Computability and Logic

5.1. ABACUS MACHINES 49
Figure 5-4. Addition.
In case no assumption is made about the contents of register p at the beginning, the
operation accomplished by this program is the following:
[m] + [n] → n
[m] + [p] → m
0 → p.
Here, as always, the vertical order represents a sequence of moves, from top to bottom.
Thus, p is emptied after the other two moves are made. (The order of the first two moves
is arbitrary: The effect would be the same if their order were reversed.)
5.4 Example (Multiplication). The numbers to be multiplied are in distinct boxes m 1 and
m 2 ; two other boxes, n and p, are empty to begin with. The product appears in box n. The
program is indicated in Figure 5-5.
Figure 5-5. Multiplication.
Instead of constructing a flow chart de novo, we use the block diagram of the preceding
example a shorthand for the flow chart of that example. It is then straightforward to draw
the full flow chart, as in Figure 5-5(b), where the m of the preceding example is changed to
m 2 . The procedure is to dump [m 2 ] stones repeatedly into box n, using box m 1 as a counter: