Computable reducibility of equivalence relations and an effective jump operator
aa r X i v : . [ m a t h . L O ] M a y COMPUTABLE REDUCIBILITY OF EQUIVALENCE RELATIONS AND ANEFFECTIVE JUMP OPERATOR
JOHN D. CLEMENS, SAMUEL COSKEY, AND GIANNI KRAKOFFA
BSTRACT . We introduce the computable FS-jump, an analog of the classical Friedman–Stanley jump in the context of equivalence relations on N . We prove that the computableFS-jump is proper with respect to computable reducibility. We then study the effect of thecomputable FS-jump on computably enumerable equivalence relations (ceers). §
1. I
NTRODUCTION
The backdrop for our study is the notion of computable reducibility of equivalencerelations. If E , F are equivalence relations on N we say E is computably reducible to F ,written E ≤ F , if there exists a computable function f : N → N such that for all n , n ′ n E n ′ ⇐⇒ f ( n ) F f ( n ′ ) .The notion goes back at least to [GG01]; see also [FF09, FFH +
12, CHM12] as well as nu-merous other authors including those cited below.Computable reducibility of equivalence relations may be thought of as a computableanalog to Borel reducibility of equivalence relations on standard Borel spaces. Here if E , F are equivalence relations on standard Borel spaces X , Y we say E is Borel reducible to F ,written E ≤ B F , if there exists a Borel function f : X → Y such that x E x ′ ⇐⇒ f ( x ) Ff ( x ′ ) . We refer the reader to [Gao09] for the basic theory of Borel reducibility.One of the major goals in the study of computable reducibility is to compare the rel-ative complexity of classification problems on a countable domain. In this context, if E ≤ F we say that the classification up to E -equivalence is no harder than the classifi-cation up to F -equivalence. For instance, classically the rank 1 torsion-free abelian groups(the subgroups of Q ) may be classified up to isomorphism by infinite binary sequencesup to almost equality. Since this classification may be carried out in a way which is com-putable in the indices, there is a computable reduction from the isomorphism equivalence Mathematics Subject Classification.
Primary 03D25, 03D30, Secondary 03D65, 03F15.
Key words and phrases. computable reducibility, ceers, hyperarithmetic equivalence relations. relation on c.e. subgroups of Q to the almost equality equivalence relation on c.e. binarysequences.A second major goal in this area is to study properties of the hierarchy of equivalencerelations with respect to computable reducibility. The full hierarchy is of course quitecomplex, for instance, it is shown in [Bar19, Theorem 4.5] that it is at least as complex asthe Turing degree order. In a portion of this article we will pay special attention to thesub-hierarchy consisting of just the ceers. An equivalence relation E on N is called a ceer if it is computably enumerable, as a set of pairs. Ceers were introduced in [GG01], andtheir structure has been studied in subsequent works such as [ALM +
14, AS18a, AS18b].As with other complexity hierarchies, it is natural to study operations such as jumps.One of the most important jumps in Borel complexity theory is the Friedman–Stanleyjump, which is defined as follows. If E is a Borel equivalence relation on the standardBorel space X , then the Friedman–Stanley jump of E , denoted E + , is the equivalence relationdefined on X N by x E + x ′ ⇐⇒ { [ x ( n )] E : n ∈ N } = { [ y ( n )] E : n ∈ N } .Friedman and Stanley showed in [FS89] that the jump is proper , that is, if E is a Borelequivalence relation, then E < B E + . Moreover they studied the hierarchy of iterates ofthe jump and showed that any Borel equivalence relation induced by an action of S ∞ isBorel reducible to some iterated jump of the identity.In this article we study a computable analog of the Friedman–Stanley jump, called thecomputable FS-jump, in which the arbitrary sequences x ( n ) are replaced by computableenumerations φ e ( n ) . In Section 2 we will give the formal definition of the computableFS-jump, and establish some of its basic properties.In Section 3 we show that the computable FS-jump is proper, that is, if E is a hyperarith-metic equivalence relation, then E < E + . We do this by showing that any hyperarithmeticset is many-one reducible to some iterated jump of the identity, and establishing roughbounds on the descriptive complexity of these iterated jumps.In Section 4 we study the effect of the computable FS-jump on ceers. We show that if E isa ceer with infinitely many classes, then E + is bounded below by the identity relation id on N , and above by the equality relation = ce on c.e. sets. This leads to a natural investigationof the structure that the jump induces on the ceers, analogous to the study of the structurethat the Turing jump induces on the c.e. degrees. For instance, we may say that a ceer E is high for the computable FS-jump if E + is computably bireducible with = ce . At the close OMPUTABLE REDUCIBILITY OF EQUIVALENCE RELATIONS AND AN EFFECTIVE JUMP OPERATOR 3 of the section, we begin to investigate the question of which ceers are high and which arenot.In the final section we present several open questions arising from these results.
Acknowledgement.
This work includes a portion of the third author’s master’s thesis[Kra19]. The thesis was written at Boise State University under the supervision of the firstand second authors. §
2. B
ASIC PROPERTIES OF REDUCIBILITY AND THE JUMP
In this section we fix some notation, introduce the computable FS-jump, and expositsome of its basic properties.In this and future sections, we will typically use the letter e for an element of N whichwe think of as an index for a Turing program. We will use φ e for the partial computablefunction of index e , and W e for the domain of φ e . Definition 2.1.
Let E be an equivalence relation on N . The computable FS-jump of E is theequivalence relation on indices of c.e. subsets of N defined by e E + e ′ ⇐⇒ { [ φ e ( n )] E : n ∈ N } = { [ φ e ′ ( n )] E : n ∈ N } .Furthermore we define the iterated jumps E + n inductively by E + = E + and E +( n + ) =( E + n ) + .We acknowledge there is a notational conflict here, as we have used the symbol E + forboth the FS-jump and the computable FS-jump. But this should be no cause for confusion,since from this point forward we will use the symbol E + exclusively for the computableFS-jump.We are now ready to establish some of the basic properties of the computable FS-jump.In the following, we let id denote the identity equivalence relation on N . Proposition 2.2.
