Cone Avoidance of Some Turing Degrees

We study the cone avoidance and the upper cone avoidance of two substructures of m-introimmune Turing degrees. We show that the substructure of the m-introimmune Turing degrees satisfies the cone avoidance property, and that the substructure of the computably enumerable m-introimmune Turing degrees satisfies the upper cone avoidance property.


Introduction
The study of sets of natural numbers with no subsets of higher Turing degrees started with Soare (Soare 1969) and continued with Jockusch (Jockusch 1973) and Simpson (Simpson 1978).From their works we know that such sets exist and that they cannot be arithmetic.On the other hand, for stronger reducibilities ≤ r there are arithmetic sets with no subsets of higher ≤ r -degrees (Cintioli and Silvestri 2003), (Ambos-Spies 2003).We recall that a reducibility is stronger if it is strictly contained in the Turing reducibility.
Given one of the common reducibility ≤ r studied in computability theory, a possible way to define the property of having no subsets of higher ≤ r -degrees is that of being r-introimmune, a property introduced in (Cintioli and Silvestri 2003) We know that for some stronger reducibilities ≤ r there are arithmetic r-introimmune sets.In particular, there are mintroimmune sets in the class Π 0 1 (Cintioli 2005), where m stands for the many-one reducibility ≤ m .This suggests to study which properties satisfies the substructure of the computably enumerable (c.e.) m-introimmune Turing degrees.Let I m = {a : a is a c.e. Turing degree containing a m-introimmune set}, and let I m = (I m , ≤) be the substructure of the c.e. m-introimmune Turing degrees.For example, we know of this substructure that it has no minimum and it has maximum (Cintioli 2005).Moreover, a portion of I m is an upper semilattice (Cintioli 2014) .In section 3 we continue the study of the substructure I m by considering the upper cone avoidance property.The upper cone avoidance property is one of the main properties studied in the structure of the c.e. Turing degrees, and we prove that I m satisfies this property.
In section 4 we consider the substructure J m = (J m , ≤) of the m-introimmune Turing degrees, that is the structure where J m = {a : a is a Turing degree containing a m-introimmune set}.We know that J m has minimal elements and that it does not have maximum.We consider here the cone avoidance property, and we prove that J m satisfies this property.

Notations and Preliminaries
For the notations we refer to the Soare's book (Soare, 1987).Letter N denotes the set of natural numbers.Given two sets A, B ⊆ N, the complement of A is denoted by A, while A − B denotes the difference of A and B. For every set A ⊆ N and for every natural number n, A n = A ∩ {0, 1, . . ., n}.We identify the characteristic function of a set A with A itself, that is A(x) = 1 if x ∈ A and A(x) = 0 if x A, for every x ∈ N. We use the standard enumeration of all the Turing functionals Φ 0 , Φ 1 , . .., where for every e ∈ N and for every X ⊆ N Φ X e is the partial function computated by the e-th oracle Turing machine with the aid of the oracle X.We write Φ A e (x) ↓ if the e-th oracle Turing machine with oracle A halts on input x.For every natural numbers e, s, x and for every oracle X we define Φ X e,s (x) := Φ X e (x) if there exists t ≤ s such that the e-th oracle Turing machine on input x with oracle X halts in exactly t steps; in this case we say that Φ X e,s (x) is defined and we write Φ X e,s (x) ↓; we say that Φ X e,s (x) is undefined otherwise.We fix also an acceptable enumeration f 0 , f 1 , . . . of all the unary Turing computable functions.W 0 , W 1 , . . . is the corresponding enumeration of all the computably enumerable (c.e.) sets.For every e, s, x ∈ N we define f e,s (x) := f e (x) if there exists t ≤ s such that the e-th Turing machine on input x halts in exactly t steps; in this case we say that f e,s (x) is defined and we write f e,s (x) ↓; we say that f e,s (x) is undefined otherwise.For every e, s ∈ N, W e,s is the finite approximation of W e given by performing s steps in the enumeration of W e .Given two sets A, B ⊆ N: -A is many-one reducible to B, in short A ≤ m B, if there exists a total computable function f : N → N such that for every x ∈ N, x ∈ A if and only if f (x) ∈ B.
-A is Turing reducible to B, in short A ≤ T B, if there exists e ∈ N such that Φ B e = A.
The Turing degree of a set A ⊆ N is the equivalence class {B ⊆ N : A ≡ T B}, where A ≡ T B if and only if A ≤ T B and B ≤ T A. The Turing reducibility ≤ T induces the partial ordering ≤ on the Turing degrees in the obvious way.Turing degrees, or simply degrees, are denoted with the lower case letters a, b, . ... Two degrees a, b are incomparable if a ̸ ≤ b and b ̸ ≤ a, and in this case we write a|b.0 is the Turing degree of the computable sets.The jump of degree a is a ′ .In particular, 0 ′ is the Turing degree of the set {x ∈ N : Φ ∅ x (x) ↓}.Given a Turing degree a, (i) the upper cone of a is the set {b : a ≤ b}, and (ii) the cone of a is the set {b : a ≤ b} ∪ {c : c ≤ a}.

