aa r X i v : . [ m a t h . L O ] J a n FIXPOINTS AND RELATIVE PRECOMPLETENESS
ANTON GOLOV AND SEBASTIAAN A. TERWIJN
Abstract.
We study relative precompleteness in the context of thetheory of numberings, and relate this to a notion of lowness. We intro-duce a notion of divisibility for numberings, and use it to show that forthe class of divisible numberings, lowness and relative precompletenesscoincide with being computable.We also study the complexity of Skolem functions arising from Ar-slanov’s completeness criterion with parameters. We show that for suit-ably divisible numberings, these Skolem functions have the maximalpossible Turing degree. In particular this holds for the standard num-berings of the partial computable functions and the c.e. sets. Introduction
A numbering is a surjective function γ : ω → S to a given set of objects S .The computability theoretic study of numberings was initiated by Ershov [8]in a series of papers. For a given numbering γ , we can consider two functionsto be equivalent if their values are mapped by γ to the same objects in S . Akey notion in the theory is the notion of precompleteness, which says thatevery partial computable (p.c.) function has a total extension modulo thenumbering. Among other things, Ershov showed that Kleene’s recursiontheorem [12] holds for any precomplete numbering. The recursion theoremis then the special case for the numbering of the p.c. functions. The re-cursion theorem has been generalized in several other ways. For example,Arslanov [4] generalized the recursion theorem from the computable func-tions to the class of functions computable from a Turing incomplete c.e.set. This gives a completeness criterion for c.e. sets: A c.e. set is Turingcomplete if and only if it can compute a fixed point free function. Seliv-anov [15] proved that Arslanov’s theorem also holds for any precompletenumbering. (This result was also discussed in Barendregt and Terwijn [5],who considered fixed point theorems in the more general setting of partialcombinatory algebras.) Another generalization of the recursion theorem, si-multaneously generalizing Arslanov’s theorem and a theorem of Visser [19],was given in Terwijn [17]. This is a result about the standard numberingof the computably enumerable (c.e.) sets, and it is currently open whetherthis also holds for every precomplete numbering. Variations of Arslanov’scompleteness criterion that arise by considering different kinds of fixpointswere studied in Jockusch et al. [11].In this paper, we study relative precompleteness, that is, for a given ora-cle A we consider which numberings γ have the property that every A -p.c. Date : February 1, 2021.2010
Mathematics Subject Classification.
Key words and phrases. fixpoint theorems, precomplete numberings. function has a total A -computable extension modulo γ . We introduce anotion of lowness (Definition 2.2), and show that for a certain class of divis-ible numberings that includes the standard numberings of the p.c. functionsand the c.e. sets, lowness and A -precompleteness are equivalent to A beingcomputable (Theorem 3.7).We then proceed to study the complexity of Skolem functions that arisefrom Arslanov completeness criterion with parameters. It was shown in Ter-wijn [18] that these Skolem functions in general cannot be computable. Herewe use the notions of lowness and divisibility from the first part of the paperto show that they have the maximal possible Turing degree (Corollary 4.4).Our notation from computability theory is mostly standard. In the fol-lowing, ϕ n denotes the n -th partial computable (p.c.) function, in somestandard numbering of the p.c. functions. Similarly, W n denotes the n -thcomputably enumerable (c.e.) set, in some standard numbering of the c.e.sets. We denote these numberings as ϕ ( − ) and W ( − ) respectively. Partialcomputable functions are denoted by lower case Greek letters and (total)computable functions by lower case Roman letters. ω denotes the naturalnumbers. We write ϕ e ( n ) ↓ if this computation is defined and ϕ e ( n ) ↑ other-wise. We let h e, n i denote a computable pairing function. For unexplainednotation we refer to Odifreddi [14] and Soare [16].2. Turing degrees, up to a numbering
The theory of numberings originates from Ershov [8].
