Introenumerability, autoreducibility, and randomness

We define $\Psi$-autoreducible sets given an autoreduction procedure $\Psi$. Then, we show that for any $\Psi$, a measurable class of $\Psi$-autoreducible sets has measure zero. Using this, we show that classes of cototal, uniformly introenumerable, introenumerable, and hyper-cototal enumeration degrees all have measure zero. By analyzing the arithmetical complexity of the classes of cototal sets and cototal enumeration degrees, we show that weakly 2-random sets cannot be cototal and weakly 3-random sets cannot be of cototal enumeration degree. Then, we see that this result is optimal by showing that there exists a 1-random cototal set and a 2-random set of cototal enumeration degree. For uniformly introenumerable degrees and introenumerable degrees, we utilize $\Psi$-autoreducibility again to show the optimal result that no weakly 3-random sets can have introenumerable enumeration degree. We also show that no 1-random set can be introenumerable.


Introduction
In 1959, Friedberg and Rogers [4] introduced enumeration reducibility.A set A ⊆ ω is enumeration reducible to another set B ⊆ ω if there is a c.e. set W such that A = {x : (∃y) x, y ∈ W and D y ⊆ B}, where {D y } y∈ω gives a computable listing of all finite sets.We call the c.e. set W that witnesses this reduction an enumeration operator and write A = W (B). The degree structure induced by enumeration reduction ≤ e consists of the enumeration degrees.We can identify subsets of ω with infinite strings in the Cantor space 2 ω .Therefore, we can consider the measure of different classes of enumeration degrees (often abbreviated by e-degrees), including cototal e-degrees, uniformly introenumerable e-degrees, introenumerable e-degrees, and hyper-cototal e-degrees.
Given a set A of natural numbers and any number n, we may ask whether the membership of n in A can be determined using the oracle A without asking "is n in A".If so, A has a kind of self-reducibility.The notion of autoreducibility introduced by Trakhtenbrot [12] in 1970 is a formalization of this idea.A set A is said to be autoreducible if there is a Turing functional Φ such that for any n, A(n) = Φ A−{n} (n).We will generalize the autoreduction notion by defining Ψ-autoreducibility for any autoreduction procedure Ψ, Date: February 14, 2024.
which is a function from ω × 2 ω to {0, 1}.The classes of enumeration degrees mentioned above all have natural autoreducibility by replacing the Turing functional with different autoreduction procedures.Next, we will show that any measurable class of Ψ-autoreducible sets has measure zero for any Ψ.Then, we use this property of classes of Ψ-autoreducible sets to show that the classes of above e-degrees all have measure zero.
Intuitively, given a set A ⊆ ω, or equivalently an infinite string in 2 ω , it is random if it is hard to compress or one cannot predict the next bit or it has no rare properties.In 1966, Martin-Löf introduced a randomness notion using the latter idea that a random set is in no effective measure zero set in [8].
Generally, a set is random if it avoids a particular kind of null classes.Such null classes can be arithmetical as above or even go beyond arithmetical.
Since our classes of e-degrees have measure zero, sufficiently random sets must avoid such measure zero classes.Therefore, we can ask questions about what level of randomness the above sets or e-degrees can reach, and what level of randomness the above sets or e-degrees must avoid.We answer such questions for cototal sets, cototal e-degrees, uniformly introenumerable sets, uniformly introenumerable e-degrees, introenumerable sets, and introenumerable e-degrees.For references for randomness notions, see [2] or [10].
We start by giving the definitions of the sets and e-degrees we mentioned.First, a set A is total if A ≤ e A. It is named total because the degree of a total set is the degree of the graph of a total function.In [1], the notion of cototality is given by reversing the relationship between A and A.
Definition 1.2.An infinite set X is uniformly introenumerable if there is an enumeration operator Γ such that for every infinite subset Y of X, Γ(Y ) = X.
In [7], Jockush introduced the notion of uniform introenumerability.The definition of uniform introenumerability we give here is slightly different by using an enumeration operator instead of a c.e. operator, though the two definitions were shown to be equivalent in [6] by Greenberg et al.Recently, Goh et al. [5] also showed that Jockush's notion of (non-uniform) introenumerability is equivalent to the following notion: Definition 1.3.An infinite set X is introenumerable if, for every infinite subset Y of X, there is an enumeration operator Γ such that Γ(Y ) = X.
In [11], Sanchis introduced a reduction that is related to hyperarithmetical reduction and only uses positive information about membership in the set: Definition 1.4.Let A and B be sets such that, for some c.e. set W , the following relation holds: x ∈ B if and only if Then we say that B is hyper-enumeration reducible to A and write this relation: B ≤ he A.
Theorem 1.6.The relationship of enumeration degrees of the above notions is the following: Introenumerable → Hyper-cototal.
Remark 1.7.The solid arrows are strict.For proof of the first arrow, see [9].The third arrow and the strictness of the first arrow are proved in [5] by Goh et al.It is still unknown whether there is a set of introenumerable e-degree that does not have uniformly introenumerable e-degree.

Measure of Classes with Autoreduction
In this section, we define Ψ-autoreducible sets given an autoreduction procedure Ψ and show that any measurable class of Ψ-autoreduction sets has measure zero.Next, we apply the autoreducibility of hyper-cototal edegrees to show that the measure of the class of such e-degrees is zero.
Here, we say that the function Ψ is an autoreduction procedure.
Next, to show that the measure of a class of Ψ-autoreducible sets is zero, we use the Lebesgue density theorem.Theorem 2.2.Fix an autoreduction procedure Ψ, a measurable class S of Ψ-autoreducible sets has measure zero.
Proof.Suppose a class S of Ψ-autoreducible sets has positive measure.By the Lebesgue density theorem, for any ε > 0, there is a string 4 along with the corresponding string σ.Consider an n ∈ ω larger than |σ|.Define subsets P i (i = 0, 1) of S as follows: P i = {X ∈ S : Ψ(n, X − {n}) = i}.Since P 0 and P 1 partition S, one of them must have the following relative measure: . Without loss of generality, assume that such subset is P 0 .Now, consider the set > 0. So, P 2 ∩ S is not empty.For any . This is a contradiction.Therefore, S has measure zero.
Remark 2.3.In this theorem, the assumption that the class S is measurable is necessary.Consider the finite difference equivalence classes: two sets A and B are in the same equivalence class if and only (A − B) ∪ (B − A) is finite.Now, we can define a class S 0 that contains exactly one element from each of the equivalence classes.It is not difficult to see that S 0 is not measurable.We can define a function Ψ 0 such that if A ∈ S 0 and n ∈ ω, then Ψ 0 (n, A − {n}) = A(n).It is well-defined because, for any B ∈ S 0 and B − {n} = A − {n}, Ψ 0 (n, B − {n}) has to equal A(n) by the definition of S 0 .Therefore, S 0 is a class consisting of Ψ 0 -autoreducible sets that does not have measure zero since it is not measurable.Now we use the above theorem to show that the measure of the class of hyper-cototal e-degrees is zero.First, we discuss the autoreducibility of hyper-cototal sets.
Lemma 2.4.For every hyper-cototal set A, there is a Ψ such that A is Ψ-autoreducible.
Proof.Suppose A is hyper-cototal and there is some hyper-enumeration operator Then, we can define Ψ(n, X) := ∆ X (n).
In fact, each set of hyper-cototal degree is Ψ-autoreducible for some autoreduction procedure Ψ as well.
Lemma 2.5.Any set in the class of hyper-cototal e-degrees is a hyper-cototal set.
Proof.In [11], Sanchis proved that If A ≤ e B, then A ≤ he B and A ≤ he B. Suppose A has hyper-cototal e-degree and A ≡ e B, where B is a hypercototal set.Then, A ≡ he B ≤ he B ≡ he A.
Next, in order to apply Theorem 2.2 to show that the measure of the classes of hyper-cototal e-degrees is 0, we first need to show that the class of hyper-cototal e-degrees is measurable by analyzing the arithmetical complexity of Notice that n ∈ Γ A and n ∈ Γ A are Π 1 1 and Σ 1 1 respectively for a hyperenumeration operator Γ by Definition 1.4.So, the class of hyper-cototal edegrees is the difference of two Π 1  1 classes.Recall that Π 1 1 sets are measurable.
Therefore, the class of hyper-cototal e-degrees is measurable.Now, we use the results from above to see that the class of hyper-cototal e-degrees has measure zero.
