The Degree Structures Induced by the Arithmetical Hierarchy of Countably Continuous Functions

Home , Pointclass

Inside the Muchnik Degrees II: The Degree Structures induced by the Arithmetical Hierarchy of Countably Continuous Functions

K. Higuchi

Department of Mathematics and Informatics, Chiba University, 1-33 Yayoi-cho, Inage, Chiba, Japan

T. Kihara∗ School of Information Science, Japan Advanced Institute of Science and Technology, Nomi 923-1292, Japan

Abstract It is known that infinitely many Medvedev degrees exist inside the Muchnik degree Π0 of any nontrivial 1 subset of Cantor space. We shed light on the fine structures inside these Muchnik degrees related to learnability and piecewise computability. As for Π0 ∆0 nonempty 1 subsets of Cantor space, we show the existence of a finite- 2-piecewise Π0 Π0 degree containing infinitely many finite-( 1)2-piecewise degrees, and a finite-( 2)2- ∆0 Π0 piecewise degree containing infinitely many finite- 2-piecewise degrees (where ( n)2 ff Π0 denotes the di erence of two n sets), whereas the greatest degrees in these three Γ Π0 “finite- -piecewise” degree structures coincide. Moreover, as for nonempty 1 subsets Π0 of Cantor space, we also show that every nonzero finite-( 1)2-piecewise degree includes infinitely many Medvedev (i.e., one-piecewise) degrees, every nonzero countable- ∆0 2-piecewise degree includes infinitely many finite-piecewise degrees, every nonzero Π0 ∆0 ∆0 finite-( 2)2-countable- 2-piecewise degree includes infinitely many countable- 2-piecewise Π0 degrees, and every nonzero Muchnik (i.e., countable- 2-piecewise) degree includes in- Π0 ∆0 finitely many finite-( 2)2-countable- 2-piecewise degrees. Indeed, we show that any ∆0 nonzero Medvedev degree and nonzero countable- 2-piecewise degree of a nonempty Π0 1 subset of Cantor space have the strong anticupping properties. Finally, we obtain an elementary difference between the Medvedev (Muchnik) degree structure and the finite-Γ-piecewise degree structure of all subsets of Baire space by showing that none of the finite-Γ-piecewise structures are Brouwerian, where Γ is any of the Wadge classes mentioned above. Π0 Keywords: 1 class, Medvedev degree, Borel measurable function, countable continuity 2010 MSC: 03D30, 03D78, 03E15, 26A21, 68Q32

∗Corresponding author Email addresses: [email protected] (K. Higuchi), [email protected] (T. Kihara)

Preprint submitted to Annals of Pure and Applied Logic January 18, 2014 1. Summary

1.1. Introduction This paper is a continuation of Higuchi-Kihara [29]. Our objective in this paper is to investigate the degree structures induced by intermediate notions between the Medvedev reduction (uniformly computable function) and Muchnik reduction (nonuniformly computable function). We will shed light on a hidden, but extremely deep, Π0 structure inside the Muchnik degree of each 1 subset of Cantor space. In 1963, Albert Muchnik [46] introduced the notion of Muchnik reduction as a partial function on Baire space that is decomposable into countably many computable functions. Such a reduction is also called a countably computable function, σ-computable function, or nonuniformly computable function. The notion of Muchnik reduction has Π0 been a powerful tool for clarifying the noncomputability structure of the 1 subsets of Cantor space [57–59, 61]. Muchnik reductions have been classified in Part I [29] by introducing the notion of piecewise computability. Remarkably, many descriptive set theorists have recently focused their attention on the concept of piecewise definability of functions on Polish spaces, in association with the Baire hierarchy of Borel measurable functions (see [43, 44, 55]). Roughly speaking, if Γ is a pointclass (in the Borel hierarchy) and Λ is a class of functions (in the Baire hierarchy), a function is said to be Γ-piecewise Λ if it is decomposable into countably many Λ-functions with Γ domains. If Γ is the class of all closed sets and Λ is the class of all continuous functions, it is simply called piecewise continuous (see for instance [32, 36, 45, 50]). The notion of piecewise continuity is known to be ∆0 Γ Λ equivalent to the 2-measurability [32]. If is the class of all sets and is the class of all continuous functions, it is also called countably continuous [44] or σ-continuous [54]. Nikolai Luzin was the first to investigate the notion of countable-continuity, and today, many researchers have studied this concept, in particular, with an important dichotomy theorem (see [51, 64]). ∆0 Our concepts introduced in Part I [29], such as 2-piecewise computability, are indeed the lightface versions of piecewise definability. This notion is also known to ff ∆0 be equivalent to the e ective 2-measurability [50]. See also [5, 19, 38] for more information on effective Borel measurability. To gain a deeper understanding of piecewise definability, we investigate the Medvedev- and Muchnik-like degree structures induced by piecewise computable notions. This also helps us to understand the notion of relative learnability since we have observed a close relationship between lightface piecewise definability and algorithmic learning in Part I [29]. In Part II, we restrict our attention to the local substructures consisting of the de- Π0 grees of all 1 subsets of Cantor space. This indicates that we consider the relative piecewise computably (or learnably) solvability of computably-refutable problems. When a scientist attempts to verify a statement P, his verification will be algorithmi- cally refuted whenever it is incorrect. This falsifiability principle holds only when P is Π0 Π0 represented as a 1 subset of a space. Therefore, the restriction to the 1 sets can be regarded as an analogy of Popperian learning [11] because of the falsifiability principle.

2 Π0 From this perspective, the universe of the 1 sets is expected to be a good playground of Learning Theory [31]. Π0 N The restriction to the 1 subsets of Cantor space 2 is also motivated by several Π0 other arguments. First, many mathematical problems can be represented as 1 subsets Π0 of certain topological spaces (see Cenzer and Remmel [15]). The 1 sets in such spaces have become important notions in many branches of Computability Theory, such as Recursive Mathematics [23], Reverse Mathematics [60], Computable Analysis [65], Effective Randomness [21, 48], and Effective Descriptive Set Theory [42]. For Π0 N these reasons, degree structures on 1 subsets of Cantor space 2 are widely studied from the viewpoint of Computability Theory and Reverse Mathematics. In particular, many theorems have been proposed on the algebraic structure of the Π0 Medvedev degrees of 1 subsets of Cantor space, such as density [13], embeddability of distributive lattices [3], join-reducibility [2], meet-irreducibility [1], noncuppability [12], non-Brouwerian property [28], decidability [16], and undecidability [56] (see also [30, 57–59, 61] for other properties on the Medvedev and Muchnik degree structures). Π0 The 1 sets have also been a key notion (under the name of closed choice) in the study of the structure of the Weihrauch degrees, which is an extension of the Medvedev degrees (see [6–8]). Among other results, Cenzer and Hinman [13] showed that the Medvedev degrees Π0 of the 1 subsets of Cantor space are dense, and Simpson [57] questioned whether the Π0 Muchnik degrees of 1 subsets of Cantor space are also dense. However, this question remains unanswered. We have limited knowledge of the Muchnik degree structure of Π0 ffi the 1 sets because the Muchnik reductions are very di cult to control. What we know is that as shown by Simpson-Slaman [62] and Cole-Simpson [17], there are infinitely Π0 many Medvedev degrees in the Muchnik degree of any nontrivial 1 subsets of Cantor space. Now, it is necessary to clarify the internal structure of the Muchnik degrees. In Part II, we apply the disjunction operations introduced in Part I [29] to understand the inner structures of the Muchnik degrees induced by various notions of piecewise computability.

1.2. Results In Part I [29], the notions of piecewise computability and the induced degree structures are introduced. Our objective in Part II is to study the interaction among the P/F F Π0 F structures of -degrees of nonempty 1 subsets of Cantor space for notions of piecewise computability listed as follows.

• <ω Π0 NN decp [ 1] also denotes the set of all partial functions on that are decompos- Π0 able into finitely many partial computable functions with 1 domains. • <ω Π0 NN decd [ 1] denotes the set of all partial functions on that are decomposable Π0 into finitely many partial computable functions with ( 1)2 domains, where a Π0 ff Π0 ( 1)2 set is the di erence of two 1 sets. • <ω ∆0 NN decp [ 2] denotes the set of all partial functions on that are decomposable ∆0 into finitely many partial computable functions with 2 domains.

3 • ω ∆0 NN decp [ 2] denotes the set of all partial functions on that are decomposable ∆0 into countably many partial computable functions with 2 domains. • <ω Π0 NN decd [ 2] denotes the set of all partial functions on that are decomposable Π0 into finitely many partial computable functions with ( 2)2 domains. • <ω Π0 ω ∆0 NN decd [ 2]decp [ 2] denotes the set of all partial functions on that are decom- ∆0 Π0 posable into finitely many partial 2-piecewise computable functions with ( 2)2 Π0 ff Π0 domains, where a ( 2)2 set is the di erence of two 2 sets. • ω Π0 NN decp [ 2] denotes the set of all partial functions on that are decomposable Π0 into countably many partial computable functions with 2 domains. The relationship among these notions is summarized as follows.

<ω P/dec [Π0] — — d 2 P/ <ω Π0 P/ <ω Π0 P/ <ω ∆0 P/ <ω Π0 ω ∆0 P/ ω Π0 decp [ 1]— decd [ 1]— decp [ 2] decd [ 2]decp [ 2]— decp [ 2] — P/ ω ∆0 decp [ 2] — In Part I [29], we observed that these degree structure are exactly those induced by the (α, β|γ)-computability. • 1 NN [CT ]1 denotes the set of all partial computable functions on . 1 N • [CT ]<ω denotes the set of all partial functions on N learnable with bounded mind changes.

1 N • [CT ]ω|<ω denotes the set of all partial functions on N learnable with bounded errors.

1 N • [CT ]ω denotes the set of all partial learnable functions on N . • <ω NN [CT ]1 denotes the set of all partial k-wise computable functions on for some k ∈ N.

<ω N • [CT ]ω denotes the set of all partial functions on N learnable by a team. • ω NN [CT ]1 denotes the set of all partial nonuniformly computable functions on (i.e., all functions f satisfying f (x) ≤T x for any x ∈ dom( f )). α α As in Part I [29], each degree structure P/[CT ]β|γ is abbreviated as Pβ|γ. Then, we have the following relationship among these notions.

<ω P — — 1 P1 P1 P1 P<ω Pω 1 — <ω — ω|<ω ω — 1 — 1 Pω —

We will see that all of the above inclusions are proper. Beyond the properness of these inclusions, there are four LEVELs signifying the diﬀerences between two classes NN 1 ω F and G of partial functions on (lying between [CT ]1 and [CT ]1 ) listed as follows.

4 1. There is a function Γ ∈ F \ G. N 2. There are sets X, Y ⊆ N such that F has a function ΓF : X → Y, but G has no function ΓG : X → Y. Π0 , ⊆ N Γ → Γ 3. There are 1 sets X Y 2 such that F has a function F : X Y, but G has no function ΓG : X → Y. Π0 ⊆ N Π0 ⊆ N 4. For every special 1 set Y 2 , there is a 1 set X 2 such that F has a function ΓF : X → Y, but G has no function ΓG : X → Y. The LEVEL 1 separation just represents F ⊈ G. Clearly, 4 → 3 → 2 → 1. Note Σ0 , ⊆ NN Π0 that the LEVEL 2 separation holds for no 1 sets X Y , since 1 is the first level in the arithmetical hierarchy which can define a nonempty set S ⊆ NN without Π0 computable element. Such a 1 set is called special, i.e., a subset of Baire space is special if it is nonempty and contains no computable points. As mentioned before, Simpson-Slaman [62] (see Cole-Simpson [17]) showed that the LEVEL 4 separation 1 ω ∈ Pω holds between [CT ]1 and [CT ]1 , that is, every nonzero Muchnik degree a 1 contains ∈ P1 infinitely many Medvedev degrees b 1. Π0 In section 2, we use the consistent two-tape disjunction operations on 1 subsets of Cantor space introduced in Part I [29] to obtain LEVEL 3 separation results. • ▽ Π0 n is the disjunction operation on 1 sets induced by the two-tape Brouwer- Heyting-Kolmogorov-interpretation with mind-changes < n. • ▽ Π0 ω is the disjunction operation on 1 sets induced by the two-tape Brouwer- Heyting-Kolmogorov-interpretation with finitely many mind-changes. • ▽ Π0 ∞ is the disjunction operation on 1 sets induced by the two-tape Brouwer- Heyting-Kolmogorov-interpretation permitting unbounded mind-changes.

1 By using these operations, we obtain the LEVEL 3 separation results for [CT ]1, 1 1 <ω Π0 , ⊆ N [CT ]<ω,[CT ]ω|<ω, and [CT ]1 . We show that there exist 1 sets P Q 2 such that all of the following conditions are satisﬁed. Γ1 ▽ → ▽ 1. (a) There is no computable function 1 : P 2Q P 1Q; 1 (b) There is a function Γ<ω : P▽2Q → P▽1Q learnable with bounded mind- changes. 1 2. (a) There is no function Γ<ω : P▽ωQ → P▽1Q learnable with bounded mind- changes; 1 (b) There is a function Γω|<ω : P▽ωQ → P▽1Q learnable with bounded errors. 1 3. (a) There is no function Γω|<ω : P▽∞Q → P▽1Q learnable with bounded errors; Γ<ω ▽ → ▽ (b) There is a 2-wise computable function 1 : P ∞Q P 1Q. The above conditions also suggest how does degrees of diﬃculty of our disjunction operations behave. In contrast to the above results, in section 3, we will see that the hierarchy be- 1 <ω Π0 N tween [CT ]<ω and [CT ]1 collapses for homogeneous 1 subsets of Cantor space 2 . 1 1 <ω In other words, the LEVEL 4 separations fail for [CT ]<ω,[CT ]ω|<ω, and [CT ]1 . For other classes, is the LEVEL 4 separation successful?

5 To archive the LEVEL 4 separations, we use dynamic disjunction operations de- veloped in Part I [29]. 7→ ⌢ Π0 ⊆ N 1. The concatenation P P P of two 1 sets P 2 indicates the mass problem “solve P by a learning proof process with mind-change-bound 2”. 2. Every iterated concatenation along a well-founded tree indicates a learning proof process with an ordinal bounded mind changes. 7→ ▼ Π0 ⊆ N 3. The hyperconcatenation P P P of two 1 sets P 2 is deﬁned as the iterated concatenation of P along the corresponding ill-founded tree of P. These operations turn out to be extremely useful to establish the LEVEL 4 separation results. Some of these results will be proved by applying priority argument inside some learning proof model of P. 1 1 7→ ⌢ 1. The LEVEL 4 separation succeeds for [CT ]1 and [CT ]<ω, via the map P P P. <ω 1 2. The LEVEL 4 separation succeeds for [CT ]1 and [CT ]ω, via the map ∪ P 7→ (P⌢P⌢ ... (m times) ... ⌢P⌢P). m∈N

1 <ω 3. The LEVEL 4 separation succeeds for [CT ]ω and [CT ]ω , via the map P 7→ P▼P. <ω ω 7→ 4. The LEVEL 4 separation succeeds for [CT ]ω and [CT ]1 , via the map P Deg(d P), where Deg(d P) denotes the Turing upward closure of P. The method that we use to show the ﬁrst and the third items also implies that any ∈ P1 ∈ P1 nonzero a 1 and a ω have the strong anticupping property, i.e., for every nonzero a ∈ P, there is a nonzero b ∈ P below a such that a ≤ b ∨ c implies a ≤ c. Indeed, these strong anticupping results are established via concatenation ⌢ and hyperconcatenation ▼. P1 |= ∀ , ≤ ⌢ ∨ → ≤ 1. 1 ( a c)(a (a a) c a c). 1 2. Pω |= (∀a, c)(a ≤ (a▼a) ∨ c → a ≤ c).

In section 5, we apply our results to sharpen Jockusch’s theorem [33] and Simp- son’s Embedding Lemma [58]. Jockusch showed the following nonuniform computability result for DNRk, the set of all k-valued diagonally noncomputable functions. Γ1 → 1. There is no (uniformly) computable function 1 : DNR3 DNR2. Γω → 2. There is a nonuniformly computable function 1 : DNR3 DNR2. This result will be sharpened by using our learnability notions as follows.

1 1. There is no learnable function Γω : DNR3 → DNR2. Γ<ω → ∈ N 2. There is no k-wise computable function 1 : DNR3 DNR2 for k . Γ1 → ▼ 3. There is a (uniformly) computable function 1 : DNR3 DNR2 DNR2. Hence, <ω there is a function Γω : DNR3 → DNR2 learnable by a team of two learners. Finally, we employ concatenation and hyperconcatenation operations to show that D1 D<ω D<ω neither <ω nor 1 nor ω are Brouwerian. Hence, these degree structures are not elementarily equivalent to the Medvedev (Muchnik) degree structure.

6 1.3. Notations and Conventions For any sets X and Y, for convenience, we say that f is a function from X to Y (written f : X → Y) if the domain dom( f ) of f includes X, and the image of X under f is included in Y. We also use the notation f :⊆ X → Y to denote that f is a partial function from X to Y, i.e., the image of dom( f ) ∩ X under f is included in Y. For basic terminology in Computability Theory, see Soare [63]. For σ ∈ N

1 In some contexts, a function Φ is called partial computable if it can be extended to some Φe. In this paper, we identify each partial computable function with such a Φe.

7 1.4. Notations from Part I 1.4.1. Functions Every partial function Ψ :⊆ N

mclΨ(σ) = {n < |σ| : Ψ(σ ↾ n + 1) , Ψ(σ ↾ n)}. ∪ = ↾ ∈ NN We also deﬁne mclΨ( f ) n∈N mclΨ( f n) for any f . Then, #mclΨ( f ) denotes the number of times that the learner Ψ changes her/his mind on the informant f . Moreover, the set of indices predicted by a learner Ψ on the informant σ is deﬁned by

indxΨ(σ) = {Ψ(σ ↾ n): n ≤ |σ|}. ∪ = ↾ ∈ NN We also define indxΨ( f ) n∈N indxΨ( f n) for any f . We say that a partial N N N function Γ :⊆ N → N is identified by a learner Ψ on g ∈ N if limn Ψe(g ↾ n) Φ = Γ Γ converges, and limn Ψe(g↾n)(g) (g). We also say that a partial function is identified by a learner Ψ if it is identified by Ψ on every g ∈ dom(Γ). In Part I [29, Definition 2], we introduced the seven notions of (α, β|γ)-computability for a partial function Γ :⊆ NN → NN listed as follows:

1. Γ is (1, 1)-computable if it is computable. 2. Γ is (1, < ω)-computable if it is identified by a learner Ψ with sup{#mclΨ(g): g ∈ dom(Γ)} < ω. 3. Γ is (1, ω| < ω)-computable if it is identified by a learner Ψ with sup{#indxΨ(g): g ∈ dom(Γ)} < ω. 4. Γ is (1, ω)-computable if it is identified by a learner. 5. Γ is (< ω, 1)-computable if there is b ∈ N such that for every g ∈ dom(Γ), Γ(g) = Φe(g) for some e < b. 6. Γ is (< ω, ω)-computable if there is b ∈ N such that for every g ∈ dom(Γ), Γ is identified by Ψe for some e < b on g. 7. Γ is (ω, 1)-computable if it is nonuniformly computable, i.e., Γ(g) ≤T g for every g ∈ dom(Γ).

