Abstract
The article investigates a system of polymorphically typed combinatory logic which is equivalent to Gödel’s T. A notion of (strong) reduction is defined over terms of this system and it is proved that the class of well-formed terms is closed under both bracket abstraction and reduction. The main new result is that the number of contractions needed to reduce a term to normal form is computed by an ε 0-recursive function. The ordinal assignments used to obtain this result are also used to prove that the system under consideration is indeed equivalent to Gödel’s T. It is hoped that the methods used here can be extended so as to obtain similar results for stronger systems of polymorphically typed combinatory terms. An interesting corollary of such results is that they yield ordinally informative proofs of normalizability for sub-systems of second-order intuitionist logic, in natural deduction style.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Altenkirch, T., T. Coquand: A finitary subsystem of the polymorphic lambda calculus. In: S. Abramsky (ed.), Typed Lambda Calculi and Applications (Lecture Notes in Computer Science 2044) Springer, pp. 22–28 (2001)
Arai T.: An introduction to finitary analyses of proof figures. In: Cooper, S.B., Truss, J.K. (eds) Sets and Proofs, pp. 1–26. Cambridge University Press, Cambridge (1999)
Barendregt H.P. et al.: Lambda calculi with types. In: Abramsky, S. (ed) Handbook of Logic in Computer Science, vol 2, pp. 117–309. Oxford University Press, Oxford (1992)
Beckmann A., Weiermann A.: Analysing Gödel’s T by means of expanded head reduction trees. Math. Logic Q. 46, 517–536 (2000)
Coquand T.: Completeness theorems and λ-Calculus. In: P. Urzyczyn (ed.) Typed Lambda Calculi and Applications (Lecture Notes in Computer Science 3461) Springer, pp. 1–9 (2005)
Curry H.B., Feys R.: Combinatory Logic, vol I. North-Holland, Amsterdam (1958)
Curry H.B., Hindley J.R., Seldin J.P.: Combinatory Logic, vol II. North-Holland, Amsterdam (1972)
Hinata S.: Calculability of primitive recursive functionals of finite type. Sci. Rep. Tokyo Kyoiku Daigaku, Section A 9, 218–235 (1967)
Howard W.A.: Assignment of ordinals to terms for primitive recursive functionals of finite type. In: Kino, A., Myhill, J., Vesley, R.E. (eds) Intuitionism and Proof Theory, pp. 443–458. North-Holland, Amsterdam (1970)
Howard W.A.: Ordinal analysis of terms of finite type. J. Symbol. Logic 45, 493–504 (1980)
Howard W.A.: Computability of ordinal recursion of type level two. In: Richman, F. (ed) Constructive Mathematics (Lecture Notes in Mathematics 873), pp. 87–104. Springer, Berlin (1981)
Leivant D.: Peano’s lambda calculus. In: Anderson, C.A., Zeleny, M. (eds) Logic, Meaning and Computation, pp. 313–330. Kluwer, Dordrecht (2001)
Rose H.E.: Subrecursion: Functions and Hierarchies. Clarendon Press, Oxford (1984)
Schütte K.: Proof Theory, translated by J. N. Crossley Springer, Berlin (1977)
Sørensen M.H.B., Urzyczyn P.: Lectures on the Curry–Howard Isomorphism. Elsevier, Amsterdam (2006)
Takeuti G.: On the fundamental conjecture of GLC I. J. Math. Soc. Jpn. 7, 249–275 (1955)
Takeuti G.: On the fundamental conjecture of GLC V. J. Math. Soc. Jpn. 10, 121–134 (1958)
Takeuti G.: Proof Theory. 2nd edn. North-Holland, Amsterdam (1987)
Terlouw J.: On definition trees of ordinal recursive functions. J. Symbol. Logic 47, 395–403 (1982)
Troelstra A.S., Schwichtenberg H.: Basic Proof Theory. 2nd edn. Cambridge University Press, Cambridge (2000)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Stirton, W.R. How to assign ordinal numbers to combinatory terms with polymorphic types. Arch. Math. Logic 51, 475–501 (2012). https://doi.org/10.1007/s00153-012-0277-8
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00153-012-0277-8
Keywords
- Proof theory
- Combinatory logic
- Primitive recursive functionals
- Gödel’s T
- Ordinal analysis
- Second-order logic