Proof.Suppose the class of hyper-cototal e-degrees has positive measure.
Because there are only countably many hyper-enumeration operators, there exists a Γ such that the class of hyper-cototal e-degrees witnessed by this operator has positive measure.However, any set in this class would be Γ-autoreducible by Lemma 2.4.Now, applying Theorem 2.2 gives us a contradiction.By the relationship between the e-degrees mentioned above in Theorem 1.6, we see that the measure of these classes are all zero.

Bounds of Randomness
Notice that, for any class of measure zero, sufficiently random sets avoid it.So, we now discuss what level of randomness these e-degrees could and could not have.In this section, all necessary background knowledge of randomness is from Nies' book [10].We first discuss the class of cototal sets and the class of cototal e-degrees.where Γ e 's are enumeration operators.Therefore, the class of cototal sets is a union of Π 0 2 classes.By Lemma 2.6, all such classes have measure zero.Because any weakly 2-random set avoids all null Π 0 2 classes, weakly 2-random sets are not cototal.
To see that weak 2-randomness is optimal, we show that the 1-random Chaitin's Ω is a cototal set.Theorem 3.2.There exists a 1-random cototal set.
Proof.Because Ω is left-c.e., there is a non-descending computable sequence {q n } of rationals such that Ω = lim n→∞ q n .For any enumeration of Ω, we can enumerate Ω using this computable sequence.First, to determine whether 0 is in Ω or not, either for some n, we see the dyadic expansion of q n starts with 1 or we see 1 enter Ω.Only for the first case, we enumerate 0 in Ω.Then, we can iteratively do this process for each nature number in order.Eventually, we obtain an enumeration of Ω.Therefore, Ω ≤ e Ω.
For the class of cototal e-degrees, we first discuss what level of randomness is enough to avoid them.Theorem 3.3.Weakly 3-random sets do not have cototal e-degree.
Proof.Notice that the class of cototal e-degrees defined by an enumeration operator Γ e is Since D ∩ K A = ∅ and D ∩ K A = ∅ are Π 0 1 and Σ 0 1 respectively, the class of cototal e-degrees defined by Γ e is Π 0 3 .Since each of these classes is null, weakly 3-random sets avoid them all.So, we conclude that weakly 3-random sets do not have cototal e-degree.
Next, we see that weak 3-randomness is optimal by showing that there is a 2-random set of cototal e-degree even though any cototal set cannot be weakly 2-random.Theorem 3.4.There exists a 2-random set of cototal e-degree.
In [1], it was shown that every Σ 0 2 set has cototal e-degree.So, there exists M such that M ≡ e L and M ≥ e M .Then, Hence, we have a cototal set that is enumeration equivalent to Ω ∅ ′ .
In the proofs above, we did not use autoreducibility since it is enough to analyze the arithmetical complexities of the class of cototal sets and the class of cototal e-degrees to show the optimal level of randomness the sets in these classes must avoid.However, a similar analysis would not work for the classes of (uniform) introenumerable sets or e-degrees.We can verify the complexity of the collection of uniformly introenumerable e-degrees: We suspect that the class of uniformly introenumerable e-degrees is Π 1 1 -complete.This was shown to be true for the class of uniformly introreducible sets in [6].Assuming that there is no simpler definition, the analysis we used for cototal e-degrees would not work.Instead, for each set A of uniform introenumerable e-degree, we show Ψ-autoreducibility for some autoreduction procedure Ψ so that we can apply Theorem 2.2 again.Theorem 3.5.Weakly 3-random sets do not have uniformly introenumerable e-degree.
Proof.We will show that uniformly introenumerable e-degrees are contained in a countable union of measure zero Π 0 3 classes.To do this, we show that each set A of uniformly introenumerable e-degree is Ψ-autoreducible for some Ψ.Since A has uniformly introenumerable e-degree, there is a set B, enumeration operators Φ, Γ, and ∆ such that A = ∆(B), B = Φ(A), and for any infinite subset C of B, Γ(C) = B. Let Note that n has to be in A when Φ(A−{n}) is finite.So, A is Ψ-autoreducible.Now we consider the class of Ψ-autoreducible sets: This is a Π 0 3 class.By Theorem 2.2, this is a null class.Because weakly 3-random sets cannot be in any Π 0 3 null class, weakly 3-random sets do not have uniformly introenumerable e-degree.