α α Let [CT ]β (resp. [CT ]β|γ) denote the set of all (α, β)-computable (resp. (α, β|γ)- computable) functions. If F be a monoid consisting of partial functions under com- N position, P(N ) is preordered by the relation P ≤F Q indicating the existence of a function Γ ∈ F from Q into P, that is, P ≤F Q if and only if there is a partial function Γ :⊆ NN → NN such that Γ ∈ F and Γ(g) ∈ P for every g ∈ Q. Let D/F and P/F P NN / ≡ Π0 N / ≡ Π0 N denote the quotient sets ( ) F and 1(2 ) F , respectively. Here, 1(2 ) denotes the set of all nonempty Π0 subsets of 2N. For P ∈ P(NN), the equivalence N 1 α class {Q ⊆ N : Q ≡F P} ∈ D/F is called the F -degree of P. If F = [CT ]β|γ for α α α some α, β, γ ∈ {1, < ω, ω}, we write ≤β|γ, Dβ|γ, and Pβ|γ instead of ≤F , D/F and P/F . ≤1 ≤ω The preorderings 1 and 1 are equivalent to the Medvedev reducibility [41] and the Muchnik reducibility [46], respectively.

8 In Part I [29, Theorem 26 and Proposition 27], we showed the following equiva- lences:

P1 = P/ <ω Π0 P1 = P/ <ω ∆0 P1 = P/ ω ∆0 <ω decd [ 1] ω|<ω decp [ 2] ω decp [ 2] P<ω = P/ <ω Π0 P<ω = P/ <ω Π0 ω ∆0 Pω = P/ ω Π0 1 decd [ 2] ω decd [ 2]decp [ 2] 1 decp [ 2]

Here, for a pointclass Λ, a function Γ :⊆ NN → NN is finite (countable, resp.) Λ-piecewise computable if there is a finite Λ-cover {Xi}i<ω (a uniform Γ-cover {Xi}i∈ω, resp.) of dom( f ) such that Γ ↾ Xi is computable for any i ∈ N, and the set of all Λ <ω Λ finite (countable, resp.) -piecewise computable functions is denoted by decp [ ] ω Λ <ω Π0 Π0 (decp [ ]). We denote by decd [ n] the set of all finite n-layerwise computable func- <ω Π0 Π0 tion (see Part I [29, Section 2.5]), which is equivalent to decp [( n)2], where ( n)2 is ff Π0 the complexity of the di erences of two n sets. α This observation allows us to think of each degree structure Pβ|γ as a piecewise- degree structure in the following sense. P1 Π0 1. 1 is the Medvedev degrees of 1 sets. P1 Π0 Π0 2. <ω is the finite-( 1)2-piecewise degrees of 1 sets. P1 ∆0 Π0 3. ω|<ω is the finite- 2-piecewise degrees of 1 sets. P1 ∆0 Π0 4. ω is the countable- 2-piecewise degrees of 1 sets. P<ω Π0 Π0 5. 1 is the finite-( 2)2-piecewise degrees of 1 sets. P<ω Π0 ∆0 Π0 6. ω is the finite-( 2)2-countable- 2-piecewise degrees of 1 sets. Pω Π0 7. 1 is the Muchnik degrees (or equivalently, the countable- 2-piecewise degrees) Π0 of 1 sets.

1.4.2. Sets To deﬁne the disjunction operations in Part I [29, Deﬁnition 29], we introduced some auxiliary notions. Let I ⊆ N be a set. Fix σ ∈ (I × N)

Moreover, the number of times of mind-changes of (the process reconstructed from a record) σ ∈ (I × N)

mc(σ) = #{n < |σ| − 1 : (σ(n))0 , (σ(n + 1))0}.

Here, for x = (x0, x1) ∈ I × N, the first (second, resp.) coordinate x0 (x1, resp.) is ∈ × N N = denoted∪ by (x)0 ((x)1, resp.). Furthermore, for f (I ) , we define pri( f ) ↾ ∈ = ↾ n∈N pri( f n) for each i I, and mc( f ) limn mc( f n), where if the limit does not exist, we write mc( f ) = ∞. In Part I [29, Definition 33, 36 and 55], we introduced the disjunction operations. N Fix a collection {Pi}i∈I of subsets of Baire space N . ∨ ⟦ ⟧ = { ∈ × N N ∃ ∈ ∈ = } 1. i∈I Pi Int f (I ) : (( i I) pri( f ) Pi)& mc( f ) 0 .

9 ∨ ⟦ ⟧ = { ∈ × N N ∃ ∈ ∈ < } 2. ∨i∈I Pi LCM[n] f (I ) : (( i I) pri( f ) Pi)& mc( f ) n . ⟦ ⟧ = { ∈ × N N ∃ ∈ ∈ } 3. i∈I Pi CL f (I ) :( i I) pri( f ) Pi . As in Part I, we use the notation write(i, σ) for any i ∈ N and σ ∈ N

write(i, σ) = i|σ| ⊕ σ = ⟨(i, σ(0)), (i, σ(1)), (i, σ(2)),..., (i, σ(|σ| − 1))⟩. σ / This string indicates the instruction to write the string∪ on the i-th tape in the one two- , = , ↾ = N ⊕ tape model. We also use the notation write(i f ) n∈N write(i f n) i f for any f ∈ NN. Π0 N In Part II, we are mostly interested in the degree structures of 1 subsets of 2 . As mentioned in Part I [29], the consistent disjunction operations are useful to study such local degree structures. The consistency set Con(Ti)i∈I for a collection {Ti}i∈I of trees is deﬁned as follows. = { ∈ × N N ∀ ∈ ∀ ∈ N ↾ ∈ }. Con(Ti)i∈I f (I ) :( i I)( n ) pri( f n) Ti

{ } Π0 Then we use the following modiﬁed deﬁnitions. Fix a collection Pi i∈I of 1 subsets of Baire space NN and n ∈ ω ∪ {ω}. [` ] ∨ 1. Pi = ⟦ ∈ Pi⟧LCM[n] ∩ Con(TP )i∈I. [`n i]∈I ∨i I i = ⟦ ⟧ ∩ ∈ 2. ∞ i∈I Pi i∈I Pi CL Con(TPi )i I. ∈ ∈ { , } Here TPi is the corresponding tree for Pi for every i I. If i 0 1 , then we simply write P0▽nP1, P0▽ωP1, and P0▽∞P1 for these notions. In Part II, we use the following notion.

Deﬁnition 1. Pick any ∗ ∈ N∪{ω}∪{∞}. For each disjunctive notions ▽∗ and collection { } NN ⊆ N

ρ∈LP ⌢ ⌢ We also write TP TQ for the corresponding tree of P Q. Moreover, the commutative ⌢ ⌢ concatenation P▽Q is defined as (P Q)⊕(Q P). Let P and {Qn}n∈N be computable col- Π0 N ρ lection of 1 subsets of 2 , and let n denote the length-lexicographically n-th leaf of the corresponding computable tree of P. Then, we define the infinitary concatenation and recursive meet [3] as follows: ∪ ⊕ ⌢{ } = ∪ ρ ⌢ , −→ = ⌢{ } . P Qi i∈N P n Qn i∈NQi CPA Qi i∈N n

10 Here, CPA is a Medvedev complete set, which consists of all complete consistent extensions of Peano Arithmetic. The Medvedev completeness of CPA ensures that for Π0 ⊆ N Φ → any nonempty 1 subset P 2 , a computable function : CPA P exists. In Part I, we studied the disjunction and concatenation operations along graphs. Π0 N ▼ For nonempty 1 subsets P and Q of 2 , the hyperconcatenation Q P of Q and P is deﬁned by the iterated concatenation of P’s along the ill-founded tree TQ, that is,        ∪ ⊓   ▼ =   ⌢⟨τ ⟩ ⌢  . Q P   TP (i)  TP τ∈TQ i<|τ|

Note that, after writing this paper, Kihara [37] gave effective topological interpre- tations of some of these constructions. Π0 Remark. Recall from Section 1.3 that a corresponding tree of a 1 set is assumed to Π0 be uniquely determined when an index of the 1 set is given. Indeed, most of our above definitions obviously depend on our choice of indices (hence, corresponding trees) of Π0 given 1 sets, that is, most of operations introduced above are defined on subtrees of N

Contents

1 Summary 2 1.1 Introduction ...... 2 1.2 Results ...... 3 1.3 Notations and Conventions ...... 7 1.4 Notations from Part I ...... 8 1.4.1 Functions ...... 8 1.4.2 Sets ...... 9

2 Degrees of Diﬃculty of Disjunctions 12 2.1 The Disjunction ⊕ versus the Disjunction ∪ ...... 13 2.2 The Disjunction ∪ versus the Disjunction ▽ ...... 13 2.3 The Disjunction ▽ versus the Disjunction ▽ω ...... 15 2.4 The Disjunction ⊕▽ω versus the Disjunction ▽`∞ ...... 16 2.5 The Disjunction versus the Disjunction ∞ ...... 17

11 3 Contiguous Degrees and Dynamic Infinitary Disjunctions 20 3.1 When the Hierarchy Collapses ...... 20 3.2 Concatenation, Dynamic Disjunctions, and Contiguous Degrees . . . 24 3.3 Infinitary Disjunctions along the Straight Line ...... 28 3.4 Infinitary Disjunctions along ill-Founded Trees ...... 31 3.5 Infinitary Disjunctions along Infinite Complete Graphs ...... 34

4 Applications and Questions 37 4.1 Diagonally Noncomputable Functions ...... 37 4.2 Simpson’s Embedding Lemma ...... 39 4.3 Weihrauch Degrees ...... 40 4.4 Some Intermediate Lattices are Not Brouwerian ...... 42 4.5 Open Questions ...... 45

Bibliography 46

2. Degrees of Diﬃculty of Disjunctions

The main objective in this section is to establish LEVEL 3 separation results among our classes of nonuniformly computable functions by using disjunction operations introduced in Part I [29, Sections 3 and 5]. We have already seen the following inequali- Π0 , ⊆ N ties for 1 subsets P Q 2 in Part I [29, Section 5.1]. ⊕ ≥1 ∪ ≥1 ▽ ≥1 ▽ ≥1 ▽ . P Q 1 P Q 1 P Q 1 P ωQ 1 P ∞Q As observed in Part I [29, Section 4], these binary disjunctions are closely related ≤1 ≤<ω ≤1 ≤1 ≤<ω to the reducibilities 1, tt,1, <ω, ω|<ω, and 1 , respectively. We employ rather Π0 exotic 1 sets constructed by Jockusch and Soare to separate the strength of these disjunctions. We say that a set A ⊆ NN is an antichain if it is an antichain with respect to the Turing reducibility ≤T . In other words, f is Turing incomparable with g, for any two distinct elements f, g ∈ A. A nonempty closed set A ⊆ NN is perfect if it has no isolated point. Π0 N Theorem 2 (Jockusch-Soare [35]). There exists a perfect 1 antichain in 2 . A stronger condition is sometimes required. For a set P ⊆ NN and an element N ≤ g ∈ N , let P T g denote the set of all element of P which are Turing reducible to g. N ≤ g N Then, a set A ⊆ N is⊕ antichain if and only if A T = {g} for every g ∈ A. A set P ⊆ N ≤ is independent if P T D = D for every ﬁnite subset D ⊂ P. Π0 Theorem 3 (see Binns-Simpson [3]). There exists a perfect independent 1 subset of 2N.

On the other hand, in Section 3.1, we will see that our hierarchy of disjunctions collapses for homogeneous sets, which may be regarded as an opposite notion to antichains and independent sets.

12 2.1. The Disjunction ⊕ versus the Disjunction ∪ We first separate the strength of the coproduct (the intuitionistic disjunction) ⊕ and the union (the classical one-tape disjunction) ∪. This automatically establish the 1 <ω ⊆ NN LEVEL 3 separation result between [CT ]1 and [Ctt]1 . Recall that a set P is special if it is nonempty and it contains no computable points. , Π0 N Π0 N Lemma 4. Let P0 P1 be 1 subsets of 2 , and let Q be a special 1 subset of 2 . ≤T f ⊕g ≤T f ≤T g ≤T f Assume that there exist f ∈ P0 and g ∈ P1 with Q = Q ∪ Q such that Q ≤ N N T g ≰1 ⊗ ∪ ⊗ and Q are separated by open sets. Then Q 1 (P0 2 ) (2 P1). ≤1 ⊗ N ∪ N ⊗ Φ Proof. Suppose that Q 1 (P0 2 ) (2 P1) via a computable functional . Then N N f ⊕ g ∈ (P0 ⊗ 2 ) ∪ (2 ⊗ P1). By our choice of f and g, Φ( f ⊕ g) must belong ≤ ⊕ ≤ ≤ ≤ ≤ to Q T f g = Q T f ∪ Q T g. By our assumption, Q T f and Q T g are separated by two N ≤ ≤ disjoint open sets U, V ⊆ 2 . That is, Q T f ⊆ U, Q T g ⊆ V, and U ∩ V = ∅. Therefore, either Φ( f ⊕ g) ∈ Q ∩ U or Φ( f ⊕ g) ∈ Q ∩ V holds. In any case, there exists an open neighborhood [σ] ∋ Φ( f ⊕ g) such that [σ] ⊆ U or [σ] ⊆ V. Without loss of generality, we can assume [σ] ⊆ U. We pick initial segments τ0 ⊂ f and τ1 ⊂ g with Φ(τ0 ⊕ τ1) ⊇ ⌢ N N N σ. Then (τ0 0 ) ⊕ g ∈ (P0 ⊗ 2 ) ∪ (2 ⊗ P1), and it is Turing equivalent to g. However ⌢ N ≤T g ≤T g this is impossible because Φ(τ0 0 ⊕ g) ∈ [σ], and [σ] ∩ Q ⊆ U ∩ Q = ∅. □ Π0 , ⊆ N ∪ <1 ⊕ Corollary 5. 1. There are 1 sets P Q 2 such that P Q 1 P Q. Π0 , ⊆ N ≡<ω <1 2. There are 1 sets P Q 2 such that P tt,1 Q and P 1 Q. Proof. (1) Let R be a perfect independent Π0 subset of 2N. Set P = 2N ⊗ R and N 1 ≤ = ⊗ ⊕ ≡1 , ∈ , T f = { } Q R 2 . Note that P Q 1 R. Pick f g R such that f g. Then R f , ≤ ≤ ⊕ ≤ ≤ N R T g = {g}, and R T f g = R T f ⊔ R T g = { f, g}. Since 2 is Hausdorff, two points ⊕ ≡1 ≰1 ∪ f and g are separated by open sets. Thus, P Q 1 R 1 P Q by Lemma 4. (2) ⊕ ≡<ω ∪ <1 ⊕ □ P Q tt,1 P Q 1 P Q. Remark. One can adopt the unit interval [0, 1] as our whole space instead of Cantor N space 2 . Then, P0 † P1 := (P0 × [0, 1]) ∪ ([0, 1] × P1) is connected as a topological space. If P0 ⊆ [0, 1] is homeomorphic to Cantor space, then the connected space P0 † P0 is sometimes called the Cantor tartan. The above proof shows that every Π0 ⊆ , , † perfect independent 1 set R [0 1] is not (1 1)-reducible to the obtained tartan R R, while these sets are (< ω, 1)-tt-equivalent. Note that the tartan plays an important role on the constructive study of Brouwer’s fixed point theorem (see [10]).

2.2. The Disjunction ∪ versus the Disjunction ▽ We next separate the strength of the union ∪ and the concatenation (the LCM disjunction with mind-change-bound 2) ▽. Moreover, we also see the LEVEL 3 separation <ω 1 between [Ctt]1 and [CT ]<ω. , Π0 N Π0 N Lemma 6. Let P0 P1 be 1 subsets of 2 , and let Q be a special 1 subset of 2 . ≤T f ≤T g Assume that there exist f ∈ P0 and g ∈ P1 such that any h ∈ Q and Q are ≰1 ⌢ separated by open sets. Then Q 1 P0 P1.

13 ≤1 ⌢ Φ Proof. Suppose that Q 1 P0 P1 via a computable functional . By our choice of ⌢ N f ∈ P0 ⊆ P0 P1, there must exist an open set U ⊆ 2 such that Φ( f ) ∈ Q ∩ U and ≤ Q T g ∩ U = ∅. Since U is open there exists a clopen neighborhood [σ] ∋ Φ( f ) such that [σ]∩ Q ⊆ U. We pick an initial segment τ ⊂ f with Φ(τ) ⊇ σ. Since f ∈ P0 holds, τ ∈ ρ ∈ τ ρ⌢ ∈ ⌢ ρ⌢ we have that TP0 , and we pick LP0 extending . Then g P0 P1, and g ⌢ ⌢ ≤ ≤1 Φ Φ ρ T g is Turing equivalent to g. So, if Q 1 P0 P1 via , then ( g) must belong to Q . ⌢ ≤ ≤ However this is impossible because Φ(ρ g) ∈ [σ], and [σ]∩ Q T g ⊆ U ∩ Q T g = ∅. □ Π0 , ⊆ N ⌢ <1 ∪ <1 ⊕ Corollary 7. There are 1 sets P Q 2 such that P Q 1 P Q 1 P Q. Proof. Assume that R be a perfect Π0 antichain of 2N. Set P = 2N ⊗ R and Q = N 1 ≤ ≤ R ⊗ 2 . Pick f, g ∈ R such that f , g. Then R T f = { f } and R T g = {g} since R is ≤ N N antichain. Therefore, (P ∪ Q) T X ⊆ ({X} ⊗ 2 ) ∪ (2 ⊗ {X}) for each X ∈ { f, g}. For ≤T f N N h = h0 ⊕ h1 ∈ (P ∪ Q) , we have h0 , g and h1 , g. Thus, h < (2 ⊗ {g}) ∪ ({g} ⊗ 2 ), and note that (2N ⊗ {g}) ∪ ({g} ⊗ 2N) is closed. Then, there is an open neighborhood N ≤ U ⊆ 2 such that h ∈ U and U ∩ (P ∪ Q) T g = ∅, since P ∪ Q is regular, and ≤ N N ≤ ≤ (P ∪ Q) T g ⊆ (2 ⊗ {g}) ∪ ({g} ⊗ 2 ). Namely, any h ∈ (P ∪ Q) T f and (P ∪ Q) T g are ∪ ≰1 ⌢ □ separated by some open set. Consequently, by Lemma 6, we have P Q 1 P Q. One can establish another separation result for the concatenation. Recall from [12] ⊆ NN ext that a closed set P is immune if TP contains no inﬁnite c.e. subset. In [12] it Π0 is shown that the class of non-immune 1 subsets of Cantor space is downward closed P1 in the Medvedev degrees 1. This property also holds in a coarser degree structure. In P<ω P1 Part I [29, Section 2.4] we have seen that tt,1 is an intermediate structure between 1 1 and P<ω. Π0 N ≤<ω Lemma 8. Let P and Q be 1 subsets of 2 . If P is not immune, and Q tt,1 P, then Q is not immune. ext ≤<ω Proof. Let V be an inﬁnite c.e. subset of TP . Assume that Q tt,1 P holds via n truth- {Γ } Γ table functionals i i

14 Π0 , ⊆ N ⊓ <1 ⌢ Proposition 10. There are 1 sets P Q 2 such that P Q 1 P Q. ⊓ ≤1 → Π0 , ⊆ NN ⊆ N Proof. It is easy to see that P Q 1 P Q for any 1 sets P Q . Let R 2 0 be a Π antichain. Then we divide R into four parts, P0, P1, P2, and P3. Put P = P3, 1 ⌢ ⌢ ⌢ ⌢ and Q = (⟨0, 1⟩ P2 P0) ∪ (⟨1⟩ P2 P1). Without loss of generality, we may assume ⟨ ⟩ ∈ → ≤1 ⊓ Φ that 0 TP. Suppose that P Q 1 P Q via a computable function . Choose ⌢ ⊓ ⌢ → g ∈ P2. Then we have ⟨0, 1⟩ g ∈ P Q. Therefore, Φ(⟨0, 1⟩ g) ∈ P Q must contain ♯, since P = P3 has no element computable in g ∈ P2. Thus, there is n ∈ N such that Φ(⟨0, 1⟩⌢(g ↾ n)) contains ⟨♯, i⟩ as a substring for some i < 2. Fix such i. Then, ⌢ → Φ(⟨0, 1⟩ (g ↾ n)) ∈ P (Q ∩ [⟨i⟩]). We extend g ↾ n to some leaf ρ of P2. Choose ⌢ ⌢ ⊓ ⌢ ⌢ ⌢ hk ∈ Pk for each k < 2. Then, ⟨0, 1⟩ ρ h0 ∈ Q ⊆ P Q, and ⟨0, 1⟩ ρ h1 ∈ ⟨0⟩ Q ⊆ ⊓ ⌢ ⌢ → P Q. Thus, Φ(⟨0, 1⟩ ρ hk) must belongs to P (Q ∩ [⟨i⟩]), for each k < 2. However → ⌢ ⌢ P (Q ∩ [⟨i⟩]) has no element computable in ⟨0, 1⟩ ρ h1−i. A contradiction. □ ⊓ ≡1 ⌢ Π0 , ⊆ NN Proposition 11. P Q ω P Q holds for every 1 sets P Q . → 1 ⊓ ⊓ Proof. It suﬃces to show that P Q ≤ω P Q. Given f ∈ P Q, our learner Ψ ﬁrst → guesses that f is also a correct solution to P Q. If f ↾ n < TP happens, we know that ( f ↾ m)⌢♯⌢ f ↼m ∈ P→Q for some m ≤ n, where note that f = ( f ↾ m)⌢ f ↼m holds for each m ∈ N. Thus, the learner Ψ can guess a correct number m ≤ n such that ( f ↾ m)⌢♯⌢ f ↼m ∈ P→Q with at most n mind-changes. □

2.3. The Disjunction ▽ versus the Disjunction ▽ω Let Ψ be a learner (i.e., a total computable function Ψ : N

We say that P0 and P1 are everywhere (ω, 1)-incomparable if P0 ∩ [σ0] is Muchnik ∩ σ ∩ σ ≰ω ∩ σ < incomparable with P1 [ 1] (that is, Pi [ i] 1 P1−i [ 1−i] for each i 2) whenever [σi] ∩ Pi , ∅ for each i < 2.

2The set of m-changing points is closedly related to the m-th derived set obtained from the notion of discontinuity levels ([19, 26, 27, 40]). See also Part I [29, Section 5.3] for more information on the relationship between the notion of mind-changes and the level of discontinuity. 3 In the sense of the identiﬁcation in the limit [24], the learner Ψ is said to be Popperian if ΦΨ(σ)(∅) is total for every σ ∈ N

15 , ω, Π0 N ρ Theorem 12. Let P0 P1 be everywhere ( 1)-incomparable 1 subsets of 2 , and be any binary string. For any (1, ω)-computable function Γ identiﬁed by a learner Ψ, −1 N ⌢ ♡ the closure of mcΨ(≥ m) ∪ Γ (N \ P0 ⊕ P1) ∪ APΨ includes ρ (P0▽nP1) with respect ⌢ ♡ N to the relative topology on ρ (P0▽m+nP1) (as a subspace of Baire space N ). ρ⌢τ ρ⌢ ▽ τ Proof. Fix a string 0 which is extendible in the heart of (P0 nP1). Then, pri( 0) ∈ ∩ τ ≰ω must be extendible in Pi. Fix fi Pi [pri( 0)] witnessing P1−i 1 Pi for each i < 2, i.e., P1−i contains no fi-computable element. Such fi exists, by everywhere ω, = τ ⌢ ∗ < ( 1)-incomparability. Assume that fi pri( 0) fi for each i 2 and that the last τ τ = τ−⌢ , < declaration along 0 is j0, i.e., 0 0 ( j0 k) for some k 2. Then we can proceed the following actions.

• Extend τ to g = τ ⌢write( j , f ∗ ) ∈ ρ⌢(P ▽ P ). 0 0 0 0 j0 0 n 1 • > |τ | Φ ↾ = Wait for the least s0 0 such that Ψ(g0↾s0)(g0 s0; 0) j0. ⌢ ∗ ⌢ • Extend g ↾ s to g = (g ↾ s ) write( j , f ) ∈ ρ (P ▽ + P ), where j = 0 0 1 0 0 1 j1 0 n 1 1 1 1 − j0. • > Φ ↾ = − Wait for the least s1 s0 such that Ψ(g1↾s1)(g1 s1; 0) 1 j0.

If both s0 and s1 are defined, then this action forces the learner Ψ to change his mind. In other words, g1 ∈ mcΨ(≥ 1). Assume that sl is undefined for some l < 2 Note ≡ = < that gl T f jl , since pr jl (gl) f jl and pr1− jl (gl) is finite, for each l 2. Therefore, Γ < − ⌢ ∈ (gl) (1 jl) P1− jl since P1− jl has no gl-computable element. In this case, gl −1 N ⌢ ♡ −1 N Γ (N \ P0 ⊕ P1). Hence, in ρ (P0▽n+1P1) , the closure of mcΨ(≥ 1) ∪ Γ (N \ ⌢ ♡ ⌢ ♡ P0 ⊕ P1) ∪ APΨ includes ρ (P0▽nP1) . By iterating this procedure, in ρ (P0▽m+nP1) , −1 N we can easily see that the closure of mcΨ(≥ m) ∪ Γ (N \ P0 ⊕ P1) ∪ APΨ includes ⌢ ♡ ρ (P0▽nP1) . □ Corollary 13. Π0 , ⊆ N ▽ <1 ▽ 1. There exists 1 sets P Q 2 such that P ωQ <ω P Q. Π0 , ⊆ N ≡1 <1 2. There exists 1 sets P Q 2 such that P ω|<ω Q and P <ω Q. Π0 N Proof. (1) Let P be a perfect 1 antichain in 2 of Theorem 2. Fix a clopen set C such that P0 = P ∩ C , ∅, and P1 = P \ C , ∅. Then every f ∈ P0 and g ∈ P1 are Turing incomparable. Therefore, P0 and P1 are everywhere (ω, 1)-incomparable. ρ Π0 ⊆ Let n denote the n-th leaf of the tree TCPA of a Medvedev complete 1 set CPA 2N. Fix a (1, m)-computable function Γ identified by a learner Ψ. By Theorem 12, ⌢ −1 ω ρm+1 (P0⊕▽m+1P1) intersects with mcΨ(≥ m + 1) ∪ Γ (ω \ P0 ⊕ P⊕1). Thus, P0 ⊕ ≰1 −→ ▽ ▽ ≤1 −→ ▽ P1 <ω ⊕ n (P0 nP1). Additionally, we easily have P0 ωP1 <ω n (P0 nP1). = −→ ▽ = ⊕ Π0 □ (2) P n (P0 nP1) and Q P0 P1 are 1.

2.4. The Disjunction ▽ω versus the Disjunction ▽∞ By the similar argument, we can separate the strength of the concatenation ▽ω and the classical disjunction ▽∞.

16 (1,1)-computable v Y (1,1)-computable P ⊕ Q YYYidY Yl YYYYYYY, v YYY (1,<ω)-computable <ω, P ∪ Q YYYYidY ( 1)-truth-table Yl YYYYYYY, v YYY YY (1,ω)-computable (1,<ω)-computable P▽Q Yl YYidYYY YYYYYYY, v Y ▽ YYY (1,ω|<ω)-computableP ωQ Yl Y YidYYYY YYYYYY, (<ω,1)-computable P▽∞Q

Π0 , ⊆ N Figure 1: The two-tape (bounded-errors) model of disjunctions for independent 1 sets P Q 2 .

, ω, Π0 N Theorem 14. Let P0 P1 be everywhere ( 1)-incomparable 1 subsets of 2 . For any −1 ♡ (1, ω)-computable function Γ, the complement of Γ (P0 ⊕ P1) is dense in (P0▽∞P1) (as a subspace of Baire space NN).

Proof. Fix a learner Ψ which identiﬁes the (1, ω)-computable function Γ. Fix any ♡ clopen set [τ] intersecting with the heart of (P0▽∞P1). Assume that [τ] ∩ (P0▽∞P1) −1 N contains no element of Γ (N \ P0 ⊕ P1) ∪ APΨ. By Theorem 12, mcΨ(≥ n) is dense ▽ ∩ τ = τ⌢ ∩ τ ▽ ∩ τ and open in the heart of (P0 ∞P1) [ ]∩ ((P0 [pr0( )]) ∞(P1 [pr1( )])). As τ ∩ ▽ ♡ Π0 ≥ τ ∩ ▽ ♡ [ ] (P0 ∞P1) is 1, the intersection n∈N mcΨ( n) is dense in [ ] (P0 ∞P1) , N −1 by Baire Category Theorem. Hence, N \ Γ (P0 ⊕ P1) intersects with any nonempty ♡ clopen set [τ] with [τ] ∩ (P0▽∞P1) , ∅. □ Corollary 15. Π0 , ⊆ N ≡<ω ≰1 1. There exist 1 sets P Q 2 such that P 1 Q holds but Q ω P holds. Π0 , ⊆ N ≡<ω <1 2. There exist 1 sets P Q 2 such that P ω Q holds but P ω Q holds. Π0 N Proof. Let P be a perfect 1 antichain in 2 of Theorem 2. Fix a clopen set C such that P0 = P ∩ C , ∅, and P1 = P \ C , ∅. Then P0 and P1 are everywhere (ω, 1)- incomparable. Fix a (1, ω)-computable function Γ identiﬁed by a learner Ψ. By Theo- NN \ ⊕ ▽ ♡ Π0 , ⊆ N ▽ rem 14, (P0 P1) is dense in (P0 ∞P1) . For 1 sets P0 P1 2 , both P0 ∞P1 ⊕ Π0 ▽ ≤1 ⊕ □ and P0 P1 are 1, and P0 ωP1 1 P0 P1. ⊕ ` 2.5. The Disjunction versus the Disjunction ∞

By the similar argument, we can separate inﬁnitary disjunctions. A⊕ sequence {xi}i∈N NN of elements of is Turing independent if xi is not computable in j,i x j for each N i ∈ N. A collection {Pi}i∈I of subsets of N is pairwise everywhere independent if, for { σ } ∩ σ , ∅ ∈ any collection∏ [ i] i∈I of clopen sets with Pi [ i] for each i ⊕I, there is a choice { } ∈ ∩ σ xi i∈I i∈I(Pi [ i]) such that Pi has no element computable in j∈I\{i} x j for each i ∈ I. { } Π0 Theorem 16. Let Pi i<2t be a pairwise everywhere independent collection of 1 subsets of 2N, and let ρ be any binary string. For any (t, ω)-computable function Γ, the −1 ⌢ complement of Γ (P0 ⊕ · · · ⊕ P2t−1) is dense in the heart of ρ (P0▽∞ ... ▽∞P2t−1) (as

17 a subspace of Baire space NN). Indeed, for any nonempty interval I in the heart of ⌢ ⌢ ♡ −1 ρ (P0▽∞ ... ▽∞P2t−1), there is g ∈⊗ρ (P0▽∞ ... ▽∞P2t−1) ∩ I \ Γ (P0 ⊕ · · · ⊕ P2t−1) ∗ ∈ which is computable in some g k<2t−1 Pk.

Proof. Assume that the (t, ω)-computable function Γ is identiﬁed by a team {Ψi}i |τ | such that ΦΨ ↾ (g ↾ s ; 0) ∈ E for some 0 0 e(g0 s0) 0 0 δ( j0;e) e < 2t.

t • If such s0 exists, then enumerate all such e < 2 into an auxiliary set Ch0, and deﬁne δ( j1) as follows:  δ < , δ = ( j0; e) if e Ch0 ( j1; e)  1 − δ( j0; e) if e ∈ Ch0.

⌢ ∗ ⌢ • Extend g0 ↾ s0 to g1 = (g0 ↾ s0) write( j1, f ) ∈ ρ (P0▽∞ ... ▽∞P2t−1), where ∑ j1 = t−1 e · δ j1 e=0 2 ( j1; e).

These actions force each learner Ψe ⊕with e ∈ Ch0 to change his mind whenever the Ψ ∈ N learner e want to have an element of i<2t Pi. Fix u . Assume that ju, gu, su, and Chu has been already deﬁned, and the following induction hypothesis at stage u is satisﬁed. • ⊆ < t ∈ ρ⌢ ▽ ... ▽ ♡ ∩ ρ⌢τ pre(gu) fe for any e 2 , hence, gu (P0 ∞ ∞P2t−1) [ 0].

• {sv}v≤u is strict increasing, and Chu , ∅. • ∈ Φ ↾ < t For each e Chu, if Ψe(gu↾su)(gu su; 0) converges to some value k 2 , then k ∈ Ee δ( ju;e) It is easy to see that u = 0 satisﬁes the induction hypothesis. At stage u + 1 ∈ N, we proceeds the following actions.

• Deﬁne δ( ju+1) as follows:  δ < , δ = ( ju; e) if e Chu ( ju+1; e)  1 − δ( ju; e) if e ∈ Chu. ∑ ⌢ (u+1) t−1 e • Extend g ↾ s to g + = (g ↾ s ) write( j + , f ), where j + = = 2 · u u u 1 u u u 1 ju+1 u 1 e 0 δ u+1 = ↾ ⌢ (u+1) = ρ⌢ ( j + ; e), and f satisﬁes pr (g + ) pr (g s ) f f + . u 1 ju+1 u 1 ju+1 u u ju+1 ju 1

18 • > Φ ↾ ∈ e Wait for the least s + s such that Ψ + ↾ + (g + s + ; 0) E for u 1 u e(gu 1 su 1) u 1 u 1 δ( ju+1;e) some e < 2t.

t • If such su+1 exists, then enumerate all such e < 2 into Chu+1, By our action, it is easy to see that u + 1 satisﬁes the induction hypothesis. As the ⌢ ♡ set ρ (P0▽∞ ... ▽∞P2t−1) is closed (with respect to the Baire topology) and {su}u∈N is ⌢ ♡ strictly increasing, the sequence {gu}u∈N converges to some g ∈ ρ (P0▽∞ ... ▽∞P2t−1) . Let I(g) ⊆ 2t be the set of all e < 2t such that pr (g) is total. ⊕ e ≤ Claim. g T e∈I(g) fe.

Note that g = g[ f0,..., f2t−1] is eﬀectively constructed uniformly in a given collection { fk}k<2t . In other words, there is a (uniformly) computable function Θ mapping { fk}k<2t to Θ({ fk}k<2t ) = g = g[ f0,..., f2t−1]. Then, it⊕ is easy to see⊕ that the function Θ ⌢ N ⌢ N { } ∈ ∪{pr } ∈ t\ ≤ ⊕ pr maps fe e I(g) ⊕e(g) 0 e 2 I(g) to g. Hence, g T e∈I(g) fe e∈2t\I(g) e(g) 0 . ≤ ⌢ N ∈ Therefore, g T e∈I(g) fe as desired, since pre(g) 0 is computable for any e 2t \ I(g).

Let Γe denote the (1, ω)-computable function identiﬁed by Ψe, that is, Γe(α) = Φ α α ∈ NN limn Ψ(α↾n)( ) for any . We consider the following two cases.

Case 1 (e ∈ Chu for finitely many u ∈ N). Fix u such that e < Chv for any v > u. e > ΦΨ ↾ ↾ = For each v u, e(g su)(g su; 0) does not converges to an element of Eδ( j ;e) e e ⌢ v E . By our definition, for each k < E , pr (g) ⊂ ρ fk is finite. By previous δ( ju;e) ⊕ δ( ju;e) k ≤ claim, g T e,k fe. Thus, by independence, Pk has no g-computable element. If ΦΨ ↾ ↾ ↑ ∈ N ∈ Ψ Ψ ↾ e(g su)(g su; 0) ∩ for any u , then g AP e . If limn e(g n) does not ∈ ≥ Φ converge, then g m∈N mcΨe ( m). Otherwise, limn Ψe(g↾n)(g; 0) converges to some e ⌢ value k < E . As Φlim Ψ (g↾n)(g) is g-computable, we see Φlim Ψ (g↾n)(g) < k Pk. δ( ju;e) n ⊕e n e ∈ NN \ Γ−1 Consequently, g e ( k<2t Pk).

Case 2 (e ∈ Chu for inﬁnitely many u ∈ N). We enumerate an inﬁnite increasing sequence {u[n]}n∈N, where u[n] is the n-th element such that e ∈ Chu. As e ∈ e Ch , we have ΦΨ ↾ (g ↾ u[n]; 0) ∈ E . By our action, δ( j + ; e) = u[n] e(g u[n]) δ( ju[n];e) u[n 1] e e δ( j + ; e) , δ( j ; e). This implies E ∩ E = ∅. However, we must u[n] 1 u[n] δ( ju[n+1];e) δ( ju[n];e) e have ΦΨ ↾ + (g ↾ u[n + 1]; 0) ∈ E , since e ∈ Ch + . This forces the e(g u[n 1]) δ( ju[n+1];e) u[n 1] Ψ learner∩ e to change his mind. By iterating this procedure, we eventually obtain ∈ ≥ g m∈N mcΨ ( m). e ∩ ⊕ ⊕ ∈ NN \ Γ−1 ∈ NN \ Γ−1 Consequently, g e∈N( e ( < t Pk)). Thus, g e ( < t Pk). For ⌢ k 2 ⌢ k 2 any τ0 such that ρ τ0 which is extendible in the heart⊕ of ρ (P0▽∞ ... ▽∞P2t−1), we can τ NN \ Γ−1 construct such g extending 0. Therefore, e ( k<2t Pk) intersects any nonempty ρ⌢ ▽ ... ▽ ♡ NN \ Γ−1 ⊕ · · · ⊕ interval in (P0 ∞ ∞P2t−1) . In other words, e (P0 P2t−1) is dense ⌢ ♡ in ρ (P0▽∞ ... ▽∞P2t−1) as desired. □ The following theorem by Jockusch-Soare [35, Theorem 4.1] is important. ∏ { i } Theorem 17 (Jockusch-Soare [35]). There is a computable sequence n Pn i∈N of Π0 N { } nonempty homogeneous∏ 1 subsets of 2 such that xi i∈N is Turing independent for ∈ i ∈ N any choice xi n Pn, i .

19 Π0 Clearly any such 1 set contains no element of a PA degree, a Turing degree of a complete consistent extension of Peano Arithmetic. Accordingly, every element of such a Π0 set computes no element of a Medvedev complete Π0 set CPA. 1 ⊕ 1 Π0 ⊆ N ∈ N −→ ▽ ... ▽ <<ω Corollary⊕ 18. There are sets Pn 2 , n , such that t (P0 ∞ ∞Pt) ω −→ 1 t Pt.

Proof. Fix the computable sequence {Pi}i∈N⊕of Theorem⊕ 17. Then {Pi}i∈N is pair- −→ ≤<ω −→ ▽ ... ▽ wise everywhere independent. Assume that t Pt ω t (P0 ∞ ∞Pt) via a , ω Γ ρ (t )-computable function . Let n denote the n-th∪ leaf of the tree TCPA of a Medvedev Π0 N Γ−1 ρ ⌢ complete 1 subset of 2 . By Theorem 16, ( k<2t k Pk) is dense in the heart of ρ ⌢ ▽ ... ▽ ∈ ρ ⌢ ▽ ... ▽ 2t (P0∪∞ ∞P2t−1). In particular, there is g 2t (⊗P0 ∞ ∞P2t−1) such that Γ < ρ ⌢ ∗ ≤ (g) k<2t k Pk which is computable∪ in some g T k<2t Pk.⊕ By our choice of { } Γ ∪ Γ < −→ □ Pi i∈N, (g) computes no element of k≥2t Pk CPA. Thus, (g) t Pt. Corollary 19. { } Π0 N 1. There exists a computable[` sequence] P⊕n n∈N of 1 subsets of Cantor space 2 , <<ω such that the condition ∞ n Pn ω n Pn is satisﬁed. 0 N ω <ω 2. There exist Π sets P, Q ⊆ 2 such that P ≡ Q holds but P <ω Q holds. 1 [` ]1 ⊕ ≤1 −→ ▽ ... ▽ <<ω Proof.⊕ (1) By⊕ Corollary 18. the condition [`∞ n]Pn 1 ⊕t (P0 ∞ ∞Pt) ω −→ ≤1 = ≤1 −→ ▽ ... ▽ t ⊕Pt 1 n Pn is satisﬁed. (2) Put P ∞ n Pn 1 t (P0 ∞ ∞Pt) and = −→ , Π0 Q t Pt. Then P and Q are (1 1)-equivalent to 1 subsets as seen in Part I [29, <<ω ≡ω Section 5.2]. By Theorem 18, P ω Q, and Q 1 P as seen in Part I [29, Sections 4 and 5.2]. □

3. Contiguous Degrees and Dynamic Inﬁnitary Disjunctions 3.1. When the Hierarchy Collapses Π0 We have already observed the following hierarchy, for pairwise independent 1 sets P, Q ⊆ 2N. ⊕ >1 ∪ >1 ▽ >1 ▽ >1 ▽ ≡<ω ⊕ . P Q 1 P Q 1 P Q <ω P ωQ ω|<ω P ∞Q 1 P Q

Homogeneity is an opposite notion∏ of antichain (and independence). Recall that ⊆ NN = ⊆ N ∈ N S is homogeneous if S n S n for some S n , n . Every antichain is degree-non-isomorphic everywhere. On the other hand, every homogeneous set S is degree-isomorphic everywhere, that is to say, S ∩ C is degree-isomorphic to S ∩ D for any clopen sets C, D ⊆ NN with S ∩ C , ∅ and with S ∩ D , ∅4. The next observation is that every finite-piecewise computable method of solving a Π0 Π0 homogeneous 1 mass problem can be refined by a finite-( 1)2-piecewise computable ≤1 ≤<ω method. That is to say, our hierarchy between <ω and 1 collapses for homogeneous Π0 , < ω 1 sets, modulo the (1 )-equivalence.

4An anonymous referee pointed out that the notion of degree-isomorphic everywhere is related to the notion of fractal in the study of Weihrauch degrees [9, 53]. The (reverse) lattice embedding d of the Medvedev degrees into the Weihrauch degrees has the property that a subset P of Baire space is degree-isomorphic everywhere if and only if d(P) is a fractal.

20 Π0 ⊆ NN ⊆ NN Theorem 20. For every homogeneous 1 set S and for any set Q , if <ω 1 S ≤ Q then S ≤<ω Q. 1 ∏ = Π0 ⊆ N ≤<ω Proof. Let S x Fx for some 1 sets Fx . Assume S 1 Q via the bound b. That is, for every g ∈ Q there exists an index e < b such that Φe(g) ∈ S . Let us begin defining a learner Ψ who changes his mind at most finitely often. Fix g ∈ Q. The learner Ψ first sets A0 = {e ∈ N : e < b}. By our assumption, we have Φe(g) ∈ S for some e ∈ A0. Then the learner Ψ challenges to predict the solution algorithm e < b such that Φe(g) ∈ S by using an observation g ∈ Q. He begins the 1-st challenge. On the (s + 1)-th challenge of Ψ, inductively assume that, the learner have already defined a set As ⊆ A0. Let v be a stage at which the s + 1-th challenge of Ψ on g begins. In this challenge, the learner Ψ uses the two following computable functionals Γ and ∆.

• For a given argument x, Γ(x, s+1) outputs the least ⟨e(x), t(x)⟩ such that e(x) ∈ As and Φe(x)(g ↾ t(x); x) ↓ if such ⟨e(x), t(x)⟩ exists.

• If Γ(x, s + 1) = ⟨e(x), t(x)⟩, then ∆(g; x, s + 1) = Φe(x)(g ↾ t(x); x).

Set ∆s+1(g; x) = ∆(g; x, s + 1). Clearly, an index d(s + 1) of ∆s+1 is calculated from s + 1. Then the learner Ψ(g ↾ v) outputs d(s + 1) on the (s + 1)-th challenge. Hence ΦΨ(g↾v)(g; x) = Φd(s+1)(g; x) = Φe(x)(g ↾ t(x); x) for any x. He does not change his ′ mind until the beginning stage v of the next challenge, i.e., ΦΨ(g↾v′′)(g) = ΦΨ(g↾k)(g) for k ≤ v′′ < v′. The next challenge might begin when it turns out that Ψ’s prediction on his (s + 1)-th challenge is incorrect, namely:

• ΦΨ(g↾v)(g ↾ u) ↾ n < TS,u for some n < u at some stage u > v.

Here TS is a corresponding computable tree of S . For each x ∈ N, ﬁx a decreasing { } Π0 ⊆ { , } approximation Fx,s s∈N of a 1 set Fx 0 1 , uniformly in x. In this case, there exists x < n such that the following condition holds.

ΦΨ(g↾v)(g ↾ u; x) = ∆s+1(g; x) = Φe(x)(g; x) < Fx,s.

For such a least x, the learner removes e(x) from As, that is, let As+1 = As \{e(x)}. If As+1 , ∅ then the learner Ψ begins the (s + 2)-th challenge at the current stage u. The construction of the learner Ψ is completed. An important point of this construction is that the learner never uses an index rejected on some challenge. This makes the prediction on g ∈ Q of the learner Ψ converge. Claim. Ψ changes his mind at most b times.

Whenever Ψ changes, As must decrease. However #A0 = b.

Claim. For every g ∈ Q it holds that Φlim Ψ(g↾s)(g) ∈ S . s ∏ ∈ g ⊆ ∈ Φ ∈ = For g Q, let B A0 be the set of all e A0 such that ∩e(g) S x Fx. By g g ⊆ the deﬁnition of∩A0, clearly B is not empty. Moreover, B s As holds, since e is Φ < Φ N → N removed from s As only when e(g; x) Fx for some x. Thus, Ψ(g↾v)(g): is total for every stage v. This means that, if ΦΨ(g↾v)(g) < S , then the learner Ψ will Φ ↾ < < know his mistake at some stage∩u, i.e., Ψ(g↾v)(g u; x) Fx,u for some x u. Then some index is removed from s As. However, this occurs at most b times. Thus, Φ ∈ □ lims Ψ(g↾s)(g) S .

21 α, β, γ ∈ { , < ω, ω} α, β|γ Π0 Let 1 . We say that a ( )-degree a of a nonempty 1 subset N α, β|γ ≤ α, β|γ Π0 of 2 is ( )-complete if b a for every ( )-degree b of a nonempty 1 subset N Π0 ⊆ N α, β|γ α, β|γ of 2 . If a 1 set P 2 has an ( )-complete ( )-degree, then it is also called (α, β|γ)-complete. Π0 N , < ω , ω| < ω Corollary 21. A 1 subset of 2 is (1 )-complete if and only if it is (1 )- complete if and only if it is (< ω, 1)-complete.

Proof. Let DNR2 denote the set of all two-valued diagonally noncomputable functions, where a function f : N → 2 is diagonally noncomputable if f (e) , Φe(e) for any index Π0 , e. This set is clearly homogeneous, and 1. Moreover, it is (1 1)-complete (hence (α, β|γ)-complete for any α, β, γ ∈ {1, < ω, ω}). Therefore, we can apply Theorem 20 with S = DNR2. □ Π0 , ⊆ N ⊕ ≡1 ▽ Corollary 22. There are 1 sets P Q 2 such that P Q <ω P ∞Q. Indeed, if P ≡1 ⊕ ≡1 ▽ is homogeneous and Q 1 P, then P Q <ω P ∞Q is satisfied. Π0 N ⊕ Proof. Let P be any homogeneous 1 subset of 2 . Then P P is also homogeneous. As seen in Part I [29, Section 4], there is a (2, 1)-computable function from P▽∞P to ⊕ ⊕ ≤<ω ▽ ⊕ ≤1 ▽ P P, hence P P 1 P ∞P. Thus, by Theorem 20, P P <ω P ∞P. Recall ≡1 ▽ ≡1 ▽ ≡1 from Part I [29, Proposition 38] that Q 1 P implies P ∞P 1 P ∞Q. Hence, Q 1 P ⊕ ≡1 ⊕ ≤1 ▽ ≡1 ▽ □ implies P Q 1 P P <ω P ∞P 1 P ∞Q. It is natural to ask whether our hierarchy of disjunctive notions for homogeneous Π0 , 1 sets also collapses modulo∏ the (1 1)-equivalence. The answer is negative. We say that a homogeneous set n Fn is computably bounded if there is a computable function l : N → N such that Fn ⊆ {0,..., l(n)} for any n ∈ N. Clearly, every homogeneous subset of Cantor space 2N is computably bounded. Cenzer-Kihara-Weber-Wu [12] introduced the notion of immunity for closed sets. A closed subset P of Cantor space N ext 2 is immune if TP has no infinite computable subset. ⊆ N Π0 , ,..., ⊆ NN Theorem 23. Let P 2 be a non-immune 1 set,∪ and S 0 S 1 S m be special Π0 ≰1 computably bounded homogeneous 1 sets. Then i≤m S i 1 P. ∪ ∏ ext i ≤1 Proof. Let V0 be an infinite c.e. subtree of TP . Assume that i≤m n Fn 1 P via a Φ < { i } Π0 computable functional , where, for each i m, Fn n∈ω is a uniformly∏ 1 sequence { , ,... } ext Π0 i of subsets of 0 1 li . Let S i denotes the corresponding 1 tree of n Fn, and let = {ρ ∃τ ∈ ext ∃ ρ = τ⌢⟨ ⟩ < ext} ff Li :( S i )( i) i S i , for each∏i. Note that Li di ers from the i set of leaves of the corresponding computable tree of n Fn. We first consider the set Φ = {ρ ∈ ∃σ ∈ Φ σ ⊇ ρ} < ≤ Li Li :( Vi) ( ) , where Vi for 0 i m will be defined in the Φ below construction. Note that L0 is computably enumerable. There are three cases: Φ 1. L0 is infinite; ∏ Φ Φ 0 2. L0 is finite, hence ([V0]) is a subset of n Fn; 3. otherwise. ρ ∈ Φ > + ρ ∈ 0 (Case 1): For any n, there exists L0 of height n 1, and (n) ∏Fn. From Φ 0 any computable enumeration of L0 ∏we can calculate a computable path of n Fn. This = 0 contradicts the specialness of S 0 n Fn.

22 σ ∈ (Case 2): There exists a finite number k such that, for every string ∏V0 of height > Φ σ ext = 0 k, ( ) belongs to S i . This also contradicts the specialness of S 0 n Fn. (Case 3): There exists infinitely many strings σ ∈ V0 such that Φ(σ) extends some Φ Φ ρ ∈ Φ string of L0 . Since L0 is finite, by the pigeon hole principle, there exists 0 L0 such that Φ(σ) extends ρ for infinitely many σ ∈ V0. Fix such ρ0, and let V1 = {σ ∈ V0 : Φ σ ⊇ ρ} ext ( ) . Then the downward closure of V1 is an infinite c.e. subtree of TP , and Φ([V1]) ∩ S 0 = ∅. By iterating this procedure, we win the either of the cases 1 or 2 for some i ≤ m. The reason is that, if the case 3 occurs∪ for j, then V j+1∪is defined∏ as an infinite c.e. ext Φ ∩ = ∅ i ≤1 ≤ subtree of T∪P such that ([V1]) ( i≤ j S i) . Since i≤m n Fn 1 P [Vm], i.e., Φ ⊆ □ ([Vm]) i≤m S i, the case 3 does not occur for m. , Π0 N , Corollary 24. Let P Q be any nonempty 1 subsets of 2 , and S T be special com- Π0 ∪ ≰1 ⌢ putably bounded homogeneous 1 sets. Then S T 1 P Q. ⌢ ∪ ≰1 ⌢ □ Proof. Clearly P Q is not immune. Thus, Theorem 23 implies S T 1 P Q. To understand degrees of difficulty of disjunctive notions, and to discover new eas- ier (possibly infinitary) disjunctive notions, it is interesting to discuss contiguous degrees.

∗ ∗ ∗ 3 α Deﬁnition 25. Let (α, β, γ), (α , β , γ ) ∈ {1, < ω, ω} , and assume that ≤β|γ is not ﬁner α∗ α ∗ ∗ ∗ α than ≤β∗|γ∗ . An (α, β|γ)-degree aβ|γ is (α , β |γ )-contiguous if aβ|γ contains at most one ∗ ∗ ∗ α (α , β |γ )-degree, that is to say, for any representatives A, B ∈ aβ|γ, we have that A is (α∗, β∗|γ∗)-equivalent to B. Corollary 26. , < ω < ω, Π0 N 1. There is a (1 )-contiguous ( 1)-degree of 1 sets of 2 . , < ω Π0 Π0 2. Every (1 )-degree which contains a homogeneous 1 set or a 1 antichain is not (1, 1)-contiguous. , ω| < ω Π0 , < ω 3. Every (1 )-degree of 1 antichains is not (1 )-contiguous. < ω, Π0 , ω 4. Every ( 1)-degree of 1 antichains is not (1 )-contiguous (hence, is not (1, ω| < ω)-contiguous). Proof. (1) This follows from Theorem 20. , < ω Π0 (2) If d is a (1 )-degree of a homogeneous 1 set S , then d contains S and ▽ ≡1 ▽ ▽ <1 ∪ = S S , since S <ω S S . However, S S 1 S S S by Corollary 24. If d is a , < ω Π0 × N ∪ N × ▽ (1 )-degree of a 1 antichain P, then d contains (P 2 ) (2 P) and P P, since ≡1 × N ∪ N × ▽ <1 × N ∪ N × P <ω (P 2 ) (2 P). However, P P 1 (P 2 ) (2 P) holds by Lemma 6. Π0 ∩ ⊕ \ ≡1 (3) Note that, for any 1 set P and any clopen set C, it holds that (P C) (P C) 1 , ω| < ω Π0 P. Let d be a (1 )-degree of a 1 antichain P. Fix a clopen set⊕C such that = ∩ , ∅ = \ , ∅ ⊕ ⇀ ▽ P0 P C , and⊕P1 P C . Then d⊕contains P0 P1 and n (P0 nP1), ⊕ ≡1 ⇀ ▽ ⇀ ▽ <1 ⊕ since P0 P1 ω|<ω n (P0 nP1). However, n (P0 nP1) <ω P0 P1 holds by Corollary 13. < ω, Π0 (4) Let d be a ( 1)-degree of a 1 antichain P. Fix a clopen set C such that P0 = P ∩ C , ∅, and P1 = P \ C , ∅. Then d contains P0 ⊕ P1 and P0▽ωP1, since ⊕ ≡<ω ▽ ▽ < ⊕ □ P0 P1 1 P0 ωP1. However, P0 ωP1 l P0 P1 holds by Corollary 15.

