I've met two different definitions of total recursive functions. One has 0; +1; projections as basic functions and composition; primitive recursion; minimalization as production rules. Another has +; ×; projections; the indicator function of < as basic functions and composition; minimalization as production rules. I know the second one doesn't need primitive recursion. Is primitive recursion indispensable to the first definition? If this is true, then one of +,× and I$_<$ can't be generated from zero, sucessor and projections with composition and minimalization only. Indeed I've tried to do so without success but I want to know if this is truly the case. Thanks.
Is primitive recursion necessary for recursive functions?
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computability
recursion
1 Answers
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All the initial functions $0, x+1$ and $I^n_k(x_1, \dots, x_n)$ essentially depend (at most) on one variable. Assume that we have $f(x_1, \dots, x_n), g_1(x_1, \dots, x_m), \dots, g_n(x_1, \dots, x_m)$ and they depend essentially on $x_{i_0}$ and $x_{i_1}, \dots, x_{i_n}$ respectively. Then their composition $$f(g_1(x_1, \dots, x_m), \dots, g_n(x_1, \dots, x_m))$$ depends essentially only on the essential variable of $g_{i_0}$, that is $x_{i_{i_0}}$.
Assume $g(x_1, \dots, x_n, y)$ depends essentially only on one variable and let $$f(x_1, \dots, x_n) = \mu y[g(x_1, \dots, x_n, y) = 0].$$ It is clear that in this case $f$ also depends on the same variable (or doesn't depend on its variables at all, if $g$ depends only on $y$).
The above shows that any function generated by using composition and minimalization from the initial functions depends essentially only on one variable. Hence, for instance, $x + y$ can't be generated this way.