NiesBookWithErrata

1.2 Relative computational complexity of sets 9If X is computable, Y ≠ ∅, and Y ≠ N, then X ≤ m Y : choose y 0 ∈ Y andy 1 ∉ Y . Let f(n) =y 0 if n ∈ X, and f(n) =y 1 otherwise. Then X ≤ m Y via f.Thus, disregarding ∅ and N, the computable sets form the least many-one degree.For each set Y the class {X : X ≤ m Y } is countable. In particular, there is nogreatest many-one degree. However, ∅ ′ is the most complex among the c.e. setsin the sense of ≤ m :1.2.2 Proposition. A is c.e. ⇔ A ≤ m ∅ ′ .An index for the many-one reduction as a computable function can be obtainedeffectively from a c.e. index for A, and conversely.Proof. ⇐: If A ≤ m ∅ ′ via h, then A = dom(Ψ) where Ψ(x) ≃ J(h(x)) (recallthat J(e) ≃ Φ e (e)). So A is computably enumerable.⇒: We claim that there is a computable function g such that{{e} if n ∈ A,W g(e,n) =∅ else.For let Θ(e, n, x) converge if x = e and n ∈ A. By a three-variable versionof the Parameter Theorem 1.1.2, there is a computable function g such that∀e, n, x [Θ(e, n, x) ≃ Φ g(e,n) (x)]. By Theorem 1.1.6, there is a computable functionh such that W g(h(n),n) = W h(n) for each n. Thenn ∈ A ⇒ W h(n) = {h(n)} ⇒ h(n) ∈∅ ′ , andn ∉ A ⇒ W h(n) = ∅ ⇒ h(n) ∉∅ ′ .The uniformity statements follow from the uniformity of Theorem 1.1.6.1.2.3 Definition. A c.e. set C is called r-complete if A ≤ r C for each c.e. set A.Usually ≤ m implies the reducibility ≤ r under consideration. Then, since ∅ ′ is m-complete, a c.e. set C is r-complete iff ∅ ′ ≤ r C. An exception is 1-reducibility,which is more restricted than ≤ m : we say that X ≤ 1 Y if X ≤ m Y via a one-onefunction f.Exercises.1.2.4. The set ∅ ′ is 1-complete. (This will be strengthened in Theorem 1.7.18.)1.2.5. (Myhill) X ≡ 1 Y ⇔ there is a computable permutation p of N such thatY = p(X). (For a solution see Soare 1987, Thm. I.5.4.)Turing reducibilityMany-one reducibility is too restricted to serve as an appropriate measure forthe relative computational complexity of sets. Our intuitive understanding of “Yis at least as complex as X” is: X can be computed with the help of Y (or, “Xcan be computed relative to Y ”). If X ≤ m Y via h, then this holds via a veryparticular type of relative computation procedure: on input x, compute k = h(x)and output 1 (“yes”) if k ∈ Y , and 0 otherwise. To formalize more general waysof relative computation, we extend the machine model by a one-way infinite✷