23 3.2. Concatenation, Dynamic Disjunctions, and Contiguous Degrees We next show the non-existence of nonzero (1, 1)-contiguous (1, < ω)-degree, that 1 1 is, we will see the LEVEL 4 separation between [CT ]1 and [CT ]<ω. Indeed, we show the strong anti-cupping result for (1, 1)-degrees inside every nonzero (1, < ω)-degree via the concatenation operation. The following theorem is one of the most important and nontrivial results in this paper. Π0 , ⊆ N ⌢ , Theorem 27. For any nonempty 1 sets P Q 2 ,Q P does not (1 1)-cup to P. That ⊆ NN ≤1 ⌢ ⊗ ≤1 is to say, for any R , if P 1 (Q P) R then P 1 R. Proof. We ﬁrst note that P and Q may be assumed to be special. If P is not special, the assertion is trivial. If Q has a computable element, then Q⌢P also has a computable ⌢ ⊗ ≡1 element. In this case, (Q P) R 1 R, and then the assertion is obvious. Therefore, we may assume that Q is special. Let TP and TQ be corresponding trees of P and Q, and let LP and LQ denote all leaves of TP and TQ, respectively. Note that TQ is inﬁnite since Q is special. For a tree T ⊆ 2

24 l : 2

As a consequence of the previous lemma, Dg turns out to be an inﬁnite g-c.e. sub- g tree of TP without dead ends for any g ∈ P. Hence, we can compute a path through D uniformly in g as follows. g Lemma 31. D has a g-computable path of TP uniformly in g ∈ R. Proof. The set of all inﬁnite paths through a c.e. tree of N

25 ∪ Ψ Ψ = ∆ Γ ∈ NN Then we define a computable functional as (g) ( n (g)) for any g . ≤1 □ This witnesses P 1 R as desired. Π0 ⊆ N Π0 ⊆ N Corollary 32. For every special 1 set P 2 , there exists a 1 set Q 2 such that <1 ≡1 Q 1 P and Q <ω P. ⌢ <1 ≡1 □ Proof. By Theorem 27, we have P P 1 P <ω P. α Definition 33. Fix α, β, γ ∈ {1, < ω, ω}. An (α, β|γ)-degree a ∈ Pβ|γ has the strong α anticupping property if there is a nonzero (α, β|γ)-degree b ∈ Pβ|γ such that, for any (α, β|γ)-degree c, if a ≤ b ∨ c, then a ≤ c. ∈ P1 Corollary 34. Every nonzero a 1 has the strong anticupping property. Proof. Fix P ∈ a. Let b be the (1, 1)-degree of P⌢P. Then, by Theorem 27, for any (1, 1)-degree c, if a ≤ b ∨ c, then a ≤ c. □ Π0 ⊕ ∪ For 1 sets, if P and Q are disjoint, then P Q is equivalent to P Q modulo , P1 = P/ <ω Π0 Π0 the (1 1)-equivalence, since 1 decp [ 1]. However, if P and Q are not 1, the above claim is false, in general. Π0 ⊆ N Π0 ⊆ N Proposition 35. For any special 1 set P 2 , there exists a ( 1)2 set Q 2 such ∩ = ∅ ∪ <1 ⊕ that P Q but P Q 1 P Q. ⌢ Proof. Put Q = (P P) \ P. For any g ∈ Q, there is a leaf ρ ∈ LP such that ρ ⊂ g. So we ρ ∈ ↼|ρ| ≤1 wait for such a leaf LP. Then g belongs to P. Hence, P 1 Q. Thus, we have ≤1 ⊕ ∪ = ⌢ <1 □ P 1 P Q, while P Q P P 1 P by Theorem 27. Definition 36. The operation ▽ : P(NN) → P(NN) is defined as the map sending P to P▽ = CPA⌢P, where CPA denotes the set of all complete consistent extensions of , Π0 N Peano Arithmetic, and it is a (1 1)-complete 1 subset of 2 . By the previous theorem, the derived set P▽ does not (1, 1)-cup to P whenever P is Π0 ▽ <1 1. In particular, we have P 1 P. Recall from Part I [29, Proposition 38] that the ▽ P1 → P1 1 ▽ = 1 ▽ operator : 1 1 introduced by (deg1(P)) deg1(P ) is well-defined. Moreover, P1 ≤ ▽ = { ∈ P1 ≤ ▽} 1( 1 ) a 1 : a 1 is a principal prime ideal consisting of tree-immune- Π0 ⊆ N ext free Medvedev degrees [12]. Here, recall that a 1 set P 2 is tree-immune if TP contains no infinite computable subtree. Then, we also observe the following. Π0 , , , ⊆ N ⌢ ≤1 ⌢ Proposition 37. Fix 1 sets P0 P1 Q0 Q1 2 , and assume that P0 P1 1 Q0 Q1. ≤1 ≤1 Then, either P0 1 Q0 or P1 1 Q1 holds. Moreover, if P0 is tree-immune and Q0 is ≤1 nonempty, then P1 1 Q1. Proof. Assume that P ⌢P ≤1 Q ⌢Q via a computable function Φ. If Φ(ρ) ∈ T ext for 0 1 1 0 1 P0 ρ ∈ Φ ∈ ∈ ≤1 Φ ρ < ext any leaf LQ0 , then (g) [TP0 ] for any g [Q0], i.e., P0 1 Q0. If ( ) TP ρ ∈ Φ ρ0 for some leaf LQ0 , then there are only finitely many strings of TP0 extending ( ). ⌢ ∩ Φ ρ , Thus, [TP0 TP1 ] [ ( )] is essentially a sum of finitely many P1’s, hence it is (1 1)- ⌢ equivalent to P1. Since a computable functional Φ maps ρ Q1 to the above class, ≤1 Φ ρ < ext ρ ∈ obviously, P1 1 Q1. If P0 is tree-immune, then ( ) TP for some leaf LQ0 , Φ 0 since otherwise the image of TQ0 under is included in TP0 , and clearly it is infinite ≤1 □ and computable. Therefore, we must have P1 1 Q1.

26 Corollary 38. The operator ▽ : a 7→ a▽ is injective. Hence, ▽ provides an order- , P1 Π0 N preserving self-embedding of the (1 1)-degrees 1 of nonempty 1 subsets of 2 . Proof. By Cenzer-Kihara-Weber-Wu [12], CPA is tree-immune. Therefore, by Propo- ⌢ = ▽ ≤1 ▽ = ⌢ ≤1 □ sition 37, CPA Q Q 1 P CPA P implies Q 1 P. P1 P1 ≤ ▽ It is natural to ask whether the image of 1 under the operator is exactly 1( 1 ). Unfortunately, it turns out to be false. Π0 ⊆ N Proposition 39. There exists a non-tree-immune 1 set Q 2 such that no nonempty Π0 , ⊆ N ≡1 ⌢ ▽ P1 → P1 ≤ ▽ 1 sets P0 P1 2 satisfy Q 1 P0 P1. In particular, the operator : 1 1( 1 ) is not surjective. { } Π0 N ⊕Proof. Let Qn n∈N be a computable sequence of nonempty 1 subsets of 2 such that = ⌢{ } n∈N Qn forms a Turing antichain. Define Q Q0 Qn+1 n∈N. Suppose that there Π0 , ⊆ N ≡1 ⌢ exist nonempty sets P0 P1 2 with Q 1 P0 P1. Choose computable functions ⌢ 1 ⌢ Φ : Q → P0 P1 and Ψ : P0 P1 → Q. Since {Qn}n∈N forms a Turing antichain, Ψ ◦ Φ is an identity function on Q. Consider two cases. The first case is that Φ(Q) ⊆ P0. In this case, Ψ(P0) = Q since Ψ ◦ Φ is identity. ext Ψ Thus, every string in TQ is extended by some string in (TP0 ). Moreover, he condition ext ext ext ext T ⊆ T ⌢ implies Ψ(T ) ⊆ T . Therefore, Ψ(T ) = T . Hence T is a P0 P0 P1 P0 Q P0 Q Q computable tree without leaves. But this is impossible since Q contains no computable elements. The second case is that Φ(Q) ⊈ P0, that is, there exists f ∈ Q such that Φ( f ) ∈ ρ⌢ ρ ≡ Φ Ψ ◦ Φ P1, where is a leaf of TP0 . We have f T ( f ) since is identity. Note = ρ ⌢ ρ ∈ ∈ that we may assume that f k fk for some leaf k TQ0 and fk Qk, since even if f ∈ Q0 the string Φ( f ↾ n) extends ρ for sufficiently large n, and replace f with a string extending f ↾ n which is removed from Q0. On the one hand, f is the only σ ∈ element in Q computable in f . On the other hand, every TP0 always extends to an σ ∈ Φ σ element of P which is Turing equivalent to f . Thus, for every TP0 , the string ( ) ⌢ must be compatible with ρk. Hence, Ψ(P) ⊆ ρk Qk. This contradicts the property that Ψ ◦ Φ(Q) = Q. □ Let O denote Kleene’s system of ordinal notations (see [52]). As usual, this system | · | ⊆ N → ωCK Π1 involves a representation O : 1 of computable ordinals with a 1 domain |·| = O | | = | a| = | | + | · e| = |Φ | Φ N → N dom( ) , where 0 O 0, 2 O a O 1, and 3 5 O supn e(n) O if e : is total and strictly increasing. Recall from Part I [29, Definition 62] that P(a) is the a-th derivative of P, i.e., the a-th iterated concatenation starting from P, for every a ∈ O. Π0 ⊆ N , ∈ O < (b) Proposition 40. For any special 1 set P 2 , if a b and a O b, then P does , (a) ⊆ NN (a) ≤1 (b) ⊗ (a) ≤1 not (1 1)-cup to P , i.e., for any set R , if P 1 P R then P 1 R. < a ≤ (b) ≤1 (2a) Proof. The assumption a O b implies 2 O b. Therefore, we have P 1 P . By Theorem 27, P(2a) does not (1, 1)-cup to P(a). Thus, P(b) does not (1, 1)-cup to P(a). □

Fix any notation omega ∈ O such that |Φomega(n)|O = n for each n ∈ N. Note that |omega|O = ω. Π0 N Π0 ⊆ N ≤1 Proposition 41. Let P be a special 1 subset of 2 . For any 1 set R 2 , if P <ω (omega) 1 P ⊗ R, then P ≤<ω R.

27 Π0 , ⊆ N ≤1 Proof. As seen in Part I [29, Section 2.4], for every 1 sets P Q 2 , P <ω implies ≤1 (omega) ⊗ Π0 ≤1 (omega) ⊗ ≤1 (omega) ⊗ P tt,<ω Q. Since P R is 1, P <ω P R implies P tt,<ω P R, and then there is a (1, n)-truth-table function Γ : P(omega) ⊗ R → P for some n ∈ N. In ⌢ (Φomega(n+1)) particular, Γ :(ρn+1 P ) ⊗ R → P, where ρn+1 is the (n + 1)-th leaf of TP. By modifying Γ, we can easily construct a (1, n)-truth-table function Θ : P(n+1) ⊗ R → P. Assume that Θ is (1, n)-truth-table via n many total computable functions Θ0,..., Θn−1.

(b) Proof. Let g(e, b) be an index of Pe . Then, the desired conditions follow from Propo- sition 40. □

For any reducibility notion r, and any ordered set (I, ≤I), a sequence {ai}i∈I of r- degrees is r-noncupping if, for any i

1 Corollary 43. For any nonzero (1, ω)-degree a ∈ Pω, there is a (1, 1)-noncupping , α < ωCK □ computable sequence of (1 1)-degrees inside a of arbitrary length 1 .

3.3. Inﬁnitary Disjunctions along the Straight Line 1 1 We next see the LEVEL 4 separation between [CT ]ω|<ω and [CT ]ω. Indeed, we < ω, , ω show the non-existence of` a ( 1)-contiguous∪ (1 )-degree. We introduce the LCM { } = ⌢ ... ⌢ disjunctions of Pi i∈N as n∈N Pn n∈N(P0 Pn). This is a straightforward` in- = ∈ N ﬁnitary` iteration of the concatenations. If Pn P for all n , we write P instead of n Pn. { } Π0 N Proposition` 44. Let Pi i∈N be a computable collection of nonempty 1 subsets of 2 . , Σ0 N Then n Pn is (1 1)-equivalent to a dense 2 set in Cantor space 2 . N Proof. Let S denote the set {g ∈ {0, 1, ♯} :(∃n ∈ N)(count(g) = n & tail(g) ∈ Pn)}, = { ∈ N = ♯} Σ0 { , , ♯}N where count(g) # n` : g(n) . Then, S is clearly a 2 subset of 0 1 , and ≡1 σ ∈ { , , ♯}

28 Proposition 46 (Lewis-Shore-Sorbi [39]). No somewhere dense set in Baire space (1, 1)-cup to a closed set in Baire space. In other words, for any somewhere dense set ⊆ NN ⊆ NN ⊆ NN ≤1 ⊗ ≤1 □ D , any closed set C , and any set R , if C 1 D R then C 1 R. Proposition 47. For any somewhere dense set D ⊆ NN and any special closed set ⊆ NN ≰<ω C , we have C 1 D. ∪ { } N = N Proof. If Di i

The n-th bounded learner will` be diagonalized above the n-th leaf of the spine`TP, = ⊆ Π0 Σ0 where note that P [TP] P. To make a desired 1 set inside the 2 set P, we need to specify upper bounds of mind-changes to diagonalize all bounded learners. Unfortunately, we cannot give a computable sequence of such upper bounds. However, our ﬁnite injury construction will specify such upper bounds by a left-c.e. way, which will be called a timekeeper.

Definition 49. A sequence ⟨tn⟩n∈N of finite strings is a timekeeper if there is a uniformly c.e. collection of finite sets, {Vn}n∈N, such that, for any n ∈ N, |tn| = |Vn| and tn(i) is given as the stage at which the i-th element is enumerated into Vn, for each i < |tn|. Definition 50. For a finite string τ ∈ N

Proposition 51. If τ(m) = 0 for each m < |τ|, then P(τ) = P(|τ|+1). Proof. Straightforward from the deﬁnition. □

Lemma 52. For any timekeeper ⟨tn⟩n∈N, the following conditions hold. ` (tn) (|tn|+1) ⌢ (tn) 1. P ⊆ P . Hence, P {P }n∈N ⊂ P.

29 ⌢ (tn) Π0 { (tn)} Π0 2. P is 1, uniformly in n. Hence, P P n∈N is 1. Proof. (1) Straightforward. (2) We construct a computable tree T (tn) corresponding (tn) N ⌢ ⌢ ⌢ ⌢ to P . Each σ ∈ 2 can be represented as σ = ρ0 ρ1 ... ρk τ, where ρm ∈ LP (tn) for any m ≤ k, and ⟨⟩ , τ ∈ TP. Then σ ∈ T if and only if tn(k) holds by stage ⌢ ⌢ ⌢ (tn) (tn) (tn) |ρ0 ρ1 ... ρk|. Then T is a computable tree, and clearly P = [T ]. □ Remark. The delayed derivative construction is useful to bound the complexity of the ⌢ (|tn|+1) set, since the recursive meet P {P }n∈N of the standard derivatives along a time- { } Π0,∅′ keeper tn n∈N is only assured to be 1 . Π0 Π0 Proof of Theorem 48. Let Q be a special 1 set, and P be a given 1 set. By a uniformly computable procedure, from P, we will construct a timekeeper {tn}n∈N. The b b ⌢ (tn) desired class P will be given by P = P {P }n∈N. Requirements. We need to ensure, for all n ∈ N, the following:

1 b Rn : Q ≤<ω P via n → (∃∆n) ∆n ∈ Q. N Here, ∆n ranges over computable elements of 2 .

Action of an Rn-strategy. Fix an effective enumeration {ρn : n ∈ N} of all leaves of TP. An Rn-strategy uses nodes extending the n-th leaf ρn of TP, and it constructs a finite sequence tn[s], a sequence τn[s] of strings, and a computable function ∆n. For any n, put tn[0] = ⟨⟩, and τn[0] = ρn at stage 0. An Rn-strategy acts at stage s + 1 if the following condition holds: ∃ρ ∈ s ∃ < Φ τ ⌢ρ ∈ Φ τ ⌢ρ ⊋ Φ τ . ( TP)( e n) e( n[s] ) TQ & e( n[s] ) e( n[s]) + ρ ∈ s ρ∗ ∈ If an Rn-strategy acts at stage s 1 then, for a witness TP, we pick LP ⌢ ∗ ⌢ extending ρ. Then let us define τn[s + 1] = τn[s] ρ , tn[s + 1] = tn[s] ⟨|τn[s + 1]|⟩, and ∆e,n ↾ l = Φe(τn[s + 1]), where l is the length of Φe(τn[s + 1]). Otherwise, + = τ + = τ , 7→ τ tn[s 1] tn[s], i[s 1] i[s]. Note that the mapping∪ (n m) n(m) is partial = b computable. At the end of the construction, set tn s tn[s]. As mentioned above, P b ⌢ (tn) is defined by P = P {P }n∈N.

Claim. An Rn-strategy acts at most ﬁnitely often for each n. ∪ τ = τ ∆ = Clearly n s n[s] is a computable string. If Rn acts inﬁnitely often, then e,n Φe(τn) ∈ Q for some e < n by our choice of τn. Since Φe(τn) is computable, Q contains a computable element. However, this contradicts our assumption that Q is special. Therefore, we concludes the claim. As a corollary, ⟨tn⟩n∈N is a timekeeper. 1 b Claim. P ≰<ω P. ∪ Let τ = τ [s]. By induction we show that τ ∈ ρ ⌢T ext . First we have the n s n n n P(tn) following observation: τ = ρ ∈ ρ ⌢ ext = ρ ⌢ ext ⊆ ρ ⌢ ext . n[0] n n TP n TP(tn↾0) n TP(tn[0]) τ ∈ ρ ⌢ ext τ + = τ ⌢ρ∗ ρ∗ ∈ + = Assume n[s] n T (tn[s]) . If n[s 1] n[s] for LP then tn[s 1] ⌢ P ⌢ ⟨|τ + |⟩ τ + ∈ ρ + ↾| | |τ + | ≥ + tn[s] n[s 1] . In particular n[s 1] n LP(tn[s 1] tn[s] ) and n[s 1] tn[s

30 (tn[s+1]) ⌢ 1](|tn[s]|). Hence, by the definition of P , it is easy to see that τn[s + 1] P ⊆ ⌢ + ⌢ ⌢ ρ (tn[s 1]) τ + ∈ ρ ext τ ∈ ρ ext n P . Thus, n[s 1] n T (tn[s+1]) . So we obtain n n T (tn) and by our P ⌢ P construction of τn there is no ρ ∈ P and e < n such that Φe(τn ρ) ⊋ Φe(τn). Since Φ (τ ) is a finite string, for any g ∈ ρ ⌢T ext ⊂ Pˆ extending τ , Φ (g) is also a finite e n n P(tn) n e 1 b string. Consequently, this g witnesses that P ≰<ω P. □ ` ` Π0 ⊆ N <<ω ≡1 Corollary 53. 1. For every special 1 set P 2 , we have P 1 P ω P. Π0 ⊆ N Π0 ⊆ N <<ω ≡1 2. For every special 1 set P 2 there exists a 1 set Q 2 with Q 1 P ω Q. ` 1 b 1 Proof. By applying Theorem 48 to Q = P, we have P ≰<ω P ≥<ω P. Moreover, ⊕ b <<ω ≡1 ⊕ b □ P P 1 P ω P P.

