It is harder to motivate the inclusion of the function \(x \# y\) in \(\mathcal{F}_0\) and of the condition \(f(x,y) \leq k(x,y)\) in Definition 4.1 on pre-theoretical grounds alone. For these features of the definition of \(\mathcal{F}\) can be see to have precisely the effect of placing a polynomial bound on the auxiliary functions which can be computed during the sort of length-bounded recursion just described. On the other hand, such indirect reference to polynomial rates of growth is avoided in similar functional characterizations of \(\textbf{FP}\) due to Leivant (1994) (using a form of positive second-order definability over strings) and Bellantoni and Cook (1992) (using a structural modification of the traditional primitive recursion scheme).