Computability and Logic

5
Abacus Computability
Showing that a function is Turing computable directly, by giving a table or flow chart
for a Turing machine computing the function, is rather laborious, and in the preceding
chapters we did not get beyond showing that addition and multiplication and a few
other functions are Turing computable. In this chapter we provide a less direct way of
showing functions to be Turing computable. In section 5.1 we introduce an ostensibly
more flexible kind of idealized machine, an abacus machine, or simply an abacus. In
section 5.2 we show that despite the ostensible greater flexibility of these machines, in
fact anything that can be computed on an abacus can be computed on a Turing machine.
In section 5.3 we use the flexibility of these machines to show that a large class of
functions, including not only addition and multiplication, but exponentiation and many
other functions, are computable on a abacus. In the next chapter functions of this class
will be called recursive, so what will have been proved by the end of this chapter is that
all recursive functions are Turing computable.
5.1 Abacus Machines
We have shown addition and multiplication to be Turing-computable functions, but
not much beyond that. Actually, the situation is even a bit worse. It seemed appropriate,
when considering Turing machines, to define Turing computability for functions
on positive integers (excluding zero), but in fact it is customary in work on other
approaches to computability to consider functions on natural numbers (including
zero). If we are to compare the Turing approach with others, we must adapt our
definition of Turing computability to apply to natural numbers, as can be accomplished
(at the cost of some slight artificiality) by the expedient of letting the number n
be represented by a string of n + 1 strokes, so that a single stroke now represents zero,
two strokes represent one, and so on. But with this change, the adder we presented in
the last chapter actually computes m + n + 1, rather than m + n, and would need to be
modified to compute the standard addition function; and similarly for the multiplier.
The modifications are not terribly difficult to carry out, but they still leave us with
only a very few examples of interesting effectively computable functions that have
been shown to be Turing computable. In this chapter we greatly enlarge the number of
examples, but we do not do so directly, by giving tables or flow charts for the relevant
Turing machines. Instead, we do so indirectly, by way of another kind of idealized
machine.
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