For any equivalence relations E and F on N we have: (a) E ≤ E + . (b) If E has only finitely many classes, then E < E + . (c) If E ≤ F then E + ≤ F + .Proof. (a) Let f be a computable function such that for all e we have that φ f ( e ) is the con-stant function with value e . (To see that there is such a computable function f , one caneither “write a Turing program” for the machine indexed by f ( e ) or employ the s-m-n the-orem. In the future we will not comment on the computability of functions of this nature.)Then e E e ′ if and only if [ e ] E = [ e ′ ] E , or in other words, if and only if f ( e ) E + f ( e ′ ) . OMPUTABLE REDUCIBILITY OF EQUIVALENCE RELATIONS AND AN EFFECTIVE JUMP OPERATOR 4 (b) Note that if E has n classes, then E + has 2 n classes.(c) This is similar to [GG01, Theorem 8.4]. Let f be a computable reduction from E to F .Let g be a computable function such that φ g ( e ) ( n ) = f ( φ e ( n )) . Then it is straightforwardto verify that g is a computable reduction from E + to F + . (cid:3) In the following, we let E ⊕ F denote the equivalence relation defined on N × {
0, 1 } by ( m , i )( E ⊕ F )( n , j ) iff ( i = j = ) ∧ ( m E n ) or ( i = j = ) ∧ ( m F n ) . Finally, welet E × F denote the equivalence relation defined on N × N by ( m , n )( E × F )( m ′ , n ′ ) iff m E m ′ ∧ n F n ′ . Proposition 2.3. ( E ⊕ F ) + is computably bireducible with E + × F + .Proof. For the forward reduction, given an index e for a function into N × {
0, 1 } , let φ e ( n ) = m if φ e ( n ) = ( m , 0 ) and let φ e ( n ) = m if φ e ( n ) = ( m , 1 ) ; φ e i is undefinedotherwise. Then the map e ( e , e ) is a reduction from ( E ⊕ F ) + to E + × F + . Forthe reverse reduction, given a pair of indices ( e , e ) we define φ e ( n ) = ( φ e ( n ) , 0 ) and φ e ( n + ) = ( φ e ( n ) , 1 ) . Once again it is easy to verify ( e , e ) e is a reduction from E + × F + to ( E ⊕ F ) + . (cid:3) In the next result we will briefly consider the connection between the computable FS-jump and the restriction of the classical FS-jump to c.e. sets. To begin, we recall that forany n , F n denotes the n th iterated classical FS-jump of id . The equivalence relation F isBorel bireducible with the equality relation = on N N . The equivalence relation F is Borelbireducible with the equivalence relation E set defined on ( N N ) N by x E set y ⇐⇒ { x ( n ) : n ∈ N } = { y ( n ) : n ∈ N } .Below it will be convenient to regard the F n ’s as equivalence relations on P ( N ) . Tobegin, we officially define F as the equality relation on P ( N ) . Next, let h· , ·i be the usualpairing function N → N , and let A ( n ) denote the n th “column” of A , that is, A ( n ) = { p ∈ N : h n , p i ∈ A } . We then officially define A F B iff { A ( n ) : n ∈ N } = { B ( n ) : n ∈ N } .Similarly for all n we can officially define F n on P ( N ) by means of a fixed uniformlycomputable family of bijections between N n and N .Next, recall from [CHM12] that for any equivalence relation E on P ( N ) we can defineits restriction to c.e. sets E ce on N by e E ce e ′ ⇐⇒ W e E W e ′ .We are now ready to state the following. Proposition 2.4.
For any n, we have that id + n is computably bireducible with ( F n ) ce . OMPUTABLE REDUCIBILITY OF EQUIVALENCE RELATIONS AND AN EFFECTIVE JUMP OPERATOR 5
Proof sketch.
It is sufficient to show that for any n we have that (( F n ) ce ) + is computablybireducible with ( F n + ) ce . For notational simplicity, we briefly illustrate this just in thecase when n =
1. For the reduction from (( F ) ce ) + to ( F ) ce , we define f to be a computablefunction such that for all n we have ( W f ( e ) ) ( n ) = W φ e ( n ) . For the reduction from ( F ) ce to (( F ) ce ) + , we define g to be a computable function such that for all n we have W φ g ( e ) ( n ) =( W e ) ( n ) . (cid:3) To conclude the section, we define transfinite iterates of the computable FS-jump. Thetransfinite jumps allow one to extend results such as the previous proposition into thetransfinite, and they also play a key role in the next section. For the definition, recall thatKleene’s O consists of notations for ordinals and is defined as follows: 1 ∈ O is a notationfor 0, if a ∈ O is a notation for α then 2 a is a notation for α +
1, and if for all n we have φ e ( n ) is a notation for α n with the notations increasing in O with respect to n , then 3 · e isa notation for sup n α n . We refer the reader to [Sac90] for background on O . Definition 2.5.
We define E + a for a ∈ O recursively as follows. E + = EE + b = ( E + b ) + E + · e = { ( h m , x i , h n , y i ) : ( m = n ) ∧ ( x E + φ e ( m ) y ) } We remark that it is straightforward to extend Proposition 2.4 into the transfinite asfollows. Given a notation a ∈ O for α , we may use a to define an equivalence relation F a on P ( N ) which is Borel bireducible with the α -iterated FS-jump F α . We then have that id + a is computably bireducible with ( F a ) ce . We do not know, however, whether id + a and id + a ′ are computably bireducible when a and a ′ are different notations for the same ordinal.The following proposition will be used in the next section. Proposition 2.6.
If E + ≤ E then for any a ∈ O we have E + a ≤ E.Proof.
We proceed by recursion on a ∈ O . It follows from our hypothesis together withProposition 2.2(b) that E has infinitely many classes. By Proposition 4.3 (below), we have id ≤ E ++ and hence id ≤ E . It follows from this that E × id ≤ E ++ (for instance if h : id ≤ E then define h ′ : E × id ≤ E ++ by arranging for W h ′ ( e , n ) = {
0, 1 } , φ h ′ ( e , n ) ( ) = a code for { e } , and φ h ′ ( e , n ) ( ) = a code for { h ( n ) , h ( n + ) } ). Hence we have E × id ≤ E , and we mayfix a computable reduction function g : E × id ≤ E . OMPUTABLE REDUCIBILITY OF EQUIVALENCE RELATIONS AND AN EFFECTIVE JUMP OPERATOR 6
Now let f : E + ≤ E and define uniformly f a : E + a ≤ E by letting f be the identity map, f a = f ◦ f a , and f · e as follows. If d is an E + · e code, φ d ( i ) = h n i , j i i with j i an E + φ e ( n i ) code for each i , then f · e ( d ) = f ( m ) , where φ m ( i ) = g ( f φ e ( n i ) ( j i ) , n i ) . (cid:3) §
3. P
ROPERNESS OF THE JUMP
In this section we establish the following result.
Theorem 3.1.
If E is a hyperarithmetic equivalence relation on N , then E < E + . The proof will proceed by showing that iterated jumps of the identity have cofinal de-scriptive complexity. Specifically, we will show that every hyperarithmetic set is many-one reducible to id + a for some a ∈ O . The following notion will be fundamental to theinduction used. Definition 3.2.
We write e ⊆ E e ′ if the following holds: ∀ n [ φ e ( n ) ↓ → ∃ m ( φ e ′ ( m ) ↓ ∧ φ e ( n ) E φ e ′ ( m ))] .Note that we have e E + e ′ iff e ⊆ E e ′ and e ′ ⊆ E e .We next introduce the hyperarithmetic sets by means of a characterization in terms ofrecursive Borel codes. Definition 3.3. A recursive Borel code is a pair ( T , f ) where T is a recursive well-foundedtree on N so that t a n ∈ T for all n for non-terminal nodes t , and f is a computablefunction from the terminal nodes of T to N . Given a recursive Borel code ( T , f ) , the set B ( T , f ) is defined by recursion on t ∈ T as follows. If t is a terminal node, then B t ( T , f ) = ran φ f ( t ) , and if t is not a terminal node, then B t ( T , f ) = { n : ∀ p ∃ q ( n ∈ B t a h p , q i ( T , f )) } .We let B ( T , f ) = B ∅ ( T , f ) .Note that there are many different presentations of recursive Borel codes, all of whichgive the same collection of sets, and we have selected the above presentation as the bestone for our proof. Theorem 3.4.
The set B is hyperarithmetic if and only if there is a recursive Borel code ( T , f ) such that B = B ( T , f ) . This follows from the fact that a set is hyperarithmetic if and only if it is ∆ , togetherwith the Kleene Separation Theorem (see, e.g., [Sac90, Chapter II] and [Mil95, Theorem27.1]).In the following, we will say that e is an index for an enumeration of the c.e. set W ifran φ e = W . Before proving our key lemma about hyperarithmetic sets, we start with a OMPUTABLE REDUCIBILITY OF EQUIVALENCE RELATIONS AND AN EFFECTIVE JUMP OPERATOR 7 special case, which illustrates the key idea of enumerating c.e. supersets of a given setwhich we will use in the proof of the main result. This case also serves as the base forsharper complexity bounds discussed at the end of this section. We will repeatedly utilizethe fact that { [ e ] E + : e ⊇ E e } = { [ W i ∪ e ] E + : i ∈ N } , where W i ∪ e is an index foran enumeration of ran φ e ∪ W i . The analogous statement with ⊆ E replacing ⊇ E does nothold, as illustrated in Proposition 3.8. Lemma 3.5.
Let P be Π . Then there is e ∈ N and a recursive h so that ∀ n [ P ( n ) ⇐⇒ h ( n ) (= ce ) + e ] , and h ( n ) ⊆ = ce e for all n.Proof. Choose i with P ( n ) ⇐⇒ ∀ p ∃ q ∀ m φ i ( h p , q , m , n i ) ↓ , so that P ( n ) ⇐⇒ ∀ p ∃ q {h p , q , m , n i : m ∈ N } ⊂ W i .Letting W g ( p , q , n ) = W i ∪ {h p , q , m , n i : m ∈ N } , we then have P ( n ) ⇐⇒ ∀ p ∃ q W g ( p , q , n ) = W i ,with W g ( p , q , n ) ⊃ W i for all p , q , n . Then P ( n ) ⇐⇒ ∀ p ∃ q { W i ∪ W g ( p , q , n ) : i ∈ N } = { W i ∪ W i : i ∈ N } ,with { W i ∪ W g ( p , q , n ) : i ∈ N } ⊂ { W i ∪ W i : i ∈ N } for all p , q , n . Hence P ( n ) ⇐⇒ ∀ p { W i ∪ W g ( p , q , n ) : i ∈ N , q ∈ N } = { W i ∪ W i : i ∈ N } ,with { W i ∪ W g ( p , q , n ) : i ∈ N , q ∈ N } ⊂ { W i ∪ W i : i ∈ N } for all p , q , n , and equalityholding only when there is q with W g ( p , q , n ) = W i . Letting W ∗ p = { p } × W , we have P ( n ) ⇐⇒ { ( W i ∪ W g ( p , q , n ) ) ∗ p : i , q , p ∈ N } = { ( W i ∪ W i ) ∗ p : i , p ∈ N } ,so, with h ( n ) such that φ h ( n ) ( h i , q , p i ) is an index for an enumeration of ( W i ∪ W g ( p , q , n ) ) ∗ p and e such that φ e ( h i , p i ) is an index for an enumeration of ( W i ∪ W i ) ∗ p , we have P ( n ) ⇐⇒ h ( n ) (= ce ) + e ,with h ( n ) ⊆ = ce e for all n . (cid:3) We now prove the key lemma for establishing properness of the jump.
Lemma 3.6.
Let ( T , f ) be a recursive Borel code. Then there is a T ∈ O so that B ( T , f ) ≤ m id + a T .Proof. For notational convenience, let B t = B t ( T , f ) for t ∈ T . We will recursively define a t ∈ O and let E t = id + a t and establish by effective induction on t ∈ T that B t ≤ m E + t . To OMPUTABLE REDUCIBILITY OF EQUIVALENCE RELATIONS AND AN EFFECTIVE JUMP OPERATOR 8 carry out the induction, we show that there are recursive maps t a t , t h t , and t e t so that n ( h t ( n ) , e t ) is a many-one reduction of B t to E + t satisfying for all n :(a) B t ( n ) ⇐⇒ h t ( n ) E + t e t , and(b) h t ( n ) ⊆ E t e t .For t a terminal node we have B t = ran φ f ( t ) and we set a t = E t = id and E + t is = ce . Fixa single e t for all terminal t so that ran φ e t = N , and let h t ( n ) be such that ran φ h t ( n ) = N if n ∈ ran φ f ( t ) and ran φ h t ( n ) = ∅ if n / ∈ ran φ f ( t ) . This satisfies (a) and (b).Now let t be a non-terminal node. We have B t ( n ) ⇐⇒ ∀ p ∃ q B t a h p , q i ( n ) , so B t ( n ) ⇐⇒ ∀ p ∃ q h t a h p , q i ( n ) E + t a h p , q i e t a h p , q i ,where h t a h p , q i ( n ) ⊆ E t a h p , q i e t a h p , q i for all p , q , and n . We first absorb the existential quan-tifier. We start by defining ˜ a t , p ∈ O , an iterated jump E t , p = id + ˜ a t , p , e t , p , and h ′ t , p so that h ′ t , p ( q , n ) and e t , p uniformly witness B t a h p , q i ≤ m E + t , p and satisfy (a) and (b). We first adjustordinal ranks to produce a sequence increasing in q so that we can take their supremum in O . Let ˜ a t , p ,0 = a t a h p ,0 i and ˜ a t , p , q + = ˜ a t , p , q + O a t a h p , q + i + O
1. Then let ˜ a t , p = · i t , p where φ i t , p ( q ) = ˜ a t , p , q for all q . Observe that if ψ : E ≤ F then the map ˜ ψ : E + ≤ F + as producedin the proof of Proposition 2.2(c) will satisfy e ⊆ E e ′ ⇐⇒ ˜ ψ ( e ) ⊆ F ˜ ψ ( e ′ ) . Hence wecan uniformly replace E t a h p , q i , e t a h p , q i , and h t a h p , q i by id + ˜ a t , p , q , ˜ e t , p , q , and a map ˜ h t , p , q whilemaintaining conditions (a) and (b).We now let h ′ t , p ( q , n ) be an index for an enumeration of {h q , φ ˜ h t , p , q ( n ) ( i ) i : i ∈ N } ∪ {h q ′ , φ ˜ e t , p , q ′ ( i ) i : q ′ = q ∧ i ∈ N } and let e t , p be an index for an enumeration of {h q ′ , φ ˜ e t , p , q ′ ( i ) i : q ′ , i ∈ N } .We then have B t ( n ) ⇐⇒ ∀ p ∃ q h ′ t , p ( q , n ) E + t , p e t , p ,with h ′ t , p ( q , n ) ⊆ E t , p e t , p for all p , q , and n , as desired. Then B t ( n ) ⇐⇒ ∀ p ∃ q { [ e ] E + t , p : e ⊇ E t , p h ′ t , p ( q , n ) } = { [ e ] E + t , p : e ⊇ E t , p e t , p } ,with { [ e ] E + t , p : e ⊇ E t , p h ′ t , p ( q , n ) } ⊃ { [ e ] E + t , p : e ⊇ E t , p e t , p } for all p , q , and n . Now let j ( t , p , q , n ) be such that φ j ( t , p , q , n ) ( i ) is an index