Inside the Muchnik Degrees I: Discontinuity, Learnability, And

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0 many computable functions with Π1 domains. In this paper, we introduce various concepts of dynamic- approximability, and then, we characterize these concepts as static-approximability. Now, we focus our attention on the concepts lying between (uniform) computability and nonuniform computability. In 1950-60th, Medvedev [46] and Muchnik [48] introduced the degree structure induced by uniform and nonuniform computability to formulate semantics for the intuitionistic propositional calculus based on Kolmogorov’s idea of interpreting each proposition as a problem. The degree structure induced by the Medvedev (Muchnik) reduction forms a Brouwer algebra (the dual of a Heyting algebra), where the (intuitionistic) disjunction is interpreted as the coproduct of subsets of Baire space. Our objective is to reveal the hidden relationship between the hierarchy of nonuniformly computable functions and the hierarchy of disjunction operations. When a certain suitable disjunction-like operation such as the coproduct is introduced, we will see that one can recover the associated degree structure from the disjunction operation. As a consequence, we may understand the noncomputability feature of functions by observing the degree-theoretic behavior of associated disjunction operations. This phenomenon can be explained by using the terminology of Galois connections or adjoint functors. For instance, one can introduce a disjunction operation on Baire space using the limit-BHK interpretation of Limit Com- putable Mathematics [31] (abbreviated as LCM), a type of constructive mathematics based on Learning Theory, whose positive arithmetical fragment is characterized as Heyting arithmetic with the recursive 0 ω-rule and the Σ1 law of excluded middle [6, 73]. Then, the “limit-BHK disjunction” includes all the information about the reducibility notion induced by learnable functions on Baire space. Furthermore, in this paper, we introduce more complicated disjunction-like operations using BHK- like interpretations represented as “dynamic proof models” or “nested models”. For instance, a dynamic disjunction along a well-founded tree realizes the concept of learnability with ordinal-bounded mind changes, and a dynamic disjunction along an ill-founded tree realizes the concept of decomposability 0 into countably many computable functions along a Σ2 formula. We also interpret these results in the context of the Weihrauch degrees and Wadge-like games. We introduce a partial interpretation of nonconstructive principles including LLPO and LPO in the Weihrauch degrees and characterize the noncomputability/discontinuity level of nearly computable functions using these principles.

1.2. Results. In section 2, we introduce the notion of (α, β|γ)-computability for partial functions on NN, for each ordinal α, β, γ ≤ ω. Then, the notion of (α, β|γ)-computability induces just seven classes closed under composition. C 1 N • [ T ]1 denotes the set of all partial computable functions on N . C 1 N • [ T ]<ω denotes the set of all partial functions on N learnable with bounded mind changes. C 1 N • [ T ]ω|<ω denotes the set of all partial functions on N learnable with bounded errors. C 1 N • [ T ]ω denotes the set of all partial learnable functions on N . C <ω N • [ T ]1 denotes the set of all partial k-wise computable functions on N for some k ∈ N. C <ω N • [ T ]ω denotes the set of all partial functions on N learnable by a team. C ω N • [ T ]1 denotes the set of all partial nonuniformly computable functions on N (i.e., all functions f satisfying f (x) ≤T x for any x ∈ dom( f )). We will see that the following inclusions hold. C <ω ⊂ [ T ]1 ⊂ C 1 C 1 C 1 C <ω C ω [ T ]1 ⊂ [ T ]<ω ⊂ [ T ]ω|<ω [ T ]ω ⊂ [ T ]1 ⊂ C 1 [ T ]ω ⊂ These notions are characterized as the following piecewise computability notions, respectively. 1 N • decp[−] also denotes the set of all partial computable functions on N . INSIDE THE MUCHNIK DEGREES I 3

<ω 0 N • decd [Π1] denotes the set of all partial functions on N that are decomposable into finitely many 0 0 0 partial computable functions with (Π1)2 domains, where a (Π1)2 set is the difference of two Π1 sets. <ω 0 N • decp [∆2] denotes the set of all partial functions on N that are decomposable into finitely many 0 partial computable functions with ∆2 domains. ω 0 N • decp [Π1] denotes the set of all partial functions on N that are decomposable into countably 0 many partial computable functions with Π1 domains. <ω N • decp [−] denotes the set of all partial functions on N that are decomposable into finitely many partial computable functions. <ω ω 0 N • decp decp [Π1] denotes the set of all partial functions on N that are decomposable into finitely 0 many partial Π1-piecewise computable functions. ω N • decp [−] denotes the set of all partial functions on N that are decomposable into countably many partial computable functions. <ω ⊂ decp [−] ⊂ 1 <ω 0 <ω 0 <ω ω 0 ω decp[−] ⊂ decd [Π1] ⊂ decp [∆2] decp decp [Π1] ⊂ decp [−] ⊂ ω 0 decp [Π1] ⊂ In Section 3, we formalize the disjunction operations. Medvedev interpreted the intuitionistic disjunction as the coproduct (direct sum) ⊕ : P(NN) × P(NN) → P(NN). We will introduce the following ∗ N N N disjunction operations ~·∨·∗ : P(N ) × P(N ) → P(N ): • ~·∨·3 P NN LCM[n] is the disjunction operation on ( ) induced by the backtrack BHK-interpretation with mind-changes < n. • ~·∨·2 P NN LCM is the disjunction operation on ( ) induced by the two-tape BHK-interpretation with finitely many mind-changes. • ~·∨·3 P NN LCM is the disjunction operation on ( ) induced by the backtrack BHK-interpretation with finitely many mind-changes. ~ 2 N • ·∨· CL is the disjunction operation on P(N ) induced by the two-tape BHK-interpretation permitting unbounded mind-changes. ⊕ ~·∨·3 Then, the direct sum is characterized as the LCM disjunction without mind-changes LCM[1]. In section 5, we also introduce more complicated disjunction operations, which will play key roles in Part II. In section 4, we study the interaction between the disjunction operations and the learnable/piecewise computable functions. We will construct new operations by iterating the disjunction operations introduced in Section 3 in the following way:

~ (m) 2 ≥ ≥ m∈N P CL ~ 3 ~ (m) 2 L W ~ (m)~ 3 2 ~ (m) 2 P ≥ m∈N P ∨ P LCM[m] ≥ m∈N P LCM m∈N P ∨ P LCM CL ≥ m∈N P CL L L W ≥ 3 ≥ L W S W ~P ∨ PLCM Every such operation induces a functor from the associated Medvedev/Muchnik-like degree structure to the Medvedev degree structure. The main result is that every such functor is left adjoint to the canonical map from the Medvedev degree structure onto the associated degree structure. In section 6, we will see that how our classes of nonuniformly computable functions relate to the arithmetical hierarchy of non-intuitionistic principles such as the law of excluded middle (LEM), the lessor limited principle of omniscience or de Morgan’s law (LLPO), and the double negation elimination (DNE). The arithmetical hierarchy of non-intuitionistic principles is illustrated as follows: Σ0-LLPO — — 2 0 0 0 0 HA — Σ1-LEM — ∆2-LEM Σ2-LEM — Σ3-DNE — 0 Σ2-DNE — 4 K. HIGUCHI AND T. KIHARA

Here, Γ-LEM represents the sentence ϕ ∨ ¬ϕ for Γ-sentences ϕ; Γ-LLPO represents the sentence ¬(ϕ ∧ ψ) → ¬ϕ ∨¬ψ for Γ-sentences ϕ, ψ; and Γ-DNE represents the sentence ¬¬ϕ → ϕ for Γ-sentences ϕ. We interpret these principles as partial multi-valued functions on NN, and then we characterize our notions of nonuniform computability by using these principles in the context of the Weihrauch degrees. We also characterize our notions by Wadge-like games. 1.3. Notations and Conventions. For any sets X and Y, 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 range range( f ) of 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 domain dom( f ) of f is included in X, and the range rng( f ) of f is also included in Y. For basic terminology in Computability Theory, see Soare [68]. For σ ∈ N

Contents 1. Summary 1 1.1. Introduction 1 1.2. Results 2 1.3. Notations and Conventions 4

1 In some context, a function Φ is sometimes called partial computable if it can be extended to some Φe. In this paper, however, we do not need to distinguish our definition as being different from this definition. INSIDE THE MUCHNIK DEGREES I 5

2. Nonuniformly Computable Discontinuous Functions 5 2.1. Piecewise Computable Functions 5 2.2. Seven Classes of Nonuniformly Computable Functions 8 2.3. Degree Structures and Brouwer Algebras 10 2.4. Falsifiable Problems and Total Functions 12 2.5. Learnability versus Piecewise Computability 13 3. Strange Set Constructions 16 3.1. Medvedev’s Semantics for Intuitionism 16 3.2. Disjunction Operations Based on Learning Theory 17 4. Galois Connection 21 4.1. Decomposing Disjunction by Piecewise Computable Functions 21 4.2. Galois Connection between Degree Structures 23 0 4.3. Σ2 Decompositions 24 5. Going Deeper and Deeper 26 5.1. Falsifiable Mass Problems 26 5.2. Compactified Infinitaly Disjunctions 29 5.3. Infinitary Disjunctions along well-Founded Trees 30 5.4. Infinitary Disjunctions along any Graphs 32 5.5. Infinitary Disjunctions along ill-Founded Trees 35 5.6. Nested Infinitary Disjunctions along ill-Founded Trees 37 6. Weihrauch Degrees and Wadge Games 41 6.1. Weihrauch Degrees and Constructive Principles 41 6.2. Duality between Dynamic Operations and Nonconstructive Principles 45 6.3. Borel Measurability, and Backtrack Games 47 Bibliography 51 References 51

2. Nonuniformly Computable Discontinuous Functions 2.1. Piecewise Computable Functions. Our main objective in the paper is to study the intermediate notion of (uniform) computability and nonuniform computability. The concept of nonuniform computability can be rephrased as countable computability, i.e., partial functions that are decomposable into countably many computable functions. One can expect that the class of nonuniformly computable functions is classified on the basis of the least cardinality and least complexity of the decomposition (see also Pauly [53]). For instance, if a partial function Γ :⊆ NN → NN is decomposable into k many computable functions, we say that it is k-wise computable or (k, 1)-computable, and if Γ is decomposable into countably many (finitely many, resp.) computable functions with uniformly Λ-definable domains, we say that it is countable (finite, resp.) Λ-piecewise computable, where Λ is a lightface pointclass. An important subclass of the piecewise computable functions consists of partial functions that are identifiable in the limit ([29]). The relationship between the computability with trial-and-error (limit 0 computability or effective learnability) and the subhierarchy of the level ∆2 has been common knowledge among recursion theorists since the last fifty years or so (see also Shoenfield [62], Gold [29], Putnam [55], and Ershov [27]). A basic observation (see Theorem 26) regarding the concept of type-two learnability N 0 (see also de Brecht-Yamamoto [24, 25]) is that a partial function on N is Π1-piecewise computable if and only if it is identifiable in the limit or learnable in the following sense: a partial function Γ :⊆ NN → NN will be called learnable or (1,ω)-computable if there is a computable function Ψ :⊆ N

Φlimn→∞ Ψ( f ↾n)( f ) = Γ( f ) for every f ∈ dom(Γ), where recall that {Φe}e∈N is a ﬁxed enumeration of all partial computable functions. Such a Ψ is called a learner. 6 K. HIGUCHI AND T. KIHARA

Proposition 1. There is an eﬀective enumeration {Ψe}e∈N of all learners that consists of total functions

The set {Ψe}e∈N in Proposition 1 is referred as the eﬀective enumeration of all learners, and Ψe is called the e-th learner. Remark. We urge the reader not to confuse the notions Ψ(σ) and Φ(σ) for a learner Ψ and a computable function Φ (on NN). In the former case, Ψ(σ) simply denotes the output (the inference) of the learner Ψ based on the current input σ. In the latter case, however, we use σ as an initial segment of some oracle information, and so really Φ(σ) denotes a string hΦ(σ; 0), Φ(σ; 1), Φ(σ; 2),...i. Notation. Let Ψ : N

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

indxΨ(σ) = {Ψ(σ ↾ n) : n ≤ |σ|}. N We also deﬁne indxΨ( f ) = n∈N indxΨ( f ↾ n) for any f ∈ N . S We now introduce various subclasses of nonuniformly computable functions on NN based on Learning Theory. Deﬁnition 2. Let D be a subset of Baire space NN, and α, β, γ ≤ ω be ordinals. A function Γ : D → NN is (α, β|γ)-computable if there is a set I ⊆ N of cardinality α such that, for any g ∈ D, there is an index e ∈ I satisfying the following three conditions.

(1) (Learnability) limn Ψe(g ↾ n) converges, and Φlimn Ψe(g↾n)(g) =Γ(g).

(2) (Mind-Change Condition) #mclΨe (g) = #{n ∈ N : Ψe(g ↾ n + 1) , Ψe(g ↾ n)} < β.

(3) (Error Condition) #indxΨe (g) = #{Ψe(g ↾ n) : n ∈ N}≤ γ. C α If γ = ω, then we simply say that Γ is (α, β)-computable for (α, β|γ)-computable function Γ. Let [ T ]β C α (resp. [ T ]β|γ) denote the set of all (α, β)-computable (resp. (α, β|γ)-computable) functions. Hereafter, the symbol < ω will be used in referring to “some natural number n”. For instance, Γ is said to be (< ω, 2| <ω)-computable if there are a, c ∈ N such that it is (a, 2|c)-computable. Remark. Some of (α, β|γ)-computability notions are related to learnability notions: Every (1,< ω)- computable function is learnable with bounded mind-changes; every (1,ω| <ω)-computable function is learnable with bounded errors; every (1,ω)-computable function is learnable; every (< ω, 1)-computable function is k-wise computable; and every (< ω,ω)-computable function is team-learnable. The concept of learnability in the context of real number computation has been studied by several researchers including Chadzelek-Hotz [21], Ziegler [81, 80], and de Brecht-Yamamoto [24, 25]. The notion of mind-change INSIDE THE MUCHNIK DEGREES I 7

Table 1. Seven Classes of Nonuniformly Computable Functions

C 1 [ T ]1 (Uniformly) computable C 1 [ T ]<ω Learnable with bounded mind changes C 1 [ T ]ω|<ω Learnable with bounded errors C 1 [ T ]ω Learnable C <ω [ T ]1 k-wise computable for some k ∈ ω C <ω [ T ]ω Learnable by a team C ω [ T ]1 Nonuniformly computable

C <ω C <ω ⊆ [ T ]1 = [ T ]ω|<ω ⊆ C 1 C 1 C 1 C <ω C ω C ω [ T ]1 ⊆ [ T ]<ω ⊆ [ T ]ω|<ω [ T ]ω ⊆ [ T ]1 = [ T ]ω ⊆ C 1 [ T ]ω ⊆ Table 2. Seven monoids of nonuniformly computable functions

is also related to the level of discontinuity studied by several researchers, for instance, Hertling [33], and Hemmerling [32]. See also Section 5.3 for more information on the relationship between the notion of mind-changes and the level of discontinuity. The notion of k-wise computability has been also studied by, for example, Pauly [53] and Ziegler [83].

Thus, clearly, limn ψ(g ↾ n) converges to Φlimn Ψ(g↾n)(g) =Γ(g). Assume that Γ(g) = limn ψ(g ↾ n) for any g ∈ dom(Γ) for some ψ satisfying the condition in Propo- N N

Φlimn Ψ(g↾n)(g) = n≥|σ| ψ(g ↾ n) = limn∈N ψ(g ↾ n) =Γ(g), since {ψ(g ↾ n)}n≥|σ| is an increasing sequence of strings. S Corollary 4 (de Brecht-Yamamoto [24]). A partial function Γ :⊆ NN → NN is (1,ω)-computable if and only if there is a computable sequence {Γn}n∈N of partial computable functions which converges pointwise to Γ on dom(Γ) with respect to the discrete topology on NN. Proof. By Proposition 3. 8 K. HIGUCHI AND T. KIHARA

2.2. Seven Classes of Nonuniformly Computable Functions. We ﬁrst check several basic properties of (α, β|γ)-computability to show the following theorem stating that the classes obtained from Deﬁnition 2 closed under composition are exactly the classes listed in Table 1. C α C 1 C 1 C 1 Theorem 5. {[ T ]β|γ : α, β, γ ∈ N ∪{< ω,ω}} contains just seven monoids, [ T ]1, [ T ]<ω, [ T ]ω|<ω, C <ω C 1 C <ω C ω [ T ]1 , [ T ]ω, [ T ]ω , and [ T ]1 . Proposition 6. Let Γ be a partial function on Baire space NN.

(1) If Γ is (α0, β0|γ0)-computable, α0 ≤ α1, β0 ≤ β1, and γ0 ≤ γ1, then Γ is (α1, β1|γ1)-computable. (2) Γ is (α, 1)-computable if and only if Γ is (α, β|1)-computable. (3) Γ is (α, β)-computable if and only if Γ is (α, β|β)-computable. (4) Γ is (1, 1)-computable if and only if Γ is computable. (5) Γ is (ω, 1)-computable if and only if Γ is (ω,ω)-computable if and only if Γ is nonuniformly computable, i.e., Γ(g) ≤T g for any g ∈ dom(Γ), where recall that ≤T denotes the Turing reducibility.

Proof. The items (1) and (2) easily follow from the deﬁnitions. The item (3) follows from #indxΨ(g) − 1 ≤ #mclΨ(g). (4) If Γ is computable via Φe, then Γ is (1, 1)-computable via the singleton {i(e)}, where Ψi(e)(σ) = e

extendible to an element of dom(Γ), since #mclΨe (g) = 0 for any g ∈ dom(Γ). Therefore, Γ is computable via ΦΨe(hi). (5) If Γ is nonuniformly computable, then Γ is (ω, 1)-computable via {i(e)}e∈N, where Ψi(e)(σ) = e for any σ ∈ N

e e The algorithm Φk proceeds as follows for g. Recall that indxΨ (g) represents the set of all indices e e e occurring in hypothesis of the learner Ψ . We have an effective enumeration d0(g), d1(g),... of all indices e e contained in indxΨe (g) uniformly in g. Then, we set Φ (g) = Φ e (g) if d (g) is defined. For any k dk (g) k e g ∈ D, there is e < m such that lims Ψ (g ↾ s) converges to some correct computation d of Γ(g), i.e., e e Φd(g) =Γ(g). Since #indxΨ (g) < n, we have dk(g) = d for some k < n. Thus, for any g ∈ D, there are e e e e < m and k < n such that Φk(g) =Γ(g). Therefore, if ik is an index of Φk for each e < m and k < n, then e Γ is (m · n, 1)-computable via an upper bound max{ik : e < m & k < n}. Corollary 8. Γ is (< ω,ω| <ω)-computable if and only if Γ is (< ω, 1)-computable. Proof. Every (< ω,ω| <ω)-computable function Γ is (m,ω|n)-computable for some m, n <ω. Therefore, by Proposition 7, Γ is (m · n, 1)-computable. In particular, Γ is (< ω, 1)-computable, since m · n <ω. N Proposition 9. For each i < 2, let Γi be a partial (αi, βi|γi)-computable function on Baire space N , where αi, βi, γi ≤ ω are ordinals. Then Γ1 ◦ Γ0 is (α0 ∗ α1, β0 ∗ β1|γ0 ∗ γ1)-computable, where ∗ is the multiplication as the cardinals, or equivalently, κ ∗ λ = min{κ · λ, ω} for ordinals κ, λ ≤ ω. Γ | {Ψi } Proof. For each i < 2, since i is (αi, βi γi)-computable, there is a collection of learners, j j<αi and a i cover {U } j<α of dom(Γi) such that Γi( f ) = Φ i ( f ↾ n) and #mcl i ( f ) < βi and #indx i ( f ) < j i limn Ψ j( f ↾n) Ψ j Ψ j i ∗ γi, for any j < αi and f ∈ U j. Fix j < α0 and k < α1. Then Ψ j,k(σ) is defined as follows. Let 0 0 + J(σ) be the longest interval [r, |σ|) satisfying Ψ j (σ ↾ r) = Ψ j (σ), and define J (σ) = J(σ) \{r}. If + ∗ 1 #(mclΨ1 ∩ J (σ)) < β1 and #(indxΨ1 ∩ J(σ)) < γ1, then put Ψ (σ) = Ψ (ΦΨ0 (σ)). Otherwise, put k k j,k k j (σ) INSIDE THE MUCHNIK DEGREES I 9

∗ ∗ − ∗ Ψ (σ) = Ψ (σ ). For given σ, we compute an index Ψ j,k(σ), where ΦΨ j,k (σ)( f ) = ΦΨ (σ))(Φ 0 ( f )) j,k j,k j,k Ψ j (σ) for any f . Note that f ∈ dom(Γ1 ◦ Γ0) if and only if f ∈ dom(Γ0) and Γ0( f ) ∈ dom(Γ1). Therefore, for such f , 0 1 0 there are j <α0 and k <α1 such that f ∈ U j and Γ0( f ) ∈ Uk . Assume that f ∈ dom(Γ1 ◦ Γ0) ∩ U j and 1 ∗ Γ ( f ) ∈ U . It is easy to see that Ψ is computable, #mlcΨ∗ ( f ) < β ∗ β and #indxΨ∗ ( f ) < γ ∗ γ . 0 k j,k j,k 0 1 j,k 0 1 0 0 Moreover, there exist s and e0 such that Ψ j ( f ↾ t) = Ψ j ( f ↾ s) = e0 for any t ≥ s. Fix such s. Since 1 + Φe0 ( f ) = Γ0( f ) ∈ U , for any t ≥ s, #(mclΨ1 ∩ J ( f ↾ t)) < β1 and #(indxΨ1 ∩ J( f ↾ t)) < γ1, since k k k 1 ∗ 1 J( f ↾ t) = J( f ↾ s) and by our choice of Ψk. Therefore, limn Ψ j,k( f ↾ n) converges to limn Ψk (Γ0( f ↾ n)). 1 1 However, there exist u ≥ s and e1 such that Ψk (Γ0( f ↾ v)) = Ψk (Γ0( f ↾ u)) = e1 for any v ≥ u, since {Γ0( f ↾ u)}u≥s is an increasing sequence of strings and Γ0( f ) ∈ dom(Γ1). Here Φe1 (Γ0( f )) = Γ1(Γ0( f )). Thus,

∗ Φlimn Ψ j,k( f ↾n)( f ) = Φlimn Ψ ( f ↾n)(Φ 0 ( f )) = Φ Ψ1 Γ ↾ (Γ0( f )) =Γ1(Γ0( f )). j,k limn Ψ j ( f ↾n) limn k ( 0( f ) n)

Consequently, Γ1 ◦ Γ0 is (α0 ∗ α1, β0 ∗ β1|γ0 ∗ γ1)-computable, via {Ψ j,k} j<α0,k<α1 . C α Corollary 10. [ T ]β|γ forms a monoid under composition, for any α, β, γ ∈{1,< ω,ω}. Proof. Straightforward from Proposition 9. C 1 C 1 C 1 Proposition 11. [ T ]<ω is the smallest monoid including [ T ]2; [ T ]ω|<ω is the smallest monoid includ- C 1 C <ω C 2 C <ω ing [ T ]ω|2. [ T ]1 is the smallest monoid including [ T ]1; [ T ]ω is the smallest monoid including C 2 [ T ]ω. Proof. The first result is known, and indeed, it has also been proved in Mylatz’s PhD thesis, but we also give a proof here for the sake of completeness. We first show that every (1, n + 1)-computable function Γ can be represented as Γ=Γ1 ◦ Γ0 for some (1, n)-computable function Γ0 and (1, 2)-computable function Γ1. Let Ψ be a learner for Γ. We define a learner Ψ0 for Γ0 and a learner Ψ1 for Γ1. For

g ⊕ h 7→ ΦΨ(σ)(g), i.e., ΦΨ1(σ⊕τ)(g ⊕ h) = ΦΨ(σ)(g). Since Γ is (1, n + 1)-computable, and by the deﬁnition ∗ N of σ , it is easy to see that Γ1 is (1, 2)-computable. For g ∈ N , if #mclΨ(g) < n, then

Γ1(Γ0(g)) =Γ1(g ⊕ Γ(g)) =Γ(g).

If #mclΨ(g) = n, then Γ1(Γ0(g)) =Γ1(g ⊕ ΦΨ(g∗)(g)) =Γ(g). Consequently, Γ=Γ1 ◦ Γ0 as desired. We next show that every (1,ω|n+1)-computable function Γ can be represented as Γ=Γ1 ◦Γ0 for some (1,ω|n)-computable function Γ0 and (1,ω|2)-computable function Γ1. Assume that Ψ is a learner for Γ, σ σ σ and we enumerate #indxΨ(σ) as {im}m≤|σ|. Here, if m < n then Ψ guesses im before Ψ guesses in on some σ σ τ

N via a collection {∆i}i≤n of partial computable functions. For g ∈ N , if Γ(g) = ∆i(g) for some i < n, then Γ0(g) = g ⊕ ∆i(g). Otherwise, we set Γ0(g) = g ⊕ ∆0(g). Then, clearly Γ0 is (n, 1)-computable via N {λg.g ⊕ ∆i(g)}i

2.3. Degree Structures and Brouwer Algebras. We will see some intuitionistic feature of our classes of nonuniformly computable functions. Deﬁnition 12. Let F be a monoid consisting of partial functions Γ :⊆ NN → NN under composition. N Then, P(N ) is preordered by the relation P ≤F Q indicating the existence of a function Γ ∈ F from N N Q into P, that is, P ≤F Q if and only if there is a partial function Γ :⊆ N → N such that Γ ∈ F and N 0 N Γ(g) ∈ P for every g ∈ Q. Let D/F and P/F denote the quotient sets P(N )/ ≡F and Π1(2 )/ ≡F , 0 N 0 N N respectively. Here, Π1(2 ) denotes the set of all nonempty Π1 subsets of 2 . For P ∈ P(N ), the N equivalence class {Q ⊆ N : Q ≡F P} ∈ D/F is called the F -degree of P. C α Recall from Corollary 10 that F = [ T ]β|γ forms a monoid for every α, β, γ ∈{1,< ω,ω}. C α α α α Notation. If F = [ T ]β|γ for 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 Remark. By Proposition 6 (4) and (5), the preorderings ≤1 and ≤1 are equivalent to the Medvedev reducibility [46] and the Muchnik reducibility [48], respectively. We also introduce the truth-table versions of Deﬁnition 2. Deﬁnition 13. Let D be a subset of Baire space NN, and α, β, γ ≤ ω be ordinals. A function Γ : D → NN is (α, β|γ)-truth-table if there are a set I ⊆ N of cardinality α, and a collection {p(e, k) : e ∈ I & k < N min{β, γ}} of indices of truth-table functionals (i.e., dom(Φp(e,k)) = N ) such that

C 1 C 1 C 1 C 1 C <ω C <ω C ω 2ℵ0 Proposition 14. ℵ0 = #[ r]1 = #[ r]<ω = #[ r]ω|<ω = #[ r]ω < #[ r]1 = #[ r]ω = #[ r]1 = 2 , for each r ∈{tt, T}. INSIDE THE MUCHNIK DEGREES I 11

C 1 C 1 Proof. Every learner Ψ determines just one learnable function Γ ∈ [ T ]ω. Therefore, [ T ]ω is countable. C ω 2ℵ0 N NN 2ℵ0 For non-uniform computability, we first see #[ T ]1 ≤ 2 since #(N ) = 2 by cardinal arithmetic. On the other hand, every function Γ : NN → {0N, 1N} is (< ω, 1)-truth-table via two constant truth-table N N N C <ω 2ℵ0 functionals Γ0( f ) = 0 and Γ1( f ) = 1 for any f ∈ N . Therefore, #[ tt]1 ≥ 2 . ∈ { } Dα Dα Pα Pα Proposition 15. For each α, β, γ 1,< ω,ω , the order structures β|γ, tt,β|γ, β|γ, and tt,β|γ form lattices with top and bottom elements. Proof. It is easy to see that the product ⊗ and the sum ⊕ form supremum and infimum operations in these 1 structures. Moreover, every degree structure has top and bottom elements since it is coarser than D1, that has top and bottom elements. If a lattice (L, ≤, ∨, ∧) has the top element 1, the bottom element 0, and max{c : c ∧ a ≤ b} (denoted by a →L b) exists for any a, b ∈ L, then L = (L, ≤, ∨, ∧, →L, 0, 1) is called a Heyting algebra. An algebra L = (L, ≤, ∨, ∧, →, ⊥, ⊤) is a Brouwer algebra if its dual Lop = (L, ≥, ∧, ∨, ←, ⊤, ⊥) is a Heyting algebra. 1 ω Recall that the Medvedev lattice D1 and the Muchnik lattice D1 form Brouwer algebras [46, 48]. 1 1 Proposition 16. The degree structures Dω and Dtt,ω are Brouwerian. 1 Proof. We just give a proof for Dω, although it is straightforward to modify the proof for the truth-table version. N 1 N N N Set B(P, Q) = {R ⊆ N : P ≤ω Q⊗R}. We need to a construct a function β : P(N )×P(N ) → P(N ) N such that β(P, Q) = min B(P, Q) for any P, Q ⊆ N . Let Λe denote the e-th (1,ω)-computable function, i.e., Λe(g) = Φlimn Ψe(g↾n)(g) for any g ∈ dom(Λe). Define β as follows. a N β(P, Q) = {e g ∈ N : (∀ f ∈ Q) Λe( f ⊕ g) ∈ P}. N It is easy to see that β(P, Q) ∈ B(P, Q) for any P, Q ⊆ N . If R ∈ B(P, Q), say Λe : Q ⊗ R → P, then a 1 clearly e g ∈ β(P, Q) for any g ∈ R. Thus, β(P, Q) ≤1 R. 1 1 <ω <ω In contrast, we will show in Part II that neither D<ω, nor Dω|<ω, nor D1 , nor Dω form Brouwer 1 1 <ω <ω algebras. In the meantime, the following modifications of D<ω, Dω|<ω, D1 , and Dω look more natural than our original definitions, from the viewpoint of constructive mathematics. Indeed, in Proposition 20, we will see that these modifications form Brouwer algebras. Definition 17. Let D be a subset of Baire space NN, and α, β, γ ≤ ω be ordinals, or eff. We generalize the (α, β|γ)-computability as follows. If α = eff, then we revise the word “for any g ∈ D, there is N e ∈ I” to the word “there is a partial computable function B0 :⊆ N → N such that, for any g ∈ D,

there is e < B0(g)”. If β = eff, then we revise the mind change condition as #mclΨe (g) < B1(g), where N B1 is a partial computable function from N to N. If γ = eff, then we revise the error condition as indx NN N ≤α # Ψe (g) < B2(g), where B2 is a partial computable function from to . For new notions, β|γ, α α Dβ|γ, and Pβ|γ are also defined as the usual way. Proposition 18. Suppose that, if τ = eff, then let τ∗ mean the symbol < ω, and otherwise, set τ∗ = τ. Then, every (α, β|γ)-computable function with a compact domain is (α∗, β∗|γ∗)-computable. −1 Proof. By continuity of B0, B1, and B2 in Definition 17, {Bi ({e})}e∈N for each i < 3 is an open cover of D. Hence, by compactness of D, we have the desired condition. 1 1 1 1 eff <ω eff <ω Corollary 19. Peff = P<ω; Pω|eff = Pω|<ω; P1 = P1 ; and Pω = Pω . 0 N That is to say, for Π1 subsets of Cantor space 2 , no new reducibility notion is constructed from Definition 17. However, from the perspective of intuitionistic caluculus, our new notions in Definition 17 have nice features. 12 K. HIGUCHI AND T. KIHARA

1 1 eff eff Proposition 20. Deff, Dω|eff, D1 , and Dω are Brouwerian. N α Proof. Fix α, β, γ ∈{1,< ω, eff,ω}, and set B(P, Q) = {R ⊆ N : P ≤β|γ Q ⊗ R}. We need to construct a N N N N function β : P(N ) × P(N ) → P(N ) such that β(P, Q) = min B(P, Q) for any P, Q ⊆ N . Let Λe denote N the e-th (1,ω)-computable function, and Θe be the e-th partial computable function from N to N. Put changee(g) = #{n ∈ N : Λe(g ↾ n + 1) , Λe(g ↾ n)}, and errore(g) = #{Λe(g ↾ n) : n ∈ N}. Then, a {(e, d) g : (∀ f ∈ Q) Λe( f ⊕ g) ∈ P & #mclΛ ( f ⊕ g) < Θd( f ⊕ g)},  e = eff  if (α, β, γ) (1, ,ω),  a {(e, d) g : (∀ f ∈ Q) Λe( f ⊕ g) ∈ P & #indxΛe ( f ⊕ g) < Θd( f ⊕ g)},   if (α, β, γ) = (1,ω, eff), =  β(P, Q)  a {d g : (∀ f ∈ Q)(∃e < Θ( f ⊕ g)) Φe( f ⊕ g) ∈ P},  if (α, β, γ) = (eff, 1,ω),  {dag : (∀ f ∈ Q)(∃e < Θ( f ⊕ g)) Λ ( f ⊕ g) ∈ P},  e  if (α, β, γ) = (eff,ω,ω),   It is easy to see that β(P, Q) ∈ B(P, Q) for any P, Q ⊆ NN. For the minimality, if R ∈ B(P, Q), we have a 1 suitable d and e such that (d, e) g ∈ β(P, Q) for any g ∈ R. Thus, β(P, Q) ≤1 R. 1 1 eff eff Remark. Unfortunately, neither Peff nor Pω|eff nor P1 nor Pω form Brouwer algebra (see Part II). 2.4. Falsifiable Problems and Total Functions. In Part II, we will mainly pay attention to the behavior 0 N of nonuniform computability on Π1 subsets of Cantor space 2 . Such a restriction has an interesting 0 feature by thinking of Π1 sets as falsifiable mass problems. Consider a learner Ψ identifies a (1,ω)- computable function Γ : Q → P. On an observation σ ∈ N

≤<ω {Ψ } ∈ (4,5) Assume that P tt,ω|<ω X via n Popperian learners, i i

1 1 1 1 <ω 1 1 <ω 1 Corollary 22. Ptt,1 = P1; Ptt,<ω = Ptt,ω|<ω = Ptt,ω|<ω = P<ω; and Ptt,ω = Ptt,ω = Pω. Hence, {Pα Pα ∈{ }} P1 P<ω P1 P1 P<ω P1 P<ω β|γ, tt,β|γ : α, β, γ 1,< ω,ω consists of at most nine lattices: 1, tt,1, <ω, ω|<ω, 1 , ω, ω , ω ω Ptt,1, and P1 . 1 1 One can interpreted ≤1 (≤ω, resp.) as computable reducibility with no (ﬁnitely many, resp.) mind- 1 changes. We see how ≤ω behaves like a dynamical-approximation procedure. 0 N N ω Proposition 23. For any Π1 set P ⊆ N and any set Q ⊆ N ,P ≤1 Q if and only if

(∃Ψ)(∀ f ∈ Q) Φlim infn Ψ( f ↾n)( f ) ∈ P. Here Ψ ranges over all learners (i.e., computable functions from N

