Computability and Logic

5.3. THE SCOPE OF ABACUS COMPUTABILITY 57
5.3 The Scope of Abacus Computability
We now turn from showing that particular functions are abacus computable to showing
that certain processes for defining new functions from old, when applied to old abacuscomputable
functions, produce new abacus-computable functions. (These processes
will be explained and examined in more detail in the next chapter, and readers may
wish to defer reading this section until after that chapter.)
Now we initially indicated that to compute a function of r arguments on an abacus,
we must specify r registers or boxes in which the arguments are to be stored initially
(represented by piles of rocks) and we must specify a register or box in which the
value of the function is to appear (represented by a pile of rocks) at the end of
the computation. To facilitate comparison with computations by Turing machines in
standard form, we then insisted that the input or arguments were to be placed in the
first r registers, but left it open in which register n the output or value would appear:
it was not necessary to be more specific, because the simulation of the operations
of an abacus by a Turing machine could be carried out wherever we let the output
appear. For the purposes of this section, we are therefore free now to insist that the
output register n, which we have heretofore left unspecified, be specifically register
r + 1. We also wish to insist that at the end of the computation the original arguments
should be back in registers 1 through r. In the examples considered earlier this last
condition was not met, but those examples are easily modified to meet it. We give
some further, trivial examples here, where all our specifications are exactly met.
5.7 Example (Zero, successor, identity). First consider the zero function z, the one-place
function that takes the value 0 for all arguments. It is computed by the vacuous program:
box 2 is empty anyway.
Next consider the successor function s, the one-place function that takes any natural
number x to the next larger natural number x + 1. It is computed by modifying the program
in Example 5.3, as shown in Figure 5-14.
Figure 5-14. Three basic functions.
Initially and finally, [1] = x; initially [2] = 0; finally, [2] = s(x). Finally consider identity
function id m n , the n-place function whose value for n arguments x 1, ..., x n is the mth one
among them, x m . It is computed by the program of the same Example 5.3. Initially and
finally, [1] = x 1 , ...,[n] = x n ; initially, [n + 1] = 0; finally [n + 1] = x m .