Computability and Logic

58 ABACUS COMPUTABILITY
Three different processes for defining new functions from old can be used to expand
our initial list of examples. A first process is composition, also called substitution.
Suppose we have two 3-place functions g 1 and g 2 , and a 2-place function f . The
function h obtained from them by composition is the 3-place function given by
h(x 1 , x 2 , x 3 ) = f (g 1 (x 1 , x 2 , x 3 ), g 2 (x 1 , x 2 , x 3 )).
Suppose g 1 and g 2 and f are abacus computable according to our specifications, and
we are given programs for them.
↓ ↓ ↓
f ([1], [2]) → 3 g 1 ([1], [2], [3]) → 4 g 2 ([1], [2], [3]) → 4
↓ ↓ ↓
We want to find a program for h, to show it is abacus computable:
↓
h([1], [2], [3]) → 4 .
↓
The thing is perfectly straightforward: It is a matter of shuttling the results of
subcomputations around so as to be in the right boxes at the right times.
First, we identify five registers, none of which are used in any of the given programs.
Let us call these registers p 1 , p 2 , q 1 , q 2 , and q 3 . They will be used for temporary
storage. In the single program which we want to construct, the 3 arguments are stored
initially in boxes 1, 2, and 3; all other boxes are empty initially; and at the end, we
want the n arguments back in boxes 1, 2, 3, and want the value f (g 1 ([1], [2], [3]),
g 2 ([1], [2], [3])) in box number 4. To arrange that, all we need are the three given
programs, plus the program of Example 5.2 for emptying one box into another.
We simply compute g 1 ([1], [2], [3]) and store the result in box p 1 (which figures
in none of the given programs, remember); then compute g 2 ([1], [2], [3]) and store
the result in box p 2 ; then store the arguments in boxes 1, 2, and 3 in boxes q 1 , q 2 ,
and q 3 , emptying boxes 1 through 4; then get the results of the computations of g 1
and g 2 out of boxes p 1 and p 2 where they have been stored, emptying them into
boxes 1 and 2; then compute f ([1], [2]) = f [g 1 (original arguments), g 2 (original
arguments)], getting the result in box 3; and finally, tidy up, moving the overall result
of the computation from box 3 to box 4, emptying box 3 in the process, and refilling
boxes 1 through 3 with the original arguments of the overall computation, which were
stored in boxes q 1 , q 2 , and q 3 . Now everything is as it should be. The structure of the
flow chart is shown in Figure 5-15.
Another process, called (primitive) recursion, is what is involved in defining multiplication
as repeated addition, exponentiation as repeated multiplication, and so on.
Suppose we have a 1-place functions f and a 3-place function g. The function h
obtained from them by (primitive) recursion is the 2-place function h given by
h(x, 0) = f (x)
h(x, y + 1) = g(x, y, h(x, y)).