Computability and Logic

PROBLEMS 61
Figure 5-17. Minimization.
emptied and a single rock place in box 2, preparatory to the computation of f (x, 1).
If this value is 0, the program halts, with the correct value, h(x) = 1, in box 2.
Otherwise, another rock is placed in box 2, and the procedure continues until such time
(if any) as we have a number y of rocks in box 2 that is enough to make f (x, y) = 0.
The extensive class of functions obtainable from the trivial functions considered
in the example at the beginning of this section by the kinds of processes considered
in the rest of this section will be studied in the next chapter, where they will be given
the name recursive functions. At this point we know the following:
5.8 Theorem. All recursive functions are abacus computable (and hence Turing
computable).
So as we produce more examples of such functions, we are going to be producing
more evidence for Turing’s thesis.
Problems
5.1 Design an abacus machine for computing the difference function . − defined by
letting x . −y = x − y if y < x, and = 0 otherwise.
5.2 The signum function sg is defined by letting sg(x) = 1ifx > 0, and = 0
otherwise. Give a direct proof that sg is abacus computable by designing an
abacus machine to compute it.
5.3 Give an indirect proof that sg is abacus computable by showing that sg is
obtainable by composition from functions known to be abacus computable.
5.4 Show (directly by designing an appropriate abacus machine, or indirectly) that
the function f defined by letting f (x, y) = 1ifx < y, and = 0 otherwise, is
abacus computable.
5.5 The quotient and the remainder when the positive integer x is divided by the
positive integer y are the unique natural numbers q and r such that x = qy + r
and 0 ≤ r < y. Let the functions quo and rem be defined as follows: rem(x, y) =
the remainder on dividing x by y if y ≠ 0, and = x if y = 0; quo(x, y) = the
quotient on dividing x by y if y ≠ 0, and = 0ify = 0. Design an abacus machine
for computing the remainder function rem.
5.6 Write an abacus-machine flow chart for computing the quotient function quo
of the preceding problem.