Computability and Logic

5.2. SIMULATING ABACUS MACHINES BY TURING MACHINES 55
Figure 5-11. Abbreviated s− flow chart.
1 on the tape and the nth block, repositions all strokes but the rightmost in the nth
block immediately to the right of the leftmost 1, erases the rightmost 1, and then halts
scanning the leftmost 1. In both cases, the effect is to place the leftmost 1 in the block
representing the value just where the leftmost 1 was initially. Again readers may wish
to try to fill in the details of the design for themselves, as an exercise. (Our design
will be given shortly.)
The proof that all abacus-computable functions are Turing computable is now
finished, except for the two steps that we have invited readers to try as exercises. For
the sake of completeness, we now present our own solutions to these exercises: our
own designs for the second and third stages of the construction reducing an abacus
computation to a Turing computation.
For the second stage, we describe what goes into the boxes in Figure 5-11. The
top block of the diagram contains a chart identical with the material from node 1a to
sa (inclusive) of the s+ flow chart. The arrow labelled 1:R from the bottom of this
block corresponds to the one that goes right from node sa in the s+ flow chart.
The ‘Is [s] = 0?’ box contains nothing but the shafts of the two emergent arrows:
They originate in the node shown at the top of that block.