for an enumeration of W i ∪ ran φ h ′ ( t , p , q , n ) and let j ( t , p ) be such that φ j ( t , p ) ( i ) is an index for an enumeration of W i ∪ ran φ e t , p . We then have B t ( n ) ⇐⇒ ∀ p ∃ q j ( t , p , q , n ) E ++ t , p j ( t , p ) , OMPUTABLE REDUCIBILITY OF EQUIVALENCE RELATIONS AND AN EFFECTIVE JUMP OPERATOR 9 with j ( t , p , q , n ) ⊇ E + t , p j ( t , p ) for all p , q , and n , and therefore B t ( n ) ⇐⇒ ∀ p ∃ q { [ e ] E ++ t , p : e ⊇ E + t , p j ( t , p , q , n ) } = { [ e ] E ++ t , p : e ⊇ E + t , p j ( t , p ) } ,with { [ e ] E ++ t , p : e ⊇ E + t , p j ( t , p , q , n ) } ⊂ { [ e ] E ++ t , p : e ⊇ E + t , p j ( t , p ) } for all p , q , and n . We nowclaim that B t ( n ) ⇐⇒ ∀ p { [ e ] E ++ t , p : ∃ q e ⊇ E + t , p j ( t , p , q , n ) } = { [ e ] E ++ t , p : e ⊇ E + t , p j ( t , p ) } ,with { [ e ] E ++ t , p : ∃ q e ⊇ E + t , p j ( t , p , q , n ) } ⊂ { [ e ] E ++ t , p : e ⊇ E + t , p j ( t , p ) } for all p , q , and n . To seethis, note that the left-to-right inclusion is immediate from the previous step. so we justneed to verify that if equality holds then there is q with { [ e ] E ++ t , p : e ⊇ E + t , p j ( t , p , q , n ) } = { [ e ] E ++ t , p : e ⊇ E + t , p j ( t , p ) } . But if equality holds then [ j ( t , p )] E + t , p is an element of the left-hand side, so there is q with j ( t , p ) ⊇ E + t , p j ( t , p , q , n ) . Since j ( t , p ) ⊆ E + t , p j ( t , p , q , n ) for all q ,we then have j ( t , p ) E ++ t , p j ( t , p , q , n ) .Then, letting ˜ ( t , p , n ) be such that φ ˜ ( t , p , n ) ( h i , q i ) is an index for an enumeration of W i ∪ ran φ j ( t , p , q , n ) and ˜ ( t , p ) be such that φ ˜ ( t , p ) ( i ) is an index for an enumeration of W i ∪ ran φ j ( t , p ) we have B t ( n ) ⇐⇒ ∀ p ˜ ( t , p , n ) E +++ t , p ˜ ( t , p ) ,with ˜ ( t , p , n ) ⊆ E ++ t , p ˜ ( t , p ) for all p and n .Now we absorb the universal quantifier. As we did with E t a h p , q i , we can replace E t , p bya higher iterate of the jump, so that the iterates are increasing in O with respect to p , toget an effective sequence a t , p and ˜ E t , p , ˜ j ′ , and ˜ j ′ so that B t ( n ) ⇐⇒ ∀ p ˜ ′ ( t , p , n ) ˜ E + t , p ˜ ′ ( t , p ) ,with ˜ ′ ( t , p , n ) ⊆ ˜ E t , p ˜ ′ ( t , p ) for all p and n . Let i t be such that φ i t ( p ) = a t , p for all p , b t = · i t , and F t = id + b t . We have B t ( n ) ⇐⇒ {h p , e i : p ∈ N ∧ e ⊇ ˜ E t , p ˜ j ′ ( t , p , n ) } = {h p , e i : p ∈ N ∧ e ⊇ ˜ E t , p ˜ j ′ ( t , p ) } ,with {h p , e i : p ∈ N ∧ e ⊇ ˜ E t , p ˜ j ′ ( t , p , n ) } ⊃ {h p , e i : p ∈ N ∧ e ⊇ ˜ E t , p ˜ j ′ ( t , p ) } for all n .Let c ( t , p , n ) be such that φ c ( t , p , n ) ( i ) is an index for an enumeration of W i ∪ ran φ ˜ j ′ ( t , p , n ) and c ( t , p ) be such that φ c ( t , p ) ( i ) is an index for an enumeration of W i ∪ ran φ ˜ j ′ ( t , p ) , and let d ( t , n ) be such that φ d ( t , n ) ( h p , i i ) = h p , φ c ( t , p , n ) ( i ) i and d ( t ) be such that φ d ( t ) ( h p , i i ) = h p , φ c ( t , p ) ( i ) i . Then B t ( n ) ⇐⇒ d ( t , n ) F + t d ( t ) , OMPUTABLE REDUCIBILITY OF EQUIVALENCE RELATIONS AND AN EFFECTIVE JUMP OPERATOR 10 with d ( t , n ) ⊇ F t d ( t ) for all n . Finally, let E t = F + t and let h t be such that φ h t ( n ) ( i ) isan index for an enumeration of W i ∪ ran ( φ d ( t , n ) ) and e t such that φ e t ( i ) is an index for anenumeration of W i ∪ ran φ d ( t ) . Then we have for all n that B t ( n ) ⇐⇒ h t ( n ) E + t e t and h t ( n ) ⊆ E t e t . This completes the induction. (cid:3) We are now ready to conclude the proof of the main theorem of the section.
Proof of Theorem 3.1.
By Proposition 2.2(b) we can assume that E has infinitely many classes.Thus by Proposition 4.3 below we have id ≤ E ++ . Now suppose E + ≤ E , so id ≤ E andhence by Proposition 2.6 id + a ≤ E + a ≤ E for all a ∈ O . But now E ∈ Σ α for some com-putable α , so let P be Π α with P / ∈ Σ α . Since P is hyperarithmetic, there is a ∈ O with P ≤ m id + a which would mean P ≤ m E , a contradiction. (cid:3) The proof of Lemma 3.6 does not give optimal bounds on the number of iterates of thejump required. With a bit more care, we can show that every Π α set is reducible to id + a for some a ∈ O with | a | = α . Similar techniques to the above show: Lemma 3.7.
Suppose E × id ≤ E, Q is Π α and there is e and a computable h so that ∀ n [ Q ( n ) ⇐⇒ h ( n ) E + e ] , with h ( n ) ⊆ E e for all n. Let P ( n ) ⇐⇒ ∀ p ∃ q Q ( h p , q , n i ) be Π α + . Then there is f and acomputable g so that ∀ n [ P ( n ) ⇐⇒ g ( n ) E +++ f ] , with g ( n ) ⊆ E ++ f for all n. An analogous result holds for limit stages. We believe that the optimal bound should bethat every Π · α set is reducible to id + a for some a ∈ O with | a | = α . We can show by an adhoc argument that (= ce ) ++ is Π -complete, for instance. Our induction technique requirestwo iterates of the jump at each step in order to reverse the direction of set containmenttwice. We would prefer to use ⊆ E rather than ⊇ E throughout, but we do not see how toeffectively enumerate c.e. subsets of a given c.e. set up to E + -equivalence, whereas we canenumerate c.e. supersets. The natural attempt to do this fails as shown in the followingexample. Proposition 3.8.
