I. 3.1 (Partial) Recursive Functions and Sets
I.4 Reducibility Notions
I.4.3 E w and the Embedding Lemma
Particular attention is given to the weak degrees of nonempty Π01subsets of{0,1}N, which we call Π01classes, serving a similar role inDwas the collectionETof r.e. Turing degrees (i.e., Turing degrees of r.e. sequences) inDT.
Definition I.4.15. Ew∶= {degw(P) ∣P⊆ {0,1}Nis Π01}.
One motivation forEw in comparison toET is that specific, natural examples of weak degrees inEw can be given while there are no known specific, natural r.e. degrees aside from 0and 0′. See [25] and [24] for additional details.
Proposition I.4.16. SupposeP⊆ {0,1}N is given. The following are equivalent.
(a) P isΠ01.
(b) There exists a recursive treeT⊆ {0,1}∗ such thatP is the set of paths throughT. (c) There existse∈Nsuch that P= {X ∈ {0,1}N∣ϕXe(0)↑}.
Although our interests lie chiefly within Ew, the weak degrees we consider are often most naturally represented by mass problems which are not Π01subsets of{0,1}N. The following result shows that this issue can be side-stepped as long as the mass problem is sufficiently low in the arithmetical hierarchy.
Proposition I.4.17(Embedding Lemma). [27, Lemma 17.1]SupposeP is a nonemptyΠ01 subset of{0,1}N andS is aΣ0 subset ofNN. Then there exists a nonempty Π0 subsetQof {0,1}N such thatQ≡ P∪S.
One particular case which is especially well-behaved is withrecursively bounded Π01 classes.
Definition I.4.18. Supposeh∶N→ (0,∞)is a computable function. We write hn∶= {σ∈Nn∣ ∀i<n(σ(i) <h(i))},
h∗∶= {σ∈N∗∣ ∀i< ∣σ∣ (σ(i) <h(i))} = ⋃
n∈N
hn, hN∶= {X∈NN∣ ∀i(X(i) <h(i))}.
In other words,hn is the set ofh-bounded strings of lengthn,h∗ is the set of allh-bounded strings, and hNis the set ofh-bounded infinite sequences.
Lemma I.4.19. The subspace topology onhN⊆NN has a basis {JσKh∣σ∈h∗}, where forσ∈h∗ we define
JσKh∶= {X∈hN∣σ⊂X}.
Proof. For allσ∈h∗,hN∩JσK=JσKh. Ifσ∈N∗∖h∗, thenhN∩JσK= ∅.
Proposition I.4.20. (well-known) hN is recursively homeomorphic to{0,1}N.
Proof. Define ψ∶h∗ → {0,1}∗ recursively as follows: ψ(⟨⟩) = ⟨⟩ and given ψ(σ) has been defined, let ψ(σ⌢⟨i⟩) ∶= ψ(σ)⌢⟨1⟩i⌢⟨0⟩ for each i < h(∣σ∣) −1 and ψ(σ⌢⟨h(∣σ∣) −1⟩) ∶= ψ(σ)⌢⟨1⟩h(∣σ∣)−1. We make the following observations: (i) for all σ, σ′ ∈ h∗, σ ⊆ σ′ if and only if ψ(σ) ⊆ ψ(σ′), and (ii) for all σ ∈ h∗, Jψ(σ)K2= ⋃i<h(∣σ∣)Jψ(σ⌢⟨i⟩)K2.
Now define Ψ∶hN→ {0,1}Nby Ψ(X) ∶= ⋃n∈Nψ(X↾n). Observation (i) above implies Ψ is well-defined and injective, while observation (ii) implies Ψ is surjective. Given τ ∈ {0,1}∗, Ψ−1[JτK2] = ⋃{JσKh∣ψ(σ) ⊇τ}, showing Ψ is continuous. Conversely, givenσ∈h∗, Ψ[JσKh] =Jψ(σ)K2– that Ψ[JσKh] ⊆Jψ(σ)K2is immediate, while the reverse inclusion follows from observation (ii) above – and hence Ψ is an open map. Since Ψ is clearly recursive, it is a recursive homeomorphism.
