Computability and Logic

5.2. SIMULATING ABACUS MACHINES BY TURING MACHINES 53
Figure 5-8. Correspondence between boxes and tape.
R 4 in the figure). The 1 is mandatory if there are any strokes further to the right on the
tape, and is mandatory initially for empty argument boxes. The blank is mandatory
initially for R r+1 , R r+2 , .... Then at any stage of the computation we can be sure
that when in moving to the right or left we encounter two successive blanks, there
are no further strokes to be found anywhere to the right or left (as the case may be)
on the tape. The exact portion of the tape that represents a box will wax and wane with
the contents of that box as the execution of the program progresses, and will shift to
the right or left on the tape as stones are added to or removed from lower-numbered
boxes.
The first step in our method for converting abacus flow charts into equivalent
Turing-machine flow charts can now be specified: replace each s+ node (consisting
of a node marked s+ and the arrow leading from it) by a copy of the s+ flow chart
shown in Figure 5-9.
Figure 5-9. The s+ flow chart.
The first 2(s − 1) nodes of the s+ chart simply take the Turing machine across the
first s − 1 blocks of strokes. In the course of seeking the sth block, the machine substitutes
the 1-representation for the B-representation of any empty boxes encountered
along the way.
When it enters the node sa, the Turing machine has arrived at the sth block. Then
again substitutes the 1-representation for the B-representation of that box, if that box
is empty. On leaving node sb, the machine writes a stroke, moves 1 square right, and
does one thing or another (node x) depending on whether it is then scanning a blank
or a stroke.