Abstract
This article investigates a theory of type assignment (assigning types to lambda terms) called ETA which is intermediate in strength between the simple theory of type assignment and strong polymorphic theories like Girard’s F (Proofs and types. Cambridge University Press, Cambridge, 1989). It is like the simple theory and unlike F in that the typability and type-checking problems are solvable with respect to ETA. This is proved in the article along with three other main results: (1) all primitive recursive functionals of finite type are representable in ETA; (2) every term typable in ETA has a unique normal form; (3) there is a function defined by \({{\varepsilon}_0}\)-recursion which takes every typable term to a natural number which is an upper bound to the lengths of all βη-reduction sequences starting with that term.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Altenkirch T., Coquand T.: A finitary subsystem of the polymorphic lambda calculus. In: Abramsky, S. (ed.) Typed Lambda Calculi and Applications (Lecture Notes in Computer Science 2044), pp. 22–28. Springer, Berlin (2001)
Barendregt H.P.: The Lambda Calculus: Its Syntax and Semantics. 2nd edn. North-Holland, Amsterdam (1984)
Barendregt H.P. et al.: Lambda calculi with types. In: Abramsky, S. (ed.) Handbook of Logic in Computer Science II, pp. 117–309. Oxford University Press, Oxford (1992)
Beckmann A.: Exact bounds for lengths of reduction sequences in typed λ-calculus. J. Symbol. Log. 66, 1277–1285 (2001)
Curry H.B., Feys R.: Combinatory Logic I. North-Holland, Amsterdam (1958)
Curry H.B. et al.: Combinatory Logic II. North-Holland, Amsterdam (1972)
Girard J.-Y. et al.: Proofs and Types. Cambridge University Press, Cambridge (1989)
Gödel K.: Über eine Bisher noch nicht Benützte Erweiterung des Finiten Standpunktes. Dialectica 12, 280–287 (1958)
Hindley J.R.: Basic Simple Type Theory. Cambridge University Press, Cambridge (1997)
Hindley J.R., Seldin J.P.: Lambda Calculus and Combinators: An Introduction. Cambridge University Press, Cambridge (2008)
Kfoury A.J., Wells J.B.: A direct algorithm for type inference in the rank-2 fragment of the second-order λ-calculus. LISP Pointers 7, 196–207 (1994)
Leivant D.: Peano’s lambda calculus. In: Anderson, C.A., Zeleny, M. (eds.) Logic, Meaning and Computation, pp. 313–330. Kluwer, Dordrecht (2001)
Rathjen M. : The realm of ordinal analysis. In: Cooper, S.B., Truss, J.K. (eds.) Sets and Proofs, pp. 219–279. Cambridge University Press, Cambridge (1999)
Rose H.E.: Subrecursion: Functions and Hierarchies. Clarendon Press, Oxford (1984)
Schwichtenberg H.: An upper bound for reduction sequences in the typed λ-calculus. Arch. Math. Log. 30, 405–408 (1991)
Schütte, K.: Proof Theory translated by J. N. Crossley. Springer, Berlin (1977)
Stirton W.R.: How to assign ordinal numbers to combinatory terms with polymorphic types. Arch. Math. Log. 51, 475–501 (2012)
Troelstra A.S., Schwichtenberg H.: Basic Proof Theory. 2nd edn. Cambridge University Press, Cambridge (2000)
Wells J.B.: Typability and type checking in system F are equivalent and undecidable. Ann. Pure Appl. Log. 98, 111–156 (1999)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Stirton, W.R. A decidable theory of type assignment. Arch. Math. Logic 52, 631–658 (2013). https://doi.org/10.1007/s00153-013-0335-x
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00153-013-0335-x
Keywords
- Type assignment
- Lambda calculus
- Primitive recursive functionals
- Type reconstruction
- Typability
- Type checking
- Reduction sequences