Computability, Unsolvability, Randomness
Math 497A: Homework #4
Stephen G. Simpson
Due Monday, September 24, 2007
In this set of exercises we explore the structure of the Turing degrees.
Given two Turing degrees aand b, we know that the least upper bound
sup(a,b) always exists. Exercises 2, 4, and 7 below show that the greatest
lower bound inf(a,b) sometimes exists and sometimes does not exist.
For any Turing oracle fwe have
f0=Hf={x|ϕ(1),f
x(0) ↓} = the Halting Problem relative to f.
We kn ow tha t f0is a complete Σ0
1set relative to the oracle f. For any Turing
degree a=deg
T(f) we define
a0=deg
T(f0) = the Turing jump of a.
Clearly a<a0holds for all a. Thus, starting with any Turing degree a,we
have an ascending sequence of Turing degrees
a<a0<a00 <···<a(n)<a(n+1) <···
In particular, starting with the zero Turing degree 0, we have the ascending
sequence
0<00<000 <···<0(n)<0(n+1) <···
corresponding to the arithmetical hierarchy.
1. Given Turing oracles fand g, prove that the following conditions are
pairwise equivalent:
(a) f≤Tg
(b) Hf≤mHg
(c) all partial f-recursive functions are partial g-recursive
(d) all total f-recursive functions are g-recursive.
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