Computability and Logic

50 ABACUS COMPUTABILITY
We remove a stone from box m 1 before each dumping operation, so that when box m 1 is
empty we have
[m 2 ] + [m 2 ] +···+[m 2 ] ([m 1 ] summands)
stones in box n.
5.5 Example (Exponentiation). Just as multiplication is repeated addition, so exponentiation
is repeated multiplication. The program is perfectly straightforward, once we arrange
the multiplication program of the preceding example to have [m 2 ] · [n] → n. How that is to
be accomplished is shown in Figure 5-6.
Figure 5-6. Exponentiation.
The cumulative multiplication indicated in this abbreviated flow chart is carried out in
two steps. First, use a program like Example 5.4 with a new auxiliary:
[n] · [m 2 ] → q
0 → n.
Second, use a program like Example 5.2:
[q] + [n] → n
0 → q.
The result gives [n] · [m 2 ] → n. Provided the boxes n, p, and q are empty initially, the
program for exponentiation has the effect
[m 2 ] [m1] → n
0 → m 1
in strict analogy to the program for multiplication. (Compare the diagrams in the preceding
example and in this one.)
Structurally, the abbreviated flow charts for multiplication and exponentiation
differ only in that for exponentiation we need to put a single stone in box n at the
beginning. If [m 1 ] = 0wehaven = 1 when the program terminates (as it will
at once, without going through the multiplication routine). This corresponds to the
convention that x 0 = 1 for any natural number x. But if [m 1 ] is positive, [n] will
finally be a product of [m 1 ] factors [m 2 ], corresponding to repeated application of