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Stratified polymorphism and primitive recursion. (English) Zbl 0935.03024
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
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