The lower elementary functions (also called Skolem elementary functions) are functions generated from the successor, modified subtraction, projection functions by the operations of composition and summation. Are the lower elementary functions closed under limited recursion? More precisely, if $g(k_0,\dots,k_{n-1})$, $h(k_0,\dots,k_{n+1})$, and $j(k_0,\dots,k_n)$ are lower elementary, is the function $f(k_0,\dots,k_n)$ defined below lower elementary? \begin{align*} f(k_0,\dots,k_{n-1},0) & = g(k_0,\dots,k_{n-1}),\\ f(k_0,\dots,k_{n-1},i+1) & = h(k_0,\dots,k_{n-1},i,f(k_0,\dots,k_{n-1},i)),\\ f(k_0,\dots,k_n) & \leqslant j(k_0,\dots,k_n). \end{align*}
This is an open question in the book Subrecursion by H.E. Rose (see p. 119, Open Problem 2), but I want to know whether it is still open now.
Another related open question in the book (see p. 121, Open Problem 3) is that in the definition of the lower elementary functions, whether we can replace the operation of summation by the operation of limited minimum (and at the same time add the multiplication function to the basic functions). Is this question still open now?