Upper Cone Avoidance
In this section we consider the substructure I m = (I m , ≤).Of this structure we know that it has no minimum and it has 0 ′ as maximum (Cintioli 2005).It is not a lattice because there is a minimal pair in I m (Cintioli 2011), and we do not know if it is an upper semilattice.Continuing with the research of the properties of I m , we show that it satisfies the upper cone avoidance property, that is, for every degree c ∈ I m there exists a ∈ I m such that c ̸ ≤ a.The upper cone avoidance property of I m is an immediate corollary of Theorem 3.2.
Theorem 3.2 For every noncomputable c.e. degree c there exists a c.e. degree a ∈ I m such that c ̸ ≤ a.
Proof.Let c be a noncomputable c.e. degree and let C ∈ c be a c.e. set.Let {C s } s≥0 be a uniformly recursive sequence of finite sets such that C = ∪ s≥0 C s , with C s ⊆ C s+1 for every s ≥ 0.

Strategy
We construct by a finite-injury priority argument a c.e. set A = ∪ s≥0 A s , where A s is the finite set constructed by the end of stage s, with A 0 = ∅ and A s ⊆ A s+1 for every s ≥ 0. For every e ∈ N the set A will satisfy the following requirements: In our construction we combine the strategy of to make C ̸ ≤ T A with the strategy of to make the degree of A in I m .At each stage s of the construction we try to fulfil one requirement, if possible, among those not yet fulfilled.Requirements {P 4e } e≥0 are positive, because to fulfil them we enumerate elements in A. Requirements {N 4e+1 , N 4e+3 } e≥0 are negative, because to fulfil them we try to keep elements out of A. Requirements {N 4e+2 } e≥0 are positive and negative, because to fulfil them we enumerate elements in A and we try to keep elements out of A. From now on letter R will denote any requirement.At every stage there could be requirements requiring attention.A requirement R m has higher priority than R n if m < n.We call a requirement R m active at the stage s if m is the minimum index such that R m requires attention at stage s.

Injured Requirements
At every stage s we try to satisfy a requirement R m .If m 4e for every e ∈ N, then we try to satisfy R m by restraining some element to be enumerated in A. To this end, we use a restraint function r : All the positive requirements {P 4e } e≥0 cannot be injured, so the restraint function is useless for them and we define r(4e, s) = −1 for every e, s ∈ N.

