The hierarchy of equivalence relations on N under …jdh.hamkins.org/wp-content/uploads/2012/03/Comp-equiv...Borel equivalence relations Many naturally arising equivalence relations,

Introduction, Equivalences of c.e. sets Below equality, =ce Classifications of c.e. structures Computability relations

The hierarchy of equivalence relations on Nunder computable reducibility

Joel David Hamkins

The City University of New YorkCollege of Staten Island of CUNY and

The CUNY Graduate Center

CUNY MAMLS, March 2012

Equivalence relations on N, CUNY MAMLS 2012 Joel David Hamkins, New York


This talk includes joint work with Sam Coskey and RussellMiller. Our paper is

S. Coskey, J. D. Hamkins, R. Miller, “The hierarchy ofequivalence relations on the natural numbers undercomputable reducibility,” forthcoming.

A preprint is available on my web page: http://jdh.hamkins.org



Other researchers, related workSome of the ideas have arisen independently several times.

Claudio Bernadi and Andrea Sorbi, “Classifying positiveequivalence relations,” JSL 48(3):529–538, 1983.Su Gao and Peter Gerdes, “Computably enumerableequivalence relations,” Studia Logica, 67(1):27–59, 2001.Julia F. Knight, Sara Miller, M. Vanden Boom, “Turingcomputable embeddings,” JSL 72(3):901–918, 2007.Sam Buss, Yijià Chen, Jörg Flum, Sy-David Friedman andMoritz Müller, “Strong isomorphism reductions incomplexity theory,” JSL, 2011.Akaterina B. Fokina, Sy-David Friedman, “On Sigma-1-1equivalence relations over the natural numbers,” MLQ, toappear.



Borel equivalence relationsMany naturally arising equivalence relations, such asisomorphisms of various countable structures (groups, graphs,fields, linear orders, isometries of separable Banach spaces),can be viewed as equivalence relations, often Borel relations,on a standard Borel space.

The classification problems for those structures is the problemof assigning invariants to the equivalence classes.

The concept of Borel reducibility, due to H. Friedman and L.Stanley, allows us to compare the relative difficulty ofclassification problems, to allow us to say that a classificationproblem is wild or tame.

The subject has been enormously successful, a very fruitfulinteraction of logic with the rest of the mathematics.



Borel reducibilityOne equivalence relation is Borel reducible to another, E ≤B F ,if there is a Borel function f such that

a E b ⇐⇒ f (a) F f (b).

Thus, f maps E-classes to F -classes and thereby provides aclassification of E using F -classes.

In this case, the E-classification problem is no harder than theF -classification problem.

Goal of the Borel theory: to study the hierarchy of suchclassification problems under Borel reducibility (andmathematicians care).



Computable reducibilityWe aim to adapt the Borel theory to the computability context,with equivalence relations on N.

Namely, E is computably reducible to F , written E ≤c F , whenthere is a computable function f : N→ N such that

n E m ⇐⇒ f (n) F f (m).

Versions of this notion have arisen several times independently:Bernadi, Sorbi; Gao, Gerdes; Fokina, Friedman; Coskey,Hamkins, Miller.

Our new focus: natural relations on c.e. structures. Theresulting theory is an attractive blend of methods and ideasfrom computability theory, descriptive set theory and otherparts of mathematics.



Reducibility is not relative computabilityAlthough E ≤c F implies E ≤T F via

n E m ⇐⇒ f (n) F f (m),

nevertheless the Turing degree of E as a set of pairs does notdetermine it’s place in the hierarchy of reducibility.

The Borel theory has many examples of low-complexityequivalence relations with wild classification problems; and thesame for computable reducibility.

Much of computable model theory is about thecomparative complexity of individual structures.Computable reducibility is about the comparative difficultyof classification problems.



Computable relations are at the bottom

Theorem

The computable equivalence relations on N are exactly thosecomputably bireducible with =N or with =n.

=n is equality on {0,1, . . . ,n − 1}, with rest of N in [n − 1].

Proof.

If E has ≥ n classes, then =n reduces to E . And if E iscomputable (or even just c.e. or co-c.e.) with n classes, thenconversely. If E is Π01 with infinitely many classes, then =Nreduces to E . And computable E reduces to =N.

Thus, the computable equivalence relations are classified bytheir number of equivalence classes.



A copy of the many-one degreesFor A ⊆ N, consider the relation EA,Ac with two classes:

A Ac

If A is c.e., this has complexity ∆02.

Theorem

The relation EA,Ac reduces to EB,Bc if and only if A is many-onereducible to B or to Bc .

Thus, the hierarchy contains essentially a copy of the many-onedegrees, even among the relations with only two classes.



A copy of the 1-degreesFor A ⊆ N, consider the relation EA with classes:

A Ac

Thus, n EA m iff n,m ∈ A or n = m. If A is c.e., then so is EA.

Theorem

If A,B ⊆ N are c.e. non-computable, then EA ≤c EB iff A ≤1 B.

Proof.

A 1-reduction of A to B also reduces EA to EB. Conversely, areduction of EA to EB gives rise to a 1-reduction A ≤1 B.



c.e. relations incomparable with =N

Corollary

There exist c.e. relations incomparable with =N.

Proof.

Consider the relation EA where A is a simple set (c.e. co-infiniteset whose complement contains no infinite c.e. sets). Since Ais not computable, neither is EA, and it follows that EA is notreducible to =N. On the other hand, if f is a computablereduction from =N to EA then there exists n ∈ N such thatf (Nr {n}) ⊆ Ac . But this is a contradiction, since f (Nr {n}) isc.e. and infinite.