µe < |σ| [Φ (σ) ↾ (l(σ−, e) + 1) ∈ T ] if such e exists, Ψ(σ) =  e P |σ| otherwise  l(σ−, e) + 1 if e = Ψ(σ), l(σ, e) =  − l(σ , e) otherwise.  ω  By our assumption P ≤1 Q, lim infn Ψ( f ↾ n) exists for all f ∈ P. Thus, the desired condition Φlim infn Ψ( f ↾n)( f ) ∈ Q holds. N 0 Remark. Recall that a subset of 2 is Π1 if and only if it is the set of all inﬁnite paths through a computable subtree of 2

N Definition 24. For a class Λ of subsets of Baire space N , we say that a collection {Qi}i∈I is uniformly N Λ if the set {(i, f ) ∈ I × N : f ∈ Qi} belongs to Λ. A partition or a cover {Qi}i∈I of Q is (uniformly) Λ ∗ ∗ if there is a (uniform) Λ collection {Qi }i∈I such that Qi = Q ∩ Qi for any i ∈ I. We say that {Qi}i∈I is ∗ ∗ ∗ a (uniform) Λ layer of Q if there is a uniform Λ collection {Qi }i∈I such that Qi ⊆ Qi+1 for each i ∈ I, ∗ ∗ {Qi }i∈I covers Q, and Qi = Q ∩ Qi . We also say that {Qi}i∈I is a (uniform) Λ d-layer of Q if there is a ∗ ∗ ∗ ∗ (uniform) Λ layer {Qi }i∈I of Q such that Qi = Qi \ Qi−1 for any i ∈ I, where Q−1 = ∅. Remark. The terminology “layer” comes from the concept of layerwise computability in algorithmic randomness theory (see Hoyrup-Rojas [35]). Definition 25. Let F be a class of partial functions on NN. For X ∈ ω∪{< ω,ω} and x ∈{p, c, d}, a partial N N X functions Γ :⊆ N → N is of class decx [Λ]F if there is a uniform Λ partition (if x = p), uniform cover (if x = c) or uniform d-layer (if x = d), {Qi}i∈I, of dom(Γ) such that Γ ↾ Qi is contained in F uniformly in i ∈ I, where I = X if X ∈ ω ∪{ω} and I ∈ ω if X =< ω. If F is the class of all partial computable X X functions, we simply write decx [Λ] instead of decx [Λ]F . Moreover, if Λ is the class of all subsets of X X X X Baire space, then we write decx [−] and decx F instead of decx [Λ] and decx [Λ]F , respectively. If we X does not assume uniformity in the definition, we say that Γ is of decx [Λ]F . 0 0 0 X X X X If Λ ∈{Σn, Πn, ∆n}n∈N, for every X ∈{< ω,ω}, we have decp [Λ] ⊆ decc [Λ] ⊆ decd [Λ] ⊆ decp [(Λ)2]. ω 0 ω 0 Here a set is (Λ)2 if it is the difference of two Λ sets. Note that decp [Πn] = decc [Σn+1] holds for every n ∈ N. Our seven concepts of nonuniform computability listed in Table 1 can be characterized as classes of piecewise computable functions. Theorem 26. Let k be any finite number. C 1 k 0 (1) [ T ]k = decd[Π1]. C 1 k 0 k 0 (2) [ T ]ω|k = decx[∆2] = decc[Σ2] for any x ∈{p, c, d}. C 1 ω 0 ω 0 ω 0 (3) [ T ]ω = decx [Π1] = decx [∆2] = decc [Σ2] for any x ∈{p, c, d}. C k k (4) [ T ]1 = decx[−] for any x ∈{p, c, d}. C k k ω 0 k ω 0 k ω 0 (5) [ T ]ω = decydecx [Π1] = decydecx [∆2] = decydecc [Σ2] for any x, y ∈{p, c, d}. C ω ω (6) [ T ]1 = decx [−] for any x ∈{p, c, d}.

Table 3. Seven Classes of Nonuniformly Computable Functions

C 1 <ω 0 0 [ T ]<ω decd [Π1] finite (Π1)2-piecewise computable C 1 <ω 0 0 [ T ]ω|<ω decp [∆2] finite ∆2-piecewise computable C 1 ω 0 0 [ T ]ω decp [Π1] Π1-piecewise computable C <ω <ω [ T ]1 decp [−] finite piecewise computable C <ω <ω ω 0 0 [ T ]ω decp decp [Π1] finite piecewise Π1-piecewise computable C ω ω [ T ]1 decp [−] countably computable

s) R(m, g ↾ t)} for every m ∈ N. We set Ψ(σ) = e(min({m : R(m,σ)}∪{k − 1})). Since dom(Γ) is covered by {Qm}m

N 0 Proposition 27. Let P and Q be subsets of N , where P is Πn for n ≥ 2. Let k be any finite number. C k k 0 (1) There is Γ : Q → P with Γ ∈ [ T ]1 if and only if there is Γ : Q → P with Γ ∈ decd[Πn]. C <ω <ω 0 ω 0 (2) There is Γ : Q → P with Γ ∈ [ T ]ω if and only if there is Γ : Q → P with Γ ∈ decd [Πn]decp [Π1]. C ω ω 0 (3) There is Γ : Q → P with Γ ∈ [ T ]1 if and only if there is Γ : Q → P with Γ ∈ decd [Πn]. <ω <ω 0 <ω <ω 0 ω 0 ω ω 0 Hence, P1 = P/decd [Π2], Pω = P/decd [Π2]decp [Π1], and P1 = P/decd [Π2]. Here, recall from 0 Definition 12 that P/F denotes the F -degree structure of nonempty Π1 subsets of Cantor space. ω Proof. For (3), we assume that P ≤1 Q. Every partial computable function Φe can be assumed to have a 0 −1 0 0 Π2 domain De. Then, Qe = d≤e(Dd ∩ Φd [P]) is Πn, and {Qe}e∈N forms a Πn layer. Moreover, it is not hard to see that Φe maps everyS element of Qe \ Qe−1 into P. <ω 2 ω 0 For (2), we assume that P ≤ω Q is witnessed by two functions Γ ∈ decpdecp [Π1] by Theorem i 26. Then there is a collection of partial computable functions {Γn}i<2,n∈N and a partition {Ei}i<2 of Q i 0 i and collections {Qn}n∈N of pairwise disjoint Π1 sets that covers Ei and Γ agrees with Γn on the domain i ∗ 0 0 −1 N 0 Ei ∩ Qn for every i < 2 and n ∈ N. Then, E1 = n∈N(Qn ∩ (Γn) [N \ P]) is Σn and included in E1. ∗ ∗ 0 ∗ NS ∗ i Thus, {E0, E1} forms a Πn d-layer, where E0 = N \ E1. It is not hard to see that Γ agrees with Γn on the ∗ i domain Q ∩ Ei ∩ Qn for every i < 2 and n ∈ N. <ω 0 Remark. It is not hard to see that decp [Π1] is exactly the class of all partial computable functions, 0 because, given a finite Π1 partition {Qi}i

3. Strange Set Constructions 3.1. Medvedev’s Semantics for Intuitionism. To introduce useful set constructions, let us return back to Medvedev’s original idea. To formulate semantics for the intuitionistic propositional calculus (IPC), Kolmogorov tried to interpret each proposition as a problem. Medvedev [46] formalized his idea by interpreting each proposition p as a mass problem ~p ⊆ NN. Under the interpretation: (1) A proof π is a dynamical process represented by an inﬁnite sequence of natural numbers, i.e., π ∈ NN. (2) ~p is the set of all proofs of a proposition p, i.e., ~p ⊆ NN. (3) A proposition p is provable if p has a computable proof, i.e., ~p ⊆ NN contains a computable element.

To prove the disjunction p0 ∨ p1, we need to algorithmically decide which part is valid, i.e., we first declare one part to be valid and then construct a witness for this part. Consequently, p0 ∨ p1 is provable under that interpretation if and only if we can algorithmically construct an element of ~p0 ∨ p1 = a ~p0 ⊕ ~p1 = {hii f : i < 2 & f ∈ ~pi}. Generally, let Form denote the all propositional formulas. Medvedev’s idea is defining a mass-problem-interpretation of IPC by a function ~· : Form → P(NN) as in Definition 28.

Definition 28. We say that a function ~· : Form → P(NN) is a Medvedev interpretation if it satisfies the following six conditions. (1) ~⊤ contains a computable element. (2) ~⊥ = ∅. (3) ~ϕ ∧ ψ = ~ϕ ⊗ ~ψ = { f ⊕ g : f ∈ ~ϕ & g ∈ ~ψ}. (4) ~ϕ ∨ ψ = ~ϕ ⊕ ~ψ = {h0ia f : f ∈ ~ϕ} ∪ {h1iag : g ∈ ~ψ}. a (5) ~ϕ → ψ = ~ϕ→~ψ = {e g | Φe(g ⊕∗) : ~ϕ → ~ψ}. (6) ~¬ϕ = ~ϕ →⊥. N N Here, Φ(g ⊕∗) denotes the partial function λ f.Φ(g ⊕ f ) :⊆ N → N , and recall that Φe is the e-th partial computable function on NN. Arithmetical quantifications can also be interpreted as follows.

7. ~∃nϕ(n) = n∈N~ϕ(n). 8. ~∀nϕ(n) = L ~ϕ(n). Nn∈N 1 As mentioned in Section 2.3, Medvedev [46] showed that the quotient algebra D1 called the Medvedev lattice is Brouwerian under Medvedev’s interpretation (Deﬁnition 28). Following him, Muchnik [48] ω showed that D1 called the Muchnik lattice is Brouwerian. Usually, the Medvedev reducibility is written 1 ω as ≤M or ≤s rather than ≤1, and the Muchnik reducibility is written by ≤w rather than ≤1 . Remark. 1 ω (1) Both of the Medvedev lattice D1 and the Muchnik lattice D1 provide sound and complete semantics for Jankov’s Logic KC = IPC + ¬p ∨ ¬¬p, the intuitionistic propositional logic with the weak law of excluded middle, which is also called De Morgan logic. The Medvedev lattice and the Muchnik lattice are extensively studied from the aspect of Intermediate Logic. See Sorbi-Terwijn [71] and Hinman [34]. (2) Forty years after the pioneering work by Muchnik, the Muchnik reducibility become useful in the context of Reverse Mathematics (see Simpson [66]). The reason is that the Muchnik reducibility ω ≤1 is strongly associated with the provability relation in RCA, the recursive comprehension 0 N axiom. Then, the Muchnik degrees of Π1 subsets of 2 might be seen as instances of WKL, the weak K¨onig’s lemma. For example, by using a result of Binns and Simpson [8] for the 0 N Muchnik degrees of Π1 subsets of 2 , Mummert [49] obtains an embedding theorem about the Lindenbaum algebra between RCA0 and WKL0. INSIDE THE MUCHNIK DEGREES I 17

0 N (3) For more basic results about the Medvedev and Muchnik degrees of Π1 subsets of 2 , see Simp- son [63, 64, 65, 67]. There are lots of research on the algebraic structure of the Medvedev 0 N degrees of Π1 subsets of 2 , such as density [19], embeddability of distributive lattices [8], join- reducibility [7], meet-irreducibility [2], noncuppability [18], decidability [22], and undecidability [61]. The structure of Weihrauch degrees, an extension of the Medvedev degrees, has also been widely studied as a computable-analysistic approach to (Constructive) Reverse Mathematics (see [11, 13, 12]). 3.2. Disjunction Operations Based on Learning Theory. Hayashi [30, 31] introduced Limit Com- putable Mathematics (LCM), an extended constructive mathematics based on Learning Theory. Like the BHK-interpretation for intuitionistic logic, there is a limit-BHK interpratation for Limit Computable ~·i → P NN Mathematics. We introduce three mass-problem-interpretations LCM : Form ( ) of LCM based on the limit-BHK interpretation. To formulate a mass-problem-style interpretation of LCM, imagine the following dynamic proof models.

The one-tape model is defined as follows: When a verifier Ψ tries to prove that “P0 or P1”, a tape Λ is given. At each stage, Ψ declares 0 or 1, and writes one letter on the tape Λ. • Intuitionism: Ψ does not change his declaration, say i ∈ {0, 1}, and the infinite word written on the tape Λ witnesses the validity of Pi. • LCM: the sequence of declarations of Ψ converges, say i ∈ {0, 1}, and the infinite word written on the tape Λ witnesses the validity of Pi. • Classical: any declaration of Ψ is nonsense, and the infinite word written on the tape Λ witnesses the validity of P0 or P1. The two-tape model is follows: When a verifier Ψ tries to prove “P0 or P1”, two tapes Λ0 and Λ1 are given. At each stage, Ψ declares 0 or 1, say i, and he writes one letter on the tape Λi. • Intuitionism: For either i < 2, the word written on Λ1−i is empty, and the infinite word written on Λi witnesses the validity of Pi. • LCM: For either i < 2, the word written on Λ1−i is finite, and the infinite word written on Λi witnesses the validity of Pi. • Classical: For either i < 2, the infinite word written on Λi witnesses the validity of Pi. The backtrack-tape model is follows: When a verifier Ψ tries to prove that “P0 or P1”, a cell , and two infinite tapes Λ, ∆ are given. The cell is called the declaration, Λ is called the working tape, and ∆ is called the record tape. At each stage, the verifier Ψ works as follows. (1) If no letter is written on the declaration , then Ψ declares 0 or 1 and this is written on the declaration and the record tape ∆. (2) When some letter is written on the declaration , the verifier Ψ chooses one letter k from N ∪{♯}, and his choice k is written on the record tape ∆. (a) In the case k , ♯, it expresses that Ψ writes the letter k on the working tape Λ. (b) In the case k = ♯, it expresses that Ψ erases all letters from the declaration and the working tape Λ. • Intuitionism: Ψ does not choose ♯, hence he does not change his declaration, say i, and the infinite word written on the tape Λ witnesses the validity of Pi. • LCM: Ψ chooses ♯ at most finitely often, hence the sequence of declarations of Ψ converges, say i, and the infinite word written on the tape Λ witnesses the validity of Pi. • Classical: No classical counterpart. To give formal definitions of these dynamic proof models, we introduce some auxiliary definitions. Definition 29 (Notations for One/Two-Tape Models). Let I ⊆ N be a set of indices of working tapes. A pair (x0, x1) ∈ I × N indicates the instruction to write the letter x1 ∈ N on the x0-th tape. Then every

of instructions (i(0), n(0)), (i(1), n(1)),..., (i(s − 1), n(s − 1)). Fix σ ∈ (I × N)

(1) If σ = h(1, 3), (1, 1), (0, 4), (0, 15), (1, 9), (0, 26), (0, 5)i, then the projections of σ are pr0(σ) = h4, 15, 26, 5i, and pr1(σ) = h3, 1, 9i. Moreover, mc(σ) = 3. (2) If τ = h0, 2, 7, 18, 28,♯, 1, 8, 2, 8, 45, 9,♯, 0, 4, 52, 35, 3, 6i, then the tail of τ is tail(τ) = τ↼13 = h0, 4, 52, 35, 3, 6i. N Definition 32 (One-Tape Disjunctions). Let P0 and P1 be subsets of Baire space N . 1 N (1) ~P0 ∨ P1Int = i<2({i }⊗ Pi). ~ ∨ 1 =S { ∈ N ∀∞ = }⊗ (2) P0 P1 LCM i<2( f 2 : ( n) f (n) i Pi). ~ 1 S N (3) P0 ∨ P1 CL = i<2(2 ⊗ Pi). S Here, iN denotes the infinite sequence consisting of i’s, i.e., iN = hi, i, i,..., i, i, i,...i. N Definition 33 (Two-Tape Disjunctions). Let P0 and P1 be subsets of Baire space N . 2 N (1) ~P0 ∨ P1Int = { f ∈ (2 × N) : ((∃i < 2) pri( f ) ∈ Pi)& mc( f ) = 0}. ~ ∨ 2 = { ∈ × N N ∃ ∈ ∞} (2) P0 P1 LCM f (2 ) : (( i < 2) pri( f ) Pi)& mc( f ) < . ~ 2 N (3) P0 ∨ P1 CL = { f ∈ (2 × N) : (∃i < 2) pri( f ) ∈ Pi}. N Definition 34 (Backtrack Disjunctions). Let P0 and P1 be subsets of Baire space N . ~ 3 N ↼1 , (1) P0 ∨ P1 Int = { f ∈ (N ∪{♯}) : tail( f ) ∈ Ptail( f ;0) & (∀n) f (n) ♯}. ~ ∨ 3 = { ∈ N ∪{ } N ↼1 ∈ ∀∞ , } (2) P0 P1 LCM f ( ♯ ) : tail( f ) Ptail( f ;0) & ( n) f (n) ♯ . In Definition 34, for example, the string τ = h♯iahiiaσ represents the record that a verifier Ψ erased all letters from tapes (this action is indicated by ♯), declared that Pi is valid, and wrote the word σ on the working tape. That is to say, tail(τ; 0) = i is the current declaration of the verifier and tail(τ)↼1 = σ is the current word written on the working tape. Remark. Note that we always have to choose a new symbol ♯ which has not been already used, since we may need to distinguish the new ♯ from other symbols and other ♯’s used in other disjunctions. Formally, we can assume that all objects in our paper are elements of NN, subsets of NN, or (partial) INSIDE THE MUCHNIK DEGREES I 19

functions on NN by setting 0• = ♯, (n + 1)• = n, and f •(n) = f (n•) for every n ∈ N. For instance, ~ ∨ 3 ~ ∨ 3• ∈ NN • ∈ ~ ∨ 3 P0 P1 LCM is always interpreted as the set P0 P1 LCM of all f such that f P0 P1 LCM, ~ ∨ ~ ∨ 3 3 ~ ∨ ~ ∨ 3• 3• ∈ NN and then Q P0 P1 LCM LCM is interpreted as Q P0 P1 LCM LCM of all f such that • ∈ ~ ∨ 3 f P0 P1 LCM. Then, note that outer ♯’s are automatically distinguished from inner ♯’s contained in ∈ ~ ∨ ~ ∨ 3• 3• ~ ∨ 3 ~ ∨ 3• f Q P0 P1 LCM LCM. Hereafter, P0 P1 LCM is identiﬁed with P0 P1 LCM. Notation. Hereafter, we frequently use the notation write(i,σ) for any i ∈ N and σ ∈ N

(1) The one-tape LCM disjunction of P0 and P1 with mind-changes-bound n is deﬁned as follows. 1 1 N N ~P0 ∨ P1LCM[n] = ~P0 ∨ P1LCM ∩{ f ∈ 2 : #{n ∈ N : f (n + 1) , f (n)} < n}⊗ 2 .

(2) The two-tape LCM disjunction of P0 and P1 with mind-changes-bound n is deﬁned as follows. 2 2 N ~P0 ∨ P1LCM[n] = ~P0 ∨ P1LCM ∩{ f ∈ (2 × N) : mc( f ) < n}.

(3) The backtrack-tape LCM disjunction of P0 and P1 with mind-changes-bound n is deﬁned as follows. 3 3 N ~P0 ∨ P1LCM[n] = ~P0 ∨ P1LCM ∩{ f ∈ (N ∪{♯}) : #{k ∈ N : f (k) = ♯} < n}. Proposition 37. Let P, Q be subsets of Baire space NN. ⊕ ≡1 ~ ∨ i ∈{ } (1) P Q 1 P Q LCM[1] for each i 1, 2, 3 . ~ ∨ 2 ≡1 ~ ∨ 3 ~ 2 ≡1 ~ ∨ 3 = (2) P P LCM[2] 1 P P LCM[2]. Indeed, i

P ∪ Q P ⊕ Q ≡ ≡ ~ 1 ~ 1 ~ 1 ~ 1 P ∨ Q CL ≤ (≡) P ∨ Q LCM ≤ (≡) P ∨ Q LCM[2] ≤ (≡) P ∨ Q Int ≤ ≤ ≤ ≡ ~ 2 ~ 2 ~ 2 ~ 2 P ∨ Q CL ≤ P ∨ Q LCM ≤ P ∨ Q LCM[2] ≤ P ∨ Q Int ≤ ≤ ( )≡ ≡ ~ ∨ 3 ≤ ~ ∨ 3 ≤ ~ ∨ 3 P Q LCM P Q LCM[2] P Q Int Table 4. Degrees of diﬃculty of disjunctions, where ≤ and ≡ denote the Medvedev reducibility and equivalence, and (≡) denotes the Medvedev equivalence when P = Q

−−a ↼1 ↼1 a If σ = σ hm, ni for some m, n ∈ N, then we have Γtail(σ;0)(tail(σ) ) = Γtail(σ;0)(tail(σ) ) η for some η ∈ N

4. Galois Connection 4.1. Decomposing Disjunction by Piecewise Computable Functions. The main theorem in this sec- α tion (Theorem 40) states that our degree structures Dβ|γ (Definition 12) are completely characterized by the disjunction operations (Definitions 32, 33, and 34). Proposition 39 (Untangling). Let P, Q be subsets of Baire space NN. | Γ ~ ∨ 1 → ⊕ (1) There is a (1, n 2)-truth-table function : P Q LCM[n] P Q. | Γ ~ ∨ 2 → ⊕ (2) There is a (1, n 2)-computable function : P Q LCM[n] P Q. Γ ~ ∨ 3 → ⊕ (3) There is a (1, n)-computable function : P Q LCM[n] P Q. ~ 1 (4) There is a (1,ω|2)-truth-table function Γ : P ∨ Q LCM → P ⊕ Q. | Γ ~ ∨ 2 → ⊕ (5) There is a (1,ω 2)-computable function : P Q LCM P Q. Γ ~ ∨ 3 → ⊕ (6) There is a (1,ω)-computable function : P Q LCM P Q. Γ ~ ∨ 1 → ⊕ (7) There is a (2, 1)-truth-table function : P Q CL P Q. ~ 2 (8) There is a (2, 1)-computable function Γ : P ∨ Q CL → P ⊕ Q. a Proof. For the items (1), (4), and (7), we consider the truth-table functionals ∆0 : f ⊕ g 7→ 0 g and a ~ 1 ∆1 : f ⊕ g 7→ 1 g. By the definition of P ∨ Q CL, obviously ∆0( f ⊕ g) ∈ P ⊕ Q or ∆1( f ⊕ g) ∈ P ⊕ Q for ⊕ ∈ ~ ∨ 1 ∆ ∆ ⊕ ∈ × N

a For the items (2), (5), and (8), we consider the partial computable functions ∆0 : f 7→ 0 pr0( f ) and a ~ 2 ∆1 : f 7→ 1 pr1( f ). By the deﬁnition of P ∨ Q CL, obviously ∆0( f ) ∈ P ⊕ Q or ∆1( f ) ∈ P ⊕ Q for ∈ ~ ∨ 2 ∆ ∆ ∈ × N

a g a a a g a a a  g  g ∆(g) = 0 ΦΨ(hi)(g ↾ m ) ♯ 0 ΦΨ(g↾m +1)(g ↾ m ) ♯ 0 ΦΨ(g↾m +1)(g). 0  i i  k−1  j

Fix g ∈ Q. Note that reindexΨ(g ↾ s) < n for each s ∈ N, since #indxΨ(g) < n. Thus, we have N ∆(g) ∈ (n × N) . Moreover, mc(∆(g)) < ∞, since Ψ is a learner converging on Q. Thus, lims Ψ(g ↾ s) and hence lims reindexΨ(g ↾ s) converge. Therefore, prlims reindexΨ(g↾s)(∆(g)) = Φlims Ψ(g↾s)(g) ∈ P. Hence, ~ (n) 2 1 P LCM ≤1 Q. W (3) By similar argument used in proof of (1). <ω (4) Assume that P ≤1 Q via a ﬁnite collection {Φe}e

Pick g ∈ Q. Then, by our assumption, Φlimn Ψi(g↾n)(g) ∈ P for some i < b. Then tail(pri(∆(g))) ∆ ↼1 = Φ ∈ ∆ ∈ ~ (m)~ ∨ 3 2 converges, and tail(pri( (g))) limn Ψi(g↾n)(g) P. Thus, (g) P P LCM CL. W ≤ω ∆ ~ (m) 2 ≤1 (6) Assume that P 1 Q. We need to construct a computable function witnessing m∈N P CL 1 Q. Assume that ∆(σ−) has been already deﬁned. Deﬁne ∆(σ) as follows. S W

∆(σ) = ∆(σ−)a  write(e, newΦ (σ)) a(write(|σ|, Φ (σ))).  e  |σ| e<|σ|    Note that pre(∆(g)) = Φe(g). Thus, for any g ∈ Q, we have pre(∆(g)) ∈ P for some e ∈ N. In other ~ (m) 2 ≤1 ∆ words, m∈N P CL 1 Q via . S W N N N Remark. Given an operation O : P(N )×P(N ) → P(N ), one can introduce the reducibility notion ≤O (n) 1 (1) (n+1) (n) by defining P ≤O Q as O (P) ≤1 Q for some n ∈ N, where O (P) = P and O (P) = O(P, O (P)). Then, Theorem 40 indicates that our reducibility notions induced by seven monoids in Theorem 5 are also induced from corresponding disjunction operations. 4.2. Galois Connection between Degree Structures. N N N Remark. For degree structures Du and Dr on P(N ), each operator O : P(N ) → P(N ) induces the N new operator Our : Du → Dr defined by Our(degu(P)) = degr(O(P)) for any P ⊆ N . We identify O with Our whenever Our is well-defined. Recall that every partially ordered set can be viewed as a category. ω 1 1 ω ω ω Sorbi [70] showed that Deg : D1 → D1 is left-adjoint to id : D1 → D1 , and id ◦ Deg : D1 → D1 is identity, where Deg(P) denotesd the Turing upward closure of P ⊆ NN. d Definition 41. d (1) V1 (P) = ~P ∨ P3 . eff Lm∈N LCM[m] (2) V1 (P) = ~ (m) P2 . ω|eff Lm∈N LCM V1 = ~ ∨ 3 W (3) ω(P) P P LCM. eff = ~ (m) 2 (4) V1 (P) m∈N P CL. Veff = L ~W(m)~ ∨ 3 2 (5) ω (P) m∈N P P LCM CL. L W 24 K. HIGUCHI AND T. KIHARA

Vω = ~ (m) 2 (6) 1 (P) m∈N P CL. Corollary 42. S W 1 1 1 1 1 1 (1) Veff : Deff → D1 is left-adjoint to idP(NN) : D1 → Deff, and idP(NN) ◦ Veff is the identity on 1 Deff. 1 1 1 1 1 1 (2) Vω|eff : Dω|eff → D1 is left-adjoint to idP(NN) : D1 → Dω|eff, and idP(NN) ◦ Vω|eff is the 1 identity on Dω|eff. 1 1 1 1 1 1 1 (3) Vω : Dω → D1 is left-adjoint to idP(NN) : D1 → Dω, and idP(NN) ◦ Vω is the identity on Dω. eff eff 1 1 eff eff (4) V1 : D1 → D1 is left-adjoint to idP(NN) : D1 → D1 , and idP(NN) ◦ V1 is the identity on eff D1 . eff eff 1 1 eff eff (5) Vω : Dω → D1 is left-adjoint to idP(NN) : D1 → Dω , and idP(NN) ◦ Vω is the identity on eff Dω . ω ω 1 1 ω ω ω (6) V1 : D1 → D1 is left-adjoint to idP(NN) : D1 → D1 , and idP(NN) ◦ V1 is the identity on D1 . Proof. By Theorem 26.

0 4.3. Σ2 Decompositions. In computability theory, we sometimes encounter conditional branching given 0 ˜ by a Σ2 formula S ≡ ∃nS (n). That is, if S is true, one chooses a procedure p1, and if S is false, one 0 chooses another procedure p2. Thus, one may deﬁne the computability with a Σ2 conditional branching 2 0 as the class decd[Π2]. However, even if we know that S is true, we have no algorithm to ﬁnd a witness of ˜ 0 S since S (n) is Π1, while we sometimes require a witness of S . This observation motivates us to study a missing interesting subclass of the nonuniformly computable functions.

<ω 0 ω 0 2 0 ω 0 Proposition 43. decd [Π2]decp [Π1] is the smallest monoid including decd[Π2] and decp [Π1]. 2 0 ω 0 2 0 Proof. It suffices to show that every Γ ∈ decd[Π2]decp [Π1] is the composition of some Γ0 ∈ decd[Π2] ω 0 2 0 ω 0 0 0 and Γ1 ∈ decp [Π1]. For every Γ ∈ decd[Π2]decp [Π1], there exist a Π2 d-layer {D0, D1} and Π1 partitions 0 1 i i {{Pn}n∈N, {Pn}n∈N} such that Γn = Γ ↾ Di ∩ Pn is computable uniformly in i < 2 and n ∈ N, where i {Pn}n∈N is a partition of Di for every i ∈ {0, 1}. Let Γ0 : D0 ∪ D1 → D0 ⊕ D1 be the union of two a a a computable homeomorphisms D0 ≃ 0 D0 and D1 ≃ 1 D1. For instance, put Γ0(g) = i g for g ∈ Di. 2 0 0 a i a i Then Γ0 ∈ decd[Π2] since {D0, D1} is a Π2 d-layer. Define Γ1(i g) = Γn(g) for any i < 2 and g ∈ i Pn. ω 0 i i 0 Then, Γ1 ∈ decp [Π1], since {Γn}i<2,n∈N is uniformly computable, and {Pn}i<2,n∈N is uniformly Π1. Clearly i i i we have Γn ↾ Di ∩ Pn =Γ1 ◦ Γ0 ↾ Di ∩ Pn for any i < 2 and n ∈ N. Hence, Γ=Γ1 ◦ Γ0. The following concept of hyperconcatenation plays a key role in many proofs in Part II. In the next section, we will see that the hyperconcatenation can be defined as infinitary disjunction along an ill- founded tree. Definition 44 (Hyperconcatenation). For any strings σ ∈ (N ∪{pass})

content(σ−)aσ(|σ|− 1) if σ(|σ|− 1) , pass, content(hi) = hi, content(σ) =  − content(σ ) otherwise.  walk (τ−)av if τ(|τ|− 2) = ♯ & τ(|τ|− 1) = v, walk(τ ↾ 1) = hi, walk(τ) =  − walk(τ ) otherwise.  Then, the content of f ∈ (N∪{pass})N and the walk of g ∈ (N∪{♯, pass})N are deﬁned by content( f ) = n∈N content( f ↾ n) and walk(g) = n∈N walk(g ↾ n), respectively. Let P and Q be any subsets of N H SBaire space N . The hyperconcatenationS~Q∨P and the non-Lipschitz hyperconcatenation ~Q∨P 0 Σ0 Σ 2 2 INSIDE THE MUCHNIK DEGREES I 25

of Q and P are defined as follows. H N ↼1 ~Q ∨ P 0 = {g ∈ (N ∪{♯}) : walk(g) ∈ Q or tail(g) ∈ P}, Σ2 N ↼1 ~Q ∨ PΣ0 = {g ∈ (N ∪{♯, pass}) : content ◦ walk(g) ∈ Q or tail(g) ∈ P}. 2 Note that these notions are non-commutative. Theorem 45 (As the Law of Excluded Middle). The implications (b+) → (a) → (a−) ↔ (b−) hold for any P, Q, R ⊆ NN: (a) ~Q ∨ PH ≤1 R. Σ0 1 2 − 1 (a ) ~Q ∨ PΣ0 ≤ R. 2 1 + 0 (b ) There is a Σ2 sentence ϕ ≡ ∃vθ(v) with a uniform sequence {Γi}i∈N, ∆ of computable functions such that • if g ∈ R satisfies θ(v), then Γv(g; u) ↓ for any u ∈ N, and Γv(g) ∈ P. • if g ∈ R satisfies ¬θ(v), then ∆(g; u) ↓ for any u ≤ v, and [∆(g) ↾ v + 1] intersects with Q. • if g ∈ R satisfies ¬∃vθ(v), then ∆(g; u) ↓ for any u ∈ N, and ∆(g) ∈ Q. − 0 (b ) There is a Σ2 sentence ϕ ≡ ∃vθ(v) with a uniform sequence {Γi}i∈N, ∆ of computable functions such that • if g ∈ R satisfies θ(v), then Γv(g; u) ↓ for any u ∈ N, and Γv(g) ∈ P. • if g ∈ R satisfies ¬∃vθ(v), then ∆(g; u) ↓ for any u ∈ N, and ∆(g) ∈ Q. + N Proof. (b )→(a): Assume that S i = {g ∈ N : Θ(g; i) ↑} for some computable function Θ, and that 1 1

mcl♯(σ, n) = min{m ≤ |σ| : count(σ ↾ m) > n}, if such m exists. ↼mcl (σ,count(σ↾v))+1 Then set Γv(σ) = Φ(σ) ♯ ; and set ∆(σ) = λn.Φ(σ, mcl♯(σ, n)). Note that if g ∈ R ∩ S k 1 1 for some k ∈ N, then Γk(g) ∈ P; otherwise, ∆(g) ∈ Q. Therefore, P ≤1 R ∩ S v via Γv and Q ≤1 R \ S via ∆. (b−)→(a−): For each σ ∈ N

computable strictly along {S n}n∈N if there is a uniform sequence of computable functions {Γn}n∈N and ∆ such that Γ ↾ dom(Γ)\ n S n = ∆ ↾ dom(Γ)\ n S n and Γ ↾ dom(Γ)∩S n \S n−1 =Γn ↾ dom(Γ)∩S n \S n−1 and ∆(g) ↾ n is deﬁnedS for any g ∈ dom(Γ) \SS n. H Remark. Theorem 45 implies that there is a function Γ : ~Q ∨ P 0 → P ⊕ Q (Γ : ~Q ∨ P → P ⊕ Q) Σ Σ0 2 2 0 such that Γ is computable (strictly) along sequences of Π1 sets. <ω 0 ω 0 Corollary 47. decd [Π2]decp [Π1] is the smallest monoid containing all functions computable (strictly) 0 along sequences of Π1 sets. 0 Proof. Let S be the class of all functions computable (strictly) along sequences of Π1 sets. Then, clearly, 2 0 ω 0 <ω 0 ω 0 we have decd[Π2] ∪ Γ1 ∈ decp [Π1] ⊆S⊆ decd [Π2]decp [Π1]. Thus, the desired condition follows from Proposition 43.

5. Going Deeper and Deeper 5.1. Falsifiable Mass Problems. We are mostly interested in local degree structures such as Turing 0 N degrees of c.e. subsets of N and Medvedev degrees of Π1 subsets of 2 . In such cases, the straightforward ~ ∨ 2 two-tape (backtrack) notions in Definitions 33 and 34 are hard to use, since, for instance, P Q LCM[2] 0 0 may not belong to Π1 even if P and Q are Π1. This observation prompts us to define consistent two-tape disjunctions. The consistency set Con(Ti)i∈I for {Ti}i∈I is defined as follows. N Con(Ti)i∈I = { f ∈ (I × N) : (∀i ∈ I)(∀n ∈ N) pri( f ↾ n) ∈ Ti}. The notion of consistency sets has a relationship with consistent learning (see also Remark below 0 Proposition 53). The consistency sets are useful to reduce the complexity of our disjunctions to be Π1. We now introduce the following consistent modifications of our disjunctive notions. 0 N Definition 48. Let P0 and P1 denote Π1 subsets of N . 2 P0▽ωP1 = ~P0 ∨ P1LCM ∩ Con(TP0 , TP1 ). ~ 2 P0▽nP1 = P0 ∨ P1 LCM[n] ∩ Con(TP0 , TP1 ). 2 P0▽∞P1 = ~P0 ∨ P1CL ∩ Con(TP0 , TP1 ).