Definition 2.1. A numbering of a set S is a surjection γ : ω → S .For every numbering γ there is an associated equivalence relation ∼ γ on ω defined by n ∼ γ m iff γ ( n ) = γ ( m ). Thus, as noted in Bernardi andSorbi [6], the study of numberings is essentially equivalent to the study ofcountable equivalence relations.2.1. Lowness.
While our aim is to characterize the Turing degrees of par-ticular functions, in the context of numberings it is more natural to showresults about the following relation, which expresses a lowness property.
Definition 2.2.
Given a numbering γ and oracles A and B , we say that A is ( γ, B ) -low if for every A -computable function f there is a B -computablefunction g such that for all n , f ( n ) ∼ γ g ( n ) . (1)We will denote this by A γ B . We will say that A is γ -low if A γ ∅ . Afunction g satisfying (1) will be called a γ -lift of f to B .Let us now look at some basic properties and some simple examples oflowness results. Theorem 2.3.
For every γ , the relation γ is a preorder.Proof. Reflexivity is trivial. For transitivity, suppose A γ B γ C and let f be an A -computable function. Since A γ B , f has a γ -lift g to B , andsince B γ C , g has a γ -lift h to C . Since these are γ -lifts, for every n wehave that f ( n ) ∼ γ g ( n ) ∼ γ h ( n ) . (cid:3) IXPOINTS AND RELATIVE PRECOMPLETENESS 3
The following result shows that the lowness relation γ is coarser thanTuring reducibility. Proposition 2.4.
For every numbering γ and all A and B , if A T B then A γ B .Proof. Since by assumption every A -computable function is B -computable,for any A -computable f we can take f itself as its γ -lift to B . (cid:3) The converse of Proposition 2.4 does not hold by Example 2.7, since forthe numbering there A γ ∅ for every A , but not every A is computable. InTheorem 2.12 we prove a partial converse of Proposition 2.4. Corollary 2.5.
A set A is γ -low iff it is minimal with respect to the γ ordering.Proof. A is γ -low if A γ ∅ . Since ∅ T B for all B , it follows fromProposition 2.4 that A γ B for all B . (cid:3) For numberings γ and γ , we say that γ is a quotient of γ if n ∼ γ m implies that n ∼ γ m for all n and m . Proposition 2.6.
Let γ and γ be numberings and let γ be a quotient of γ . If A γ B then A γ B .Proof. Let f be A -computable and let g be its γ -lift to B . For every n , f ( n ) ∼ γ g ( n ). Since γ is a quotient of γ , it follows that f ( n ) ∼ γ g ( n ). (cid:3) Example 2.7.
Let A be an arbitrary oracle. Recall that ϕ A ( − ) is the standardnumbering of partial A -computable functions. By the S - m - n -theorem, forevery A -computable function f there exists a primitive recursive function g such that for all n , ϕ Af ( n ) = ϕ Ag ( n ) and hence A is ϕ A ( − ) -low.Since the numbering of A -c.e. sets W A ( − ) is a quotient of ϕ A ( − ) , every A is W A ( − ) -low by Proposition 2.6.2.2. Divisibility.
In order to turn results about lowness into results aboutTuring reducibility, we will make use of the following notion of divisibility .For x ∈ ω , we let [ x ] ∼ γ denote the equivalence class of x under ∼ γ . Definition 2.8.