3.4. Inﬁnitary Disjunctions along ill-Founded Trees 1 <ω We next show the LEVEL 4 separation between [CT ]ω and [CT ]ω . The follow- 1 ing theorem concerning the hyperconcatenation ▼ and the (1, ω)-reducibility ≤ω is a counterpart of Theorem 27 concerning the concatenation ▽ and the (1, 1)-reducibility ≤1 1. Π0 , ⊆ N ≤1 ▼ ⊗ Theorem 54. For every special 1 sets P Q 2 , and for any R, if P ω (Q P) R 1 then P ≤ω R holds. ▼ Proof. Let T▼ denote the corresponding computable⊓ tree for Q P. The heart of T▼, ♡ γ ∈ γ ⊆ σ ⌢⟨τ ⟩ {σ } ⊆ T▼ , is the set of all strings T▼ such that⊓ i

1 Now we assume P ≤ω (Q▼P) ⊗ R via a learner Ψ. To show the theorem it is needed 1 to construct a new learner ∆ witnessing P ≤ω R. Fix g ∈ R. ♡ ♡ Lemma 56. There exists a string ρ ∈ T▼ such that, for every τ ∈ T▼ extending ρ, we have Ψ(ρ ⊕ (g ↾ |ρ|)) = Ψ(γ) for any γ with ρ ⊕ (g ↾ |ρ|) ⊆ γ ⊆ τ ⊕ (g ↾ |τ|). Proof. If Lemma 56 were false, we could inductively define an increasing sequence {τ } τ = ⟨⟩ τ τ ⊋ τ τ ∈ ♡ i i∈ω of strings. First let 0 , and i+1 be the least i such∪ that T▼ and Ψ τ⊕ ↾ |τ|+ , Ψ ρ⊕ ↾ |ρ|+ , < τ ∈ ▼ ∪( (g ( i))) ( (g ( j))) for some i j 2. Since∪ i i Q P, clearly τ ⊕ ∈ ▼ ⊗ τ ⊕ ( i i) g (Q P) R. However, based on the observation ( i i) g, the∪ learner Ψ Ψ τ ⊕ changes his mind infinitely often. This means that his prediction limn (( i i) g) 1 diverges. This contradicts our assumption that P ≤ω (Q▼P)⊗R via the learner Ψ. Thus, our claim is verified. □

31 Lemma 56 can be seen as an analogy of an observation of Blum-Blum [4] in the theory of inductive inference for total computable functions on N. Such ρ is sometimes called a locking sequence. ﬀ Θ NN ×

≤1 ρ⌢ ⌢⟨ ⟩⌢ ⊗ { } Φ . P 1 ( P m P) g via Ψ(g↾|ρ|⊕ρ) Our proof process in Theorem 27 is effective with respect to g, m, and an index of ΦΨ(g↾|ρ|⊕ρ) which are calculated from g, ρ, and an index of Ψ. To see this, recall our m = ∪ {ρ⌢⟨ ⟩ ρ ∈ } proof in Theorem 27. Define VP TP m : LP . g,ρ,m = m ⊗ { } g,ρ,m = Φ g,ρ,m . E0 VP g ; D0 Ψ(g↾|ρ|⊕ρ)(E0 ) g,ρ,m = m⌢ g,ρ,m ⊗ { } g,ρ,m = Φ g,ρ,m . Ei+1 (VP Di ) g ; Di+1 Ψ(g↾|ρ|⊕ρ)(Ei+1 ) ∪ g,ρ,m = g,ρ,m Then, as in the proof of Theorem 27, D i∈N Di+1 is a subtree of VP, and it has no dead ends. Moreover, this construction is clearly c.e. uniformly in g, ρ, and m. Therefore, we can effectively choose an element Θ(g, ρ, m) ∈ [Dg,ρ,m] ⊆ P, uniformly in g, ρ, and m. □

1 Now, a procedure to get P ≤ω R is follows. For given g ∈ Q, on the i-th challenge of a learner ∆, the learner ∆ chooses the lexicographically i-th least pair ⟨ρ, m⟩ ∈ 2

32 ≤1 ▼ ⊗ N ≡1 ▼ ≤1 N Proof. By Theorem 54, if P ω (P P) 2 1 P P, then P ω 2 , i.e., P contains a 1 computable element. As P is special, we must have P ≰ω P▼P. As seen in Part I [29, <ω 1 <ω Section 4], P ≤ω P▼P. Therefore, for Q = P▼P, we have Q <ω P ≡ω Q. □

1 Corollary 59. Every nonzero a ∈ Pω has the strong anticupping property. Proof. Fix P ∈ a. Let b be the (1, ω)-degree of P▼P. Then, by Theorem 54, for any (1, ω)-degree c, if a ≤ b ∨ c, then a ≤ c. □

The primary motivation of the second author behind introducing the notions of Π0 N learnability reduction was to attack an open problem on 1 subsets of 2 . The problem ω, Π0 (see Simpson [57]) is whether the Muchnik degrees (( 1)-degrees) of 1 classes are , Π0 dense. Cenzer-Hinman [13] showed that the Medvedev degrees ((1 1)-degrees) of 1 classes are dense. One can easily apply their priority construction to prove densities of (1, < ω)-degrees and (< ω, 1)-degrees. The reason is that the arithmetical complexity α = { , ∈ N2 ≤α } Σ0 α, β ∈ { , , , < ω , < ω, } of Aβ (i j) : Pi β P j is 3 for ( ) (1 1) (1 ) ( 1) , where 0 N {Pe}e∈N is an eﬀective enumeration of all Π subsets of 2 . It enables us to use a priority 1 α argument directly. However, for other reductions (α, β), the complexity of Aβ seems to Π1 {⟨ , ⟩ ≤ω } Π1 be 1. For instance, Cole-Simpson [17] showed that i j : Pi 1 P j is 1-complete. This observation hinders us from using priority arguments. Hence it seems to be a hard task to prove densities of such (α, β)-degrees. Nevertheless, our disjunctive notions turn out to be useful to obtain some partial results. Π0 , ⊆ N <1 <<ω Theorem 60 (Weak Density). For nonempty 1 sets P Q 2 , if P ω Q and P ω Q Π0 ⊆ N <1 <1 then there exists a 1 set R 2 such that P ω R ω Q. 1 <ω 1 1 Proof. Assume P <ω Q and P <ω Q. Let R = (Q▼Q) ⊗ P. Then P ≤ω R ≤ω Q. 1 1 Moreover Q ≰ω P implies Q ≰ω R = (Q▼Q) ⊗ P, by non-cupping property of ▼. 1 <ω <ω On the other hand, R = (Q▼Q) ⊗ P ≰ω P since Q▼Q ≡ω Q ≰ω P. Consequently, 1 1 P <ω R = (Q▼Q) ⊗ P <ω Q. □ One can introduce a transﬁnite iteration P▼(a) of hyperconcatenation along a ∈ O (see also the nested tape model introduced in Part I [29, Section 5.6]). Π0 ⊆ N , ∈ O < ▼(b) Proposition 61. For any special 1 set P 2 , if a b and a O b, then P does ▼(a) N ▼(a) 1 ▼(b) ▼(a) 1 not (1, ω)-cup to P , i.e., for any set R ⊆ N , if P ≤ω P ⊗ R then P ≤ω R.

a ▼(b) 1 ▼(2a) Proof. The assumption a

Fix again any notation omega ∈ O such that |Φomega(n)|O = n for each n ∈ N. Recall Ψ ∈ NN Φ from Part I that a learner is eventually-Popperian if, for every f , lims Ψ( f ↾s)( f ) is total whenever lims Ψ( f ↾ s) converges. Π0 N ⊆ NN ≤<ω Proposition 62. Let P be a special 1 subset of 2 . For any set R , if P ω ▼(omega) <ω P ⊗ R by a team of eventually-Popperian learners, then P ≤ω R.

33 <ω ▼(omega) Proof. If P ≤ω P ⊗ R via a team of eventually-Popperian learners, then this reduction is also witnessed by a team of n eventually-Popperian learners, for some n ∈ N. In particular, by modifying this reduction, we can easily construct a team of <ω ▼(n+1) n eventually-Popperian learners witnessing P ≤ω P ⊗ R. In this case, it is not ▼(n) ≤1 ▼(n+1) ⊗ ▼(n) ≤1 ≤<ω hard to show P 1 P R. By Theorem 54, P ω R. Hence, P ω R is witnessed by a team of n learners, as seen in Part I [29, Proposition 75]. □

Corollary 63. For every a ∈ O there exists a computable function g such that, for any Π0 1 index e, if Pe is special then the following properties hold. 1 1. Pg(e,b) <ω Pg(e,c) holds for every c

3.5. Infinitary Disjunctions along Infinite Complete Graphs The following is the last LEVEL 4 separation result, which reveals a difference <ω ω between [CT ]ω and [CT ]1 . Π0 , ⊆ N Π0 b ⊆ N Theorem 65. For every special 1 set P Q 2 there exists a 1 set P 2 such that <ω b b Q ≰ω P and P is (ω, 1)-equivalent to P. Π0 ⊆ N Proof. We construct a 1 set P 2 by priority argument with infinitely many requirements {Pe, Ge}e∈N. Each preservation (Pe-)strategy will injure our coding (G-)strategy b b of P into P infinitely often. In other words, for each Pe-requirement, P contains an el- ⋆ ∈ ⋆ ement fe which is a counterpart of each f P, but each fe has infinitely many noises. Indeed, to satisfy the P-requirements, we need to ensure that there is no uniformly ∈ ⋆ ∈ b team-learnable way to extract the information of f P from its code fe P. Never- G ∈ ⋆ ∈ b theless, the global ( -)requirement must guarantee that f P is computable in fe P {Ψe} = via a non-uniform way. Let i i

Π0 b ⊆ N ∪ { ⋆ ∈ N ∈ } Here, the desired 1 set P 2 will be of form P fe : e & f P .

34 Construction. We will construct a computable sequence of computable trees {Ts}s∈N, { } b and∪ a computable sequence of natural numbers hs s∈N. The desired set P is defined as [ s Ts], and hs is called active height at stage s. We will ensure that the tree Ts consists of strings of length ≤ hs. The strategy for the Pe-requirement acts on some string extending the e-th leaf ρe of TCPA. We will inductively define a string γe(α, s) ∈ Ts extending ρe for each s ∈ N and α ∈ T of height ≤ s. The map α 7→ lim γ (α, s) restricted to T ext will provide a tree- P ∪ s e P ext ext b ∩ ρ isomorphism between TP and ( s Ts) , i.e., P [ e] will be constructed as the set { γ α, α ∈ } ⋆ of all infinite∪ paths of the tree generated by lims e( s): TP . In other words, fe γ α, γ α, ∈ b is defined by α⊂ f lims e( s), and each string e( s) is an approximation of ge P witnessing to satisfy the Pe requirements. We will also define a finite set Me(α, s) ⊆ b for each s ∈ N and α ∈ TP of height ≤ s. Intuitively, Me(α, s) contains any index of the learner who have been already changed his mind |α| times along any string extending α of length s, and the string γe(α, s) also plays the role of an active node for learners in Me(α, s). To satisfy the Pe-requirement, each learner in Me(α, s) can act on γe(α, s) at stage s + 1, and then he extends γe(α, s) to some new string γe(α, s + 1) of length hs, and injures all constructions of γe(β, s + 1) for β ⊋ α. We assume that, for any α ∈ TP of length s, {Me(β, s)}β⊆α is a partition of {i ∈ N : i < b}.

Stage 0. At ﬁrst, put Ts = {⟨⟩}, hs = 0, Me(⟨⟩, 0) = {i ∈ N : i < b}, and γe(⟨⟩, 0) = ρe.

Stage s + 1. At the beginning of each stage s + 1, assume that Ts and hs are given, and that Me(β, s) and γe(β, s) have been already deﬁned for each s ∈ N and β ∈ TP of ≤ , ∈ N τ ∈ N i τ height s. For each i e and each 2 , the length-of-agreement function le( ) is the maximal l ∈ N such that ΦΨe τ (τ; x) ↓ for each x < l, and ΦΨe τ (τ) ∈ T . i ( ) i ( ) Q Fix a string α ∈ TP of length s, and then each i belongs to some Me(β, s) for β ⊆ α. Ψe γ β, Ψe In this case, the learner i can act on e( s). Then, we say that the learner i requires attention along α at stage s + 1 if there exists τ ∈ Ts of length hs extending γe(β, s) such that either of the following conditions are satisﬁed. Ψe γ β, , τ σ γ β, ⊊ σ ⊆ τ 1. i changes on ( e( s) ], i.e., there is a string such that e( s) and Ψe σ− , Ψe σ i ( ) i ( ). i τ > { i σ σ ⊆ γ β, } 2. or, le( ) max le( ): e( s) .

Let Rs be the set of all α ∈ TP of length s such that some learner requires attention along α at stage s + 1. For α ∈ Rs, let m(α) be the least m such that there is a string β ⊆ α ∈ β, Ψe α of length m and an index i Me( s) such that i requires attention along at + Ψe α stage s 1. That is to say, some learner i who has already changed his mind m( ) times requires attention.

Claim. For any α, β ∈ Rs, we have that α ↾ m(α) = β ↾ m(β) holds or α ↾ m(α) is incomparable with β ↾ m(β). ∗ = {α ↾ α α ∈ } β ∈ ∗ β ∈ α , Put Rs m( ): Rs . Then, for Rs, let i( ) be the least i M(m( ) s) Ψe α ⊇ β + such that i requires attention along some of length s at stage s 1. For β ∈ ∗ Ψe + β ∈ ∗ τ β Rs, we say that i(β) acts at stage s 1. Moreover, for Rs, let ( ) be the lexicographically least string τ ∈ Ts of length hs extending γe(β, s) such that τ

35 Ψe α ⊇ β witnesses that the learner i(β) requires attention along some of length s at stage + ∗∗ ⊆ ∗ β ∈ ∗ Ψe s 1. Then Rs Rs is deﬁned as the set of all Rs such that i(β) changes on (γe(β, s), τ(β)]. β ∈ ∗∗ β, + = β, \{ β } β⌢ , + = For each Rs , put Me( s 1) Me( s) i( ) , and put Me( i s 1) β, ∪ { β } β⌢ ∈ β < ∗∗ β, + = β, Me( s) i( ) for i TP. For any Rs , put Me( s 1) Me( s). For each β ∈ ∗ β⌢σ ∈ ≤ σ ∈

⌢ hs+1−|γe(α,s+1)| Ts+1 = Ts ∪ {σ ⊆ γe(α, s + 1) 0 : α ∈ TP & |α| = s + 1 & e ∈ N}. ∪ b = b Π0 N Finally, we set P [ s∈N Ts]. Clearly, P is a nonempty 1 subset of 2 .

Lemma 66. lims γe(α, s) converges for any e ∈ N and α ∈ TP.

Proof. Note that γe(α, s) is incomparable with γe(β, s) whenever α is incomparable with β. Therefore, γe(α, s) changes only when some learner in Me(β, s) acts for some β ⊆ α. Assume that γe(α, s) changes infinitely often. Then there is β ⊆ α, t ∈ N and ∈ β, ∈ β, ≥ Ψe i Me( t) such that i Me( s) for any s t, and i(β) acts infinitely often. However, α = γ α, ∈ β, by our construction, ge lims e( s) is computable. Additionally, since i Me( s) e α α for any s ≥ t, lim Ψ (g ↾ n) exists, and Φ Ψe α↾ (g ) ∈ Q. This contradicts our n i(β) e limn i(β)(ge n) e assumption that Q is special. □ ∪ ∈ ⋆ = γ α, ⋆ For f P, put fe α⊂ f lims e( s). By this lemma, such fe exists, and we b b = ∪ { ⋆ ∈ N ∈ } ∈ N observe that P can be represented as P P fe : e & f P . For each e and α ∈ TP, we pick t(e, α) ∈ N such that γe(α, s) = γe(α, t) for any s, t ≥ t(e, α). Lemma 67. The P-requirements are satisfied. <ω b Proof. Assume that P ≤ω P via the e-th team {Ψi}i

Lemma 68. The G-requirements are satisﬁed. ﬃ ≤ ⋆ ∈ N ∈ {Ψ } Proof. It su ces to show that f T fe for any e and f P. Assume that i i Set l maxi∈He( f ) i , and u maxi∈He( f ) t(e i). For n l, to compute f (n), we wait for stage v(n) > u such that, for every i < He( f ), i ∈ Me( f ↾ m, v(n)) for some m ≥ n+1.

36 (1,1)-computable v YYY (<ω,1)-computable P Yl YYYYYYidYY YYYYYYYYY, v YYY ▽ YYY (1,ω)-computable (1,<ω)-computable Yl YYidYY P YYYYYYYYY, ` v YYY YYY (<ω,ω)-computable (1,ω)-computable P Yl YYidYY YYYYYYYYY, v YYY ▼ YYYY (<ω,ω)-computable P Yl YYYYYidYYYY YYYYYY, ` (ω,1)-computable ∞ P

Π0 ⊆ N Figure 2: The dynamic proof model for a special 1 set P 2 .

By our construction, it is easy to see that we can extract f (n) from γe( f ↾ n + 1, v(n)), by a uniformly computable procedure in n. □ <ω b b d Thus, we have Q ≰ω P by Lemma 67, and P ⊆ P ⊆ Deg(P) by Lemma 68. Thus, b Π0 ≰<ω b ≡ω □ P is a 1 set satisfying Q ω P 1 P. This concludes the proof. Π0 , ⊆ N ≤<ω d Corollary 69. For any nonempty 1 sets P Q 2 , if Q ω Deg(P) then Q contains a computable element. <ω d Proof. Assume that Q ≤ω Deg(P) is satisﬁed. Suppose that Q has no computable , ⊆ N ≰<ω b ≡ω element. Then, for P Q 2 , we obtain Q ω P 1 P by Theorem 65. Note that b ≡ω b ⊆ d ≤<ω d ≤1 b the condition P 1 P implies P Deg(P). Then, Q ω Deg(P) 1 P. It involves a contradiction. □

4. Applications and Questions

4.1. Diagonally Noncomputable Functions A total function f : N → N is a k-valued diagonally noncomputable function if f (n) < k for any n ∈ N and f (e) , Φe(e) whenever Φe(e) converges. Let DNRk denote the set of all k-valued diagonally noncomputable functions. Jockusch [33] showed that every DNRk function computes a DNR2 function. However, he also noted that there is no uniformly computable algorithm ﬁnding a DNR2 function from any DNRk function. Theorem 70 (Jockusch [33]). >1 ∈ N 1. DNRk 1 DNRk+1 for any k . ≡ω ∈ N 2. DNR2 1 DNRk for any k . Proposition 71. , ω 1 NN , 1 1. If a (1 )-degree dω of subsets of contains a (1 1)-degree h1 of homogeneous 1 , 1 sets, then h1 is the greatest (1 1)-degree inside dω. < ω, <ω Π0 N , < ω 1 2. If an ( 1)-degree d1 of 1 subsets of 2 contains a (1 )-degree h<ω of Π0 1 , < ω <ω homogeneous 1 sets, then h<ω is the least (1 )-degree inside d1 .

37 < ω, Π0 N , 3. Every ( 1)-degree of 1 subsets of 2 contains at most one (1 1)-degree of Π0 homogeneous 1 sets. Proof. For the item 1, we can see that, for any P ⊆ NN and any closed set Q ⊆ NN, if ≤1 σ ∩ σ ≤1 ∩ σ P ω Q then there is a node such that Q [ ] is nonempty and P 1 Q [ ]. That σ ≤1 ≡1 ∩ σ is, is a locking sequence. If Q is homogeneous, then P 1 Q 1 Q [ ]. The item 2 follows from Theorem 20. By combining the item 1 and 2, we see that every (< ω, 1)- Π0 N , < ω Π0 degree of 1 subsets of 2 contains at most one (1 )-degree of homogeneous 1 , Π0 □ sets which contains at most one (1 1)-degree of homogeneous 1 sets. <<ω <1 Corollary 72. DNR3 1 DNR2, and DNR3 ω DNR2. <1 Proof. By Jockusch [33], we have DNR3 1 DNR2. Thus, Proposition 71 implies the desired condition. □

By analyzing Jockusch’s proof [33] of the Muchnik equivalence of DNR2 and DNRk for any k ≥ 2, we can directly establish the (< ω, ω)-equivalence of DNR2 and DNRk for any k ≥ 2. However, one may find that Jockusch’s proof [33] is essen- Σ0 tially based on the 2 law of excluded middle. Therefore, the fine analysis of this proof structure establishes the following theorem. ▼ <1 Theorem 73. DNRk DNRk 1 DNRk2 for any k. 2 Proof. As Jockusch [33], fix a computable function z : N → N such that Φz(v,u)(z(v, u)) 2 N = ⟨Φv(v), Φu(u)⟩ for any v, u ∈ N. Note that every g ∈ (k ) determines two functions N N g0 ∈ k and g1 ∈ k such that g(n) = ⟨g0(n), g1(n)⟩ for any n ∈ N. We define a uniform sequence {Γv}v∈N, ∆ of computable functions as Γv(g; u) = g1(z(v, u)), and ∆(g; v) = g0(z(v, uv)), where uv = min{u ∈ N : g1(z(v, u)) = Φu(u) ↓}. Fix g ∈ DNRk2 . Since ⟨g0(z(v, u)), g1(z(v, u))⟩ = g(z(v, u)) , ⟨Φv(v), Φu(u)⟩, either g0(z(v, u)) , Φv(v) , , Φ , ∈ N Σ0 or g1(z(v u)) u(u) holds for any v u . We consider the following 2 sentence:

(∃v)(∀u)(Φu(u) ↓ → g1(z(v, u)) , Φu(u)).