Definition I.4.21(recursively bounded). P⊆NNisrecursively bounded, orr.b., if there exists an recursive functionh∶N→ (1,∞)such that P⊆hN.
In particular, a recursively bounded Π01 class, or ar.b. Π01 class, is a recursively bounded and Π01 subset ofNN.
Proposition I.4.22. [24, Theorem4.7] Suppose P is a r.b. Π01 class and that Ψ∶P → NN is a recursive functional.
(a) The imageΨ[P]is recursively bounded andΠ01.
(b) Ψextends to a total recursive functional Ψ∶˜ NN→NN.
Corollary I.4.23. Suppose P is a r.b. Π01 class. Then there exists a Π01 subset Q of {0,1}N which is recursively homeomorphic toP.
Proof. Let Ψ be the recursive homeomorphism defined in the proof of Proposition I.4.20. Then Proposi- tion I.4.22(a) shows that the imageQofP under Ψ is another Π01class.
CHAPTER II
COMPLEXITY, AVOIDANCE, AND DEPTH
In Chapter I we gave brief definitions of COMPLEX(f) and DNR(p) for f an order function and p a nondecreasing computable function, as well as alluded to a variation on DNR which we termed LUA.
In Section II.1, we define Martin-L¨of randomness through three paradigms as a precursor to their general- izations which give rise tof-randomness and strongf-randomness for any computable functionf∶ {0,1}∗→R, with Martin-L¨of randomness corresponding to(λσ.∣σ∣)-randomness. The classes COMPLEX(f)are defined and shown to lie in Ew. Finally, we list some of the properties of prefix-free and conditional prefix-free complexity that we will use later.
In Section II.2, we show that the effect of the growth rate of p on degwDNR(p) depends explicitly on the choice of admissible enumeration used, and motivate the definition of the class Avoidψ(p) for any computablep∶N→ (1,∞)and partial recursiveψ. Linearly universal partial recursive functions are defined, followed by defining LUA(p)as the union of the classes Avoidψ(p)asψ ranges over those linearly universal partial recursive functions. Basic and technical results are covered for the linearly universal partial recursive functions, LUA(p), and Avoidψ(p)more generally.
In Section II.3, we formally define the notion of being fast-growing and slow-growing for an order function and address the problem “Given a recursive sequence of fast-growing (resp., slow-growing) order functions
⟨pk⟩k∈N, find fast-growing (resp., slow-growing) order functionsq+andq−such thatpk≤domq+andq−≤dompk
for allk∈N.” Towards that end, for the slow-growing case we prove that such aq−always exists and that a q+ exists with additional hypotheses on⟨pk⟩k∈N(Proposition II.3.2), but that aq+need not exist in general (Example II.3.3). On the other hand, for the fast-growing case we can prove thatq+ always exists and that q− exists with additional hypotheses on thepk’s and the sequence⟨pk⟩k∈N(Proposition II.3.5).
To better understand the extra hypothesis of requiring∑∞n=0p(n)−1 not only be finite but also recursive, we prove the following equivalence:
Proposition II.3.10. Suppose p∶N→ (0,∞) is a fast-growing order function and let p∶ [0,∞) → (0,∞)be any continuous nonincreasing extension ofp. Then∑∞n=0p(n)−1is a recursive real if and only if∫
∞
0 p(x)−1dx is a recursive real.
Section II.4 introduces the notion of depth for r.b. Π01 classes, of which our interest is based on strong general properties of deep r.b. Π01 classes and the fact that the classes LUAp are deep exactly when p is slow-growing. Depth is shown to be well-behaved with respect to≤s while slightly less well-behaved for≡w.
We end the section with a discussion of the applicability of ‘depth’ to subsets ofNN which are not r.b. Π01 classes.