Let n ω . A numbering γ is said to be ( n, A ) -divisible ifthere exist A -computable sequences ( x i ) i Let γ and γ be numberings and let γ be a quotient of γ . If γ is ( n, A ) -divisible then so is γ .Proof. The same witnesses can be used and the requirements are easily ver-ified. (cid:3) A. GOLOV AND S. A. TERWIJN Example 2.10. For an arbitrary oracle A , the numbering W A ( − ) is ( ω, A )-divisible since we can take the enumerations W Ax i = { i } and W Ae i = { d ∈ ω : i ∈ W Ad } . It is then easily verified that the conditions from Definition 2.8 are met,because when i = j then j / ∈ W Ax i . Using the fact that W A ( − ) is a quotient of ϕ A ( − ) , we also see that ϕ A ( − ) is ( ω, A )-divisible by Proposition 2.9. Proposition 2.11. For any oracles A T B and any numbering γ , if γ is ( n, A ) -divisible then it is also ( n, B ) -divisible.Proof. Trivial, since the same witnesses can be used. (cid:3) The following result is a partial converse of Proposition 2.4. Theorem 2.12. Let γ be a numbering and let A γ B . If γ is (2 , B ) -divisible then A T B .Proof. Let γ be such a numbering and let the points x, y and the sets X, Y witness the divisibility of γ . Define A -computable f by f ( n ) = ( x if n Ay if n ∈ A . Let g be a γ -lift of f to B . Note that by property (i) above, g ( n ) ∈ X ∪ Y ,and by property (ii), if g ( n ) ∼ γ x then g ( n ) Y and vice-versa. Define h as follows: h ( n ) = ( g ( n ) ∈ X g ( n ) ∈ Y . Since h is B -computable and h = χ A , A is B -computable. (cid:3) Partial computable functions and c.e. sets. Before we continue,let us consider a few more examples and investigate the consequences of The-orem 2.12 for the standard numberings ϕ A ( − ) and W A ( − ) . The contrapositiveof Theorem 2.12 gives us some examples of non-divisible numberings. Corollary 2.13. For noncomputable A , the numberings W A ( − ) and ϕ A ( − ) arenot -divisible.Proof. Immediate by the theorem together with Example 2.7. (cid:3) The following propositions will serve to characterize the behavior of γ for the numbering of the A -c.e. sets for any A . Proposition 2.14. If A is γ -low and B T A , then B is γ -low as well.Proof. Since B T A implies B γ A by Proposition 2.4, we have B γ A γ ∅ . (cid:3) Proposition 2.15. Suppose γ is (2 , A ) -divisible and A T B . If C γ B ,then C T B .Proof. Since A T B , γ is also (2 , B )-divisible by Proposition 2.11. It nowfollows from Theorem 2.12 that C T B . (cid:3) IXPOINTS AND RELATIVE PRECOMPLETENESS 5 For the numbering γ of the A -c.e. sets, we can now characterize the rela-tion γ on the upper and lower cone of A as follows: Theorem 2.16. Fix A , and let γ be one of W A ( − ) and ϕ A ( − ) . Then (i) every B T A is γ -low, (ii) for all B, C > T A , we have B γ C if and only if B T C .Proof. By Example 2.7, A is γ -low, so item (i) immediately follows fromProposition 2.14.For item (ii), as we have seen in Example 2.10, γ is ( ω, A )-divisible, hencecertainly (2 , A )-divisible. Suppose that B, C > T A . If B γ C then B T C by Proposition 2.15. The converse always holds by Proposition 2.4. (cid:3) Ceers and c.e. numberings. Every numbering γ gives rise to theequivalence relation ∼ γ . A particularly interesting class of numberings arethose for which ∼ γ is a computably enumerable equivalence relation, typ-ically called a ceer . These were already studied by Ershov [8], who calledthem positive numberings. Every ceer, in turn, gives rise to a c.e. num-bering by sending n ∈ ω to its equivalence class. Ceers have been studiedextensively in recent years, see for example Andrews, Badaev, and Sorbi [1],Andrews et al. [2], Andrews and Sorbi [3], and the references mentioned inthese papers. Definition 2.17. A numbering γ is c.e. if the set {h x, y i : x, y ∈ ω, x ∼ γ y } is computably enumerable.Since each equivalence class of a c.e. numbering is c.e., the notion of n -divisibility is trivial for finite n provided that there are sufficiently manyequivalence classes. However, this does not extend to the infinitary case,since it may not be possible to enumerate an infinite subset of the equivalenceclasses without repetition. Proposition 2.18. If γ is a c.e. numbering with at least n ∈ ω equivalenceclasses then γ is n -divisible.Proof. Select n non- γ -equivalent elements x i and note that for all 0 i < n ,the set { y ∈ ω : x i ∼ γ y } is computably enumerable. (cid:3) Corollary 2.19. For every c.e. numbering γ with at least two equivalenceclasses, the notions A T B and A γ B are equivalent.Proof. This follows immediately from Proposition 2.4 and 2.12. (cid:3) Theorem 2.20. There exists a c.e. numbering γ with ω equivalence classesthat is not ω -divisible.Proof. We construct a c.e. set of pairs A and let ∼ γ be the least equivalencerelation containing A . The numbering γ then arises by sending every n ∈ ω to its ∼ γ -equivalence class. A. GOLOV AND S. A. TERWIJN To guarantee that γ is not ω -divisible, we ensure that for every totalcomputable function f there is some equivalence class of γ that appearsin the output of f at least twice, and thus that taking x i = f ( i ) doesnot satisfy requirement (ii) in Definition 2.8. We do this by ensuring thefollowing requirements are satisfied: R e : f = ϕ e total and injective = ⇒ ∃ i, j. h f ( i ) , f ( j ) i ∈ A. Let A s be the set of pairs enumerated up to stage s . Throughout theconstruction, we keep track of a function g s that maps n to the least re-quirement e that has contributed a value to the equivalence class of n in A s , or ∞ if there is no such requirement. We also track which requirementshave acted.At stage s = 0, set A to be empty.At stage s + 1, we say that a requirement R e needs attention if it has notacted previously and there exist i, j such that x = ϕ e,s ( i ) ↓ and y = ϕ e,s ( j ) ↓ and g s ( x ) , g s ( y ) > e .If no requirement R e , with e < s , requires attention, do nothing forthis stage. Otherwise, let e < s be least such that R e requires attention,enumerate h x, y i into A , and mark R e as having acted. This completes theconstruction.To see that the requirements are satisfied, it suffices to show that if f = ϕ e is total and injective, then R e eventually acts, since any requirementthat acts is clearly satisfied. Suppose the antecedent of R e is satisfied. Byinduction, there is some stage s after which no requirement d < e acts andthus g − s ( { , . . . , e − } ) is fixed. Since f is total and injective it is alsounbounded, and thus there exist i, j such that g s ( f ( i )) , g s ( f ( j )) > e and R e will thus eventually act.It remains to show that A induces infinitely many equivalence classesin γ , which we will do by showing every equivalence class is finite. Let x ∈ ω . If the equivalence class of x is a singleton, then it is clearly finite.Otherwise, there is some requirement R e that first extended the equivalenceclass of x . By construction, every next extension was by some requirement R d with d < e , and since there are only finitely many such requirements,the equivalence class of x was only extended finitely often and is thus finite.It follows that every equivalence class is finite, and thus that ∼ γ has in-finitely many equivalence classes. However, since every computable functionis non-injective modulo γ , it follows that γ is not ω -divisible. (cid:3) In practice, many ceers from the literature are in fact ω -divisible. We givetwo examples from Visser [19]. Example 2.21. Let M λβ be the set of λ -calculus terms modulo β -equivalenceand let γ be the numbering given by some effective coding of terms.Since β -equivalence is a c.e. relation, γ is c.e. To see that γ is ω -divisible,note that by strong normalization and the diamond property, it suffices toenumerate any infinite set of non-equal terms in normal form. This can bedone, for example, by taking the Church numerals λf.