Computability, Unsolvability, Randomness
Math 497A: Homework #2
Stephen G. Simpson
Due Monday, September 10, 2007
1. Let rbe a positive real number. Prove that ris computable if and only
if the number-theoretic function
f(n)=thenth decimal digit of r
is computable.
2. Consider the 2-place computable number-theoretic function f(x, y)=
x+y. Exhibit three different indices of f.
(By an index of a partial recursive function, we mean the G¨odel number
of some program which computes the function.)
3. If fis a computable permutation of N, prove that the inverse permu-
tation f−1is also computable.
(Here f−1(y)=xif and only if f(x)=y.Byacomputable permutation
of Nwe mean a computable 1-place function f:N→Nwhich maps
None-to-one onto N.)
4. Generalize the previous exercise as follows. Prove that if ψis a 1-place
partial recursive function which is one-to-one, then the inverse function
ψ−1is again partial recursive.
5. Consider the sets
Kn={x∈N|ϕ(1)
x(x)'n}
where n=0,1,2,.... Show that the sets K0and K1are recursively
inseparable. More generally, show that Kmand Knare recursively
inseparable for all m, n such that m6=n.
(Two sets A, B ⊆Nare said to be recursively separable if there exists a
recursive function f:N→{0,1}such that f(n) = 1 for all n∈A,and
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