Computability and Logic

60 ABACUS COMPUTABILITY
Figure 5-16. Recursion.
Initially, [1] = x, [2] = y, and [3] = [4] =···=0. We use a register number p that
is not used in the f and g programs as a counter. We put y into it at the beginning, and
after each stage of the computation we see whether [p] = 0. If so, the computation
is essentially finished; if not, we subtract 1 from [p] and go through another stage.
In the first three steps we calculate f (x) and see whether entry y was 0. If so, the
first of the pair of equations for h is operative: h(x, y) = h(x, 0) = f (x), and the
computation is finished, with the result in box 3, as required. If not, the second of
the pair of equations for h is operative, and we successively compute h(x, 1),
h(x, 2), ...(see the cyle in Figure 5-16) until the counter (box p) is empty. At that
point the computation is finished, with h(x, y) in box 3, as required.
A final process is minimization. Suppose we have a 2-place function f ; then we
can define a 1-place function h as follows. If f (x, 0), ..., f (x, i − 1) are all defined
and ̸= 0, and f (x, i) = 0, then h(x) = i. If there is no i with these properties, either
because for some i the values f (x, 0), ..., f (x, j − 1) are all defined and ≠ 0but
f (x, j) is not defined, or because for all i the value f (x, i) is defined and ≠ 0,
then h(x) is undefined. The function h is called the function obtained from f by
minimization. If f is abacus computable, so is h, with a flow chart as in Figure 5-17.
Initially, box 2 is empty, so that if f (x, 1)= 0, the program will halt with the
correct answer, h(x) = 0, in box 2. (Box 3 will be empty.) Otherwise, box 3 will be