λx.f n ( x ). IXPOINTS AND RELATIVE PRECOMPLETENESS 7 Example 2.22. Let A be an arithmetical set and let γ be the numberinginduced by the equivalence relation n ∼ γ m ⇐⇒ PA ⊢ ∀ x. ϕ An ( x ) ≃ ϕ Am ( x ) . Since we can enumerate proofs in PA, γ is a c.e. numbering. To see that γ is ω -divisible we can take ϕ Af ( n ) = n , which gives us an enumeration f ofcodes of distinct functions. Since PA is sound, these codes are not identifiedby γ .Note that by Theorem 2.12, it follows that if A noncomputable then A isnot γ -low. This is in contrast with the numbering ϕ A ( − ) , where A is always ϕ A ( − ) -low. 3. Relative precompleteness The notion of a precomplete numbering was introduced by Ershov [8].We will work with the following relativized version of precompleteness, thatwas already studied in Selivanov [15]. Definition 3.1. Given an oracle A , a numbering γ is A -precomplete if forevery partial A -computable function ψ there is a total A -computable func-tion f such that ψ ( n ) ↓ = ⇒ ψ ( n ) ∼ γ f ( n ) . for every n . We call f a totalization of ψ modulo γ . Note that ∅ -precompleteis the same as Ershov’s notion of precompleteness.The following is Ershov’s version of the recursion theorem for precompletenumberings [9]. In relativized form it reads as follows: Theorem 3.2 (Recursion theorem for precomplete numberings) . If γ is an A -precomplete numbering then every A -computable function f has a fixpointmodulo γ ; that is, for every A -computable f there is an n ∈ ω such that f ( n ) ∼ γ n . Kleene’s second recursion theorem (cf. Moschovakis [13] for the rich his-tory of this result) is a uniform, parameterized version of the first recursiontheorem. It also holds for precomplete numberings (with the same proof),and reads as follows: Theorem 3.3 (Recursion theorem with parameters) . If γ is an A -precom-plete numbering then for every binary A -computable function h there is aunary A -computable function f such that for all n ∈ ω , h ( n, f ( n )) ∼ γ f ( n ) . Arslanov [4] generalized the recursion theorem from computable func-tions to the class of functions that are computable from an incomplete c.e.set. Phrased differently: A c.e. set is Turing complete if and only if it cancompute a fixpoint free function. This is Arslanov’s completeness criterion.Selivanov [15] (see also [5]) proved that Arslanov’s theorem also holds forany precomplete numbering: Theorem 3.4 (Arslanov’s completeness criterion for precomplete number-ings) . If A is Turing incomplete and c.e. and γ is a precomplete numbering,then every A -computable function f has a fixpoint modulo γ . A. GOLOV AND S. A. TERWIJN At first glance, we may hope to prove Arslanov’s completeness criterionby showing that that if A is incomplete and c.e. and γ is precomplete, then γ is also A -precomplete. In the remainder of this section we show that thisis not the case for precomplete ω -divisible numberings. As we have seen inthe previous section, this includes the numberings W ( − ) and ϕ ( − ) . Lemma 3.5. If an A -precomplete numbering γ is -divisible and the total-ization u of the universal function modulo γ is computable, then A is γ -lowand therefore computable.Proof. Let γ be such a numbering and let u be the totalization of the univer-sal function. Suppose u is computable. Let f = ϕ Ae be a total A -computablefunction. Define a computable function g by g ( n ) = u ( e, n ) . Now for all n , g ( n ) = u ( e, n ) ∼ γ ϕ Ae ( n ) = f ( n )and thus A is γ -low. By Theorem 2.12, A is computable. (cid:3) Theorem 3.6. Let A T B . If a numbering γ is ( ω, A ) -divisible and γ is B -precomplete, then A ≡ T B .Proof. Let A T B and let u be the totalization of a partial B -computableuniversal function modulo γ . Let ( x i ) i ∈ ω and ( X i ) i ∈ ω witness that γ is( ω, A )-divisible.Let U ( e, x ) be the use of u ( e, x ) and define a B -computable function f by f ( n ) = max i nj n U ( i, j ) . If lim n →∞ f ( n ) is finite then u is computable and thus B is γ -low byLemma 3.5 and is thus computable by Theorem 2.12. It follows that A ≡ T B ≡ T ∅ .Suppose, then, that lim n →∞ f ( n ) diverges. Let D ( − ) be the canonicalnumbering of finite sets. Since f is B -computable, there is a B -computablefunction g such that D g ( n ) = { b ∈ B : b f ( n + 1) } . Let u ′ ( e, n, m ) be the computable function that acts like u ( e, n ) but withqueries to the oracle B replaced by reads from D m .Now fix e to be a code of n x g ( n ) . Note that this function is B -computable, since g is B -computable, the sequence x i is A -computable, and A T B . Note that we have u ( e, n ) ∼ γ ϕ Be ( n ) = x g ( n ) . (2)Define the partial A -computable function δ by δ ( n, m ) = least i such that u ′ ( e, n, m ) ∈ X i . For any n, m ∈ ω with n > e , if D m agrees with B on the first f ( n ) bits,then δ ( n, m ) terminates after using at most the first f ( n ) bits of D m and D δ ( n,m ) agrees with B on the first f ( n +1) bits. Namely, u ′ ( e, n, m ) = u ( e, n )since by assumption D m gives the correct answers to the oracle queriesbelow f ( n ), which includes the use of u ( e, n ), and because by (2) we have IXPOINTS AND RELATIVE PRECOMPLETENESS 9 u ( e, n ) ∼ γ x g ( n ) it follows that u ′ ( e, n, m ) ∈ X g ( n ) by the properties of thedividing sets X i . Hence δ ( n, m ) ↓ = g ( n ).Let D c be the code of the first f ( e ) bits of the oracle B . By recursion, wecan define b (0) = D c b ( k + 1) = δ ( k, b ( k )) . Now b is an A -computable sequence of canonical codes of increasing initialsegments of B , where D b ( n ) contains at least f ( n ) bits if n > e . Sincelim n →∞ f ( n ) = ∞ , eventually we can compute all of B this way. Since δ is A -p.c., it follows that B T A . (cid:3) Putting this together with earlier results, we obtain the following equiv-alence. Theorem 3.7. For any precomplete ω -divisible numbering γ , the followingare equivalent:(1) A is computable.(2) A is γ -low.(3) γ is A -precomplete.In particular, this is the case for γ one of W ( − ) and ϕ ( − ) ,Proof. If A is computable, by Proposition 2.4 it is γ -low. Since γ is alreadyprecomplete, it is also A -precomplete.If A is γ -low, then by Theorem 2.12 it is in fact computable.If γ is A -precomplete, Theorem 3.6 applies and again A is computable.As we have seen W ( − ) and ϕ ( − ) are precomplete and ω -divisible and thussatisfy the requirements of the theorem. (cid:3) Taking A noncomputable, and considering the numbering W ( − ) , we seefrom Theorem 3.7 that this numbering is precomplete, but not A -precom-plete. If moreover A is c.e. and incomplete, then by Arslanov’s completenesscriterion (Theorem 3.4) we know that every A -computable function has afixpoint for this numbering. So we see that the existence of fixpoints of A -computable functions does not, in general, give us A -precompleteness, evenfor a precomplete numbering. Corollary 3.8. There exists a set A and a numbering γ such that γ isprecomplete but not A -precomplete. We currently do not know if the converse implication also does not hold,so we ask: Question 3.9. Does there exists a set A and a numbering γ such that γ is A -precomplete but not precomplete? Skolem functions Just as the recursion theorem has a version with parameters (Theo-rem 3.3), we can formulate in the same way a parameterized version ofArslanov’s completeness criterion (Theorem 3.4). Again we can formulatethis for arbitrary precomplete numberings, and in relativized form, as fol-lows: Theorem 4.1 (Arslanov’s completeness criterion with parameters, for pre-complete numberings) . If A is a incomplete c.e. set and γ is a precompletenumbering then for every A -computable binary function h there is an A -computable