Computability and Logic

56 ABACUS COMPUTABILITY
Figure 5-12. Detail of the s− flow chart.
The ‘Return to standard position’ blocks contain replicas of the material to the
right of node x in the s+ chart: The B:L arrows entering those boxes correspond to
the B:L arrow from node x.
The only novelty is in the remaining block: ‘Find and erase the ...’ That block
contains the chart shown in Figure 5-12.
For the third stage, the mop-up chart, for n ≠ 1, is shown in Figure 5-13.
Figure 5-13. Mop-up chart.
We have proved:
5.6 Theorem. Every abacus-computable function is Turing computable.
We know from the preceding chapter some examples of functions that are not
Turing computable. By the foregoing theorem, these functions are also not abacus
computable. It is also possible to prove directly the existence of functions that are not
abacus computable, by arguments parallel to those used for Turing computability in
the preceding chapter.