Abstract
Sections 1 through 4 define, in the usual inductive style, various classes of object including one which is called the “combinatory terms of polymorphic type”. Section 5 defines a reduction relation on these terms. Section 6 shows that the weak normalizability of the combinatory terms of polymorphic type entails the weak normalizability of the lambda terms of polymorphic type. The entailment is not vacuous, because the combinatory terms of polymorphic type are indeed weakly normalizable, as is proven in Sect. 7 using Tait and Girard’s computability predicates. The remainder of the paper is devoted to arguing that interesting consequences would follow from the existence of an “ordinally informative” proof, i.e. one which uses transfinite induction over recursive ordinal numbers and otherwise finitary methods, of normalizability for as large a class of the terms as possible.
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Notes
Other than, perhaps, the attempt made in [20], which is superseded by the present treatment.
Compare a similar remark at the foot of p. 122 of [19].
On p. 18 of [2], systems of this last kind are called “de Bruijn systems”.
Why it is possible to restrict the class of open redexes to terms of the two shapes mentioned in clause (1) is explained at pp. 91–93 of [5].
If the normalizability proof presented in this section appears complicated, this is largely to be put down to the fact that the type of terms in Comp\(_{\xi ,A}\) is \(A^{\xi }\), which is not, in general, the same as A. This is due to the fact that the terms in \(\xi (\alpha )\) may have any type (not necessarily \(\alpha \)); and this in turn seems unavoidable in view of the fact that valuations were introduced in order to deal with instantiations of universal quantifiers ranging over all types. Valuable discussions of the heuristic thinking behind Girard’s proof can be found in [7, 12] (“Appendix B”) and [23] (Sect. 7.9).
For a rough definition of “ordinally informative”, see [14]. For the present purpose, “terms” has to be read in place of “proofs”
This definition is taken from p. 283 of [19].
References
Barendregt, H.P.: Lambda Calculi with types. In: Abramsky, S., et al. (eds.) Handbook of Logic in Computer Science, vol. 2, pp. 117–309. Oxford University Press, Oxford (1992)
Barendregt, H.P., Dekkers, W., Statman, R.: Lambda Calculus with Types. Cambridge University Press, Cambridge (2013)
Bunder, M.W.: Some improvements to Turner’s algorithm for bracket abstraction. J. Symb. Logic 55, 656–669 (1990)
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)
Girard, J-Y.: Une Extension de l’Interpretation de Gödel a l’Analyse. In: J. E. Fenstad (ed.): Proceedings of the Second Scandinavian Logic Symposium, , pp. 63–92. North-Holland, Amsterdam and London (1971)
Girard, J.-Y., et al.: Proofs and Types. CUP, Cambridge (1989)
Hindley, J.R.: Basic Simple Type Theory. CUP, Cambridge (1997)
Hindley, J.R., Seldin, J.P.: Lambda Calculus and Combinators: An Introduction. CUP, Cambridge (2008)
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. Symb. Logic 45, 493–504 (1980)
Prawitz, D.: Ideas and results of proof theory. In: Fenstad, J.E. (ed.) Proceedings of the Second Scandinavian Logic Symposium, pp. 235–307. North-Holland, Amsterdam (1971)
Quine, W.V.O.: Mathematical Logic, 2nd edn. Harvard University Press, Cambridge (1951)
Rathjen, M.: Recent advances in ordinal analysis: \(\Pi ^1_2\) and related systems. Bull. Symb. Logic 1, 468–485 (1995)
Rose, H.E.: Subrecursion: Functions and Hierarchies. Clarendon, Oxford (1984)
Schütte, K.: Proof Theory. Springer, Berlin (1977). (Translated by J. N. Crossley)
Schwichtenberg, H.: Elimination of higher type levels from definitions of primitive recursive functionals by means of transfinite recursion. In: Rose, H.E., Shepherdson, J.C. (eds.) Logic Colloquium -73, pp. 279–303. North-Holland, Amsterdam (1975)
Seldin, J.P.: The search for a reduction in combinatory logic equivalent to \(\lambda \beta \)-reduction. Theoret. Comput. Sci. 412, 4905–4918 (2011)
Sørensen, M.H.B., Urzyczyn, P.: Lectures on the Curry–Howard Isomorphism. Elsevier, Amsterdam (2006)
Stirton, W.R.: How to assign ordinal numbers to combinatory terms with polymorphic types. Arch. Math. Logic 51, 475–501 (2012)
Stirton, W.R.: A decidable theory of type-assignment. Arch. Math. Logic 52, 631–658 (2013)
Terlouw, J.: Reduction of higher type levels by means of an ordinal analysis of finite terms. Ann. Pure Appl. Logic 28, 73–102 (1985)
Troelstra, A.S. (ed.): Metamathematical Investigation of Intuitionistic Arithmetic and Analysis (Lecture Notes in Mathematics 344). Springer, Berlin (1973)
Troelstra, A.S., Schwichtenberg, H.: Basic Proof Theory, 2nd edn. Cambridge University Press, Cambridge (2000)
Weiermann, A.: How is it that infinitary methods can be applied to finitary mathematics? Goedel’s T: a case study. J. Symb. Logic 63, 1348–1370 (1998)
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Stirton, W.R. Combinatory logic with polymorphic types. Arch. Math. Logic 61, 317–343 (2022). https://doi.org/10.1007/s00153-021-00792-5
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DOI: https://doi.org/10.1007/s00153-021-00792-5