Computability and Logic

5.1. ABACUS MACHINES 47
be thought of as operated by a little man who is capable of carrying out two sorts of
operations: adding a stone to the box of a specified number, and removing a stone
from the box of a specified number, if there are any stones there to be removed.
The table for a Turing machine is in effect a list of numbered instructions, where
‘attending to instruction q’ is called ‘being in state q’. The instructions all have the
following form:
{ if you are scanning a blank then perform action a and go to r
(q)
if you are scanning a stroke then perform action b and go to s.
Here each of the actions is one of the following four options: erase (put a blank in the
scanned square), print (put a stroke in the scanned square), move left, move right. It
is permitted that one or both of r or s should be q, so ‘go to r’ or ‘go to s’ amounts
to ‘remain with q’.
Turing machines can also be represented by flow charts, in which the states or
instructions do not have to be numbered. An abacus machine program could also be
represented in a table of numbered instructions. These would each be of one or the
other of the following two forms:
(q) add one to box m and go to r
{ if box m is not empty then subtract one from box m and go to r
(q)
if box m is empty then go to s.
But in practice we are going to be working throughout with a flow-chart representation.
In this representation, the elementary operations will be symbolized as in
Figure 5-1.
Figure 5-1. Elementary operations in abacus machines.
Flow charts can be built up as in the following examples.
5.1 Example (Emptying box n). Emptying the box of a specified number n can be accomplished
with a single instruction as follows:
{ if box n is not empty then subtract 1 from box n and stay with 1
(1)
if box n is empty then halt.
The corresponding flow chart is indicated in Figure 5-2.
In the figure, halting is indicated by an arrow leading nowhere. The block diagram also
shown in Figure 5-2 summarizes what the program shown in the flow chart accomplishes,