Degrees of selector functions and relative computable categoricity

We study the degrees of selector functions related to the degrees in which a rigid computable structure is relatively computably categorical. It is proved that for some structures such degrees can be represented as the unions of upper cones of c.e. degrees. In addition we show that there are non-c.e. upper cones realized as the degrees in which some computable structure is relatively computably categorical.

Introduction.We start with the main definition of the notion studied through the paper.
Definition 1 Let {C i } i∈ω be a computable sequence of differences of c.e. sets, C i = A i \ B i .A selector function for {C i } i∈ω is any function f such that f (i) ∈ C i for every i ∈ ω.
As an example for a computable rigid structure A (on the domain ω) consider the sequence {C i = A i \ B i } i∈ω , where A i and B i are the following c.e. sets of existential formulae in the language of A: Then an existence of x-computable selector function for {C i } i∈ω implies the relative x-computable categoricity of the structure A, i.e., for every isomporphic copy B ∼ = A there is a (deg(B) ∪ x)-computable isomorphism from B onto A. Moreover, by the well-known result of Ash, Knight, Manasse and Slaman [1] and Chisholm [2] the inverse also holds: if a rigid computable structure A is x-computably categorical for some degree x then for some finite constant enrichment A = (A, a) the sequence { C i = A i \ B i } i∈ω defined as above will have an x-computable selector function.
It is easy to check here that the degrees of such selector functions does not actually depend on the choice of the constants a.Moreover, due a possible conjuctions of existential formulae the degrees of the selector functions in this case are the same as the degrees of the weak selector functions defined below.
Definition 2 Let {D n } n∈ω be the standard (canonical) numbering of all finite subsets of ω defined via n = x∈Dn 2 x .A weak selector function for a computable sequence of differences of c.e. sets Theorem 1 1.For every rigid computable structure A there is a computable sequence of differences of c.e. sets {C i } i∈ω such that for each degree x the structure A is relatively x-computably categorical if and only if there is an x-computable weak selector function for {C i } i∈ω .2. For every computable sequence of differences of c.e. sets {C i } i∈ω there is a rigid computable structure A such that for each degree x the structure A is relatively x-computably categorical if and only if there is an x-computable weak selector function for {C i } i∈ω .
The next theorem shows that 1-generic oracles can not compute non-trivial selector functions.
Theorem 2 If a degree x is 1-generic and there is an x-computable (weak) selector function for a computable sequence of differences of c.e. sets then there is a computable (weak) selector function for this sequence.
Corollary 1 If a computable rigid structure A is relatively x-computably categorical for a 1-generic degree x then A is relatively computably categorical.
In contrast with Theorem 2 an interesting example of computable sequence of differences of c.e. sets {C i } i∈ω with nontrivial properties of selector functions appear if we define for any pair of c.e. sets U ⊆ V This can be defined also via where Then it is easy to see that an x-computable selector function for {C i } i∈ω exists if and only if there is an x-computable set X such that U ⊆ X ⊆ V.It is clear also that so that an existence of x-computable selector function for such {C i } i∈ω is equivalent to an existence of x-computable weak selector function for {C i } i∈ω .
In the particular case U = V we easily can build a computable sequence of differences of c.e. sets with a unique selector function of the same c.e. degree as U and V .We can extend this as the following.
Theorem 3 For every finite sequence of c.e. degrees a 1 , a 2 , . . ., a k there is a pair of c.e. sets U ⊆ V such that an x-computable set X, U ⊆ X ⊆ V, exists if and only if By Theorem 1 we can code the degrees of sets X such that U ⊆ X ⊆ V into the degrees in which a computable structure is relatively computably categorical.
Corollary 2 For every finite sequence of c.e. degrees a 1 , a 2 , . . ., a k there is a computable rigid structure A such that A is relatively xcomputably categorical if and only if Theorem 3 allows to build more non-trivial and more non-uniform examples of computable sequence of differences of c.e. sets, e.g., we can apply Theorem 3 for a 0 > a 1 .By this way we can make only c.e. degrees as the least degrees of selector functions.The following theorem allows to find the least degrees of selector functions among 2-CEA non-c.e.degrees.
Theorem 4 Let e be a c.e. degree, and let F ∈ f be an e-c.e.set such that there is a for all x < y < s < t.Then there is a pair of c.e. sets U ⊆ V such that an x-computable set X, U ⊆ X ⊆ V, exists if and only if e ∪ f ≤ x.
The degrees e ∪ f satisfying the conditions of Theorem 4 form sufficiently large class of 2-CEA degrees.Indeed, Jockusch and Shore [3] have constructed a ∆ 0 2 2-CEA degree not belonging to any given uniform ∆ 0 2 class.We can note that the construction of the corresponding set F ∈ f assumes only one witness per requirement, and the requirements are satisfied in a finite injury priority manner.If a witness x enters or leaves F during the construction at a stage s then other assigned earlier witnesses y > x of lower priority are initialized at this stage, and hence F s (y) = F t (y) for t > s.. Therefore, the condition from Theorem 4 holds, so that we have proven the following statement.[4] we can not produce a 3-c.e.degree f / ∈ C.But the proof of Theorem 4 allows to repeatedly apply the arguments through the n-CEA hierararchy.Namely the proof of Theorem 4 can be adapted for e = g i and f = g i+1 , if and g 1 is a c.e. degree, g 2 is a g 1 -c.e.degree, g 3 is a g 2 -c.e.degree, . . .
Here we need only check that each g i contains an appropriate set G i ∈ g i with a ∆ 0 2 -approximation satisfying the condition from Theorem 4. The following theorem can be proved by this way but we prefer to give a direct proof, where an appropriate approximation is needed only for the final degree f = g n .
Theorem 5 For every n > 1 there are an n-c.e.degree f which is not (n − 1)-c.e. and a pair of c.e. sets U ⊆ V such that an x-computable set X, U ⊆ X ⊆ V, exists if and only if f ≤ x.Corollary 4 For every n > 1 there are a non-(n − 1)-c.e.n-c.e.degree f and a computable rigid structure A such that A is relatively x-computably categorical if and only if f ≤ x.
The rest of the paper is devoted to the proofs of the theorems above.We use the monograph [5] as the source of used notations and terminology.

The proof of Theorem 1
The proof of Part 1.By the result of Ash, Knight, Manasse and Slaman [1] and Chisholm [2] a rigid computable structure is A is relatively x-computably categorical if and only if for some tuple a from A there is a x-computable enumeration of existential formulae defining all individual elements in the (A, a).
If this never happen then we can simply define the sequence C i = ∅ which obviously has no selector function.
Otherwise, if for some a there is a collection of existential formulae defining all individual elements in the (A, a), then the problem of enumeration such a collection does not depend on the choice of a.Therefore, for a fixed a we can define the sequence The proof of Part 2. Suppose a computable sequence of differences of c.e. sets {C i } i∈ω , C i = A i \ B i = ∅, be given.Without loss of generality we can assume that B i = ∅, B i ⊆ A i , and the same degrees of selector functions: Let a and b be injective computable functions such that rng a = ∪ i∈ω A i and rng b = ∪ i∈ω B i .Also let h be a computable function such that h(i) ∈ B i for every i ∈ ω.
We construct a structure A on the domain ω in the in the language of infinitely many unary functions e 0 , e 1 , e 2 , . . ., by the following: Despite the infinity of the functional language the structure A is locally finite so that we can find a computable function c such that the finite set D c(s) is the set generated from the elements 0, 1, . . ., s. Suppose that there is there is an x-computable weak selector function f for {C i } i∈ω , i.e., D f (i) ⊆ A i and D f (i) ⊆ B i for every i ∈ ω.
Then we can build an x-computable list of existential formulae defining all elements of A : By [1] the structure A is relatively x-computably categorical.
Conversely, suppose the structure A is relatively x-computably categorical.Then by [1] for some tuple a from A there is an x-computable sequence {Φ i (x)} i∈ω of existential formulae such that i = j ⇐⇒ (A, a) |= Φ i (j).

Now if 4i /
∈ D c(s * ) then an x-computable weak selector function f for {C i } i∈ω can be defined for as follows: It is easy to see that D f (i) ⊆ A i and D f (i) ⊆ B i since otherwise we would have (A, a) |= Φ 4i (4i + 1).It is enough now to apppropriately extend the definition of f for finitely many i with 4i ∈ D c(s * ) .

The proof of Theorem 2
If x is 1-generic then there is a set X ∈ x such that for every c.e. set W ⊆ 2 <ω there is a string σ ⊂ X with the property Suppose that f = {e} X is an x-computable selector function for a computable sequence of differences of c.e. sets there is an σ ⊂ X such that τ / ∈ W for all τ ⊇ σ.

Then we can find a computable selector function for {C
The statement of Theorem 2 for weak selector functions also follows from the arguments above since the weak selector functions for {C i = A i \ B i } i∈ω are the selector function for the modified sequence 3 The proof of Theorem 3 Without loss of generality we can assume that the given c.e. degrees a 1 , a 2 , . . ., a k are all non-zero.For 1 ≤ i ≤ k let A i be a c.e. set such that A i ∈ a i .
To proceed we need the following trick similar to the Dekker's deficiency set: if A = rng a and V = rng v are infinite c.e. sets, where a and v are computable injective functions, then we can consider the c.e. subset of V : It is easy to see that for every X ⊆ ω x ∈ A ⇐⇒ x ∈ {a(0), a(1), . . ., a(s)}.
In particular, if A is not computable then the c.e. set A V is infinite.Also we have A V ≤ T A since A ↾ x ⊆ {a(0), a(1), . . ., a(n)} implies Since each c.e. set A i ∈ a i is not computable we can now consider the chain of infinite c.e. sets

The proof of Theorem 4
For a fixed c.e. set E ∈ e we fix an index e such that F = W E e = dom {e} E .We re-define the ∆ 0 2 -approximation F (x) = lim s F s (x) for s = 0, 1 by setting F 0 (x) = 1 and F 1 (x) = 0 for each x.
Note that the property again holds for the modified approximation (just because there are no tuples x < y < s ≤ 1).Then for the set Suppose now that U ⊆ Z ⊆ V .We will prove that F ≤ T E ⊕ Z considering two cases.
Case 1.There are infinitely many elements y, t ∈ F \ Z.Note that for every y there are only finitely many t such that F t (y) = 1 and F t+1 (y) = 0. Hence, the following (E ⊕ Z)-c.e. subset of F is infinite: Now if y ∈ Y then for every s ≥ y + 1 and x < y we have since otherwise F s (x) = F s+1 (x) would imply 1 = F t (y) = F (y) = F s (y) and, therefore, y, t ∈ U ⊆ Z.Thus, each new y from the infinite (E ⊕ Z)-c.e. set Y gives a possibility to find the value F (x) for each x < y.This implies F ≤ T E ⊕ Z.
Case 2. There are only finitely many elements y, t ∈ F \ Z. Then we make only finitely many errors computing F (x) = F ( y, 0 ) using the following (E ⊕ Z)-computable recursive procedure which assumes The procedure deciding whether y, t ∈ F .

If y, t /
∈ Z then answer "no".
2. If y, t ∈ Z then y, t ∈ V , and so F s (y) = 1 for some s > t.
3. Due F = W E e = lim s F s there is a w ≥ s such that either y ∈ W E e,w , or F w (y) = 1, F w+1 (y) = 0.
4. In the former case answer "yes".
5. In the last case we call recursively the procedure for y, w by the reduction y, t ∈ F ⇐⇒ y, w ∈ F .
Since for each y there are only finitely many w with F w (y) = 1, F w+1 (y) = 0 the recursion chain can not be infinite.
Therefore, in both cases we have proved F ≤ T E ⊕ Z.
Consider now the the interval of c.e. sets U ⊆ V , where U = E ⊕ U and V = E ⊕ V .Then we have U ⊆ E ⊕ F ⊆ V for the (e ∪ f )computable set E ⊕ F , and also 5 The proof of Theorem 5 Cooper proved (see [6], 12.3.6 and 12.3.7)that for every n > 1 there is an n-c.e.set F such that F ≡ T V e for every (n − 1)-c.e.set V e .The construction of F can be given via a ∆ 0 2 -approximation F (x) = lim s F s (x), F s (x) ∈ {0, 1}, such that for every x.Moreover, since each requirement deals only with one witness at once we have the property The last property again holds if we re-define the ∆ 0 2 -approximation for s = 0, 1 by setting F 0 (x) = 1 and F 1 (x) = 0 for each x.This re-definition also gives Then for the sequence of F -computable sets we have other equalities follow from c t (y) ≤ n + 1), and Also for every i the set F i is F i+1 -c.e.since y, t ∈ F i iff F t+1 (y) = F t (y), c t (y) = i, and Suppose now that U ⊆ X ⊆ V .Due F 1 ≡ T F and F n+1 = ∅ it is enough for F ≤ T X to prove that for every i.Indeed, since F i+1 ≤ T X and F i is F i+1 -c.e. the set is X-c.e.If Y i is infinite then we get F i ≤ T F ≤ T X since each y ∈ Y i computes F (x) = F y+1 (x) for x < y.Indeed, y, t ∈ F i and F s (x) = F s+1 (x), s ≥ y + 1, would imply F t (y) = F (y) = F s (y) and, therefore, y, t ∈ U ⊆ X.
Let us consider the case when the set Y i is finite.To show F i ≤ T X in this case we need only to know how to decide whether y, t ∈ F i for y, t ∈ X with c t (y) = i.But if y, t ∈ X then y, t ∈ V so that for some s = w + 1 > t we have F t+1 (y) = F t (y) = F w+1 (y) = F w (y).
Hence, F i+1 ≤ T X implies F i ≤ T X.Thus, U ⊆ X ⊆ V implies F ≤ T X, and simultaneously U ⊆ F ⊆ V holds for an F -computable set F .

Corollary 3
For every uniform ∆ 0 2 class of degrees C (e.g., C = the c.e. degrees, C = the 2-c.e.degrees, etc.) there are a 2-CEA degree f / ∈ C and a computable rigid structure A such that A is relatively x-computably categorical if and only if f ≤ x.If C = the c.e. degrees the proof produces a non-c.e.2-c.e.degree f If C = the 2-c.e.degrees by the result of Arslanov, LaForte and Slaman