Here TP0 and TP1 are corresponding trees for P0 and P1, respectively. 0 N Proposition 49. Let P and Q be Π1 subsets of N . ▽ ≡1 ~ ∨ 2 ∈ N (1) P nQ 1 P Q LCM[n] for each n . 1 ~ 2 (2) P▽ωQ ≡1 P ∨ Q LCM. 1 ~ 2 (3) P▽∞Q ≡1 P ∨ Q CL. 1 2 Proof. For each item, clearly P▽∗Q ≥1 ~P ∨ Q∗. Thus, it suﬃces to construct a computable functional 1 2 Φ witnessing P▽∗Q ≤1 ~P ∨ Q∗. Let T0 and T1 denote the corresponding computable trees for P and Q respectively. Set Φ(hi) = hi. Fix σ ∈ (2 × N)

0 (1) P▽nQ is Π1, for any n ∈ N. 0 (2) P▽ωQ is Σ2. 0 (3) P▽∞Q is Π1.

Proof. Let T0 and T1 denote the corresponding computable trees for P and Q respectively. We consider the following computable tree:

Note that TP,Q,n is uniformly computable in n, since pri(σ) and mc(σ) are computable uniformly in

▽ = a ⊕ a ≡1 a ▽ ≡1 ~ ∨ 2 Proof. (1) P P (P P) (P P) 1 P P. (2) By Proposition 49 (1), we have P 2Q 1 P Q LCM[2]. 1 Then, P▽2Q ≤1 P▽Q is witnessed by the following reduction ∆. write( f (0), f ↼1), if f ↼1 ∈ [T ], ∆ = σ(0) ( f )  ↼1 a ↼k+1 ↼1 write( f (0), f ↾ k) write(1 − f (0), f ), if (∃k ∈ N) f ↾ k ∈ Lσ(0).  Here, T0 and T1are the corresponding computable trees for P and Q respectively, and Li is the set of a a all leaves of Ti for each i < 2. Clearly, ∆ is computable. Fix hii g ∈ P▽Q. Obviously, mc(hii g) < 2. a a If g ∈ [Ti] then pri(∆(hii g)) = g ∈ [Ti], and if g = σ h for some σ ∈ Li and h ∈ [T1−i] then a a a pri(∆(hii σ h)) = h ∈ [T1−i]. Hence, ∆(hii g) ∈ P▽2Q. 1 → → 1 To see P▽Q ≤1 P▽2Q, it suﬃces to construct a computable functional Γ witnessing (P Q)⊕(Q P) ≤1 P▽2Q by Proposition 52. Set Γ(hi) = hi, and Γ(h(i, n)i) = hi, ni for any i < 2 and n ∈ N. Fix σ = σ−−ah(i, m), ( j, n)i ∈ (2 × N)

0 N Proposition 54. For Π1 sets P, Q ⊆ N and n ∈ N, 2 1 2 1 1 1 1 ~P ∨ QLCM ≤1 ~P ∨ QLCM[n+2] ≤1 ~P ∨ QCL ≤1 ~P ∨ QInt. ﬃ ▽ ≤1 ∪ ~ ∨ 1 ≡1 ∪ Proof. It su ces to show P Q 1 P Q, since P Q CL 1 P Q by Proposition 35 (5) and ~ ∨ 2 ≡1 ▽ a ⊗ a ≤1 ∪ P Q LCM[2] 1 P Q by Proposition 53 (2). Indeed, we can show that (P Q) (Q P) 1 P Q. INSIDE THE MUCHNIK DEGREES I 29

Table 5. Hierarchy of Consistent Disjunctions

1 P ⊕ Q ~P ∨ QInt Intuitionistic disujunction (= P▽1Q) ~ 1 P ∪ Q P ∨ Q CL Classical one-tape disjunction ▽ ~ ∨ 2 ≡ a = P Q P Q LCM[2] Commutative concatenation ( P Q if P Q) ▽ ~ ∨ 2 P nQ P Q LCM[n] LCM disjunction with mind-changes-bound n ~ 2 P▽ωQ P ∨ Q LCM LCM disjunction ~ 2 P▽∞Q P ∨ Q CL Classical disjunction

a 1 We construct a computable functional Φ witnessing P Q ≤1 P ∪ Q. If σ ∈ TP, then set Φ(σ) = σ. If a σ < TP, then pick a unique ρ ⊆ σ such that ρ ∈ LP, and set Φ(σ) = ρ σ for such ρ, where LP is the set of all leaves of TP. Clearly Φ is computable, and note that Φ(σ) ⊆ Φ(τ) whenever σ ⊆ τ. If g ∈ P, then a a Φ(g) = g ∈ P. If g ∈ Q \ P, then there is a unique ρ ⊂ g such that ρ ∈ LP, and Φ(g) = ρ g ∈ P Q. Thus, a 1 P Q ≤1 P ∪ Q via Φ. 5.2. Compactified Infinitaly Disjunctions. This subsection is concerned with a trick to represent infinitary disjunctive notions as effective compact sets. N Definition 55. Fix a collection {Pi}i∈I of subsets of Baire space N . N (1) ~ i∈I PiInt = { f ∈ (I × N) : ((∃i ∈ I) pri( f ) ∈ Pi)& mc( f ) = 0}. W N (2) ~ i∈I PiLCM = { f ∈ (I × N) : ((∃i ∈ I) pri( f ) ∈ Pi)& mc( f ) < ∞}. W N (3) ~ i∈I PiCL = { f ∈ (I × N) : (∃i ∈ I) pri( f ) ∈ Pi}. W N Proposition 56. Let {Pn}n∈N be an infinite collection of subsets of Baire space N . 1 a (1) ~ n∈N PnInt ≡1 n∈N Pn, where n∈N Pn = {hni f : f ∈ Pn}. ~W ≡1L~ ∨ 3 L = ∈ N (2) i,n Pi,n LCM 1 P0 P1 LCM, where Pi,n Pi for each i < 2 and n . W ~ ≥1 7→ apr ~ ≤1 Proof. (1) n∈N Pn Int 1 n∈N Pn is witnessed by f ( f (0))0 ( f (0))0 ( f ), and n∈N Pn Int 1 W L ↼1 ↼1W N n∈N Pn is witnessed by f 7→ write( f (0), f ), where recall that write( f (0), f ) = ( f (0)) ⊕ L(λn. f (n + 1)) indicates the instruction to writing the infinite word f ↼1 on the f (0)-th tape. Ξ ~ ≥1 ~ ∨ 3 (2) We first construct a computable function witnessing i,n Pi,n LCM 1 P0 P1 LCM. For ((i, n), v) ∈ (2×N)×N, we first set Ξ(h((i, n), v)i) = h((i, n), v)i. For eachW string σ = σ−−ah((i, n), v), (( j, m), w)i∈ ((2 × N) × N)

(1) ~ P ≡1 ` P . n∈N n LCM 1 ω n∈N n W 1 h i (2) ~ n∈N PnCL ≡1 `∞ n∈N Pn. W Proof. As in the proof of Proposition 49. However, the problem is that our models of infinitary disjunctions are not compact. A modification of infinitary sum was introduced by Binns-Simpson [8] to embed a free Boolean algebra into the Muchnik 0 lattice of Π1 subsets of Cantor space, and such a variation was called a recursive meet. An important 0 N feature of their modification is that it is a Π1 subset of the compact space 2 . 0 N Definition 58 (Binns-Simpson [8]). Let P and {Qn}n∈N be computable collection of Π1 subsets of 2 , and let ρn denote the length-lexicographically n-th leaf of the corresponding computable tree of P. Then, we define the infinitary concatenation and recursive meet as follows: Pa{Q } = P ∪ ρ aQ , −→ Q = CPAa{Q } . i i∈N [ n n M i∈N i i i∈N n Here, recall that CPA is a Medvedev complete set, which consists of all complete consistent extensions 0 of Peano Arithmetic. The Medvedev completeness of CPA ensures that for any nonempty Π1 subset P ⊆ 2N, a computable function Φ : CPA → P exists. 0 N −→ 1 Proposition 59. For any computable sequence {Pn}n∈N of nonempty Π subsets of 2 , Pn ≡<ω 1 L n∈N Pn. Ln∈N −→ 1 a a Proof. The condition n∈NPn ≤1 n∈N Pn is witnessed by a computable function n g 7→ ρn g. We L L−→ 1 will construct a learner witnessing n∈NPn ≥<ω n∈N Pn. Fix a computable function Φe : CPA → a L 0 L N 0 P0. Such Φe exists, since every nonempty Π1 subset of 2 is (1, 1)-reducible to CPA. We also fix a a a

Deﬁnition 60 (Transﬁnite Mind-Changes). Let (O, ≤O) denote Kleene’s system of ordinal notations (see Rogers [56]). Then for each a ∈O we introduce the a-th derivative of P ⊆ NN as follows. P P if a = 0, a  a+  + P = ~P ∨ Pb2 P = ~P ∨ Pb 2 if a = 2b,  LCM[2]  LCM[2]  Φe(n) ~ ∨ Φe(n)+2 = · e  n∈N P  P n∈N P LCM[2] if a 3 5 . L  L   e Here, we require Φe(n)

0 N a+ 1 (a) 1 Proposition 62. For any nonempty Π1 set P ⊆ 2 and any notation a ∈O, the condition P ≤1 P ≤1 Pa holds. a (b) ~ ∨ (b)2 (b) Π0 Proof. Clearly P P is (1, 1)-equivalent to P P LCM[2], since P is 1 by Proposition 61, where the ~ ∨ (Φe(n))2 ≤1 (1, 1)-equivalence follows by Proposition 37 and 53. It is easy to see that P n∈N P LCM[2] 1 a (Φe(n)) 1 (Φe(n)) L a (b) (b)a P {P }n∈N ≤1 n∈N P holds. For successor steps, it suffices to show that P P ≤W (P P). L e (b) 1 (b)a If |b|O is a finite ordinal, it is clear. If |b|O is an infinite ordinal, say b = 3 · 5 , then P ≤1 P P holds, since Φe(n) + 1 ≤O Φe(n + 1).

Notation. Every a ∈ O is often identiﬁed with the corresponding well-founded tree Ta consisting of all ﬁnite nonempty

Proposition 64. For a notation a ∈ O, a partial function Γ :⊆ NN → NN is of eﬀective discontinuity level ≤O a if and only if it is learnable via an a-bounded learner.

0 Proof. The desired equivalence is obtained from an interpretation between S b and the Σ1 set generated by the c.e. set {σ ∈ N

5.4. Infinitary Disjunctions along any Graphs. In the classical proof process, a verifier Ψ on “P0 or P1” may change his mind infinitely often. In the backtrack-tape model, this situation means that Ψ chooses the backtrack symbol ♯ infinitely many often. Then the word on Λ is eventually finite, and it verifies neither P0 nor P1. Therefore, in the model, if Ψ succeeds to verify “P0 or P1” then the backtrack symbol ♯ occurs on the record ∆ at most finitely often. Consequently, in the backtrack-tape model, classical verification coincides with LCM verification. However, we would like to cover the case that unbounded or infinitely many mind-changes occur. This may be archived by regarding the backtrack- tape model as a kind of infinitary tape model. The dynamic-tape model: Assume that a directed graph (V, E) is given, where V can be infinite, E ⊆ V × V, and an initial vertex ε ∈ V is chosen. For any v ∈ V, let adj(v) = {w ∈ V : (v, w) ∈ E}. When a verifier Ψ tries to prove that “ v∈V Pv”, infinite tapes , and Λv for v ∈ V are given. The tape is called the declaration, Λv is called theW working tape for each v ∈ V. First the letter ε is written on , and no word is written on Λv for v ∈ V. At each stage s, assume that v[s] is written on . Then the verifier Ψ executes one or the other of two following actions. (1) Ψ declares some w ∈ adj(v[s]), erases all words on , and writes w on ; or (2) Ψ writes a letter k ∈ N on the working tape Λv[s].

Assume that a veriﬁer Ψ tries to prove that “P0 or P1”.

• Intuitionism: Consider V = {ε, 0, 1}, E = {(ε, 0), (ε, 1)}, and Pε = ∅. • LCM with ordinal-bounded mind-changes: For a computable well-founded tree V = T ⊆ N

a P0, if |σ| is even, E = E(T) = {(σ, τ) ∈ T × T : (∃i ∈ N) τ = σ i}, Pσ =  P1, if |σ| is odd,  • LCM: Consider V = N; E = {(n, n + 1) : n ∈ N}; P2n = P0 for any n ∈ N; and P2n+1 = P1 for any n ∈ N. Moreover, the word written on the declaration must converge. • (V, E)-relaxed Classical: (V, E) = (V0, V1, E) is a given directed bipartite graph, and Pτ = Pi for any τ ∈ Vi and i < 2.

Deﬁnition 65 (Dynamic Disjunctions). Let G = (V, E) be a directed graph, and let {Pv}v∈V be a collection of subsets of Baire space. For E ⊆ V2, put E = E ∪ {hv, vi : v ∈ V}. We deﬁne the dynamic disjunction of {Pv}v∈V along the graph (V, E) as follows.

N Pv = f ∈ (V × N) : (∀n ∈ N) (h( f (n))0, ( f (n + 1))0i∈ E) & (∃v ∈ V) pr ( f ) ∈ Pv . _ n v o v∈(V,E) { } Π0 NN Moreover, if Pv v∈V is a computable sequence of 1 subsets of , and TPv be the corresponding tree for Pv, we also deﬁne its consistent versions.

(1) `v∈(V,E) Pv = ~ v∈(V,E) Pv ∩ Con(TPv )v∈V . W N (2) Hv∈(V,E)Pv = { f ∈ (V × N) : (∀n ∈ N) (h( f (n))0, ( f (n + 1))0i∈ E)} ∩ Con(TPv )v∈V .

Here, recall that, for x = (x0, x1), the ﬁrst coordinate x0 is denoted by (x)0. If Pv = P for any v ∈ V, then we simply write `v∈V P and Hv∈V P for `v∈(V,E) Pv and Hv∈(V,E)Pv respectively. INSIDE THE MUCHNIK DEGREES I 33

As our dynamic-tape model is an infinitary-tape model, this model may be natural to be regarded as expressing a proof process of an infinitary disjunction v∈V Pv. Therefore, we refer the model with (V, E) as an infinitary disjunction along (V, E). Later we willW introduce a more complicated model. It will be called the nested-tape model. We first see an upper and lower bound of the degrees of difficulty of these disjunctive notions, and a relationship among various models we have introduced. Let Deg(P) denote the Turing upward closure of P, i.e., Deg(P) = {g : (∃ f ≤T g) f ∈ P}, and [(V, E)] denoted the set of all N infinite paths through a graph (V, E), i.e.,d [(V, E)] = {p ∈ V : (p(n), p(n + 1)) ∈ E}. 0 Proposition 66. Let (V, E) be a computable directed graph, and {Pv}v∈V be a computable sequence of Π1 subsets of NN. 1 1 (1) Deg Pv ≤ ` Pv ≤ Pv. v∈V 1 v∈(V,E) 1 v∈V L 1 L 1 (2) Degd [(V, E)] ⊕ Pv ≤ Hv∈(V,E)Pv ≤ [(V, E)] ⊕ Pv. Lv∈V 1 1 Lv∈V d 1 a N Proof. (1) ` ∈ Pv ≤ Pv is witnessed by v f 7→ write(v, f ) = v ⊕ f . For any f ∈ ` ∈ Pv, v (V,E) 1 Lv∈V v (V,E) we have prv( f ) ∈ Pv for some v ∈ V. Thus, we have prv( f ) ≤T f , since prv is partially computable, and f ∈ dom(prv). Hence, f ∈ Deg(Pv). (2) Fix f ∈ [(V, E)] ⊕ vd∈V Pv. If f (0) = 1, then we can show the desired condition as in (1). If f is of a L 1 the form f = 0 g, we have λn.hg(n), 0i ∈ Hv∈(V,E)Pv since g ∈ [(V, E)]. Hence, Hv∈(V,E)Pv ≤1 [(V, E)] ⊕ 1 Pv. To see Deg [(V, E)] ⊕ Pv ≤ Hv∈(V,E)Pv, we inductively define a partial computable Lv∈V Lv∈V 1 function walk :⊆ (dV × N)N → VN as follows. Set walk(hi) = hi, and fix σ = σ−−ah(u, m), (v, n)i ∈ (V × N)

