Relative to the standards discussed in  Section 1.1   it seems reasonable to maintain  that the basis functions \(\mathcal{F}_0\) are feasibly computable and also that this property is preserved under composition. Now suppose that \(f(x,y)\) is defined by limited recursion on notation from \(g(x), h_0(x,y,z), h_1(x,y,z)\) and \(k(x,y)\). In this case, \(f(x,y) = h_i(x,\lceil \frac{y}{2} \rceil,f(x,\lceil \frac{y}{2} \rceil))\) depending on whether \(y \gt 0\) is even (\(i = 0\)) or odd (\(i = 1\)). From this it follows that the depth of the recursion required to compute the value of \(f(x,y)\) will be proportional to \(\log_2(y)\) – i.e. proportional to the length of \(y\)’s binary representation, as opposed to \(y\) itself as when \(f(x,y+1)\) is defined as \(h(x,y,f(x,y))\) by ordinary primitive recursion. This suggests that feasibility is also preserved when a function is defined by limited recursion on notation.