Computability and Logic

46 ABACUS COMPUTABILITY
Historically, the notion of Turing computability was developed before the age of
high-speed digital computers, and in fact, the theory of Turing computability formed
a not insignificant part of the theoretical background for the development of such
computers. The kinds of computers that are ordinary today are in one respect more
flexible than Turing machines in that they have random-access storage.ALambek or
abacus machine or simply abacus will be an idealized version of computer with this
‘ordinary’ feature. In contrast to a Turing machine, which stores information symbol
by symbol on squares of a one-dimensional tape along which it can move a single
step at a time, a machine of the seemingly more powerful ‘ordinary’ sort has access
to an unlimited number of registers R 0 , R 1 , R 2 , ..., in each of which can be written
numbers of arbitrary size. Moreover, this sort of machine can go directly to register
R n without inching its way, square by square, along the tape. That is, each register
has its own address (for register R n it might be just the number n) which allows the
machine to carry out such instructions as
which we abbreviate
put the sum of the numbers in registers R m and R n into register R p
[m] + [n] → p.
In general, [n] is the number in register R n , and the number at the right of an arrow
identifies the register in which the result of the operation at the left of the arrow is
to be stored. When working with such machines, it is natural to consider functions
on the natural numbers (including zero), and not just the positive integers (excluding
zero). Thus, the number [n] in register R n at a given time may well be zero: the
register may be empty.
It should be noted that our ‘ordinary’ sort of computing machine is really quite
extraordinary in one respect: although real digital computing machines often have
random-access storage, there is always a finite upper limit on the size of the numbers
that can be stored; for example, a real machine might have the ability to store any
of the numbers 0, 1, ..., 10 000 000 in each of its registers, but no number greater
than ten million. Thus, it is entirely possible that a function that is computable in
principle by one of our idealized machines is not computable in practice by any real
machine, simply because, for certain arguments, the computation would require more
capacious registers than any real machine possesses. (Indeed, addition is a case in
point: there is no finite bound on the sizes of the numbers one might think of adding,
and hence no finite bound on the size of the registers needed for the arguments and
the sum.) But this is in line with our objective of abstracting from technological
limitations so as to arrive at a notion of computability that is not too narrow. We seek
to show that certain functions are uncomputable in an absolute sense: uncomputable
even by our idealized machines, and therefore uncomputable by any past, present, or
future real machine.
In order to avoid discussion of electronic or mechanical details, we may imagine
the abacus machine in crude, Stone Age terms. Each register may be thought of as a
roomy, numbered box capable of holding any number of stones: none or one or two
or ..., so that [n] will be the number of stones in box number n. The ‘machine’ can