Actions to Fulfil Requirements
We describe informally the actions to fulfil requirements.To make C ̸ ≤ T A we try to satisfy N 4e+3 : Φ A e C for every e ∈ N. We employ the Sacks' agreement method.This method is based on the definition at each stage s and for every e ≤ s of the length agreement function l(e, s): The objective is to ensure that lim s→∞ l(e, s) = l e is finite for every e ∈ N, which implies Φ A e (l e ) C(l e ).We need the use function u(A s ; e, x, s) defined in the following way: u(A s ; e, x, s) is 1 + the maximum element asked to A s in the computation of Φ A s e,s (x) if Φ A s e,s (x) ↓, and u(A s ; e, x, s) is 0 if Φ A s e,s (x) is undefined.To preserve the computation of Φ A s e,s (y) for all y ≤ l(e, s) we define at each stage s the restraint function r(4e + 3, s) := max{u(A s ; e, y, s) : y ≤ l(e, s)}.
The finite-injury method will ensure that from an opportune stage s 0 onwards requirement N 4e+3 will not be injured anymore, so lim t→∞ l(e, t) = l(e, s 0 ) = l e .
This implies that for every t ≥ s 0 and Φ A e (l e ) C(l e ).To make the degree of A in I m we try to satisfy every requirement N 4e+2 , e ≥ 0. To satisfy all such requirements we use the strategy of to make A immune because of the following Proposition 3.3 proved in (Cintioli, 2011).
We will make A immune by satisfying all the requirements {P 4e } e≥0 .Thanks to the Proposition 3.3 we try to satisfy each requirement N 4e+2 by waiting for a stage s + 1 such that for some x A s is f e,s (x) ↓ x and f e,s (x) is not restrained by requirements of higher priority than N 4e+2 , that is f e,s (x) > max i<4e+2 {r(i, s)}.If such a stage is found, we enumerate f e,s (x) in A and keep out x of A by setting r(4e + 2, s) = x.In this way N 4e+2 is satisfied at stage s + 1; if N 4e+2 is not injured after stage s + 1 then f e is not a m-reduction of A to any of its co-infinite subset.

Requirements Requiring Attention
We now define formally when a requirement requires attention.For every e, s ≥ 0, we say that • requirement P 4e requires attention at stage s + 1 ≥ 4e if W e,s ∩ A s = ∅ and there is x ∈ W e,s with x > max i<4e {r(i, s)}. (1) • Requirement N 4e+1 requires attention at stage s {r(i, s)}.
Stage s + 1.If there are no requirements requiring attention, then do nothing, that this define A s+1 = A s and r(m, s + 1) := r(m, s) for every m 4e + 3, for every e ≥ 0. Otherwise, let R n be the active requirement.We distinguish three cases on R n .
• R n = P 4e .Let x be the minimum for which P 4e requires attention.Define A s+1 = A s ∪ {x}.

End construction of A
To prove that set A has the desired properties it suffices to prove that all the requirements are eventually satisfied.We prove first that all the requirements {P 4e , N 4e+1 , N 4e+2 } e≥0 require attention at most finitely often.Then we show by the Sacks' agreement method that every negative requirement N 4e+3 is fulfilled.
Lemma 3.4 For every e ∈ N, requirements P 4e , N 4e+1 and N 4e+2 require attention at most finitely often.
Proof.The proof is by complete induction on n, n 4e + 3, for every e ≥ 0. Given n, let s 0 be the minimum stage such that no requirement R m with m < n requires attention after stage s 0 .We distinguish three cases on n.
n = 4e.Let us suppose that P 4e requires attention at stage s + 1 ≥ s 0 .By hypothesis P 4e is active at stage s + 1, so an element x ∈ W e,s is enumerated in A s+1 .This means that A t ∩ W e,t ∅ for every t ≥ s + 1, therefore condition (1) will be false at every stage t ≥ s + 1 and P 4e will not require attention anymore.
n = 4e + 2. Let us suppose that N 4e+2 requires attention at stage s + 1 ≥ s 0 .By hypothesis N 4e+2 is active at stage s + 1, so an element y is enumerated in A s+1 and r(4e + 2, s + 1) = y.By hypothesis at every stage t > s + 1 no requirement R m with m < 4e + 2 will require attention, whence no element ≤ r(4e + 2, s + 1) = r(4e + 2, t) will be enumerated in A t .This means that for every t > s + 1 it holds that r(4e + 2, t) > −1, that is N 4e+2 will not require attention anymore.
End proof of Lemma 3.4 Lemma 3.5 For every e ≥ 0, requirement N 4e+3 is met.
Proof.This follows from the length agreement method of G. Sacks.Given e, let us suppose for the sake of contradiction that N 4e+3 is not met.This means that Φ A e = C, so lim s→∞ l(e, s) = ∞. (2) Let s 0 be a stage such that no requirement R n with n < 4e + 3 require attention after s 0 .The following is an informal procedure to decide C. Given any number y, decide "y ∈ C" by searching for a stage s ≥ s 0 with y < l(e, s), which exists by (2).Then, decide "y ∈ C" by computing Φ A s e,s (y).In the following claim we prove that the procedure is correct.
e,s (y).Proof.At stage s is r(4e + 3, s) = max{u(A s ; e, x, s) : x ≤ l(e, s)} = r e .Since no requirement R m with m < 4e + 3 is active after stage s, it holds that for every t ≥ s A t r e = A r e , (3) so Φ A t r e e,t (y) = Φ A r e e,t (y). (4) However, every computation of Φ A r e e,t (y) use only elements less than r e , so for every t ≥ s Φ A r e e,t (y) = Φ A e,t (y). (5) This implies that for every t ≥ s End proof of the claim.
Thus the set C is computable, contrary to the assumption that c was noncomputable.
End proof of Lemma 3.5.
It remains to prove that all the requirements {P 4e , N 4e+1 , N 4e+2 } e≥0 are met.We observe first that lim s→∞ r(4e + 3, s) exists and is finite.
Proof.Fix e ∈ N. The proof is part of the Sacks' agreement method, and it follows from the fact that N 4e+3 is satisfied and that all the requirement {P 4m , N 4m+1 , N 4m+2 } m≥0 require attention at most finitely often, that is N 4e+3 is not injured after an opportune stage.The interested reader can see also either (Soare 1987, page 123, cf. Lemma 2) or (Odifreddi 1999, page 464 et seq.) Let l e be the minimum number for which Φ e (l e ) C(l e ), which exists because N 4e+3 is met.Let s 0 be such that no requirement R m with m < 4e + 3 requires attention after s 0 and such that for every s ≥ s 0 , for every y ≤ l e , and Φ A s e,s (y) = Φ A e (y) for every y < l e .If for every s ≥ s 0 Φ A s e,s (l e ) is undefined, then lim s→∞ r(4e + 3, s) = r(4e + 3, s 0 ).
Otherwise, if s 1 is the first stage after s 0 such that Φ A s 1 e,s 1 (l e ) is defined, then for every t ≥ s 1 (∀y < l e )[Φ A t e,t (y) = Φ A e (y)] because N 4e+3 is not injured after s 1 ≥ s 0 .Moreover, by hypothesis Φ A e (l e ) C(l e ), so lim s→∞ l(e, s) = l(e, s 0 ) = l e and by the definition of r(4e + 3, s) is lim s→∞ r(4e + 3, s) = r(4e + 3, s 1 ).

End proof of Lemma 3.7
Lemma 3.8 For every e ∈ N, P 4e , N 4e+1 and N 4e+2 are met.
Proof.By Lemma 3.4 and Lemma 3.7 it follows that lim s→∞ r(n, s) exists and is finite for every n ∈ N. To prove the lemma, fix R n and let s 0 be the minimum stage after which no requirement R m with m < n requires attention.Distinguish three cases on n.
n = 4e.If W e is finite then P 4e is met, so let us suppose that W e is infinite.If at stage s 0 is W e,s 0 ∩ A s 0 ∅, then P 4e is met.Otherwise, let s + 1 > s 0 be the minimum stage such there is x ∈ W e,s with W e,s ∩ A s = ∅ and x > max{r(n, s) : n < 4e}.At stage s + 1 requirement P 4e requires attention and by hypothesis on s 0 is active.Therefore x is enumerated in A s+1 and for every t ≥ s + 1 is W e,t ∩ A t ∅, that is W e ∩ A ∅ and P 4e is met.
n = 4e + 1.If r(4e + 1, s 0 ) = x ≥ e, then by hypothesis on s 0 no requirement will enumerate x in A, so x ∈ A and N 4e+1 is met.If r(4e + 1, s 0 ) = −1, let s + 1 be the minimum stage with s + 1 ≥ max{s 0 , 4e + 1} and let x ≥ e be the minimum element not in A s .N 4e+1 requires attention and by hypothesis on s 0 is active at stage s + 1, so r(4e + 1, s + 1) = x.
After stage s + 1 ≥ s 0 requirement N 4e+1 is not injured, hence x ∈ A t for every t ≥ s + 1, that is x ∈ A and N 4e+1 is met.
n = 4e + 2. If the domain of f e is different from N then N 4e+2 is met, so let us suppose that domain of f e is N.
For the sake of contradiction let us suppose that N 4e+2 is not met, and let X ⊆ A such that A ≤ m X via f e , with |A − X| = ∞.Since for every e ≥ 0 requirement P 4e is met it follows that A is immune.By Proposition 3.3 the set { f e (x) : f e (x) x ∧ x ∈ A} is infinite.So, there are a stage s + 1 ≥ s 0 and x ∈ N such that: (i) x A s , (ii) f e,s (x) ↓ x and f e,s (x) > max{r(n, s) : n < 4e + 2}.
Requirement N 4e+2 requires attention at stage s + 1 and by hypothesis on s 0 is active.This means that f e,s (x) is enumerated in A s+1 and r(4e + 2, s + 1) = x.After stage s 0 no requirement of higher priority than N 4e+2 requires attention, so N 4e+2 is never injured, in particular x ∈ A t for every t ≥ s + 1 ≥ s 0 , that is x A. This is a contradiction, because x ∈ A and f e (x) A, which implies f e (x) X ⊆ A, contrary to the assumption that f e was a m-reduction of A to X.

