Computability, Unsolvability, Randomness
Math 497A: Homework #7
Stephen G. Simpson
Due Monday, October 15, 2007
1. Exhibit an oracle program Psuch that
ϕ(1),f
e(x)'µy (y>x∧f(y)=0)
for all f∈NNand all x∈N,wheree=#(P).
2. (a) Give an explicit example of a ∆0
4set which is neither Σ0
3nor Π0
3.
(b) Give an example of a set which cannot be classified in the arith-
metical hierarchy.
3. Let A, B, C be recursively enumerable sets with A=B∪Cand B∩C=
∅.Ifa,b,care the respective Turing degrees of A, B , C prove that
a=sup(b,c).
Note: The hardest part is to prove that B≤TAand C≤TA.Your
proof should use the assumption that A, B, C are r.e. sets. Without
this assumption, the result would not be correct.
4. Construct an infinite descending sequence of Turing degrees
a0>a1>···>an>an+1 >···
or prove that no such sequence exists.
5. Let C(σ) denote the Kolmogorov complexity of a 0,1-valued string σ.
We have seen in class that
C(σaτ)≤2C(σ)+2C(τ)+O(1)
for all 0,1-valued strings σand τ. Improve this inequality to
C(σaτ)≤C(σ)+2log
2C(σ)+C(τ)+O(1)
where log2xdenotes the base 2 logarithm of x.
Can you make further improvements?