. P. (editors), To H. B. Curry: Essays on Combinatory logic, Lambda Calculus and Formalism, pp. 580606. Academic Press.Google Scholar
Cardelli, L. and Wegner, P. 1985. On understanding types, data abstraction and polymorphism. ACM Comp. Surveys, 17 (4): 471522.CrossRefGoogle Scholar
Church, A. 1940. A formulation of the simple theory of types. J. Symbolic Logic, 5: 5668.CrossRefGoogle Scholar
Coquand, Th. 1989. An introduction to type theory. To appear in Meyer, A. R. (editor), Proc. Ecole de Printemps du LITP, Albi.Google Scholar
Coquand, Th. and Huet, G. 1988. The calculus of constructions. Information and Computation, 76: 95120.CrossRefGoogle Scholar
Curry, H. B. and Feys, R. 1958. Combinatory logic. North Holland.Google Scholar
van Daalen, D. 1980. The Language Theory of AUTOMATH. PhD. thesis, Technical University Eindhoven, Netherlands.Google Scholar
van Dalen, D. 1983. Logic and Structure. (2nd edn). Springer-Verlag.CrossRefGoogle Scholar
Fujita, K. 1989. Relationship between logic and type system. Unpublished manuscript. Research Institute of Electrical Communication, Tohoku University, Japan.Google Scholar
Geuvers, H. 1988. The Interpretation of Logics in Type Systems. Master thesis, University of Nijmegen, Netherlands.Google Scholar
Geuvers, H. 1989. Theory of constructions is not conservative over higher order logic. Manuscript. University of Nijmegen, Netherlands.Google Scholar
Geuvers, H. 1990. Type systems for higher order logic. Manuscript. University of Nijmegen, Netherlands.Google Scholar
Geuvers, H. and Nederhof, M.-J. 1991. A modular proof of strong normalization for the theory of constructions. Journal of Functional Programming 1(2): 155189.CrossRefGoogle Scholar
Girard, J.-Y. 1972. Interprétation Fonctionelle et Élimination des Coupures dans l' Arithmétique d'Ordre Supérieur. Thèse de Doctorat d'État, Université Paris VII, France.Google Scholar
Gordon, M. H., Milner, R. and Wadsworth, C. 1979. Edinburgh LCF: Lecture Notes in Computer Science, 78. Springer-Verlag.CrossRefGoogle Scholar
Harper, R., Honseil, F. and Plotkin, G. 1987. A framework for defining logics. In Proc. 2nd Symp. Logic in Computer Science, pp. 194204. Ithaca, New York.Google Scholar
Howard, W. A. 1980. The formulae-as-types notion of construction. In Hindley, J. R. and Seldin, J. P. (editors), To H. B. Curry: Essays on Combinatory logic, Lambda Calculus and Formalism, pp. 479490. Academic Press.Google Scholar
Leivant, D. 1989. Contracting proofs to programs. In: Odifreddi, P. (editor), Logic and Computer Science, pp. 279327, Academic Press.Google Scholar
Longo, G. and Moggi, E. 1988. Constructive Natural Deduction and its Modest Interpretation. Report CMU-CS-88-131, Carnegie Mellon University, Pittsburgh, USA.Google Scholar
Martin-Löf, P. 1970. A construction of the provable wellorderings of the theory of species. Unpublished Manuscript. Mathematical Institute, University of Stockholm, Sweden.Google Scholar
Martin-Löf, P. 1984. Intuitionistic Type Theory. Bibliopolis.Google Scholar
Milner, R. 1984. A proposal for standard ML. In Proc. 1984 ACM Symposium on LISP and Functional Programming, pp. 184197. Austin, Texas.CrossRefGoogle Scholar
Mostowski, A. 1951. On the rules of proof in the pure functional calculus of first order. J. Symbolic Logic, 16: 107111.CrossRefGoogle Scholar
Peremans, W. 1949. Een opmerking over intuitionistische logica. Report ZW-16. Center for Mathematics and Computer Science, Kruislaan 413, 1098 SJ Amsterdam.Google Scholar
Prawitz, D. 1965. Natural Deduction. Almqvist and Wiksell.Google Scholar
Renardel de Lavalette, G. 1987. Strictness analysis for a language with polymorphic and recursive types (preprint). Department of Philosophy, Utrecht University, Netherlands.Google Scholar
Reynolds, J. 1974. Towards a theory of type structure. In Proc. Colloque sur la Programmation. In Lecture Notes in Computer Science, 19, pp. 408425. Springer-Verlag.Google Scholar
Reynolds, J. 1985. Three approaches to type theory. In Lecture Notes in Computer Science, 185, pp. 145146. Springer-Verlag.Google Scholar
Stenlund, S. 1972. Combinators, λ-terms and proof theory. D. Reidel.CrossRefGoogle Scholar
Swaen, M. D. G. 1989. Weak and Strong Sum-elimination in Institutionistic Type Theory. PhD. thesis, University of Amsterdam, Netherlands.Google Scholar
Terlouw, J. 1988. Personal communication.Google Scholar
Turner, D. 1985. Miranda: A non-strict functional language with polymorphic types. In Jouannaud, Jean-Pierre (editor). Functional Programming Languages and Computer Architecture. Lecture Notes in Computer Science, 201, pp. 116. Springer-Verlag.Google Scholar
de Vrijer, R. 1975. Big trees in a λ-calculus with λ-expressions as types. In Proc. Symposium on λ-calculus and computer science theory, Lecture Notes in Computer Science, 37, pp. 252271. Springer-Verlag.Google Scholar
Submit a response


No Discussions have been published for this article.