Computability and Logic

48 ABACUS COMPUTABILITY
Figure 5-2. Emptying a box.
without indicating how it is accomplished. Such summaries are useful in showing how more
complicated programs can be put together out of simpler ones.
5.2 Example (Emptying box m into box n). The program is indicated in Figure 5-3.
Figure 5-3. Emptying one box into another.
The figure is intended for the case m ≠ n. (If m = n, the program halts—exits on the e
arrow—either at once or never, according as the box is empty or not originally.) In future we
assume, unless the contrary possibility is explicitly allowed, that when we write of boxes
m, n, p, and so on, distinct letters represent distinct boxes.
When as intended m ≠ n, the effect of the program is the same as that of carrying stones
from box m to box n until box m is empty, but there is no way of instructing the machine
or its operator to do exactly that. What the operator can do is (m−) take stones out of
box m, one at a time, and throw them on the ground (or take them to wherever unused
stones are stored), and then (n+) pick stones up off the ground (or take them from wherever
unused stones are stored) and put them, one at a time, into box n. There is no assurance that
the stones put into box n are the very same stones that were taken out of box m, but we
need no such assurance in order to be assured of the desired effect as described in the block
diagram, namely,
[m] + [n] → n:
the number of stones in box n after this move equals
the sum of the numbers in m and in n before the move
and then
0 → m: the number of stones in box m after this move is 0.
5.3 Example (Adding box m to box n, without loss from m). To accomplish this we must
make use of an auxiliary register p, which must be empty to begin with (and will be empty
again at the end as well). Then the program is as indicated in Figure 5-4.