A note on primitive recursive functions

Abstract

then F* can be obtained from A*, B*, and several specific functions by substitution and recursion alone. Now, by a theorem of R. M. Robinson,2 all primitive recursive functions of one variable can be obtained from S and K by successive additions, substitutions, and recursions of the form Fx = BxO. Hence if F is a primitive recursive function, then F* can be obtained from a suitable set of initial functions (including S* and K*) by substitution and recursion. We also have F=KF*J(I, I). Therefore, if K and J(I, I) are included in the initial set of functions, we can obtain not only all the star functions corresponding to primitive recursive functions but also all the primitive functions themselves. We still need to show how F* can be obtained from A* and B*. Referring to GRF, p. 713, we see that (BA)*=B*A* and (A+B)*