ﬁnite. By Pigeon Hole Principle, there is a vertex v ∈ T such that ( f (n))0 = v occurs for inﬁnitely many

n ∈ N. Then, prv( f ) must be infinite. Therefore, prv( f ) ∈ [TPv ] = Pv since f ∈ Con(TPv )v∈V . Hence, f ∈ `v∈(T,E(T)) Pv. (2) The condition P ≤1 P ⊕ P follows from Proposition 66 (1). For any f ∈ P , `v∈(V1,E1) v 1 0 1 `v∈(V1,E1) v a there is i < 2 such that ( f (n))0 = i for any n ∈ N. Thus, i f ∈ P0 ⊕ P1. The (1, 1)-equivalence of P and H P follows from the item (1) since (V , E ) is finite. `v∈(V1,E1) v v∈(V1,E1) v 1 1 (3) Clearly, P ▽ P ⊆ P . Thus, by Proposition 53 (2), P ▽P ≥1 P . For 0 2 1 `v∈(V2,E2) v 0 1 1 `v∈(V2,E2) v f ∈ P , if |( f (0)) | = 1 then Φ( f ) = f ∈ P ▽ P . If |( f (0)) | = 2, say ( f (0)) = hi, ji, then `v∈(V2,E2) v 0 0 2 1 0 0 Φ( f ) = write( j, pr ( f )) ∈ P ▽ P . Hence, P ▽P ≤1 P via the computable function Φ. The j 0 1 1 0 1 1 `v∈(V2,E2) v (1, 1)-equivalence of P and H P follows from the item (1) since (V , E ) is finite. `v∈(V1,E1) v v∈(V1,E1) v 1 1 (4) If σ is extendible to an element of P, there is a unique κ ∈ T such that σ can `v∈(Ta,E(Ta)) a |κ| be represented as i≤|κ| write(κ ↾ i, cut(σ; i)) for some sequence cut(σ) ∈ (TP) . Conversely, if a+ σ is extendible to an element of P , there is a unique κ ∈ Ta such that σ can be represented as ( κ∗(i)acut(σ; i)a♯)aκ∗(|κ| − 1) for some sequence cut(σ) ∈ (T )|κ|, where κ∗(i) indicates the i<|κ|−1 P location of κ(i) in the tree TP. The procedures to interchange these cuts are the desired (1, 1)-reductions. = ▽ ~ ∨ 2 ≡1 ▽ (5) It is easy to see that `v∈({0,1},{0,1}2) Pv P0 ∞P1. Moreover, P0 P1 CL 1 P0 ∞P1 by Proposi- tion 49. (6) For each σ = τah(i, m), ( j, n)i ∈ (N × N) 0}, − and pick the least e such that rqe(σ ) = r if such r and e exist. In this case, we say that e acts. − − − If there is no such e, we set Γ(σ) = Γ(σ ), acte(σ) = acte(σ ), and rqe(σ) = rqe(σ ). If there ∗ ↼|Φ (σ↾|act (σ)|)| a ∗ is such e, put σ = (Φe(σ)) e e , i.e., Φe(σ) = (Φe(σ ↾ |acte(σ)|)|) σ . Then we set − a ∗ ∗ Γ(σ) = Γ(σ ) write(e,σ ). Then, put rqe(σ) = −1 and acte(σ) = |σ|. For each e ∈ N \{e}, set − ∗ − acte∗ (σ) = acte∗ (σ ). Moreover, if e ≤ |σ|, rqe∗ (σ ) = −1, and |Φe∗ (σ ↾ |acte∗ (σ)|)| < |Φe∗ (σ)|, then − N declare rqe∗ (σ) = |σ|. Otherwise, put rqe∗ (σ) = rqe∗ (σ ). Fix g ∈ N . We claim that Φe(g) act infinitely often whenever Φe(g) is total. Our construction ensures that only finitely many e’s require attentions along g ↾ s for each s ∈ N. Therefore, for R = {e ∈ N : rqe(g ↾ s) > 0}, if e ∈ R, then the strategy e acts by stage s + #R, i.e., acte(g ↾ s + #R) ≥ s. Assume that e act at stage t ∈ N. Then the algorithm ∗ Γ(g ↾ t) writes the new information (g ↾ t) of Φe(g) on the e-th tape, i.e., pre(Γ(g ↾ t)) = Φe(g ↾ t). INSIDE THE MUCHNIK DEGREES I 35

/.-,()*+P 76540123Q /.-,()*+P /76540123Q i 6 b❊❊❊ ①< ❊b ❊❊ ❊ ①①① ❊ x (1) start (2) start P i❘ / Q P /.-,()*+ ❘❘❘ 76540123 8/.-,()*+ ❘❘❘ x (5) P i❘ / Q (6) P /.-,()*+ ❘❘❘ 76540123 8/.-,()*+ ❘❘❘ x P / Q P o / Q P i❘ / Q P /.-,()*+b❊❊ 76540123 /.-,()*+❊b ❊ ①76540123< /.-,()*+ ❘❘❘ 76540123 8/.-,()*+ ❊❊ ❊❊ ①① ❘❘❘ (3) (4) ① / Q x start❋❋ start /.-,()*+P b❊❊ 76540123 /.-,()*+PO ❋❋" ❊❊ /.-,()*+P o 76540123Q start start

Figure 1. The dynamical representations of disjunction operations: (1) ~P ∨ QInt (P ⊕ a ~ ∨ 2 ▽ ~ ∨ 2 ▽ ~ ∨ 3 Q); (2) P Q; (3) P Q LCM[2] (P Q); (4) P Q CL (P ∞Q); (5) P Q LCM; (6) Deg(P), the Turing upward closure of P. d Thus, eventually, we have pre(Γ(g)) = Φe(g). For any g ∈ Deg v∈N Pv , there is an index e ∈ N such S that Φe(g) ∈ Pv for some v ∈ N. Consequently, Γ(g) ∈ `v∈(Nd,N2) Pv.

0 Proposition 68. Let (V, E) be a computable directed graph, and {Pv}v∈V be a computable sequence of Π1 subsets of 2N. Then we have the following. 0 (1) `v∈(V,E) Pv is Σ3. 0 (2) Hv∈(V,E)Pv is Π1. Π0 h + i ∈ Proof. Clearly, Con(TPv )v∈V is 1. Moreover, the relation ( f (n))0, ( f (n 1))0 E is computable, N 0 0 uniformly in f ∈ (N × N) and n ∈ N. Thus, Hv∈(V,E)Pv is Π1. The relation prv( f ) ∈ Pv is Π2 in v ∈ V and f ∈ NN, since it is equivalent to the following formula.

(∀n ∈ N)(∃m ∈ N) |prv( f ↾ m)| > n & prv( f ↾ m) ∈ TPv . 0 Therefore, `v∈(V,E) Pv is Σ3. 5.5. Inﬁnitary Disjunctions along ill-Founded Trees. To study (< ω,ω)-degrees, the team-learning 0 0 proof model of P is expected to be useful. However, the model may be far from Π1 whenever P is Π1. To break out of the dilemma, the following minor modiﬁcation of consistent dynamic disjunction is helpful. N

α =  cut(α; i)ahwalk(α; i)i acut(α; |walk(α)|).   i<|walk(α)|    36 K. HIGUCHI AND T. KIHARA

Then the set Hσ∈V Pσ is characterized as follows.

1 p(e, n + 1) > p(e, n) for any index e and n. Assume that R ≤ω S via a learner Ψ. We need to construct a 1 eventually-Popperian learner ∆ witnessing R ≤ω S . At each stage s, we deﬁne a value of ∆(σ) for each s s s σ ∈ 2 . For a given σ ∈ 2 , we compute q(σ) = min({i < s : (∀τ ∈ 2 ) τ ⊇ σ → τ ∈ Ti}∪{s}), and put ∆(σ) = p(Ψ(σ), q(σ)). If f < S , then limn q( f ↾ n) diverges. Therefore, limn ∆( f ↾ n) diverges. On the

other hand, if f ∈ S , then limn q( f ↾ n) converges to some q. Then Φlimn ∆( f ↾n)( f ) = Φp(limn Ψ( f ↾n),q)( f ) = Φ ∈ ∆ ≤1 Π0 limn Ψ( f ↾n)( f ) R. Consequently, is eventually-Popperian, and witnesses R ω S . If S is 1, then we modify ∆ by setting ∆(σ) to be a ﬁxed index of a total computable function g 7→ 0ω, whenever σ extends a leaf of TS . Then, ∆ is also conﬁdent. <ω (3) If P ≤1 Q via n many computable functions {Φi}i

Here, we fix some string ρ ∈ TQ0 of length c, and we set Φi(σ; k − c) = σ(k) for each k < c. If Φi(walk(σ); |walk(σ)|− c) is undefined, then Φhyp(i, j)(τ) is undefined for any τ ⊇ σ. If the former is not cut cut − cut the case (then, |tail (σ)| = |tail (σ )|+1), then we concatenate the new values of Φ j(tail (σ)) − cut − cut to Φhyp(i, j)(σ ) if it belongs to TP0 . Formally, if Φ j(tail (σ )) ( Φ j(tail (σ)) ∈ TP0 , say cut cut − a − a Φ j(tail (σ)) = Φ j(tail (σ )) ρ, then we define Φhyp(i, j)(σ) = Φhyp(i, j)(σ ) ρ. Otherwise, we − set Φhyp(i, j)(σ) = Φhyp(i, j)(σ ). 1 ω Now assume that P0 ≤ω P1 via a computable function Φe, and Q0 ≤ Q1 via an eventually Lipschitz 1 learner Ψ with a constant c. We construct a learner ∆ witnessing Q0HP0 ≤ω Q1HP1. At first the learner ∆ guesses the index ∆(hi) = hyp(Ψ(hi), e). Fix σ ∈ N |walk(σ−)|, if either |walk(σ)| < c or walk(σ) < T ext is witnessed, the Q1 learner ∆ changes his mind (this situation occurs only finitely often). Otherwise, the learner ∆ keeps his previous guess, i.e., ∆(σ) = ∆(σ−). In this way, it is not hard to see that we may construct a learner ∆ 1 witnessing Q0HP0 ≤ω Q1HP1. 5.6. Nested Infinitary Disjunctions along ill-Founded Trees. In Part II, we employ finite iterations of the hyperconcatenation H to show that some (local) degree structures are not Brouwerian. Beyond this, 38 K. HIGUCHI AND T. KIHARA

2 1 H 1 T Hσ∈T 0 Tσ τ∈Tσ σ,τ G : 0 1 2

0 1 T 2 T T101 101,1001

1001 101

Figure 2. An example nested tape model when G is a linear order of length 3: h012i is 0 1 0 1 written on Λ; h101i is written on Λ ; h1001i is written on Λ101; then Λ, Λ , Λ101, and 2 Λ101,1001 are available.

we just note that one can iterate the hyperconcatenation H along any directed graph (V, E). However, the iteration of H does not represented by our dynamic proof model. We may introduce another new model called the nested disjunction model. ∗ 2 The nested tape model: As an example, ﬁrst we consider the nested disjunction T = H 0 H 1 [T ] σ∈T τ∈Tσ σ,τ 0 1 along the graph G = ({0, 1, 2}, {(0, 1), (1, 2)}) with the initial vertex ε = 0, where T = {T }∪{Tσ}σ∈N

Here, on the tape Λ, the veriﬁer Ψ is only allowed to write a path starting from the initial vertex ε within the graph G = (V, E). Example 76. On the nested tape model for T ∗, let α ∈ ((I ∪{}) × N)

the n-th available index along α, p(α, n) ∈ I, for each n ≤ |pr(α)|, as follows. a p(α, 0) = (ε, hi), p(α, i + 1) = (pr(α)(i), (p(α, i))1 hprp(α,i)(α)i).

Then we define the set of all indices of available tapes along α by A(α) = {p(α, n) : n ≤ |pr(α)|}. The set S (α) of successors of α is defined as follows: a S (α) = {(p, n) ∈ (I ∪{}) × N : p ∈ A(α)& prp(α) n ∈ Tp} ∪{(, v) : (pr(α)(|pr(α)|− 1), v) ∈ E}. N v Then the nested infinitary disjunction Wσ∈I[Tσ] ⊆ ((I ∪{}) × N) of {Tσ}(v,σ)∈I is defined by N Wσ∈I[Tσ] = { f ∈ ((I ∪{}) × N) : (∀n ∈ N) f (n) ∈ S ( f ↾ n)}.

We can also deﬁne Wσ∈I[Tσ] = { f ∈ Wσ∈I[Tσ] : |pr( f )| < ∞}.

Proposition 78. Assume that G = (V, E) is a computable directed graph, and {Tσ}σ∈I is a computable

Proof. Note that α 7→ A(α) is computable. Therefore, α 7→ S (α) is also computable. Thus, Wσ∈I[Tσ] is 0 Π1. max{head(β) : β ( α}, then we deﬁne − a − − − a pr˜ (α) = pr˜ (α ) w. Otherwise, set pr˜ (α) = pr˜ (α ). If pr˜ (α) , pr˜ (α ), thenp ˜(α) = p˜(α ) hhii. Otherwise, deﬁnep ˜(α) ∈ (2

P = { f ∈ Rule∀ : (∀n ∈ N) (pr˜ ( f ↾ n) ∈ V & (∀σ ∈ I) pr˜ σ( f ↾ n) ∈ Tσ)}. 0 1 Clearly, P is computably bounded, and Π1. It remains to show that P ≡1 Wσ∈I[Tσ]. We first inductively 1 −a define a computable function Φ witnessing P ≥1 Wσ∈I[Tσ]. Set Φ(hi) = hi, fix α = α w ∈ Rule, and assume that Φ(α−) has been already defined. If w ∈ {+.−}, then set Φ(α) = Φ(α−). Assume w < {+, −}. If head(α) > max{head(β) : β ( α}, then we set Φ(α) = Φ(α−)ah(, w)i. Otherwise, we set − a 1 Φ(α) = Φ(α ) h((pr˜ (α), p˜(α) ↾ h(α)), w)i. It is not hard to check P ≥1 Wσ∈I[Tσ] via Φ. 1 ∗ ∗ To prove P ≥1 Wσ∈I[Tσ], we first define a computable function head . Firstly put head (hi) = 0. −a

d = head∗(α) − head∗(α−). If d ≥ 0, then Φ(α) = Φ(α−)a +d aw. If d < 0, then Φ(α) = Φ(α−)a −−d aw. 1 It is not hard to check P ≤1 Wσ∈I[Tσ] via Φ. v v If Tσ only depends on v ∈ V, i.e., Tσ = Tv, then the nested system (I, Λ, T, G) can be viewed as the iteration of the hyperconcatenation H along the graph G. In this case, we write Wv∈(V,E)Pv for this notion.

Proposition 79. Let (V, E) be a computable directed graph, and {Pv}v∈V be a computable collection of 0 N 1 Π1 subsets of N . Then, Wv∈(V,E)Pv ≤1 Hv∈(V,E)Pv. 1 Proof. We inductively deﬁne a computable function Φ which witnesses the condition Wv∈(V,E)Pv ≤1 −−a

Let QwP be Wv∈(TQ,E(TQ))Pv, where Pv = P for each v ∈ TQ. Then, for example, the nested nested model can be introduced as the iteration of w along any directed graph (V, E). Therefore, inside the (ω, 1)- 0 N degree of any Π1 set P ⊆ 2 , one may iterate this procedure as “nested nested nested ... nested nested ... ” Actually one may iterate “nested nested nested ... nested nested ... ” along any directed graph, for example, along the corresponding tree of P. If we call it a “hypernested” model, then, of course, we may introduce models which are “hypernested hypernested”, and “hypernested hypernested hypernested”, and so on. By iterating this notion along the corresponding tree of P, we obtain a “hyperhypernested” model. Iterating this procedure, of course, we have the iteration of “hyper” along the correspoding tree INSIDE THE MUCHNIK DEGREES I 41

of P. This observation reveals to us that there are a fine structure, a deep hierarchy, and a morass inside 0 N each (ω, 1)-degree (or equivalently, each Turing upward closure) of a Π1 subset of 2 . 6. Weihrauch Degrees and Wadge Games 6.1. Weihrauch Degrees and Constructive Principles. 6.1.1. Basic Notation. We can also give a characterization of our nonuniformly computable functions in the context of the Weihrauch degrees which is a generalization of the Medvedev degrees. Then, our results could be translated into the results on the Weihrauch degrees. A partial function P :⊆ NN → P(NN) is called a multi-valued function. Then P is also written as P : NN ⇒ NN. One can think of each N multi-valued function P as a collection {P(x)}x∈dom(P) of mass problems P(x) ⊆ N , or a Π2-theorem (∀x ∈ dom(P))(∃y) y ∈ P(x). Definition 81 ([11, 13, 12, 14]). Let P :⊆ NN ⇒ NN and Q :⊆ NN ⇒ NN be multi-valued partial functions. (1) A single-valued function q :⊆ NN → NN is said to be a realizer of Q if q(x) ∈ Q(x) for any x ∈ dom(Q). (2) We say that P is Weihrauch reducible to Q (written P ≤W Q) if there are partial computable functions H, K such that K(x, q ◦ H(x)) for any x ∈ dom(P) and any realizer q of Q. x x Remark. If we think of the values P(x) and Q(x) as relativized mass problems P and Q , then P ≤W Q can be represented as the existence of partial computable functions Φ, ∆ :⊆ NN → NN satisfying Φx : Q∆(x) → Px for any x ∈ dom(Q), where Φx is the x-computable function mapping y ∈ NN to Φ(x ⊕ y). For any subset P of Baire space NN, we define ι(P) : NN ⇒ NN by ι(P)(x) = P for any x ∈ NN. Then, 1 the map ι provides an embedding of the Medvedev degrees into the Weihrauch degrees, i.e., P ≤1 Q if and only if ι(P) ≤W ι(Q). Definition 82 ([11, 13, 12, 14]). Let P, Q :⊆ NN ⇒ NN be partial multi-valued functions. (1) (Pairing) hP, Qi(x) = P(x) × Q(x). (2) (Product) (P × Q)(hx, yi) = P(x) × Q(y). (3) (Coproduct) (P Q)(0, x) = {0}× P(x); and (P Q)(1, x) = {1}× Q(x). (4) (Composition) (`P ◦ Q)(x) = {P(y) : y ∈ Q(x`)}, where x ∈ dom(P ◦ Q) if x ∈ dom(Q) and Q(x) ⊆ dom(P). S (5) (Parallelization) P(hxi : i ∈ Ni) = i∈N P(xi). b Q Let X be a computable metric space (for definition, see Weihrauch [77]). Then, A−(X) denotes the hyperspace of closed subsets of X with the upper Fell representation ψ− (see [11]). For example, P is a N N 0 computable point in the hyperspace A−(N ) (resp. A−(2 )) if and only if P is a Π1 subset of Baire space NN (resp. of Cantor space 2N). Definition 83 (Closed Choice [11, 13, 12, 14]). Let X be a computable metric space. Then, the closed choice operation of X is defined as the following partial function.