function f such that h ( n, f ( n )) ∼ γ f ( n ) . It was already stated in Arslanov [4] that this form of Arslanov’s complete-ness criterion holds for the standard numbering of c.e. sets W ( − ) . Followingthe notation in [18], we refer to f as the Skolem function, since it is theSkolemization of ∀ n ∃ x. h ( n, x ) ∼ γ x, which holds by Theorem 3.4. Theorem 4.1 can be proved by analyzing theoriginal proof of Arslanov. For completeness we include a proof here. Proof of Theorem 4.1. Let A be a incomplete, c.e. set and let γ be a pre-complete numbering. Let ˆ h be a computable approximation of h and let m be its A -computable modulus. By the properties of a modulus we have thatfor all n, x ∈ ω and all s > m ( n, x ), ˆ h ( n, x, s ) = h ( n, x ).Define ψ ( n, x, k ) = ( ˆ h ( n, x, s k ) if k ∈ ∅ ′ and s k is minimal such that k ∈ ∅ ′ s ↑ if k 6∈ ∅ ′ . Let g be a totalization of ψ modulo γ . By the recursion theorem withparameters there is a computable ˆ f ( n, k ) such that for all n, k ∈ ω ,ˆ f ( n, k ) ∼ γ g ( n, ˆ f ( n, k ) , k ) . Define r ( n ) = k where h s, k i = µ h s, k i [ m ( n, ˆ f ( n, k )) s ∧ k ∈ B ′ s ] . We claim r is total. Note that by minimality, s = s k from above. Sup-pose there exists n such that no such s and k exist. Then for all k ∈ ∅ ′ , m ( n, ˆ f ( n, k )) > s k and thus k ∈ ∅ ′ m ( n, ˆ f ( n,k )) . It follows that ∅ ′ T A , con-tradicting our assumption that A is incomplete.Since r ( n ) is total and m ( n, ˆ f ( n, r ( n ))) s r ( n ) it follows that h ( n, ˆ f ( n, r ( n )) = ˆ h ( n, ˆ f ( n, r ( n )) , s r ( n ) )and thus since r ( n ) ∈ ∅ ′ h ( n, ˆ f ( n, r ( n )) = ˆ h ( n, ˆ f ( n, r ( n )) , s r ( n ) ) ∼ γ g ( n, ˆ f ( n, r ( n )) , r ( n )) ∼ γ ˆ f ( n, r ( n )) . Therefore, f ( n ) = ˆ f ( n, r ( n )) is a fixpoint of h modulo γ . (cid:3) This shows that the Skolem functions have degree at most A . In Ter-wijn [18, Theorem 3.1] it was shown that, for the numbering W ( − ) , theSkolem functions in general cannot be computable. We will now show thatthe degree of the Skolem functions for an incomplete c.e. set A is in factequal to A . IXPOINTS AND RELATIVE PRECOMPLETENESS 11 Theorem 4.2. For every A, B and every numbering γ , if for every A -computable h there is a B -computable f such that for all n , h ( n, f ( n )) ∼ γ f ( n ) , then A γ B .Proof. Suppose that for every A -computable h there exists such a B -com-putable f . Let g be an A -computable function and define h ( n, x ) = g ( n ).Take f B -computable such that h ( n, f ( n )) ∼ γ f ( n ) for all n . It now followsthat g ( n ) = h ( n, f ( n )) ∼ γ f ( n ) . (cid:3) Corollary 4.3. Let A and B be oracles, and let γ be a (2 , B ) -divisiblenumbering. If for every A -computable h there is a B -computable f suchthat for all n , h ( n, f ( n )) ∼ γ f ( n ) , then A is B -computable.Proof. It follows from Theorem 4.2 that A γ B , and thus A T B byTheorem 2.12 (cid:3) Corollary 4.4. In particular, if A is an incomplete c.e. set, then B -compu-table Skolem functions for A -computable functions exist for the numberings W ( − ) and ϕ ( − ) if and only if A T B . In Barendregt and Terwijn [5, Question 3.4] the question was posedwhether the uniformly computable existence of fixpoints for a numberingimplies that the numbering is precomplete. This question remains open,but we note here that in general the answer is negative for the relativized version of this question. Namely, we see from Theorem 4.1, together withTheorem 3.7, that even if every A -computable family of A -computable func-tions has a fixpoint modulo γ , computable uniformly in A , this does notguarantee that γ is A -precomplete. References [1] U. Andrews, S. Badaev, and A. 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Terwijn) Radboud University Nijmegen, Department of Math-ematics, P.O. Box 9010, 6500 GL Nijmegen, the Netherlands Email address ::