Computability and Logic

44 UNCOMPUTABILITY
if p(b) ≤ p(a), we must have b ≤ a. Applying this observation to (1), we have
(2)
n + 2 j ≥ p(n)
for any n. Letting i be as in Example 4.6 above, we have
(3)
p(m + i) ≥ 2m
for any m. But applying (2) with n = m + i,wehave
(4)
m + i + 2 j ≥ p(m + i)
for any m. Combining (3) and (4), we have
(5)
m + i + 2 j ≥ 2m
for any m. Setting k = i + 2 j,wehave
(6)
m + k ≥ 2m
for any m. But this is absurd, since clearly (6) fails for any m > k. We have proved:
4.7 Theorem. The productivity function p is Turing uncomputable.
Problems
4.1 Is there a Turing machine that, started anywhere on the tape, will eventually halt
if and only if the tape originally was not completely blank? If so, sketch the
design of such a machine; if not, briefly explain why not.
4.2 Is there a Turing machine that, started anywhere on the tape, will eventually halt
if and only if the tape originally was completely blank? If so, sketch the design
of such a machine; if not, briefly explain why not.
4.3 Design a copying machine of the kind described at the beginning of the proof of
theorem 4.2.
4.4 Show that if a two-place function g is Turing computable, then so is the oneplace
function f given by f (x) = g(x, x). For instance, since the multiplication
function g(x, y) = xy is Turing computable, so is the square function f (x) = x 2 .
4.5 A universal Turing machine is a Turing machine U such that for any other Turing
machine M n and any x, the value of the two-place function computed by U for
arguments n and x is the same as the value of the one-place function computed
by M n for argument x. Show that if Turing’s thesis is correct, then a universal
Turing machine must exist.