CX :⊆ A−(X) ⇒ X, A 7→ A

Here, dom(CX) = {A ∈ A−(X) : A , ∅}. 6.1.2. Principles of Omniscience.

Deﬁnition 84. A formula is tame if it is well-formed formula constructed from symbols {⊤, ⊥, ∧, ∨, ¬, ∀n, ∃n}n∈N and one variable symbol V(n) with a number parameter n ∈ N. For any tame formula A and p ∈ NN, let A[V/p] denote the new formula obtained from A by replacing V(n) with ⊤ if p(n) = 0 and V(n) with ⊥ if p(n) , 0. Then, let TameForm denote the class of formulas of the form A −→ B for some tame formulas A and B. 42 K. HIGUCHI AND T. KIHARA

Example 85. The following formulas are contained in TameForm. 0 (1) Σ1-LEM : ⊤ −→ ∃nV(n) ∨ ¬∃nV(n). 0 (2) Σ2-LEM : ⊤ −→ ∃m∀nV(hm, ni) ∨ ¬∃m∀nV(hm, ni). 0 (3) Σ2-DNE : ¬¬∃m∀nV(hm, ni) −→ ∃m∀nV(hm, ni). 0 (4) Σ3-DNE : ¬¬∃k∀m∃nV(hk, m, ni) −→ ∃k∀m∃nV(hk, m, ni). 0 (5) Σ1-LLPO : ¬(∃nV(h0, ni) ∧ ∃nV(h1, ni)) −→ ¬∃nV(h0, ni) ∨ ¬∃nV(h1, ni). 0 (6) Σ2-LLPO : ¬(∃m∀nV(h0, m, ni)∧∃m∀nV(h1, m, ni)) −→ ¬∃m∀nV(h0, m, ni)∨¬∃m∀nV(h1, m, ni). Remark. The symbols LEM, DNE, LLPO express the law of excluded middle, the double negation elimination, and the lessor limited principle of omniscience (i.e., de Morgan’s law), respectively.

Deﬁnition 86. Given any A −→ B ∈ TameForm, we deﬁne a partial multivalued function FA−→B :⊆ NN ⇒ NN as follows: N dom(FA−→B) = {p ⊕ q ∈ N : q ∈ ~A[V/p]},

FA−→B(p ⊕ q) = ~B[V/p], where ~· : Form → P(NN) is a unique Medvedev interpretation in Deﬁnition 28 with ~⊤ = NN. One can easily see that either ~¬ϕ = NN or ~¬ϕ = ∅ holds for every arithmetical sentence ϕ in any Medvedev interpretation. Therefore, for every principle A −→ B in Example 85, its domain is {p ⊕ q ∈ NN : ~A[V/p] , ∅}, that is, we need not to use the information on q. This observation immediately implies the following proposition.

Proposition 87. The induced function FA−→B from a principle A −→ B in Example 85 is Weihrauch equivalent to the following associated partial multi-valued function A −→ B on Baire space.

0 N 0 0, if (∃n ∈ N) p(n) = 0, Σ1-LEM : N → 2, Σ1-LEM(p) =  1, otherwise.  0 N 0 (0, s), if (∀m ∈ N)(∃n > m) p(n) = 0, Σ2-LEM : N ⇒ 2 × N, Σ2-LEM(p) ∋  (1, s), if (∀n > s) p(n) , 0. 0 N 0  Σ2-DNE :⊆ N ⇒ N, Σ2-DNE(p) = {m ∈ N : (∀n > m) p(n) , 0}. 0 N 0 Σ3-DNE :⊆ N ⇒ N, Σ3-DNE(p) = {k : (∀m ∈ N)(∃n ≥ m) p(hk, ni) = 0}.

0 N 2 0 0, if (∀n ∈ N) p0(n) = 0, Σ1-LLPO :⊆ (N ) ⇒ 2, Σ1-LLPO(p0, p1) ∋  1, if (∀n ∈ N) p1(n) = 0.  0 N 2 0 0, if (∀m)(∃n > m) p0(n) = 0, Σ2-LLPO :⊆ (N ) ⇒ 2, Σ2-LLPO(p0, p1) ∋  1, if (∀m)(∃n > m) p1(n) = 0.  Here, their domains are given as follows.  0 N dom(Σ2-DNE) = {p ∈ N : (∃m ∈ N)(∀n > m) p(n) , 0}. 0 N dom(Σ3-DNE) = {p ∈ N : (∃k ∈ N)(∀m ∈ N)(∃n ≥ m) p(hk, ni) = 0}. 0 N 2 dom(Σ1-LLPO) = {(p0, p1) ∈ (N ) : (∃i < 2)(∀n ∈ N) pi(n) = 0}. 0 N 2 dom(Σ2-LLPO) = {(p0, p1) ∈ (N ) : (∃i < 2)(∀m)(∃n > m) pi(n) = 0}. 0 Remark. (1) The single-valued function Σ1-LEM is usually called the limited principle of omniscience (LPO). Brattka-de Brecht-Pauly [11] showed that a single-valued partial function f :⊆ N N N → N is (1,ω)-computable if and only if f is Weihrauch reducible to the closed choice CN for the discrete space N. Here, in their term, the (1,ω)-computability is called the computability with ﬁnitely many mind changes. INSIDE THE MUCHNIK DEGREES I 43

0 ′ (2) Σ2-LLPO is Weihrauch equivalent to the jump LLPO of LLPO in the sense of Brattka-Gherardi- Marcone [14]. They also showed that LLPO′ is Weihrauch equivalent to the Borzano-Weierstrass Theorem BWT2 for the discrete space {0, 1}. Brattka-Gherardi-Marcone [14] also pointed out that 0 the n-th jump of LLPO and LPO correspond to Σn+1-LLPO (that is, the lessor limited principle of 0 0 0 omniscience for Σn+1-formulas) and Σn+1-LEM (the law of excluded middle for Σn+1-formulas), respectively. 0 0 (3) The study of arithmetical hierarchy of semiclassical principles such as Σn-LEM, Σn-LLPO, and 0 Σn-DNE was initiated by Akama-Berardi-Hayashi-Kohlenbach [1]. In particular, on the study of the second level of arithmetical hierarhcy for semiclassical principles, see also Berardi [4] and 0 Toftdal [74]. The relationship between the learnability and Σ2-DNE has been also studied by Nakata-Hayashi [50] in the context of a realizability interpretation of limit computable mathematics. Definition 88 (Unique variant [14]). Let P : X ⇒ Y be a multi-valued function. Then UniqueP : X ⇒ Y is defined as the restriction of P up to dom(UniqueP) = {x ∈ dom(P) : #P(x) = 1}. 0 Definition 89. We define the partial multi-valued function ∆2-LEM as follows.

0 2 N 0 0, if p ∈ Toti, ∆2-LEM :⊆ N × N → 2, ∆2-LEM(i, j, p) =  1, otherwise.  0 2 N N  Here, dom(∆2-LEM) = {(i, j, p) ∈ N × N : Toti = N \ Tot j}, where Tote denotes the set of all oracles N α ∈ N such that Φe(α; n) converges for all inputs n ∈ N. 0 0 Proposition 90. ∆2-LEM is Weihrauch reducible to UniqueΣ2-LLPO. 0 0 2 N Proof. To see ∆2-LEM ≤W UniqueΣ2-LLPO, given (e0, e1, p) ∈ N × N , deﬁne H(e0, e1, p) to be a pair (x0, x1), where xi(s) = 0 if and only if the computation Φei,s+1(p) at stage s + 1 properly extends

Φei,s(p) at the previous stage. Then xi contains inﬁnitely many 0’s if and only if p is contained in Totei . 0 Note that, whenever (e0, e1, p) is contained in the domain of ∆2-LEM, H(e0, e1, p) is also contained in Σ0 = NN \ Σ0 ◦ = the domain of Unique 2-LLPO, since Tote0 Tote1 . Therefore, Unique 2-LLPO H(e0, e1, p) 0 ∆2-LEM(e0, e1, p). Theorem 91. Let f :⊆ NN → NN be a single-valued partial function. 0 (1) f is (1, 2)-computable if and only if f ≤W Σ1-LEM. 0 (2) f is (1,ω|2)-computable if and only if f ≤W ∆2-LEM. 0 (3) f is (1,ω)-computable if and only if f ≤W Σ2-DNE. 2 0 Proof. (1) Let f be a (1, 2)-computable function. By Theorem 26 (1), we have f ∈ decd[Π1]. Then, there 0 N N N is a Σ1 set S ⊆ N such that f0 = f ↾ S and f1 = f ↾ N \ S is computable. Put U = {p ∈ N : 0 0 (∃n) p(n) = 0}. Note that Σ1-LEM is the characteristic function 1U of U. By Σ1 completeness of U, we can ﬁnd a Wadge reduction (indeed, a computable function) H such that 1S = 1U ◦ H. Put K(x, i) = fi(x) for every i < 2 and x ∈ NN. Then, for every x ∈ dom( f ),

K(x, 0) = f0(x) if x ∈ S, K(x, 1U ◦ H(x)) = K(x, 1S (x)) =  K(x, 1) = f1(x) if x < S.  0 2 0  0 Conversely, we have Σ1-LEM = 1U ∈ decd[Π1] since U is Σ1. This implies that H ◦ hid, 1U ◦ Hi ∈ 2 0 decd[Π1] for every partial computable functions H and K. 2 0 (2) Let f be a (1,ω|2)-computable function. By Theorem 26 (2), we have f ∈ decd[∆2]. Then, there 0 N N are Π2 sets P0, P1 ⊆ N with P0 = N \ P1 such that f ↾ P0 and f ↾ P1 are computable. Then, we can N ﬁnd indices i, j such that P0 = Toti and P1 = Tot j. Let H be the function sending p ∈ N to (i, j, p). Put 44 K. HIGUCHI AND T. KIHARA

N 0 K(x, i) = fi(x) for every i < 2 and x ∈ N . It is not hard to see that K(x, ∆2-LEM ◦ H(x)) = f (x) for every x ∈ dom( f ). 0 0 We show the converse implication. By Proposition 90, we have f ≤W ∆2-LEM ≤W UniqueΣ2-LLPO. 0 N N Assume that f ≤W UniqueΣ2-LLPO via partial computable functions K :⊆ N × 2 → N and H :⊆ N N 2 N → (N ) . Let ei be an index of λx.K(x, i) for each i < 2. We ﬁrst compute h(σ, i) = #{n < |Hi(σ)| : Hi(σ; n) = 0}, where H(σ) = hHi(σ)ii<2. Then let c(σ) be the least i < 2 such that h(σ, k) ≤ h(σ, i) for

ΦeU◦H(x) (x) = K(x, U ◦ H(x)) = f (x) for any x ∈ dom( f ). Hence, f is (1,ω|2)-computable. 0 (3) Clearly, Σ2-DNE is Weihrauch equivalent to the closed choice CN for discrete space N. Therefore, the desired condition follows from Brattka-Brecht-Pauly [11]. Deﬁnition 92. Let Θ :⊆ NN ⇒ NN be a partial multi-valued function. A partial single-valued function N N f :⊆ N → N is Θ-computable if f is Weihrauch reducible to Θ. By CΘ, we denote the least class containing all Θ-computable functions and closed under composition. Then, for subsets P, Q of NN, we write P ≤Θ Q if f : Q → P for some f ∈ CΘ.

0 N N Theorem 93. Let P be a Π2 subset of N , and Q be any subset of N . <ω (1) P ≤1 Q if and only if P ≤Σ0-LLPO Q. ω 2 (2) P ≤ Q if and only if P ≤Σ0 Q. 1 3-DNE <ω 2 0 Proof. (1) If P ≤1 Q via two algorithms, we have a function f : Q → P with f ∈ decd[Π2] by N 0 Proposition 27 (3). Then, f0 = f ↾ Q0 and f1 = f ↾ N \ Q0 are computable for some Π2 set N 0 + N Q0 ⊆ N . Since f1 is computable, we can extend the domain of f0 to a Π2 set Q including N \ Q0. + −1 0 0 Then Q1 = Q ∩ f1 [P] is Π2 since P is Π2 and f1 is computable. It is easy to see that Q0 ∪ Q1 0 0 includes Q. Since Q0 and Q1 are Π2, they are (computably) Wadge reducible to the Π2 complete set N ∞ U = {x ∈ N : (∃ n) x(n) = 0}. That is, for every i < 2, there is a computable functions Hi such N that 1Qi = 1U ◦ Hi. Let H be a computable function sending x ∈ N to the pair (H0(x), H1(x)), and put K(x, i) = fi(x). We can easily see that 0 x ∈ Qi ↔ 1Qi (x) = 1 ↔ 1U (Hi(x)) = 1 ↔ i ∈ Σ2-LLPO(H(x)). N 0 Thus, for every realizer G :⊆ N → 2 of Σ2-LLPO, we have K(x, G ◦ H(x)) = fG◦H(x)(x) ∈ P. 0 If f : Q → P for a single-valued function f ≤W Σ2-LLPO, then there are computable functions N N 2 N N 0 H : N → (N ) and K : N × 2 → N such that K(x, G ◦ H(x)) ∈ P for any realizer G of Σ2-LLPO and any element x ∈ Q. Then K(x, i) ∈ P for some i < 2, since G ◦ H(x) < 2. Set Φe(i)(x) = K(x, i) for each <ω i < 2. Then P ≤1 Q via {Φe(i)}i<2. ω 0 (2) Assume that P ≤1 Q. It suffices to show that f : Q → P for some f ≤W Σ3-DNE. Note that 0 N the condition Φe(x) is total and belongs to P is Π2, uniformly in e ∈ N and x ∈ N . Thus, there is a computable function H : NN → NN satisfying that H(x; e, n) = 0 for infinitely many n ∈ N if and only if Φe(x) is total and belongs to P. By our assumption, there is e ∈ N such that H(x; e, n) = 0 for infinitely 0 many n ∈ N, for any x ∈ Q. Therefore, H(x) ∈ dom(Σ3-DNE) for any x ∈ Q, and, for any realizer G of 0 N N Σ3-DNE, G◦H(x) chooses e < b such that Φe(x) ∈ P. Then, for a computable function K : N ×N → N mapping (x, e) to Φe(x), we have K(x, G ◦ H(x)) = Φe(x) ∈ P. INSIDE THE MUCHNIK DEGREES I 45

0 If f : Q → P for a single-valued function f ≤W Σ3-DNE, then there are computable functions N N N N 0 H : N → N and K : N × N → N such that K(x, G ◦ H(x)) ∈ P for any realizer G of Σ3-DNE and any element x ∈ Q. Then K(x, i) ∈ P for some i ∈ N, since G ◦ H(x) < m. Set Φe(i)(x) = K(x, i) for each ω i ∈ N. Then P ≤1 Q via {Φe(i)}i∈N. 0 N <ω Theorem 94. Let P and Q be Π subsets of N . Then, P ≤ Q if and only if P ≤ 0 Q. 1 tt,1 Σ1-LLPO <ω −1 0 Proof. We assume that P ≤tt,1 Q via two truth-table functionals f0 and f1. Note that f (P) is Π1 0 −1 whenever f is total computable, and P is Π1. Then, for Qi = Q ∩ Θi (P), the domain Q is covered by 0 , Q0 ∪ Q1. By Π1 completeness of U = {x : (∀n) x(n) 0}, for every i < 2, we have a computable function Hi such that 1Qi = 1U ◦ Hi. As in the proof of Theorem 93 (2), we set H : x 7→ (H0(x), H1(x)) and K : (x, i) 7→ fi(x). Then, it is not hard to see that the condition P ≤Σ0 Q is witnessed by H and K 1-LLPO 0 If f : Q → P for a single-valued function f ≤W Σ1-LLPO, then there are computable functions N N 2 N N 0 H : N → (N ) and K : N × 2 → N such that K(x, G ◦ H(x)) ∈ P for any realizer G of Σ1-LLPO and any element x ∈ Q. Then K(x, i) ∈ P for some i < 2, since G ◦ H(x) < 2. For U = {x : (∀n) x(n) , 0}, −1 0 deﬁne Di = Hi [U], where H(x) = (H0(x), H1(x)). The computability of Hi implies that Di is Π1. N 0 Deﬁne fi : Di → N by fi(x) = K(x, i) on Di. Since Di is Π1, fi has a total computable extension Φe(i). <ω Therefore, P ≤tt,1 Q via {Φe(i)}i<2.