Meanwhile, there also exists 2-random uniformly introenumerable e-degrees because of Theorem 3.4 and the fact that every set of cototal e-degree has uniform introenumerable e-degree.
With more work, the previous result can be improved to show that weakly 3-random sets do not have introenumerable e-degree either.Theorem 3.6.No weakly 3-random set has introenumerable e-degree.
Proof.Suppose a weakly 3-random set A has introenumerable e-degree.Let B be an introenumerable set such that there are enumeration operators Φ and ∆ with A = ∆(B) and B = Φ(A).For a contradiction, we define C = i c i as an infinite subset of B such that Γ i (C) = B for any enumeration operator Γ i (here we identified strings c i with corresponding sets).When we are constructing C, we also define a set D i at each stage i.Let c 0 = ∅ and D 0 = ∅.Suppose c i and D i have been defined.By inductive assumption, Φ(A − D i ) is infinite.First, we consider whether there is an extension e of c i such that e c i Φ(A − D i ) ↾ [|c i |, ∞), and Γ i (e) − B = ∅.If so, we define c i+1 to be the least such e that contains at least one more element than c i and D i+1 = D i .If not, we consider whether there is an extension e of c i such that for some n, Φ(A If so, we define c i+1 to be the least such e that contains at least one more element than c i , and D i+1 = D i ∪ {n}.
If not, we can define similar to the proof in Theorem 3.5.Notice that A is Ψ-autoreducible and the class of Ψ-autoreducible sets is Π 0 3 .This is impossible because A is weakly 3-random.This is a contradiction.Therefore, at least one of the two cases we considered has to be true.In this way, we obtain an infinite C = i c i ⊆ B. Now we show that Γ i (C) = B for any i.For any i, if the first case we considered is true, then Γ i (C) contains an element not in B. If the second case is true, Again, by Theorems 1.6 and 3.4, we conclude that there exists 2-random introenumerable e-degree while there is no weakly 3-random introenumerable e-degree.Next, we consider the class of uniformly introenumerable sets.We use the proof ideas of Proposition 8 given by Figueira, Miller, and Nies in [3] that showed no random is autoreducible.Theorem 3.7.No 1-random set is uniformly introenumerable.
Proof.We will apply Schnorr's theorem.To do so, we will show that the initial segment of any uniformly introenumerable set A can be compressed beyond any fixed constant.
Let Γ be the enumeration operator such that Γ(B) = A for any infinite subset B of A. For introenumerable sets, we combine the methods used in Theorems 3.6 and 3.7.
Proof.Suppose there is a 1-random introenumerable set A. We prove the theorem by constructing an infinite subset B = i b i of A such that Γ i (B) =

Theorem 3 . 1 .
Weakly 2-random sets are not cototal.Proof.The class of cototal sets {A : A ≤ e A} is defined by e {A : A = Γ A e } = e {A : ∀n[n ∈ A → (∃D y ⊆ A)[ n, y ∈ Γ e ] ∧ n ∈ A → (∀y)[ n, y ∈ Γ e → D y ∩ A = ∅]]}, For each m, there is a least n m such that n m > n p for any p < m and Γ nm (0 m A ↾ [m, n m )) ↾ m = A ↾ m since A − {0, 1, ..., m − 1} is an infinite subset of A. Let c m be the number of 1's in the string A ↾ m.Now we define a prefix-free machine M that outputs A ↾ n m with input γ = 0 |σ| 1σ0 |τ | 1τ A ↾ [m, n m ), where σ, τ are binary strings corresponding to m, c m .M first obtains the length of σ by reading until the first 1 and then obtains the number m by reading |σ| many bits after the first 1.Next, M can find out c m in the same way by reading the input until τ .Now, M 's read head keeps on moving forward to read A ↾ [m, n m ) bit by bit to do the enumeration of Γ(0 m A ↾ [m, n m )) ↾ m step by step to enumerate A(x) for x between 0 and m − 1 until c m many of such A(x) is determined to be 1, which means the other bits on A ↾ m are zeros.M can output A ↾ n m by concatenation.Therefore, K(A ↾ n m ) ≤ + n m − m + 4 log(m).By Schnorr's theorem, A is not 1-random.