Recursive functions of one variable

Abstract

We say that the function F is obtained by general recursion from A, B, M, and N if (i) FA = M, FB = NF and (ii) every natural number n belongs to (R(BkA) for some k >0. If F satisfies (i), then F(BkAt) =NkMt. If A and B satisfy (ii), then every natural number n is BkAt for some k and t. Hence F is uniquely determined. Furthermore if A, B, M, and N are computable functions, then given n, we can obtain k and t effectively so F is computable. Condition (ii) is satisfied if A is the zero function 0 and B is tlle successor function S. There is a function obtained by general recursion from 0, S, M, and N if and only if M is a constant function, say M=SmO. Then Ft=Nem. Thus, iteration is included in general recursion. If G is a permutation, then F=G-1 can be obtained by general recursion from GS, GO, S, and 0 since FGS = S, FGO = OF, and every n belongs to (R((GO)kGS) for k = 0 or k =1. More generally, if F is determined by FG = H where G assumes all values, then F is obtained by general recursion from GS, GO, HS, and HO since FGS = HS, FGO =HOF, and the side condition holds as before.