What is "extended polynomials"

Question

I've heard that the functions that are definable in simply-typed $\lambda$-calculus is the class of extended polynomials.

However, it is still not clear to me what exactly is extended polynomials. Could someone explain that to me? Some pointers would be good as well.

Thanks.

Answer 1

It turns out the definition is straightforward as seen in [1]. And as an example, exponentiation is not definable in simply-type $\lambda$-calculus.

The class of extended polynomials is the smallest class of func- tions over N which contains:

1. the constant functions: 0 and 1,
2. projections,
3. addition,
4. multiplication,
5. the function ifzero(n, m, p) = if n = 0 then m else p

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1. Definable functions in the simply typed lambda-calculus by M. Zakrzewski (2007)