There are E and e so that { [ e ] E + : e ⊆ E e } 6 = { [ W i ∩ e ] E + : i ∈ N } , whereW i ∩ e is an index for an enumeration of ran φ e ∩ W i . OMPUTABLE REDUCIBILITY OF EQUIVALENCE RELATIONS AND AN EFFECTIVE JUMP OPERATOR 11
Proof.
Let E be = ce , and let A ⊂ B be c.e. sets with B − A not c.e. Let e be such thatran φ φ e ( j ) = { k } if j = k + { k , k + } if j = k + ∅ if j = e be such that ran φ φ e ( k ) = { k } if k ∈ B − A { k , k + } if k ∈ A ∅ if k / ∈ B .Then e ⊆ E e but there is no i with e E + e ∩ W i . For if there were, we would have k ∈ B − A iff ∃ x ( x ∈ W i ∧ x = φ e ( + k )) so that B − A would be c.e. (cid:3) If we consider non-hyperarithmetic equivalence relations we can find fixed points ofthe jump.
Definition 3.9.
Let ∼ = T be the isomorphism relation on computable trees.Here we can use any reasonable coding of computable trees by natural numbers. Then ∼ = T is a Σ equivalence relation which is not hyperarithmetic. Proposition 3.10. ∼ = T is a jump fixed point, i.e., ∼ = + T is computably bireducible with ∼ = T .Proof. In [FFH +
12, Theorem 2] it was shown that ∼ = T is Σ complete for computable re-ducibility, that is, ∼ = T is Σ and for every Σ equivalence relation E , E ≤∼ = T . So it sufficesto show that ∼ = + T is Σ , but this follows immediately from the fact that E + is Σ for any Σ equivalence relation E , since E + is a conjunction of E with additional natural numberquantifiers. (cid:3) We note that although every hyperarithmetic set is many-one reducible to id + a for some a ∈ O , we do not know whether every hyperarithmetic equivalence relation E satisfies E ≤ id + a for some a ∈ O . §
4. C
EERS AND THE JUMP
Recall from the introduction that E is called a ceer if it is a computably enumerableequivalence relation, and that the ceers have been studied with respect to computable re-ducibility since [GG01]. In this section, we study the relationship between the computableFS-jump and the ceers. OMPUTABLE REDUCIBILITY OF EQUIVALENCE RELATIONS AND AN EFFECTIVE JUMP OPERATOR 12
We begin with the following upper bound on the complexity of the computable FS-jump of a ceer. In the statement, recall that if E is an equivalence relation and W ⊂ N ,then W is said to be E-invariant if it is a union of E -equivalence classes. Proposition 4.1.
If E is a ceer, then E + ≤ = ce . In fact, E + is computably bireducible with therestriction of = ce to the set { e ∈ N | W e is E-invariant } .Proof. For the first statement, we define a computable function f such that W f ( e ) = [ ran φ e ] E .To see that there is such a computable function f , one can let f ( e ) be a program which, oninput n , searches through all triples ( a , b , c ) such that a ∈ ran φ e and ( b , c ) ∈ E , and haltsif and when it finds a triple of the form ( a , a , n ) . Since it is clear that e E + e ′ if and only if [ ran φ e ] E = [ ran φ e ′ ] E , we have that f is a computable reduction from E + to = ce .For the second statement, first observe that the range of the reduction function f de-fined above is contained in { e ∈ N | W e is E -invariant } . For the reduction in the reversedirection, let g be a computable function such that ran φ g ( e ) = W e . Then if W e , W e ′ are E -invariant sets, we clearly have that W e = W e ′ if and only if φ g ( e ) , φ g ( e ′ ) enumerate thesame set of E -classes, as desired. (cid:3) The next result gives a lower bound on the complexity of the computable FS-jumps ofa ceer.
Theorem 4.2.
If E is a ceer with infinitely many equivalence classes, then id < E + .Proof. We first show that id ≤ E + . To do so, we first define an auxilliary set of pairs A recursively as follows: Let ( n , j ) ∈ A if and only if for every i < j there exists m < n and ( m , i ′ ) ∈ A such that i E i ′ . It is immediate from the definition of A , the fact that E is c.e.,and the recursion theorem that A is a c.e. set of pairs.We observe that each column A ( n ) of A is an initial interval of N . It is immediate fromthe definition that the first column A ( ) is the singleton { } . Next since E has infinitelymany classes, we have that each A ( n ) is bounded. Moreover A ( n ) is precisely the interval [ j ] where j is the least value that is E -inequivalent to every element of A ( m ) for all m < n .We now define f to be any computable function such that for all n , the range of φ f ( n ) isprecisely A ( n ) . Then as we have seen, m < n implies there exists an element j in the rangeof φ f ( n ) such that j is E -inequivalent to everything in the range of φ f ( m ) . In particular, f isa computable reduction from id to E + .To establish strictness, assume to the contrary that E + ≤ id . It follows that E + iscomputable, and hence E is computable too. It further follows that id ≤ E , and hence id + ≤ E + . Finally by Theorem 3.1 we conclude that id < E + , which is a contradiction(also, the desired conclusion). (cid:3) OMPUTABLE REDUCIBILITY OF EQUIVALENCE RELATIONS AND AN EFFECTIVE JUMP OPERATOR 13
In order to put the previous result in context, we pause our investigation of ceers brieflyto consider the question of which E satisfy id ≤ E + . We first record the fact that if E isitself a jump, then id ≤ E + . Proposition 4.3.
For any E with infinitely many classes we have id ≤ E ++ .Proof. We define a reduction function f that works simultaneously for all equivalencerelations E with infinitely many classes. Given n , let f ( n ) be a code for a machine suchthat the sequence of sets S i = φ f ( n ) ( i ) consists of all n -element sets. Clearly since E ++ is reflexive we have that n = n ′ implies f ( n ) E ++ f ( n ′ ) . Conversely suppose n = n ′ ,and assume without loss of generality that n < n ′ . Then for all i ∈ N we have that [ φ f ( n ) ( i )] E + is a code for at most n -many E -classes. On the other hand since E has infinitelymany classes, there exists i ∈ N such that [ φ f ( n ) ( i )] E + is a code for exactly n ′ -many E -classes. It follows that { [ φ f ( n ) ( i )] E + : i ∈ N } 6 = { [ φ f ( n ′ ) ( i )] E + : i ∈ N } , or in other words, f ( n ) E ++ f ( n ′ ) . (cid:3) Next, we show that there exist equivalence relations E such that id E + . To describesuch an equivalence relation, we recall the following notation. If A ⊂ N then the equiva-lence relation E A is defined by m E A n ⇐⇒ m = n or m , n ∈ A .Thus the equivalence classes of E A are A itself, together with the singletons { i } for i / ∈ A . Theorem 4.4.
