By Paul B. Bailey
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It should not be difficult to see that instead of computing x 1 + y as desired, this program computes 2x 1 • Since X 1 is to be added over and over again, it is important that X 1 not be destroyed by the addition program. Here is the multiplication program, showing the macro expansion of Z 1 ~ X 1 + Y: [B] [A] Zz ~xz IF Z 2 =F 0 GOTO A GOTOE Z2 ~ Z2 - 1 zl ~x~ z3 ~ [Bz] [Az] Y IF Z 3 =F 0 GOTO A 2 GOTO £ 2 Z3 ~ Z3 - 1 Z1 ~ Z1 + 1 GOTO B2 [Ez] Macro Expansion of +Y zl ~x~ v~z 1 GOTOB Note the following: 1:- The local variable Z 1 in the addition program in (d) must be replaced by another local variable (we have used Z 3 ) because Z 1 (the other name for Z) is also used as a local variable in the multiplication program.
Hence it will follow that the set of primitive recursive functions is a proper subset of the set of computable functions. Exercises 1. Show that if hz(x) = g 2 (x, x, x), and h 3 (w, x, y, z) = h 1(giw, y), z, gi2, g4 (y, z))), then h 2 , h 2 , h 3 also belong to '??. 2. Show that the class of all total functions is a PRC class. 3. Let n > 0 be some given number, and let '?? be a class of total functions of no more than n variables. Show that ~ is not a PRC class. 44 Chapter 3 Primitive Recursive Functions 4.
We want to show that f belongs to '??. Since f is a primitive recursive function, there is a list / 1 , f 2 , ••• , fn of functions such that fn = f and each /; in the list is either an initial function or can be obtained from preceding functions in the list by composition or recursion. 4. Every primitive recursive function is computable. Proof. By the theorem just proved, every primitive recursive function • belongs to the PRC class of computable functions. In Chapter 4 we shall show how to obtain a computable function that is not primitive recursive.