Computability and Logic

5.2. SIMULATING ABACUS MACHINES BY TURING MACHINES 51
the rule x y+1 = x · x y , which is implemented by means of cumulative multiplication,
using box m 1 as a counter.
It should now be clear that the initial restriction to two elementary sorts of acts,
n+ and n−, does not prevent us from computing fairly complex functions, including
all the functions in the series that begins sum, product, power, ..., and where the
n + 1st member is obtained by iterating the nth member. This is considerably further
than we got with Turing machines in the preceding chapters.
5.2 Simulating Abacus Machines by Turing Machines
We now show that, despite the ostensible greater flexibility of abacus machines,
all abacus-computable functions are Turing computable. Before we can describe a
method for transforming abacus flow charts into equivalent Turing-machine flow
charts, we need to standardize certain features of abacus computations, as we did
earlier for Turing computations with our official definition of Turing computability.
We must know where to place the arguments, initially, and where to look, finally,
for values. The following conventions will do as well as any, for a function f of r
arguments x 1 , ..., x r :
(a) Initially, the arguments are the numbers of stones in the first r boxes, and all other
boxes to be used in the computation are empty. Thus, x 1 = [1], ..., x r = [r],
0 = [r + 1] = [r + 2] =···.
(b) Finally, the value of the function is the number of stones is some previously
specified box n (which may but need not be one of the first r). Thus,
f (x 1 , ..., x r ) = [n] when the computation halts, that is, when we come to an
arrow in the flow chart that terminates in no node.
(c) If the computation never halts, f (x 1 , ..., x r ) is undefined.
The computation routines for addition, multiplication, and exponentiation in the
preceding section were essentially in this form, with r = 2 in each case. They were
formulated in a general way, so as to leave open the question of just which boxes
are to contain the arguments and value. For example, in the adder we only specified
that the arguments are to be stored in distinct boxes numbered m and n, that the sum
will be found in box x, and that a third box, numbered p and initially empty, will
be used as an auxiliary in the course of the computation. But now we must specify
m, n, and p subject to the restriction that m and n must be 1 and 2, and p must be
some number greater than 2. Then we might settle on n = 1, m = 2, p = 3, to get a
particular program for addition in the standard format, as in Figure 5-7.
The standard format associates a definite function from natural numbers to natural
numbers with each abacus, once we specify the number r of arguments and
the number n of the box in which the values will appear. Similarly, the standard
format for Turing-machine computations associates a definite function from natural
numbers to natural numbers (originally, from positive integers to positive integers,
but we have modified that above) with each Turing machine, once we specify the
number r of arguments. Observe that once we have specified the chart of an abacus