There exists an arithmetic coinfinite set A such that id E + A .Proof. Let P be the Mathias forcing poset, that is, P consists of pairs ( s , B ) where s ⊂ N is finite, B ⊂ N is infinite, and every element of s is less than every element of B . Theordering on P is defined by ( s , B ) ≤ ( t , C ) if s ⊃ t , B ⊂ C , and s − t ⊂ C .We first show that if A c is sufficiently Mathias generic, then A satisfies id E + A . Inorder to do so, for any function f we let D f = { ( s , B ) ∈ P : ( ∃ i = j ) ( s ∪ B ) ∩ ( ran φ f ( i ) △ ran φ f ( j ) ) = ∅ } .We claim that D f is dense in P . To see this, let ( s , B ) be given. Repeatedly applying thepigeonhole principle, we can find infinitely many indices i n such that the sets ran φ f ( i n ) agree on s . In fact we only need to use i , i , i . Observe that ( ran φ f ( i ) △ ran φ f ( i ) ) c ∪ ( ran φ f ( i ) △ ran φ f ( i ) ) c ∪ ( ran φ f ( i ) △ ran φ f ( i ) ) c = N . OMPUTABLE REDUCIBILITY OF EQUIVALENCE RELATIONS AND AN EFFECTIVE JUMP OPERATOR 14
In particular we can suppose without loss of generality that B ′ = B ∩ ( ran φ f ( i ) △ ran φ f ( i ) ) c is infinite. Then ran φ f ( i ) and ran φ f ( i ) agree on both s and B ′ , and so ( s , B ′ ) ∈ D f , com-pleting the claim.Now let G ⊂ P be a filter satisfying the following conditions:(a) G meets { ( s , B ) ∈ P : | s | ≥ m } for all m ∈ N ,(b) G meets { ( s , B ) ∈ P : W e infinite → ( s ∪ B ) c ∩ W e = ∅ } for all e ∈ N , and(c) G meets D f for all computable functions f .This is possible since the sets in conditions (a) and (b) are clearly dense, and we haveshown that the D f are dense. We define the set A by declaring that A c = S { s : ( s , B ) ∈ G } .Clearly condition (a) implies that A c is infinite; we wish to show that id ( E A ) + . For thiswe will show that if f is a given computable function, then f is not a reduction from id to E + A .Suppose first that S = { i : A ∩ ran φ f ( i ) = ∅ } is infinite. We can suppose without lossof generality that the sets ran φ f ( i ) are pairwise distinct. Thus S i ∈ S ran φ f ( i ) is an infinitec.e. subset of A c . This contradicts condition (b).Suppose instead that S = { i : A ∩ ran φ f ( i ) = ∅ } is finite. Then S c = { i : A ∩ ran φ f ( i ) = ∅ } is cofinite and therefore c.e. Thus we can precompose f with a computable enumer-ation of S c to suppose without loss of generality that A ∩ ran φ f ( i ) = ∅ for all i . Next itfollows from condition (c) that there exist i = j such that A c ∩ ( ran φ f ( i ) △ ran φ f ( j ) ) = ∅ .This means that f ( i ) E + f ( j ) , so f is not a reduction from id to E + A , as desired.Finally, we can ensure A is arithmetic by enumerating the dense sets described above,inductively defining a descending sequence ( s n , B n ) meeting the dense sets, and letting A c = S s n . (cid:3) This result leaves open the question of what is the least complexity of an equivalencerelation E with infinitely many classes such that id E + . The proof above can be used toshow there is such an E which is Σ .Returning to ceers, in view of the bounds from Proposition 4.1 and Theorem 4.2, it isnatural to ask whether there is a ceer E such that E + lies properly between id and = ce . Inorder to state our results on this question, we introduce the following terminology. Definition 4.5.
A ceer E is siad to be high for the computable FS-jump if E + is computablybireducible with = ce .The term “high” is intended to be analogous with other uses in complexity theory,though we do not have an analog for the term “low”. OMPUTABLE REDUCIBILITY OF EQUIVALENCE RELATIONS AND AN EFFECTIVE JUMP OPERATOR 15
There is a large collection of ceers which are high for the computable FS-jump. To statethe result, we recall the following terminology from [AS18a]. A ceer E is said to be light ifit has finitely many equivalence classes or id ≤ E , and otherwise E is said to be dark . Proposition 4.6.
If E is a light ceer with infinitely many classes then E is high for the computableFS-jump.Proof.
This is an immediate consequence of Propositions 2.2(c), 2.4, and 4.1. (cid:3)
There also exist dark ceers E which are high for the computable FS-jump. In order todescribe such a ceer, recall that a c.e. set A ⊂ N is called simple if there is no infinite c.e. setcontained in A c . Furthermore A is called hyperhypersimple if for all computable functions f such that { W f ( n ) : n ∈ N } is a pairwise disjoint family of finite sets, there exists n ∈ N such that W f ( n ) ⊂ A . We refer the reader to [Soa16, Chapter 5] for more about theseproperties, including examples. Theorem 4.7.
Let A ⊂ N be a set which is simple and not hyperhypersimple. Then E A is a darkceer and E A is high for the computable FS-jump.Proof. It follows from [GG01, Proposition 4.5] together with the assumption that A is sim-ple that E A is dark.To see that E + A is computably bireducible with = ce , first it follows from Proposition 4.1that E + A ≤ = ce . For the reduction in the reverse direction, since A is not hyperhypersimple,there exists a computable function f such that { W f ( n ) : n ∈ N } is a pairwise disjointfamily of finite sets and for all n ∈ N we have W f ( n ) ∩ A c = ∅ . Now given an index e we compute an index g ( e ) such that φ g ( e ) is an enumeration of the set S { W f ( n ) : n ∈ W e } .Then since the W f ( n ) are pairwise disjoint and meet A c , we have W e = W e ′ if and only if A c ∩ ran φ g ( e ) and A c ∩ ran φ g ( e ′ ) are distinct subsets of A c . It follows that e = ce e ′ if andonly if g ( e ) E + A g ( e ′ ) , as desired. (cid:3) On the other hand, there also exist dark ceers E such that E is not high for the com-putable FS-jump. In order to state the results, we recall from [Soa87, Chapter X] that a c.e.subset A ⊂ N is said to be maximal if A c is infinite and for all c.e. sets W either W − A or W c − A is finite. We further note that if A is maximal then it is hyperhypersimple. Theorem 4.8.
Let A be a maximal set. If B ( A, then E + A < E + B . In particular, E A is not highfor the computable FS-jump. The proof begins with several preliminary results, which may be of independent value.
Lemma 4.9.
If A , B are c.e. sets and B ⊂ A, then E + A ≤ E + B . OMPUTABLE REDUCIBILITY OF EQUIVALENCE RELATIONS AND AN EFFECTIVE JUMP OPERATOR 16
Proof. If B is non-hyperhypersimple, then the result follows immediately from Proposi-tion 4.1 and Theorem 4.7. If B is hyperhypersimple, then by [Soa87, X.2.12] there exists acomputable set C such that B ∪ C = A . Let b ∈ B be arbitrary, and define f ( n ) = b n ∈ Cn n / ∈ C It is easy to see that f is a computable reduction from E A to E B , and hence by Proposi-tion 2.2(c) we have E + A ≤ E + B as desired. (cid:3) Corollary 4.10.