θ , Π0 ∀ Φ ↓ → , , Φ θ , Let (g v) denote the 1 sentence ( u)( u(u) g1(z(v u)) u(u)). If (g v) holds, then Γv(g; u) = g1(z(v, u)) , Φu(u) for any u ∈ N. Hence, Γv(g) ∈ DNRk. If ¬θ(g, v) holds, then uv is deﬁned. Therefore, ∆(g; v) = g0(z(v, uv)) ↓, Φv(v), since , = Φ ↓ ∆ g1(z(v uv)) uv (uv)) . Thus, (g; v) is extendible to a function in DNRk. This procedure shows that there is a function Γ : DNRk2 → DNRk that is computable strictly Π0 { } ∆ {Γ } = { θ , } along 1 sets S v v∈N via and v v∈N, where S v g : (g v) . Consequently, ▼ ≤1 DNRk DNRk 1 DNRk2 by Part I [29, Theorem 46]. ▼ ≱1 ▼ To see DNRk DNRk 1 DNRk2 , we note that DNRk DNRk is not tree-immune. By Cenzer-Kihara-Weber-Wu [12], DNRk▼DNRk does not cup to the generalized separating class DNRk2 . □ <ω Corollary 74. DNRk ≡ω DNR2 for any k ≥ 2. Indeed, for any k ∈ N, the direction <ω DNRk ≤ω DNRk2 is witnessed by a team of a conﬁdent learner and a eventually- ≡ω ≥ Popperian learner. In particular, DNRk 1 DNR2 for any k 2.

38 <ω Proof. As seen in Part I [29, Proposition 75], P ≤ω P▼P is witnessed by a team of a conﬁdent learner and a eventually-Popperian learner. Thus, Theorem 73 implies the desired condition. □

Corollary 75. There is an (< ω, ω)-degree which contains inﬁnitely many (1, 1)-degrees Π0 of homogeneous 1 sets.

Proof. By Corollary 74, the (< ω, ω)-degree of DNR2 contains DNRk for any k ∈ N, .1 , □ while DNRk 1 DNRl for k l.

4.2. Simpson’s Embedding Lemma For a pointclass Γ in a space X, we say that SEL(Γ, X) holds for (α, β)-degrees holds α, β Γ ⊆ Π0 ⊆ N for ( )-degrees if, for every set S X and for every nonempty 1 set Q 2 , Π0 ⊆ N ≡α ∪ there exists a 1 set P 2 such that P β S Q. Jockusch-Soare [34] indicates that Π0, NN ω, Π0, N SEL( 2 ) holds for ( 1)-degrees, and points out that SEL( 3 2 ) does not hold ω, N Π0 for ( 1)-degrees, since the set of all noncomputable elements in 2 is 3. Simpson’s Embedding Lemma [58] determines the limit of SEL(Γ, X) for (ω, 1)-degrees. Σ0, NN ω, □ Theorem 76 (Simpson [58]). SEL( 3 ) holds for ( 1)-degrees. Theorem 77 (Simpson’s Embedding Lemma for other degree structures). Σ0, N < ω, 1. SEL( 2 2 ) does not hold for ( 1)-degrees. Σ0, N , ω 2. SEL( 2 2 ) holds for (1 )-degrees. Π0, N , ω 3. SEL( 2 2 ) does not hold for (1 )-degrees. Π0, NN < ω, ω 4. SEL( 2 ) holds for ( )-degrees. Σ0, N < ω, ω 5. SEL( 3 2 ) does not hold for ( )-degrees. ` Π0 ⊆ N ⊆ N Σ0 Proof. (1) For any 1 set P 2 , we note` that P 2 is 2. By Theorem 48, there Π0 N < ω, Π0 N is no 1 set 2 which is ( `1)-below` P. In particular, there is no 1 set 2 which is (< ω, 1)-equivalent to P ∪ P = P. Σ0 ⊆ N { } (2) For a given 2 set S ∪2 , there is a computable∪ increasing sequence⊕ Pi i∈N Π0 = ≡1 ⊕of 1 classes such that S i∈N Pi. We need⊕ to show i∈N Pi ω i∈N P∪i, since , < ω Π0 −→ ≤1 ⊕i∈N Pi is (1 )-equivalent∪ to the 1 class i Pi. Then, it is easy to see i Pi 1 ∈ ↾ Ψ i Pi. For given f i Pi, from each initial segment f n, a learner guesses Φ = ⌢ an index of a computable function Ψ( f ↾n)∪(g) i g for the least number i such that ↾ ∈ ↾ < ∈ ∈ \ f n TPi but f n TPi−1 . For any f i Pi, for the least i such that f Pi Pi−1, ⌢ Ψ ↾ Φ Ψ ↾ = Φ Ψ ↾ ∈ limn ( f n) converges∪ to an index⊕ of limn ( f n)(g) i g. Thus, limn ( f n)(g) ⌢ = ≤1 −→ i Pi. Consequently, S i Pi ω i Pi. Π0 ⊆ N (3) Fix any special 1 set P 2 . By Jockusch-Soare [34], there is a noncom- Σ0 ⊆ N { } ⊆ N putable 1 set A such that P has no A-computable element. Then A 2 is a Π0 Σ0 ⊕ { } Π0 ffi 2 singleton, since A is 1. Therefore, P A is 2. It su ces to show that there is Π0 ⊆ N ≡1 ⊕ { } ≡1 ⊕ { } no 1 set Q 2 such that Q ω P A . Assume that Q ω P A is satisfied Π0 ⊆ N α ∈ for some 1 set Q 2 . Then Q must have an A-computable element Q. Fix a Ψ ⊕ { } ≤1 Φ α = ⌢ learner witnessing P A ω Q. Then, we have limn Ψ(α↾n)( ) 1 A, since P has no element computable in α ≤T A. We wait for s ∈ N such that Ψ(α ↾ t) = Ψ(α ↾ s) ≥ ≥ Φ α ↾ ↓= Π0 for any t s. Then, fix u s with Ψ(α↾u)( u; 0) 1. Consider the 1 set

39 Q∗ = { f ∈ Q ∩ [α ↾ u]:(∀v ≥ u) Ψ( f ↾ v) = Ψ( f ↾ u)}. Then, for any f ∈ Q∗, ⌢ 1 ∗ Φ Ψ ↾ = ΦΨ α↾ ⟨ ⟩ { } ≤ lims ( f s)( f ) ( u)( f ) must extends 1 . Thus, 1 A 1 Q via the computable Φ ∗ Π0 N function Ψ(α↾u). Since Q is special 1 subset of 2 , this implies the computability of 1⌢A which contradicts our choice of A. Π0 ⊆ NN Π0 b ⊆ NN (4) Fix a 2 set S . As Simpson’s proof, there is a 1 set S such that ≡1 b Π0 b ⊆ b▼ b ≤1 ∪ b S 1 S . We can ﬁnd a 1 set P S Q such that P 1 S Q, and P is computably Π0 ⊆ N ∪ ≤<ω b▼ ∪ ≡<ω homeomorphic to a 1 set P 2 . Since S Q ω S Q, we have S Q ω P. Π0 ⊆ N d = { ∈ N ∃ ∈ (5) For every 1 set P 2 , the Turing upward closure Deg(P) g 2 :( f ≤ } Σ0 d < ω, ω ω P) f T g of P is 3, and Deg(P) has the least ( )-degree inside deg1 (P). By Π0 N < ω, ω d □ Theorem 65, there is no 1 subset of 2 which is ( )-equivalent to Deg(P).

4.3. Weihrauch Degrees The notion of piecewise computability could be interpreted as the computability relative to the principle of excluded middle in a certain sense. Indeed, in Part I [29, Section 6], we have characterized the notions of piecewise computability as the computability relative to nonconstructive principles in the context of Weihrauch degrees. Thus, one can rephrase our separation results in the context of Weihrauch degrees as follows. Π0 N Theorem 78. The symbols P, Q, and R range over all special 1 subset of 2 , and X ranges over all subsets of NN. 1 1 1. There are P and Q ≤ P such that P ≤Σ0 Q but P ≰ Q. 1 1-LLPO 1 1 2. There are P and Q ≤ P such that P ≤Σ0 Q but P ≰Σ0 Q. 1 1-LEM 1-LLPO 1 3. For every P, there exists Q ≤ P such that P ≤Σ0 Q, whereas, for every X, if 1 1-LEM 1 P ≤Σ0 Q ⊗ X then P ≤ X. 1-DNE 1 1 4. There are P and Q ≤ P such that P ≤∆0 Q but P ≰Σ0 Q. 1 2-LEM 1-LEM 1 5. There are P and Q ≤ P such that P ≤Σ0 Q but P ≰∆0 Q. 1 2-LLPO 2-LEM 6. There is P such that, for every Q, if P ≤Σ0 Q, then P ≤Σ0 Q. 2-LLPO 1-LEM 1 7. For every P and R, there exists Q ≤ P such that P ≤Σ0 Q but R ≰Σ0 Q. 1 2-DNE 2-LLPO 1 8. For every P, there exists Q ≤ P such that P ≤Σ0 Q, whereas, for every X, if 1 2-LEM P ≤Σ0 Q ⊗ X then P ≤Σ0 X. 2-DNE 2-DNE 9. For every P and R, there exists Q ≤ P such that P ≤Σ0 Q but R ≰Σ0 Q. 3-DNE 2-LEM Proof. See Part I [29, Section 6] for the deﬁnitions of partial multivalued functions and their characterizations. (1) By Corollary 5. (2) By Corollary 9. (3) By Corollary 32. (4) By Corollary 13 (2). (5) By Corollary 15 (1). (6) By Theorem 20. (7) By Corollary 53 (2). (8) By Corollary 58. (9) By Theorem 65. □ Σ0 / Deﬁnition 79 (Mylatz [47]). The 1 lessor limited principle of omniscience with (m k) Σ0 wrong answers, 1-LLPOm/k, is the following multi-valued function. Σ0 ⊆ NN ⇒ , 7→ { < ∀ ∈ N + = }. 1-LLPOm/k : k x l k :( n ) x(kn l) 0 Σ0 = { ∈ NN , , ∈ N} Here, dom( 1-LLPOm/k) x : x(n) 0 for at most m many n .

40 Σ0 Remark. It is well-known that the parallelization of 1-LLPO1/2 is equivalent to Weak Konig’s¨ Lemma, WKL (hence, is Weihrauch equivalent to the closed choice for Cantor space, C2N ). Deﬁnition 80. ∏ ⊆ N , = 1. (Cenzer-Hinman [14]) A set P k is (m k)-separating if P n∈N Fn for { } Π0 ⊆ \ ≤ some uniform sequence Fn n∈N of 1 sets Fn k, where #(k Fn) m for any n ∈ N. 2. A function f : Nm → k is k-valued m-diagonally noncomputable in α ∈ NN ⟨ ,..., ⟩ {Φ α ⟨ ,..., ⟩ < } if the value f ( e0 em−1 ) does not belong to ei ( ; e0 em−1 ): i m m for each argument ⟨e0,..., em−1⟩ ∈ N . By DNRm/k(α), we denote the set of all k-valued functions which are m-diagonally noncomputable in α. N N 3. The (m/k) diagonally noncomputable operation DNRm/k : N ⇒ k is the multi- N valued function mapping α ∈ N to DNRm/k(α).

Remark. Clearly DNRm/k(∅) is (m, k)-separating. The structure of Medvedev degrees of (m, k)-separating sets have been studied by Cenzer-Hinman [14]. Diagonally noncomputable functions are extensively studied in connection with algorithmic randomness, for example, see Greenberg-Miller [25].

Σ0 [ Proposition 81. DNRm/k is Weihrauch equivalent to 1-LLPOm/k. ⊕ Σ0 [ ≤ ∈ N i Proof. To see 1-LLPOm/k W DNRm/k, for given (xi : i ), let et be an i∈N xi- computable index of an algorithm, for any argument, which returns l at stage s if l ∈ ∗ ∗ ∗ Ls+1 \ Ls and #Ls = t, where Ls = {l < k :(∃n) kn + l < s⊕& xi(kn + l ) , 0}. Clearly, {ei : i ∈ N & t < m} is computable uniformly in x . For any ⊕t i∈N i ∈ 7→ ⟨ i ,..., i ⟩ Σ0 [ ⟨ f DNRm/k( i∈N xi), the function i f ( e0 em−1 ) belongs to 1-LLPOm/k( xi : N m i ∈ N⟩). Conversely, for given x ∈ N , for the i-th m-tuple ⟨e0,..., em−1⟩ ∈ N , + = Φ ⟨ ,..., ⟩ < ∈ N we set xi(ks l) 1 if et ( e0 em−1 ) converges to l k at stage s for some t < m, and otherwise we set xi(ks + l) = 0. Clearly {xi : i ∈ N} is uniformly ⟨ ∈ N⟩ ∈ Σ0 [ ⟨ ∈ N⟩ ⊆ N computable in x. Then, for any li : i 1-LLPOm/k( xi : i ) k , we have < {Φ ⟨ ,..., ⟩ < } li et ( e0 em−1 ): t m by our construction. Hence, the k-valued function i 7→ li is m-diagonally noncomputable in x. □ Recall from Part I [29, Section 6] that ⋆ is the operation on Weihrauch degrees such ∗ ∗ ∗ ∗ that is deﬁned by F ⋆ G = max{F ◦ G : F ≤W F & G ≤W G}. See [53] for more information on ⋆.

Corollary 82. Let k ≥ 2 be any natural number. Σ0 [ ≰ Σ0 ⋆ Σ0 [ 1. 1-LLPO1/k W 2-DNE 1-LLPO1/k+1. Σ0 [ ≰ Σ0 ⋆ Σ0 [ 2. 1-LLPO1/k W 2-LLPO 1-LLPO1/k+1. Σ0 [ ≤ Σ0 ⋆ Σ0 [ 3. 1-LLPO1/k W 2-LEM 1-LLPO1/k+1. Proof. By Corollary 72 and Proposition 81, the item (1) and (2) are satisﬁed. It is not hard to show the item (3) by analyzing Theorem 73. □

41 Remark. By combining the results from Cenzer-Hinman [14] and our previous results, we can actually show the following. Σ0 [ ≰ Σ0 ⋆ Σ0 [ < < < ⌈ / ⌉ 1. 1-LLPOn/l W 2-DNE 1-LLPOm/k, whenever 0 n l k m . Σ0 [ ≰ Σ0 ⋆ Σ0 [ < < < ⌈ / ⌉ 2. 1-LLPOn/l W 2-LLPO 1-LLPOm/k, whenever 0 n l k m . Σ0 [ ≤ Σ0 ⋆ Σ0 [ < < < < 3. 1-LLPOn/l W 2-LEM 1-LLPOm/k, whenever 0 n l and 0 m k. Σ0 These results suggest, within some constructive setting, that the 2 law of excluded ﬃ Σ0 [ → Σ0 [ middle is su cient to show the formula 1-LLPOm/k 1-LLPOn/l, whereas neither Σ0 Σ0 the 2 double negation elimination nor the 2 lessor limited principle of omniscience is suﬃcient.

Corollary 83. DNR2 ≤Σ0 DNR3; DNR2 ≰Σ0 DNR3; DNR2 ≰Σ0 DNR3; 2-LEM 2-LLPO 2-DNE MLR ≤Σ0 DNR3; and MLR ≰Σ0 DNR3. Here, MLR denotes the set of all 2-LEM 2-DNE Martin-L¨ofrandom reals. Proof. For the ﬁrst three statements, see Corollary 72 and Theorem 73. It is easy to 1 see that MLR ≤ DNR2 ≤Σ0 DNR3. It is shown by Downey-Greenberg-Jockusch- 1 2-LEM ≰1 Millans [20] that MLR 1 DNR3. By homogeneity of DNR3 and Proposition 71, we have MLR ≰Σ0 DNR3. □ 2-DNE

4.4. Some Intermediate Lattices are Not Brouwerian Recall from Medvedev’s Theorem [41], Muchnik’s Theorem [46], and Part I [29, D1 D1 Dω Proposition 16] that the degree structures 1, ω, and 1 are Browerian. Indeed, we 1 have already observed that one can generate Dω from a logical principle so called the Σ0 D1 D1 D<ω 2-double negation elimination. Though <ω, ω|<ω and 1 are also generated from D1 certain logical principles over 1 as seen before, surprisingly, these degree structures are not Brouwerian. D1 D1 D<ω P1 P1 P<ω Theorem 84. The degree structures <ω, ω|<ω, 1 , <ω, ω|<ω, and 1 are not Brouwerian. A , = { ⊆ NN ≤1 ⊗ } B , = { ⊆ NN ≤<ω ⊗ } Put (P Q) R : Q <ω P R , and (P Q) R : Q 1 P R . Note that A(P, Q) ⊆ B(P, Q). Then we show the following lemma. Π0 , ⊆ N { } Π0 N Lemma 85. There are 1 sets P Q 2 , and a collection Ze e∈N of 1 subsets of 2 ∈ A , ∈ B , ≰ω such that Ze (P Q), and that, for every R (P Q), we have R 1 Ze for some e ∈ N. { } Π0 N Proof. By Theorem⊕ 17, we have a collection S i i∈N of nonempty 1 subsets of 2 ≰ ∈ ∈ N such that xk T j,k x j for any choice xi S i, i . Consider the following sets.

⌢ ⌢ ⌢ ⌢ P = CPA {S ⟨ , ⟩ S ⟨ , ⟩ ... S ⟨ , ⟩} ∈N, Z = S ⟨ , + ⟩, e 0 e 1 e e e  e e e 1  ⊗ , ≤ , S ⟨e,i⟩ Ze if i e ⌢  Q = CPA {Qn}n∈N, where Q⟨e,i⟩ = (P \ [ρe]) ⊗ Ze, if i = e + 1,  ∅, otherwise.

