# finitely many distinct partially n-computable unary functions

A unary function f(x) is said to be partially n-computable if it is computed by some S program P such that P has no more than n instructions, every variable in P is among X,Y,Z1,...,Zn and every label in P is among A1,...,An,E.

Prove that for every n >= 0, there are only finitely many distinct partially n-computable unary functions.

This problem is in Chapter 2, Section 4 of Computability, Complexity, and Languages by M.Davis.

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• This probably immediately follows from the fact that there are finitely many such programs $P$ and that any pair of distinct $n$-computable functions need to be computed by distinct programs. – Steven Nov 21 at 14:51
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