Let A be a maximal set. If E + ≤ E + A , then any E-invariant c.e. set contains eitherfinitely or cofinitely mane E-classes. In particular, if E + B ≤ E + A then B is maximal. In the next lemma we will use the following terminology about a function f : N → N .We say that f is = ce -invariant if W e = W e ′ implies W f ( e ) = W f ( e ′ ) , that f is monotone if W e ′ ⊂ W e implies W f ( e ′ ) ⊂ W f ( e ) , and that f is inner-regular if(1) W f ( e ) = [ n W f ( e ′ ) | W e ′ ⊂ W e and W e ′ is finite o .We are now ready to state the lemma. Lemma 4.11.
If f is a computable function, the properties = ce -invariant, monotone, and inner-regular are all equivalent.Proof. It is clear that inner-regular implies monotone, and monotone implies = ce -invariant.We therefore need only show that = ce -invariant implies inner-regular. Assume that f is = ce -invariant. The superset inclusion of Equation (1) is precisely [CHM12, Lemma 4.5].For the subset inclusion of Equation (1), we assume that x ∈ W f ( e ) and aim to showthat there exists e ′ such that W e ′ ⊂ W e , W e ′ is finite, and x ∈ W f ( e ′ ) . For this we invoke therecursion theorem to find an index e ′ for the following Turing program.On input n , program e ′ simultaneously does the following: simulate e oninput n repeatedly (each time e halts on n start the simulation over again),and simulate f ( e ′ ) on input x (we are using the recursion theorem here). Ifat any time e halts on n , and n ≤ the current time in steps, and f ( e ′ ) hasnot yet halted on x , then e ′ halts. Otherwise e ′ continues to run.We must show that W e ′ ⊂ W e , W e ′ is finite, and x ∈ W f ( e ′ ) . It is clear that W e ′ ⊂ W e . Toshow that x ∈ W f ( e ′ ) , assume to the contrary that x / ∈ W f ( e ′ ) . Then e ′ would halt on input n if and only if e halts on input n , that is, we would have W e ′ = W e . Since f is = ce -invariant, OMPUTABLE REDUCIBILITY OF EQUIVALENCE RELATIONS AND AN EFFECTIVE JUMP OPERATOR 17 we would have W f ( e ′ ) = W f ( e ) . Our assumption that x ∈ W f ( e ) would therefore imply that x ∈ W f ( e ′ ) after all.Now that we know x ∈ W f ( e ′ ) , we know that f ( e ′ ) does halt on input x . Thus there isan upper bound on the set of inputs n on which e ′ will halt. This means that W e ′ is finite,as desired. (cid:3) We now give the main ingredient to the proof of Theorem 4.8.
Lemma 4.12.
If A is maximal then E + A is self-full , i.e., if f is a computable reduction from E + A toE + A , then the range of f meets every E + A class.Proof. We can assume without loss of generality that for all e , ran φ f ( e ) is E A -invariant.Indeed, we may modify f to ensure that if f ( e ) enumerates any element of A then f ( e ) enumerates the rest of A too. Having done so, we introduce the following mild abuse ofnotation: if R = ran φ e then we will write f ( R ) for ran φ f ( e ) . Due to our assumption about f , this notation is well-defined.Our proof strategy will be to show that there exists a finitely supported permutation π of N such that for any c.e. set R , we have f ( R ) = { π ( n ) : n ∈ R } . This implies that f meets every E + A class, as desired.To begin, observe that S n f ( { n } ) is an infinite c.e. set (here we tacitly select indices for { n } uniformly), and hence intersects A , and thus contains A . Then { n : f ( { n } ) ∩ A = ∅ } is an infinite c.e. set, and so intersects A . Thus there is n ∈ A with f ( { n } ) ∩ A = ∅ , so A ⊂ f ( A ) . Next, since the sets f ( { n } ) are distinct for n / ∈ A , we have ( S n f ( { n } )) − A infinite,so the maximality of A implies S n f ( { n } ) is cofinite. Hence by the inner regularity of f we can find a finite set C such that f ( A ) ∪ ( f ( N ) − S n f ( { n } ) ⊂ f ( C ) .We now aim to construct a first approximation σ to the desired permutation π . Specif-ically, σ will be a finite support permutation such that for n / ∈ A ∪ C we have σ ( n ) ∈ f ( { n } ) − A . To start we claim that for n / ∈ A ∪ C , we have f ( { n } ) − ( f ( C ) ∪ S m = n f ( { m } )) = ∅ . For this let n / ∈ A ∪ C . Since f is a reduction and is monotone, we can find x ∈ f ( N ) − f ( N − { n } ) . Using the definition of C , the fact that n / ∈ C , and monotonicity, wehave f ( N ) − S m f ( { m } ) ⊂ f ( C ) ⊂ f ( N − { n } ) . In particular, x / ∈ f ( C ) and furthermore x ∈ S m f ( { m } ) . Again by monotonicity, x / ∈ S m = n f ( { m } ) , so we must have x ∈ f ( { n } ) ,completing the claim.We now use effective reduction to obtain a uniformly c.e. sequence B n of pairwise dis-joint sets such that B n ⊂ f ( { n } ) and for n / ∈ A ∪ C we have B n − ( f ( C ) ∪ S m = n f ( { m } )) = ∅ . We may further reduce B n to suppose that B n − f ( C ) is a singleton for all n / ∈ A ∪ C , OMPUTABLE REDUCIBILITY OF EQUIVALENCE RELATIONS AND AN EFFECTIVE JUMP OPERATOR 18 and B n is disjoint from the finite set f ( C ) − A for all n . Next observe that since C − A is fi-nite, we have f (( C − A ) c ) cofinite, so the monotonicity of f implies that | f (( C − A ) c ) c | ≥| C − A | , so we may let h be any injection C − A → f (( C − A ) c ) c . We now define: G n = A n ∈ AA ∪ { h ( n ) } n ∈ C − AA ∪ B n n / ∈ A ∪ C .Observe that G n is a uniformly c.e. sequence, since we may first check if n ∈ C − A ; ifnot, we enumerate B n into G n until we see n enumerated in A (if ever), at which point weenumerate A (which will then contain B n ) into G n . We define σ as follows: σ ( n ) = n n ∈ A the unique element of G n − A n / ∈ A This completes the definition of σ .We claim that σ is a permutation with finite support. It is immediate from the construc-ton that σ is injective. To show σ is surjective, assume k is not in the range of σ . Definethe function g ( R ) = S n ∈ R G n (it is computable in the indices) and then the sequence R = A ∪ { k } and R n + = g ( R n ) . Since k / ∈ A we have that R n − A is a singleton forall n . Moreover the singletons are distinct since σ is injective and none of the singletonscan equal k for n >
0. Applying reduction to the sequence R n , we obtain a uniformly c.e.sequence of nonempty pairwise disjoint sets, all meeting A c . This contradicts that A ishyperhypersimple (see [Soa87, Exercise X.2.16]). To see that σ has finite support, first notethat σ cannot have an infinite orbit. Otherwise, we could similarly produce a sequencewhich contradicts that A is hyperhypersimple. If σ had infinitely many nontrivial orbits,let R = { n : ( ∃ k ≥ n ) n ∈ G k } . Then A ⊂ R , and for n / ∈ A we have n ∈ R when n is theleast element of its orbit and n / ∈ R when n is the greatest element of a nontrivial orbit.Thus R − A is infinite and co-infinite, again contradicting A is maximal.We are now ready to construct π as follows. Let ˜ C = ( C − A ) ∪ supp ( σ ) . If R is disjointfrom C − A then g ( R ) ⊂ f ( R ) , therefore if R is disjoint from ˜ C we have R ⊂ f ( R ) . Bymonotonicity of f , if R is disjoint from ˜ C and cofinite, then R = f ( R ) . Thus for any k ∈ ˜ C ,we must have that f sends ˜ C c ∪ { k } to some ˜ C c ∪ { π ( k ) } , where π ( k ) ∈ ˜ C . Additionallydefine π to be the identity on ˜ C c . This completes the definition of π .Finally, let h be a computable function such that h ( R ) = { π ( n ) : n ∈ R } ; we wish toshow that for all c.e. R we have f ( R ) = h ( R ) . By monotonicity and the definition of π thisholds for all cofinite R . Let R be an arbitrary c.e. set; we may assume that R is E A -invariant. OMPUTABLE REDUCIBILITY OF EQUIVALENCE RELATIONS AND AN EFFECTIVE JUMP OPERATOR 19
Then R is the intersection of its cofinite supersets, so f ( R ) ⊂ h ( R ) . Suppose there were k ∈ h ( R ) − f ( R ) . Then k = π ( n ) for some n ∈ R , so k / ∈ f ( { n } ) ⊂ h ( { n } ) = { k } and thus f ( { k } ) = ∅ , a contradiction. (cid:3) In the following, we use the notation ∆ ( n ) for the identity equivalence relation on {
0, . . . , n − } . Proof of Theorem 4.8.