42 Here, ρe is the e-th leaf of the corresponding computable tree TCPA for CPA. To see Ze ∈ A(P, Q), choose an element f ⊕ g ∈ P ⊗ Ze. If f ↾ n ∈ TCPA or f ↾ n extends a leaf except ρe, our learner Ψ(( f ↾ n) ⊕ g) guesses an index of the identity function. If ↼|ρe| f ↾ n extends ρe, then Ψ first guesses ΦΨ(( f ↾n)⊕g)( f ⊕ g) = ( f ) ⊕ g. By continuing ⌢ 0⌢ 1⌢ ⌢ i⌢ j this guessing procedure, if f ↾ n is of the form ρe τ τ ... τ τ such that τ is a leaf of S ⟨e, j⟩ for each j ≤ i, and τ does not extend a leaf of S ⟨e, j+1⟩, then Ψ guesses 0 i ↼(|ρe|+|τ |+···+|τ |) ΦΨ(( f ↾n)⊕g)( f ⊕ g) = ( f ) ⊕ g. Note that i < e, since f ∈ P. It is easy to see 1 that Q ≤<ω P ⊗ Ze via the learner Ψ, where #{n ∈ N : Ψ(( f ⊕ g) ↾ n + 1) , Ψ(( f ⊕ g) ↾ n)} ≤ e + 1. Therefore, Ze ∈ A(P, Q). ∈ B , ≤<ω ⊗ ∈ N ⊕ ∈ ⊗ Fix R (P Q). As Q 1 P R, there is b such that, for every f g P R, we must have Φe( f ⊕ g) ∈ Q for some e < b. Suppose for the sake of contradiction ≤ω ∈ ∈ ≤ ∈ that R 1 Zb+1. Then, for any h Zb+1, we have g R with g T h. Pick f0 ρ ⌢ ⊂ ∩ ρ ∈ B , < Φ ⊕ ∈ b+1 S ⟨b+1,0⟩ P [ b+1]. Since R (P Q), there is e0 b such that e0 ( f0 g) Q. By our choice of {S n}n∈N and the property g ≤T h ∈ Zb+1 = S ⟨b+1,b+2⟩, if e , b + 1 , ⊕ Φ ⊕ or i 0, then Q⟨e,i⟩ has no ( f0 g)-computable element. Therefore, e0 ( f0 g) have ρ σ ⊂ Φ σ ⊕ ⊇ ρ to extend ⟨b+1,0⟩. Take an initial segment 0 f0 determining e0 ( 0 g) ⟨b+1,0⟩. 0 ⌢ 0⌢ Extend σ0 to a leaf τ of S b+1,0, and choose f1 ∈ ρ τ S b+1,1 ⊂ P. Again we have < Φ ⊕ ∈ Φ ⊕ ρ e1 b such that e1 ( f1 g) Q. As before, e1 ( f1 g) have to extend ⟨b+1,1⟩. However, ρ⟨b+1,1⟩ is incomparable with ρ⟨b+1,0⟩. Hence, we have e1 , e0. Again take σ ⊂ σ Φ σ ⊕ ⊇ ρ an initial segment 1 f1 extending 0 and determining e1 ( 1 g) ⟨b+1,1⟩. By iterating this procedure, we see that R requires at least b + 1 many indices ei. This ≰<ω □ contradicts our assumption. Therefore, R 1 Zb+1. 0 Proof of Theorem 84. Let P, Q, and {Ze}e∈N be Π sets in 85. Fix (α, β) ∈ {(1, < α 1 ω), (1, ω| < ω), (< ω, 1)}. To see Dβ is not Brouwerian, it suffices to show that there α α is no (α, β)-least R satisfying Q ≤β P ⊗ R. If R satisfies Q ≤β P ⊗ R, then clearly ∈ B , ≤α ≤<ω ≰α ∈ N R (P Q) since β is stronger than or equals to 1 . Then, R β Ze for some e . α α 1 Moreover, Ze ∈ A(P, Q) implies Q ≤β P⊗Ze, since ≤β is weaker than or equals to ≤<ω. α Hence R is not such a smallest set. By the same argument, it is easy to see that Pβ is Π0 □ not Brouwerian, since Ze is 1. <ω <ω Theorem 86. Dω and Pω are not Brouwrian. Moreover, the order structures in- N 0 N duced by (P(N ), ≤Σ0 ) and (the set of all nonempty Π subsets of 2 , ≤Σ0 ) are 2-LEM 1 2-LEM not Brouwrian. { } Π0 N Lemma 87.∪Let S i i≤n be a collection of 1 subsets of 2 with the property for each i ≤ n that , S has no element computable in x ∈ S . Then, there is no (n, ω)- k i k ▼ ⊕ i i computable function from i≤nS i to i≤n S i. ⊕ , ω ▼ Proof. Assume the existence of an (n )-computable function from i≤nS i to i≤n S i {Ψ } , ω which is identified by n many learners i i

43 paths through the following tree TE. ( ( ( ) )) = ▼ ▼ ... ▼ E ... . TE σ∈T E σ∈T E σ∈T E [Tn ] 0  1 n−1  T ext, if i ∈ E, Here, T E =  S i i some ﬁnite subtree of T ext, otherwise. S i

ext Here, the choice of “some finite subtree of TS ” depends on the context, and is implic- i ⊆ + ♡ itly determined when E is defined. For each E n 1, clearly S E is a closed∪ subset of ▼ ♡ + { ∗} ♡ ∗ i≤nS i. Divide S n+1 into n 1 many parts S i i≤n, where S n+1 is equal to i≤n S i , and ∗ each S i is degree-isomorphic to S i. ≤ σ ♡ For each i n, check whether there is a string extendible in S n+1 such that ♡ ∩ ∩ σ ∗ ≤ ≤ S n+1 Di [ ] is contained in S j for some j n. If yes, for such a least i n, choose a witness σ0 = σ, and put A0 = {i}, and B0 = { j}. Then, for such j ∈ B0, “some finite subtree of T ext” is choosen as the set of all strings η used in σ as a part of T ext in the S j 0 S j ext sense of the definition of ▼ ≤ S , or successors of such η in T . Note that σ is also i n i S j 0 ♡ < σ extendible in S (n+1)\{ j}. Inductively, for some s n assume that s, As, and Bs has been already defined. For each i < As, check whether there is a string σ ⊇ σs extendible in ♡ ♡ ∗ S such that S ∩Di ∩[σ] is contained in S for some j < Bs. If yes, for such (n+1)\Bs (n+1)\Bs j a least i < As, choose a witness σ0 = σ, and put As+1 = As ∪ {i}, and Bs+1 = Bs ∪ { j}. ext As before, for such j ∈ B + , “some finite subtree of T ” is choosen as the set of all s 1 S j ext ext strings η used in σ + as a part of T , or successors of such η in T . Note that σ + s 1 S j S j s 1 ♡ is also extendible in S + \ . If no such i < As exists, finish our construction of σ, A, (n 1) Bs+1 ∗ and B. Then, put A = As, B = Bs, and define σ to be the last witness σs. Put A− = n \ A and B− = (n + 1) \ B. Note that #A− + 1 = #B−, since #A = #B. ∈ ♡ ∩ σ∗ , Therefore, B contains at least⊕ one element. By our assumption, for any x S B− [ ] ∅ ∈ ∈ − − , we must have Fi(x) i≤n S i for some i A . Thus, A is nonempty. Fix a sequence α ∈ (A−)N such that, for each i ∈ A−, there are infinitely many ∗ ∗ n ∈ N such that α(n) = i. First set τ0 = σ . Inductively assume that τs ⊇ σ has been already defined. By our definition of σ∗, A and B, if ξ extends σ∗, then the set ♡ ∩ ∩ ξ ∗ ∈ − S B− Dα(s) [ ] intersects with S j for at least two j B . Therefore, we can choose ∈ ♡ ∩ ∩ τ ∩ ∗ , ∅ ∈ − = x S B− Dα(s) [ s] S j for some j B . Then, Fα(s)(x; 0) j, by our assumption { } τ∗ τ ⊆ τ∗ ⊂ τ∗ = of S i i≤n. Find a string s such that s s x and Fα(s)( s; 0) j. Again, we can ∗ ∈ ♡ ∩ ∩ τ∗ ∩ ∗ , ∅ ∈ − \{ } choose x S B− Dα(s) [ s] S k for some k B j . Then, we must have ∗ = , τ τ∗ ⊆ τ ⊂ ∗ τ = Fα(s)(x ; 0) k j. Let s+1 be a string such that s s+1 x and Fα(s)( s+1; 0) k. τ τ Ψ Therefore, between∪ s and s+1, the learner α(s) changes his mind. = τ ♡ ♡ Define y s s. Then y is contained in S B− , since S B− is closed. However, for − each i ∈ A , by our construction of y, the value Fi(y) does not converge. Moreover, for − ∗ ⊕each i < A , by our definition of A, B, and σ ⊂ y, even if Fi(y) converges, Fi(y) < , ω ▼ ⊃ ♡ ⊕i≤n S i. Consequently, there is no (n )-computable function from i≤nS i S B− to □ i≤n S i as desired. N N <ω Put J(P, Q) = {R ⊆ N : Q ≤Σ0 P⊗R}, and K(P, Q) = {R ⊆ N : Q ≤ω P⊗R}. 2-LEM Note that J(P, Q) ⊆ K(P, Q). Then we show the following lemma.

44 Π0 , ⊆ N { } Π0 N Lemma 88. There are 1 sets P Q 2 , and a collection Ze e∈N of 1 subsets of 2 ∈ J , ∈ K , ≰ω such that Ze (P Q), and that, for every R (P Q), we have R 1 Ze for some e ∈ N. { } Π0 N Proof. By Theorem⊕ 17, we have a collection S i i∈N of nonempty 1 subsets of 2 ≰ ∈ ∈ N such that xk T j,k x j for any choice xi S i, i . Consider the following sets.

⌢ P = CPA {S ⟨ , ⟩▼S ⟨ , ⟩▼ ... ▼S ⟨ , ⟩} ∈N, Z = S ⟨ , + ⟩, e 0 e 1 e e e  e e e 1  ⊗ , ≤ , S ⟨e,i⟩ Ze if i e ⌢  Q = CPA {Qn}n∈N, where Q⟨e,i⟩ = (P \ [ρe]) ⊗ Ze, if i = e + 1,  ∅, otherwise.

Here, ρe is the e-th leaf of the corresponding computable tree TCPA for CPA. To see 0 Ze ∈ J(P, Q), for f ⊕g ∈ P⊗Ze, by using Σ -LEM, check whether f does no extend ρe. ⌢ 1 If no, outputs ρe,e+1 ( f ⊕ g). If f extends ρe, it is not hard to see that an ﬁnite iteration 0 ⌢ of Σ -LEM can divide (ρe S ⟨e,0⟩▼S ⟨e,1⟩▼ ... ▼S ⟨e,e⟩) ⊗ Ze into {S ⟨e,i⟩ ⊗ Ze}i≤e. 2 ω Fix R ∈ K(P, Q). As Q ≤<ω P ⊗ R, some (b, ω)-computable function F maps P ⊗ R ≤ω ∈ into Q. Suppose for the sake of contradiction that R 1 Zb. Then, for∪ any h Zb, we ∈ ≤ ∩ ρ ⊗ { } ∩ ρ have g R with g T h. Then, F maps (P [ b]) g into Q ( i≤b∪⟨b,i⟩) by our { } ∩ ρ ⊗{ } ≡1 ▼ ⊗{ } ∩ ρ ≡1 choice⊕ of S n n∈N. Note that (P [ b]) g 1 ( i≤bS ⟨e,i⟩) g , and Q ( i≤b ⟨b,i⟩) 1 ⊗ , ω □ ( i≤b S ⟨e,i⟩) Ze. Therefore, by Lemma 87, F is not (b )-computable. { } Π0 Proof of Theorem 86. Let P, Q, and Ze e∈N be 1 sets in 88. Then, by the same argument as in the proof of Theorem 84, it is not hard to show the desired statement. □ Corollary 89. If (α, β) ∈ {(1, 1), (1, ω), (ω, 1)}, and (γ, δ) ∈ {(1, < ω), (1, ω| < ω), (< α γ ω, 1), (< ω, ω)}, then, there is an elementary diﬀerence between Dβ and Dδ , in the language of partial orderings {≤}. D1 D1 Dω Proof. Recall that the degree structures 1, ω, and 1 are Browerian, i.e., they satisfy the following elementary formula ψ in the language of partial orders.

ψ ≡ (∀p, q)(∃r)(∀s)(p ≤ q ∨ r &(p ≤ q ∨ s → r ≤ s)).

Here, the supremum ∨ is ﬁrst-order deﬁnable in the language of partial orders. On the D1 D D<ω D<ω other hand, by Theorem 84 and 86, <ω, ω|<ω, 1 , and ω are not Brouwerian, i.e., they satisfy ¬ψ. □

4.5. Open Questions Question 90 (Small Questions). 1 1. Determine the intermediate logic corresponding the degree structure Dω, where D1 Dω recall that 1 and 1 are exactly Jankov’s Logic. Π0 , ⊆ N ≤1 | | 2. Does there exist 1 sets P Q 2 with P ω Q such that there is no a -bounded Γ → ∈ O Π0 b learnable function : Q P for any a ? For a 1 set P in Theorem 48, does there exist a function Γ : Pb → P (1, ω)-computable via an |a|-bounded learner for some notation a ∈ O?

45 Π0 , ⊆ N Γ ▼ → 3. Does there exist a pair` of special 1 sets P Q 2 with a function : Q P Q ⊕ P (or Γ : Q▽∞ P → Q ⊕ P) which is learnable by a team of confident learners (or a team of eventually-Popperian learners)? Π0 N ≤1 ≤1 4. Let P0,P1,Q0, and Q1 be 1 subsets of 2 with Q0 ω Q1 and P0 ω P1. Then, 1 1 does ⟦P0 ∨ Q0⟧Σ0 ≤ω ⟦P1 ∨ Q1⟧Σ0 hold? Moreover, if Q0 ≤ω Q1 is witnessed by 2 2 1 an eventually Lipschitz learner, then does P0▼Q0 ≤ω P1▼Q1 hold? ≤ω ≤<ω ≤1 5. Compare the reducibility tt,1 and other reducibility notions (e.g., tt,1, <ω, ≤1 ≤<ω ≤1 Π0 N ω|<ω, 1 and ω) for 1 subsets of Cantor space 2 . Question 91 (Big Questions). α 1. Are there elementary differences between any two different degree structures Dβ|γ α′ α α′ and Dβ′|γ′ (Pβ|γ and Pβ′|γ′ )? ▽ D1 2. Is the commutative concatenation first-order definable in the structure 1 or P1 1? α 3. Is each local degree structure Pβ|γ first-order definable in the global degree struc- α ture Dβ|γ? 1 4. Is the structure Pω dense? 5. Investigate properties of (α, β|γ)-degrees a assuring the existence of b > a with the same (α′, β′|γ′)-degree as a. 6. Investigate the nested nested model, the nested nested nested model, and so on. 7. Does there exist a natural intermediate notion between (< ω, ω)-computability ω, Π0 (team-learnability) and ( 1)-computability (nonuniform computability) on 1 sets? 8. (Ishihara) Define a uniform (non-adhoc) interpretation (such as the Kleene re- alizability interpretation) translating each intuitionistic arithmetical sentence (e.g., (¬¬∃n∀mA(n, m)) → (∃n∀mA(n, m))) into a partial multi-valued function Σ0 ⊆ NN ⇒ NN (e.g., 2-DNE : ). Acknowledgements. The authors were partially supported by Grant-in-Aid for JSPS fellows. The second author (Kihara) would like to thank Douglas Cenzer, Hajime Ishihara, Dick de Jongh, Arno Pauly, and Albert Visser, for valuable comments and helpful discussion, and the second author also would like to thank Makoto Tatsuta and Yoriyuki Yamagata for introducing him to the syntactical study on Limit Computable Mathematics. The second author is also grateful to Sam Sanders who helped his En- glish writing. Finally, the authors would like to thank the anonymous referees for their valuable comments and suggestions.

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50 Index α [CT ]β|γ, 8 CPA, 11 α [CT ]β , 8 DNRm/k, 41 ω [CT ] , 4 DNRm/k(α), 41 <ω1 [CT ]ω , 4 DNRk, 37 <ω [CT ]1 , 4 MLR, 28 1 [CT ]<ω, 4 indxΨ, 8 1 [CT ]ω|<ω, 4 mc, 9 1 ≥ [CT ]ω, 4 mcΨ( m), 15 1 [CT ] , 4 mclΨ, 8 1 σ [σ], 7 pri( ), 9 [T], 7 write, 10 ⊕ #∨mclΨ, 8 f g, 7 ⟦ ⟧ P ≤F Q, 8 ∨i∈I Pi CL, 10 ⟦ ⟧ P ⊕ Q, 7 ∨i∈I Pi Int, 9 ⊗ ⟦ ∈ P ⟧ , 10 P Q, 7 ⊓ i I i LCM[n] ⌢ `, 7 P Q, 10 ⌢{ } ` P, 28 P Qi i∈N, 10 P▽Q, 10 ⊕n∈N Pn, 28 −→ P▽, 26 i∈NQi, 10 ⌢ (τ) [`, 7 ] P , 29 P(a), 27 [`∞] i∈I Pi, 10 P▼(a), 33 n ∈ Pi, 10 ≤ ≤α i I P T g, 12 β|γ, 8 → D/F P Q, 14 , 8 ⊓ Dα , 8 P Q, 14 β|γ † P/F , 3, 8 P0 P1, 13 ▽ Pα , 4, 8 P0 ∞P1, 10 β|γ ▽ Φ(σ; n), 7 P0 ωP1, 10 ▽ σ−, 7 P0 nP1, 10 Σ0 Q▼P, 11 1-LLPOm/k, 40 ▽ T ext, 7 ∞, 5 ♡ ▽ T∗ , 10 ω, 5 ♡ T▼ , 31 ▽n, 5 T ⌢T , 10 APΨ, 15 P Q X

51 commutative, 10 degree-isomorphic everywhere, 20 hyper-, 11 eﬀective enumeration of, 7 inﬁnitary, 10 everywhere (ω, 1)-incomparable, 15 non-commutative, 10 homogeneous, 20 consistency set, 10 independent, 12 contiguous perfect, 12 (α∗, β∗ | γ∗)-, 23 special, 5, 7, 13 uniform sequence of, 7 degree, 8 derivative, 27 recursive meet, 10 delayed, 29 diagonally noncomputable, 22 strong anticupping property, 6, 26 (m/k)-operation, 41 k-valued, 37 timekeeper, 29 k-valued m-, 41 tree, 7 dead end of, 7 heart, 10 extendible in, 7 of T▼, 31 immediate predecessor in, 7 hyperconcatenation, 11 leaf of, 7 tree-immune, 26 i-th projection, 9 Turing independent, 17 immune, 14, 22

LCM disjunction, 28 leaner m-changing point of, 15 learner, 8 anti-Popperian point of, 15 eventually-Popperian, 33 mind-change location of, 8 lessor limited principle of omniscience with (m/k) wrong answers, 40 LEVEL 1 separation, 5 LEVEL 2 separation, 5 LEVEL 3 separation, 5 LEVEL 4 separation, 5 locking sequence, 32 noncupping, 28 number of times of mind-changes, 9 pairwise everywhere independent, 17 Π0 1 set, 7 (m, k)-separating, 41 antichain, 12 computable sequence of, 7 corresponding tree of, 7