#### Title

Stratified polymorphism and primitive recursion

#### Document Type

Article

#### Publication Date

1999

#### Journal or Book Title

Mathematical Structures in Computer Science

#### Volume

9

#### Issue

4

#### First Page

507

#### Last Page

522

#### Abstract

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 in Leivant (1991) that when type abstraction in that calculus is stratified into levels, the definable numeric functions are precisely the super-elementary functions (level [script E]_{4} in the Grzegorczyk Hierarchy). We define here a second-order lambda calculus in which type abstraction is stratified to levels up to ω^{ω}, 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 < ω^{[script l]} are precisely Grzegorczyk's class [script E]_{[script l]+3} ([script l] ≥ 1). This generalizes Leivant (1991), where this is proved for [script l] = 1. Thus, the numeric functions definable in our calculus are precisely the primitive recursive functions.

#### Recommended Citation

Norman Danner and Daniel Leivant. Stratified polymorphism and primitive recursion. *Mathematical Structures in Computer Science*, 9(4):507–522, 1999.