Recall from Remark after Theorem 40 that ≤Σ0 is the reducibility relation induced by the disjunction 2 operation ~·∨·Σ0 . 2 N Theorem 95. Let P and Q be any subsets of N . Then, P ≤Σ0 Q if and only if P ≤Σ0 Q. 2 2-LEM Proof. Assume that there are two computable functions H : NN → NN and K : NN × 2 × N → NN N 0 0 such that K(x, G ◦ H(x)) ∈ P for any x ∈ Q and any realizer G : N → 2 × N of Σ2-LEM. Then the Σ2 sentence (∃v)θ(v, x) is given by (∃v)(∀n > v)H(x; n) , 0. We also deﬁne ∆(x) = K(x, h0, 0i), and Γv(x) = N 0 K(x, h1, vi), for any x ∈ N . Fix x ∈ Q. If θ(v, x) is true, then there is a realizer G of Σ2-LEM mapping H(x) to (1, v). Therefore, Γv(x) = K(x, h1, vi) = K(x, G ◦ H(x)) ∈ P. If (∀v)¬θ(v, x) is true, then there 0 is a realizer G of Σ2-LEM mapping H(x) to (0, 0). Therefore, ∆(x) = K(x, h0, 0i) = K(x, G ◦ H(x)) ∈ P. 1 Hence, by Theorem 45, we obtain ~P ∨ PΣ0 ≤ Q. 2 1 1 Conversely, we assume that ~P ∨ PΣ0 ≤ Q. Then, there are computable collection ∆, {Γv}v∈ of 2 1 0 computable functions, and a Σ2 sentence ∃vθ(v, x), as in Theorem 45. By analyzing the proof of Theorem 0 45, we may assume that this Σ2 sentence has an additional property that, if θ(v, x) is true and v ≤ u, then N θ(u, x) is also true. For any x ∈ N , put K(x, h0, ni) = ∆(x) for each n ∈ N, and K(x, h1, vi) = Γv(x). 0 N N From the Σ2 sentence ∃vθ(v, x), we can easily construct a computable function H : N → N satisfying that θ(v, x) is true if and only if H(x; n) , 0 for any n > v. Fix x ∈ Q If ∃vθ(v, x) is true, then any 0 realizer G of Σ2-LEM mapping H(x) to some (1, v) witnessing θ(v, x). Then, K(x, G ◦ H(x)) =Γv(x) ∈ P. 0 If ∀v¬θ(v, x) is true, then any realizer G of Σ2-LEM mapping H(x) to (0, s) for some s ∈ N. Then, K(x, G ◦ H(x)) = ∆(x) ∈ P. N 0 <ω Corollary 96. Let P and Q be subsets of N , where P is Π . Then, P ≤ω Q if and only if P ≤Σ0 Q. 2 2-LEM Proof. By Proposition 27 (2) and Theorem 95.

6.2. Duality between Dynamic Operations and Nonconstructive Principles. We now interpret our results in Section 4 in context of the Weihrauch degrees. Deﬁnition 97 (Le Roux-Pauly [58]). Let F, G :⊆ NN ⇒ NN be any multi-valued functions. Then, ∗ ∗ ∗ ∗ F ⋆ G = max≤W {F ◦ G : F ≤W F & G ≤W G}. 46 K. HIGUCHI AND T. KIHARA

0 C ω Σ3-DNE [ T ]1

0 C 2 Σ2-LEM [ T ]ω

0 0 C 2 C 1 Σ2-LLPO Σ2-DNE [ T ]1 [ T ]ω

0 [C ]1 ∆2-LEM T ω|2

0 C 1 Σ1-LEM [ T ]2 Figure 3. Constructive principles, and nonuniform computability.

If multi-valued functions C, D :⊆ NN ⇒ NN satisfy the condition

D ◦ E ≤W F ⇐⇒ E ≤W C ⋆ F for any multi-valued functions E, F :⊆ NN ⇒ NN, then we may think of D as the inverse of C. One could think of our disjunction operators as inverse operators of various constructive principles. Definition 98. Fix x ∈ NN. (1) ▽(x) = {y ∈ (N ∪{♯})N : #{n ∈ N : y(n) = ♯}≤ 1& tail(y) = x}. N (2) ▽ω(x) = {y ∈ (2 × N) : (∃i < 2) pri(y) = x & mc(y) < ∞}. N (3) ▽∞(x) = {y ∈ (2 × N) : (∃i < 2) pri(y) = x}. N (4) degT (x) = {y ∈ N : x ≤T y}. d (n) (n) (n) The n-th iteration of ▽ (▽ω and ▽∞) is denoted by ▽ (▽ω and ▽∞ ). Here, recall from Remark below Definition 34 that the symbol ♯ is supposed to be updated each time. For instance, ▽(2) refers to two (n) special symbols ♯0 and ♯1, and then ▽ (x) can be identified with the set of all sequences y such that y contains at most n many ♯’s and tail(y) = x. More precisely, given a partial multi-valued function E, (n) (n) 0 n every element of ▽ ◦ E(x) is of the form σ1♯σ2♯...♯σn♯y with y ∈ E(x). Thus, ▽ ◦ Σ1-LEM (x) has (n) 0 n a computable realizer, and indeed, ▽ ◦ E has a computable realizer for every E ≤W Σ1-LEM (x). We will see more general results in Proposition 99. A multi-valued function P :⊆ NN ⇒ NN is Popperian if there is a computable function r :⊆ NN×NN → N 0 N N satisfying Σ1-LEM ◦ r(x, y) = 1P(x)(y), for any x ∈ dom(P) and y ∈ N , where 1P(x) denotes the characteristic function of P(x). In other words, P is Popperian if and only if the condition y ∈ P(x) is 0 N Π1, uniformly in x ∈ dom(P) and y ∈ N . Every Popperian multi-valued function is clearly Weihrauch N reducible to the closed choice CNN of Baire space N . Proposition 99. Let E, F : NN ⇒ NN be any multi-valued functions. (n) 0 n (1) ▽ ◦ E ≤W F if and only if E ≤W Σ1-LEM ⋆ F. (n) 0 (2) ▽ω ◦ E ≤W F if and only if E ≤W UniqueΣ2-LLPOn ⋆ F. 0 (n) (3) ▽ ◦ E ≤W F if and only if E ≤W Σ2-DNE ⋆ F, where ▽ = n∈N ▽ . S Moreover, if E is Popperian, then we also have the following conditions. (n) 0 4. ▽∞ ◦ E ≤W F if and only if E ≤W Σ2-LLPOn ⋆ F. (n) 0 5. ▽∞ ◦ ▽ ◦ E ≤W F if and only if E ≤W (Σ2)2-LLPOn ⋆ F. 0 6. degT ◦ E ≤W F if and only if E ≤W Σ3-DNE ⋆ F. d Proof. (1) Assume that there are partial computable functions H :⊆ NN → NN and K :⊆ NN × NN → NN such that K(x, f ◦ H(x)) ∈ ▽(n) ◦ E(x) for any x ∈ dom(▽(n) ◦ E) and any realizer f of F. Then, for any INSIDE THE MUCHNIK DEGREES I 47

realizer f of F, we have the following condition for any x ∈ dom(E). (n) 1 K ◦ (id × f ) ◦ hid, Hi(x) = K(x, f ◦ H(x)) ∈ ▽ ◦ E(x) = `nE(x). Note that H∗ = hid, Hi : NN → NN × NN is computable, F∗ = K ◦ (id × F) : NN × NN ⇒ NN is Weihrauch reducible to F. As in the proof of Theorem 26, we can construct an (1, n)-computable function 1 γ : `n E(x) → E(x), uniformly in x ∈ dom(E). Therefore, by Theorem 91, we have a function γ ≤W 0 n ∗ ∗ ∗ ∗ Σ1-LCM satisfying γ ◦ f ◦ H (x) ∈ E(x) for any x ∈ dom(E) and any realizer f of F . Consequently, 0 n E ≤W Σ1-LEM ⋆ F. ∗ ∗ ∗ 0 n ∗ Conversely, we assume that E ≤W S ◦ F for some S ≤W Σ1-LEM and F ≤W F. Then there are computable functions H∗, K∗ such that K∗(x, H∗ ◦ f (x)) ∈ F∗(x) for any realizer f of F. From any single valued function f : NN → NN, we can eﬀectively obtain f ∗(x) = K∗(x, H∗ ◦ f (x)). Assume that ∗ 0 n ˜ ˜ ∗ ∗ ˜ ∗ S ≤W Σ1-LEM via H and K, and E ≤W S ◦ F via H and K. We consider H f (x) = H ◦ f ◦ H(x) and ∗ K f (x, i) = K(x, K˜ ( f ◦ H(x), i)). Then, we have the following condition for any x ∈ dom(E). 0 n K f (x, Σ1-LCM ◦ H f (x)) ∈ E(x). ˜ ∗ 0 n By calculating H f (x) = H ◦ f ◦ H(x), we can approximate i( f ; x) = Σ1-LEM ◦ H f (x) uniformly + (n) + in f . Therefore, we can construct F f to show ▽ ◦ E ≤W F by the following way. Set F f (hi) = hi,

6.3. Borel Measurability, and Backtrack Games. Berardi-Coquand-Hayashi [5] showed that a 1- backtrack Tarski game provides a semantics of positive arithmetical fragment of Limit Computable Math- 0 ematics (i.e., ∆2-mathematics, in the sense of Kleene realizability). A positive arithmetical formula A is true in the Limit Realizability Interpretation if and only if the ∃-player has a computable winning strategy in the 1-backtracking game bck(G(A)) associated with the Tarski game for A (for notations, see [5]). Meanwhile, Van Wesep [75] introduced backtrack game to study Wadge degrees, and Andretta [3] ∆0 used this game to characterize the 2-measurable functions (also called the ﬁrst level Borel functions) on Baire space NN. Motto Ros [47] and Semmes [60] studied more general games to study the Baire hierarchy of Borel measurable functions. The hierarchy of Borel measurable functions are deeply studied in descriptive set theory [41]. We consider the following notions for a function f on Baire space NN and a countable ordinal ξ<ω1. Σ0 −1 (1) f is a Borel function at level ξ (or a ξ+1,ξ+1 function; see [37, 38, 57, 60]) if the preimage f (A) Σ0 Σ0 N is ξ+1 for every ξ+1 set A ⊆ N . 48 K. HIGUCHI AND T. KIHARA

Σ0 (2) f is ξ+1-measurable (or equivalently, of Baire class ξ; see for instance, Kechris [41]) if the −1 Σ0 N preimage f (A) is ξ+1 for every open set A ⊆ N . N Σ0 Clearly, every level ξ Borel function on Baire space N is ξ+1-measurable. The effective hierarchy of Borel measurable functions is studied by Brattka [10] and developed by many researchers (see [23, 0 (n) 42]). An effective Σξ measurable function maps a computable point to a point of Turing degree 0 . 0 Additionally, the class of (effectively) Σξ-measurable functions does not closed under composition, in general, whereas the class of the level ξ Borel functions must be closed under composition. Our results 0 (Theorem 26) suggests that our notions of learnability is not like the effective Σξ-measurability but more like effective versions of the level ξ Borel functions, because of some results from descriptive set theory. ω Recall from Definition 25 that decp [Γ]F denotes the class of Γ-piecewise F functions. If F is the ω class of all partial continuous functions on Baire space, we abbreviate it as decp [Γ]. Jayne-Rogers [39] ω Π0 proved that decp [ 1] is exactly the class of the first level Borel functions, and Semmes [60] showed that ω Π0 f is decp [ 2] is exactly the class of the second level Borel functions. ω 0 As shown in Theorem 26 and Proposition 27, decp [Π1] is exactly the class of the learnable func- ω 0 ω tions, and the degree structure P/decp [Π2] is exactly the degree structure P1 induced from nonuniform computability. Actually, our dynamic models directly fit into the backtrack and multitape game characterization of subclasses of Borel measurable functions. We now introduce various games based on the Wadge game, the backtrack game, and the multitape game, Definition 100 (see also Motto Ros [47] and Semmes [60]). Fix a partial function f on NN, and a set X which has no intersection with N. The set X may contain pass, back♯, (move, i) for each i ∈ N. Then, we introduce various two-players games on f as follows. At every round n ∈ N, Player I chooses an element xn ∈ N, and Player II chooses an element yn ∈ N ∪ X.

I: x0 x1 x2 ... II: y0 y1 y2 ... N N A pair of inﬁnite sequences hx, yi∈ N ×(N∪X) is called a play. Fix a play hx, yi, where x = hxnin∈N and y = hynin∈N. Player I constructs an input x ∈ dom( f ) step by step, and Player II try to write a collect output f (x) on some tape, where there may be inﬁnitely many tapes {Λi}i∈N. Here, Player II can select a special symbol contained in X at each step.

• (move, i) indicates the instruction to move the head on the i-th tape Λi. • pass indicates that Player II writes no letter at this step. • back♯ indicates the instruction to delete all words on the tape under the head. Formally, we define the following notions. For each i ∈ N, the i-th content of the play y of Player II is N N a function contenti : (N ∪ X) → N which is inductively defined as follows. Set contenti(hi) = hi and tape(hi) = 0. Assume that contenti(y ↾ n) and tape(y ↾ n) have been already defined for each i ∈ N. content (y ↾ n)ahy i if y ∈ N & i = tape(y ↾ n),  i n n contenti(y ↾ n + 1) = hi if yn = back♯ & i = tape(y ↾ n),  content (y ↾ n) otherwise.  i  i if y = (move, i), tape(y ↾ n + 1) =  n tape(y ↾ n) otherwise.   N Then, for each i ∈ N, we define contenti(y) = limn∈N contenti(y ↾ n) for any y ∈ (N ∪ X) . We consider the following special rules for this game. • Player I violates the basic rule if x < dom( f ). INSIDE THE MUCHNIK DEGREES I 49

• Player II violates the basic rule if either yn ∈ {pass, (move, i) : i ∈ N} for almost all n ∈ N, or yn = back♯ for inﬁnitely many n ∈ N. • Player II violates the rule m if y contains at least m many back♯’s. • Player II violate the rule ∗ if yn ∈{(move, i) : i ∈ N} for inﬁnitely many n ∈ N.

We say that Player II wins (resp. is winnable) on the play hx, yi ∈ NN × (N ∪ X)N of the game G( f, X) if either Player II does not violate the basic rule, and f (x) = contenti(y) for the least i ∈ N with contenti(y) being total (resp. for some i ∈ N), or Player I violates the basic rule. We also say that Player II wins (resp. is winnable) on the play hx, yi of the game Gm( f, X) if Player II wins (resp. is winnable) the game G( f, X) and does not violate the rule m, and that Player II wins (resp. is winnable) the game G∗( f, X) if Player II wins (resp. is winnable) the game G( f, X) and does not violate the rule ∗. A strategy of Player II is a function ψ : N

Remark. The games G( f, P), G( f, PB), and G( f, PMω) are essentially same as the Wadge game, the backtrack game, and the multitape game, respectively. See also Motto Ros [47] and Semmes [60].

Let f be a partial function on Baire space NN.

(1) (Wadge [76]) f is continuous if and only if Player II has a winning strategy in the game G( f, P). ∆0 (2) (Andretta [3]) f is 2 if and only if Player II has a winning strategy in the game G( f, PB). Π0 (3) (Andretta, Semmes [59]) f is 2-piecewise continuous if and only if Player II has a winning strategy in the game G( f, PMω).

Theorem 101 (Game representation). Let f be a partial function on Baire space NN.

(1) f is (1, 1)-computable if and only if Player II has a computable winning strategy in the game G( f, P). (2) f is (1, m)-computable if and only if Player II has a computable winning strategy in the game Gm( f, PB). (3) f is (1,ω|m)-computable if and only if Player II has a computable winning strategy in the game G∗( f, PMm). (4) f is (1,ω)-computable if and only if Player II has a computable winning strategy in the game G( f, PB). (5) f is (m, 1)-computable if and only if Player II has a computable winnable strategy in the game G( f, PMm). (6) f is (m,ω)-computable if and only if Player II has a computable winnable strategy in the game G( f, PBMm). (7) f is (ω, 1)-computable if and only if Player II has a computable winnable strategy in the game G( f, PMω).

Proof. (2,4) We need to construct a winning strategy ψ : N

backlog(σ−) have been already defined. Then, define ψ(σ) and backlog(σ) as follows: ψ(σ−)apass if backlog(σ−) = hi, ψ(σ) =  − a − − ψ(σ ) (backlog(σ )(0)) if backlog(σ ) , hi,  backlog(σ−)↼1anewΦ (σ) if Ψ(σ) = Ψ(σ−), = Ψ(σ) backlog(σ)  − ↼1a a − backlog(σ ) back♯ ΦΨ(σ)(σ) if Ψ(σ) , Ψ(σ ).  Here, recall the notation newΦΨ(σ)(σ) defined before Theorem 40. Note that {n ∈ N : ( k ψ(x ↾ k))(n) = back♯} = mclΨ(x) for any x ∈ dom( f ). It is easy to see that ψ is a computable winningS strategy in the game G( f, PB). Assume that a computable winning strategy ψ∗ in the game G( f, PB) is given. We consider the com- ∗ putable function ψ(σ) = content0(ψ (σ)). Then {n ∈ N : ψ(x ↾ n + 1) + ψ(x ↾ n)} is finite, for any x ∈ dom( f ), since n∈N ψ(x ↾ n) contains finitely many back♯’s. Moreover, f (x) = limn ψ(x ↾ n). Thus, by Proposition 3, fSis (1,ω)-computable. (3) Assume that f is (1,ω| < ω)-computable via a learner Ψ. We inductively define a strategy ψ : N |Φe(i)(σ )|. Otherwise, for the least such i < m, the learner guesses Ψ(σ) = e(i). Clearly, #{Ψ(x ↾ n) : n ∈ N} < m for any x ∈ NN. It is easy to check that, for any x ∈ dom( f ), ∗ limn Ψ(x ↾ n) converges to e(i) for the unique i < m ensuring the totality of contenti ◦ ψ (x), and, for ∗ such i < m, we have Φlimn Ψ(x↾n)(x) = contenti ◦ ψ (x) = f (x). Consequently, f is (1,ω|m)-computable.

(5,7) For a given collection {Φi}i∈I of partial computable functions, we can easily construct a strategy

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Department of Mathematics and Informatics,Chiba University, 1-33 Yayoi-cho,Inage,Chiba, Japan E-mail address: [email protected]

School of Information Science, Japan Advanced Institute of Science and Technology,Nomi 923-1292, Japan E-mail address: [email protected]