End proof of Lemma 3.8
This also concludes the proof of the theorem.

2
Corollary 3.9 In the substructure of the c.e. m-introimmune Turing degrees holds the upper cone avoidance property.

Cone Avoidance
In this section we consider the substructure J m = (J m , ≤) of the m-introimmune degrees.Of this substructure we know that it has minimal elements and that it does not have maximum.These properties follow from results concerning cohesive degrees.First of all, in (Cintioli 2005) it has been proved that every cohesive set is m-introimmune.Then, by (Jockusch 1973) we know that if b is a cohesive degree and b ≤ a then a is a cohesive degree, from which we derive that J m does not have maximum.Moreover, it follows from (Jockusch 1973, Corollary 2) and (Cooper 1973, Theorem 1) that there are cohesive minimal degrees, so there are minimal elements in J m .We consider the cone avoidance property and we show that J m satisfies this property, that is for every b ∈ J m there exists a ∈ J m such that b ̸ ≤ a and a ̸ ≤ b.Actually, the degree a that avoids the cone of degree b is bi-m-introimmune, that is a contains a set A such that both A and A are m-introimmune.
Theorem 4.1 For every degree b > 0 there exists a bi-m-introimmune degree a < b ′ such that a|b.
In order to prove the theorem we introduce some few concepts and a result concerning them.
First of all, we say that a function f : N → N is injective almost everywhere if the set {(x, y) ∈ N 2 : x y ∧ f (x) = f (y)} is finite.We observe that a set is strongly bi-m-immune if and only if its complement is strongly bi-m-immune.It is known that a strongly bi-m-immune set with g-gaps for a dominating function g is m-introimmune.
Lemma 4.5 (Cintioli 2005) Let X be a strongly bi-m-immune set, and let g : N → N be a strictly increasing dominating function.If X has g-gaps, then X is m-introimmune.Now, we can proceed with the proof of Theorem 4.1.
Proof.Let B ∈ b, and let h : N → N be any strictly increasing dominating ∅ ′ -computable function.We construct by the finite-extension method a set A ≤ T B ′ which is strongly-bi-m-immune, Turing incomparable with B, and such that both A and A have h-gaps.So, by Lemma 4.5 A is bi-m-introimmune.The construction is by stages.At every stage s we will define a finite string σ s .The final set A will be ∪ s≥0 σ s , with σ 0 = ∅ and σ s ⊂ σ s+1 , for every s ∈ N. The set A will satisfy for every e ∈ N the following requirements.
-H e : if f e is not injective almost everywhere, then f e does not m-reduce A to any X.
Definition 4.2 A function g : N → N is dominating if for every computable total function f : N → N there exists m ∈ N such that(∀n > m)[g(n) > f (n)].Definition 4.3(Kämper 1990) Given any set D ⊆ N and given any strictly increasing function g : N → N, we say that D has g-gaps if there are infinitely many m ∈ N such that D ∩ {x : m ≤ x < g(m)} = ∅.Finally, we define the concept of strongly bi-m-immunity.Definition 4.4 (Balcázar and Schöning 1985) A set X ⊆ N is strongly bi-m-immune if and only if every m-reduction of X to any Y ⊆ N is injective almost everywhere.