General recursive function: Difference between revisions

Revision as of 21:38, 22 June 2021

Revision as of 13:54, 23 June 2021

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Primitive or "basic" functions:

#'''Constant functions C{{subsu|b=n}}{{sup|p=k}}''': For each natural number <math>n\,</math> and every <math>k\,</math><br />&nbsp;&nbsp;&nbsp;&nbsp;<math>C_n^k(x_1,\ldots,x_k) \stackrel{\mathrm{def}}{=} n</math><br />Alternative definitions use instead a '''zero function''' as a primitive function that always returns zero, and built the constant functions from the zero function, the successor function and the composition operator.

# '''Successor function S:'''<br />&nbsp;&nbsp;&nbsp;&nbsp;<math>S(x) \stackrel{\mathrm{def}}{=} x + 1\,</math>

# '''Projection function''' <math>P_i^k</math> (also called the '''Identity function'''): For all natural numbers <math>i, k</math> such that <math>1\le i\le k</math>:<br />&nbsp;&nbsp;&nbsp;&nbsp;<math>P_i^k(x_1,\ldots,x_k) \stackrel{\mathrm{def}}{=} x_i \, .</math>