Plain Bases for Classes of Primitive Recursive Functions

Abstract

A basis for a set C of functions on natural numbers is a set F of functions such that C is the closure with respect to substitution of the projection functions and the functions in F . This paper introduces three new bases, comprehending only common functions, for the Grzegorczyk classes E n with n ≥ 3. Such results are then applied in order to show that E n+1 = Kn for n ≥ 2, where {Kn} n∈N is the Axt hierarchy.Mathematics Subject Classification: 03D20, 03D55, 03B70, 68Q15.

Periods and elementary real numbers

“…Proposition 13. (Mazzanti [12]) All elementary functions can be generated from the following four functions by composition:…”

“…Let D ⊂ R ℓ be a basic open semi-algebraic subset as in (12). Now we assume that D is bounded and contained in a large cube [0, r] ℓ , r > 0.…”