Let A be maximal and B ( A . Fix any a ∈ A − B . We observe that E A −{ a } is computably bireducible with E A ⊕ ∆ ( ) . Therefore by Proposition 2.2(c) wehave that E + A −{ a } is computably bireducible with E + A × ∆ ( ) .Now assume towards a contradiction that E + B ≤ E + A . Then by Lemma 4.9 we have E + A −{ a } ≤ E + A and hence by the previous paragraph we have E + A × ∆ ( ) ≤ E + A . But if f issuch a reduction, then by Lemma 4.12 the restriction of f to either copy of E + A has rangemeeting every E + A class. But for a reduction f we cannot have this property true of bothcopies of E + A , so we have reached a contradiction. (cid:3) We record here several immediate consequences of Theorem 4.8 and its proof.
Corollary 4.13.
Let A , B be a maximal sets. ◦ If f is a computable reduction from E + A to E + A , then f corresponds to a finite-support per-mutation of E A -classes. ◦ If a ∈ A then E + A < E + A −{ a } , and if b / ∈ A then E + A ∪{ b } < E + A . ◦ E + A is part of a chain of ceers of ordertype Z with respect to computable reducibility. ◦ If | A △ B | < ∞ , then E + A ≤ E + B iff | B − A | ≤ | A − B | . ◦ If C is contained in a maximal set, then it is contained in a maximal set D such thatE + D < E + C . We conclude with a small refinement of the second statement of Theorem 4.8. Recallthat a c.e. set A is said to be quasi-maximal if it is the intersection of finitely many maximalsets. We refer the reader to [Soa87, X.3.10] for more on this notion. Theorem 4.14.
If A ⊂ N is quasi-maximal then E A is not high for the computable FS-jump.Proof. Suppose towards a contradiction that = ce ≤ E + A . First, by Proposition 4.1 thereexists a computable reduction f from = ce to = ce such that for all e , W f ( e ) is E A -invariant.Since A is simple, it follows that for all e we have A ⊂ W f ( e ) iff W f ( e ) is infinite and A ∩ W f ( e ) = ∅ iff W f ( e ) is finite.We claim that we may assume without loss of generality that for all e , we have A ⊂ W f ( e ) . To see this, we first show that there exists e such that W e is finite and A ⊂ W f ( e ) . OMPUTABLE REDUCIBILITY OF EQUIVALENCE RELATIONS AND AN EFFECTIVE JUMP OPERATOR 20
Let e be any index such that W e = N . By Lemma 4.11, f is inner-regular. Since f is areduction, it follows from Equation 1 that W f ( e ) is infinite and hence A ⊂ W f ( e ) . Furtherexamining Equation 1, together with the last sentence of the previous paragraph, we con-clude there exists e as desired. To complete the claim, let g be a computable function suchthat W g ( e ) = W e ∪ { max ( W e ) + x : x ∈ W e } . Then replacing f with g ◦ f we achieve theassumption of the claim.It follows from the claim, together with the fact that f is monotone, that the lattice ofc.e. sets modulo finite may be embedded into the lattice of c.e. sets containing A modulofinite. But the former lattice is infinite, and by [Soa87, X.3.10(a)] the latter lattice is finite,a contradiction. (cid:3) §
5. A
DDITIONAL REMARKS AND OPEN QUESTIONS
We close with some open questions and directions for further investigation.
Question . For a c.e. set A , when is E + A bireducible with = ce ?By Theorem 4.7 if A is not hyperhypersimple then E + A is high for the jump, and by The-orem 4.14 if A is quasi-maximal then E + A < = ce . The question is, if A is hyperhypersimplebut not quasi-maximal, is E + A high? One construction of such a set is given in an exercisein [Odi99, IX.2.28f].We do not know whether the choice of notation for a countable ordinal affects the iter-ated jump. Question . If a , b ∈ O with | a | = | b | , is E + a computably bireducible with E + b ?Although we saw that every hyperarithmetic set is many-one reducible to some jump ofthe identity, we do not know if every hyperarithmetic equivalence relation is computablyreducible to some iterated jump of the identity. Question . If E is hyperarithmetic, is there a ∈ O with E ≤ id + a ?For E hyperarithmetic, we have e E e ′ iff [ e ] E = [ e ′ ] E , so that E is computably reducibleto the relativized version of = ce , denoted = ce , E , considered in [Bar19]. This question isthen equivalent to asking if these relativized equivalence relations with hyperarithmeticoracles are computably reducible to iterated jumps of the unrelativized = ce .We also note that, unlike the case of the classical Friedman–Stanley jump, the equiva-lence relation E is not an obstruction. Proposition 5.1. E ce ≤ (= ce ) + . OMPUTABLE REDUCIBILITY OF EQUIVALENCE RELATIONS AND AN EFFECTIVE JUMP OPERATOR 21
Proof.
Given e , let g ( e ) be such that φ g ( e ) ( h f , m i ) is an index for an enumeration of the set S n < m ( W f ) ( n ) ∪ S n ≥ m ( W e ) ( n ) . Then e E ce e ′ if and only if g ( e ) (= ce ) + g ( e ′ ) . (cid:3) We can also ask what other fixed points exist besides ∼ = T . We note that there is nocharacterization of fixed points of the classical Friedman–Stanley jump. Question . Characterize the fixed points of the computable FS-jump.R
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