Stratified polymorphism and primitive recursion. (English) Zbl 0935.03024

Danner, Norman; Leivant, Daniel

Math. Struct. Comput. Sci. 9, No. 4, 507-522 (1999).

Summary: Natural restrictions on the syntax of the second-order (i.e., polymorphic) lambda calculus are of interest for programming language theory. One of the authors showed [D. Leivant, Inf. Comput. 93, No. 1, 93-113 (1991; Zbl 0799.68041)] that when type abstraction in that calculus is stratified into levels, the definable numeric functions are precisely the super-elementary functions (level \({\mathcal E}_4\) in the Grzegorczyk hierarchy). We define here a second-order lambda calculus in which type abstraction is stratified to levels up to \(\omega^\omega\), an ordinal that permits highly uniform (and finite) type inference rules. Referring to this system, we show that the numeric functions definable in the calculus using ranks \(<\omega^\ell\) are precisely Grzegorczyk’s class \({\mathcal E}_{\ell+3}\) \((\ell\geq 1)\). This generalizes Leivant’s result [loc. cit.], where this is proved for \(\ell= 1\). Thus, the numeric functions definable in our calculus are precisely the primitive recursive functions.

MSC:

03B40	Combinatory logic and lambda calculus
68N18	Functional programming and lambda calculus
03B70	Logic in computer science
03D20	Recursive functions and relations, subrecursive hierarchies

Keywords:

Grzegorczyk hierarchy; second-order lambda calculus; type abstraction; numeric functions; primitive recursive functions