Computability and Logic

5.3. THE SCOPE OF ABACUS COMPUTABILITY 59
Figure 5-15. Composition.
For instance, if f (x) = x and g(x, y, z) = z + 1, then
h(x, 0) = f (x) = x = x + 0
h(x, 1) = g(x, 0, x) = x + 1
h(x, 2) = g(x, 1, x + 1) = (x + 1) + 1 = x + 2
and in general h(x, y) = x + y. Suppose f and g are abacus computable according
to our specifications, and we are given programs for them:
↓
f ([1]) → 2 g 1 ([1], [2], [3]) → 4 .
↓
↓
We want to find a program for h, to show it is abacus computable
↓
h([1], [2]) → 3 .
↓
The thing is easily done, as in Figure 5-16.
↓