Universal c.e. relationThere is a universal c.e. equivalence relation.

Theorem

There is a c.e. relation Uce to which every c.e. equivalencerelation is reducible.

Proof.

Let Ee be the eth c.e. equivalence relation (take symmetric,transitive reflexive closure of We). Define (e,a) Uce (e′,a′) iffe = e′ and a Ee a′. This is c.e., and Ee reduces viaa 7→ (e,a).



Orbit equivalence relationsThe orbit equivalence relation of the action of a group Γ on N is

x EΓ y iff ∃γ ∈ Γ y = γx .

Theorem

The c.e. equivalence relations are precisely the orbitequivalence relations induced by computable actions ofcomputable groups.

Proof.

Every such orbit relation is c.e. Conversely, if E is c.e., let Γ bea computable copy of the free group Fω with generators xi . If(n,m) ∈ E at stage s, let x〈s,n,m〉 act by swapping n and m only.This is a computable action with orbit E .



Summary of the hierarchy so far

c.e.comp

=1

=2

=N···

Uce

=ce

EAEA,Ac



Equivalences of c.e. sets

Equivalence relations on c.e. setsA deeper theory arises in the context of equivalence relationson c.e. sets, such as isomorphism relations on c.e. graphs andother c.e. structures. We begin by adapting the key relations ofthe Borel theory by considering them as relations on c.e. sets.

An equivalence relation E on the c.e. sets gives rise to

e Ece e′ ⇐⇒ We E We′ ,

For example, the equality relation on c.e. sets,

e =ce e′ ⇐⇒ We = We′ .

This is a Π02-complete set of pairs. Indeed, it has a Π02-complete

class [e], where We = N.




Equality on c.e. sets, =ce

Theorem

Every c.e. equivalence relation lies properly below =ce.

Proof.

Let E be an arbitrary c.e. relation. Define a reduction from E to=ce by mapping each n to a program f (n) which enumerates[n]E .

Conversely, there can be no reduction from =ce to E since =ce

is Π02-complete and E is c.e.




Almost equality, Ece0The almost-equality relation E0 similarly plays an essential rolein the Borel theory.

A E0 B ⇐⇒ symmetric difference A 4 B is finite

Theorem

Equality =ce lies strictly below almost-equality Ece0 .

Proof.

To reduce =ce to Ece0 , amplify differences: given e, let Wf (e)have (n,0), (n,1), (n,2), . . . whenever n ∈We. Thus,We 6= We′ =⇒ Wf (e)4Wf (e′) is infinite. Conversely, there is nocomputable reduction from Ece0 to =

ce, since Ece0 isΣ03-complete, having a class equivalent to COF, whilst =

ce isjust Π02.




Analogue of Silver, Glimm-Effros dichotomy?

To what extent will the computable reducibility hierarchy mirrorthe structure of Borel reducibility?

For example, one might hope for an analogues of Silver’stheorem, that =ce is minimal in some large class. Unfortunately,this fails, as we shall see.

Are there any relations lying properly between =ce and Ece0 ?More generally, is there a form of the Glimm-Effros dichotomyin this context? In other words, is there a large collection ofequivalence relations E such that if =ce lies strictly below E ,then Ece0 lies below E?




More c.e. analogues of Borel relations

We may code subsets of N× N with subsets of N.

(An) E1 (Bn) iff for almost all n, An = Bn.A E2 B iff

∑n∈A4B 1/n



The complete diagram of Borel reducibilities on left.

=

E0

E1 E2 E3

Eset Z0

=ce

Ece1 ∼ Ece0

Ece2 Ece3

Eceset Zce0

The c.e. analogues on right. (Not sure if this is complete.)




Theorem

The positive reductions of the previous diagram hold.

Proof.

Use classical Borel reductions, which are computable here.E0 reduces to E3 by the map A 7→ N× A.E0 reduces to E1 via A 7→ { 〈x , y〉 | y ≥ x & y ∈ A }, so thatthe column x equals A− {0, . . . , x − 1}.E0 ≤ E2 via A 7→

⊔n∈A In, where

∑i∈In 1/i ≥ 1.

E0 ≤c Z0 via f (A) =⋃

n∈A[2n,2n+1).

E3 ≤ Z0 via (An) 7→⋃πn(f (An)), where In has density 1/2n

and πn : In ∼= N.To reduce E3 ≤c Eset, for each s ∈ 2



E3 does not reduce

Theorem

Ece3 is not computably reducible to any of Ece0 , E

ce1 or E

ce2 .

Proof.

A direct arithmetic complexity argument.

First, Ece0 , Ece1 , and E

ce2 are all easily Σ

03.

Meanwhile, Ece3 is not Σ03, and indeed, has a Π

03-complete

class, the set of indices e for c.e. subsets of N× N with everycolumn finite.




Ece1 bireducible with Ece0

We were very surprised by the following:

Theorem

Ece1 ≤ Ece0 . Hence, E

ce1 and E

ce0 are computably bireducible.

(See our paper for the proof.)

This is a deviation from the situation in the Borel theory, whereE0


Complexity below equality =ce

In contrast to the Borel theory, in the computable theory there isa rich collection of equivalence relations properly belowequality =ce.

This is a departure from the Borel theory, where Silver’stheorem implies that = is continuously reducible to every Borelequivalence relation (with uncountably many classes).

We shall also show that there are relations on c.e. sets whichare computably incomparable with =ce.



Same minimum, Same maximum relations

Same minimum:

e Ecemin e′ ⇐⇒ min(We) = min(We′)

Same maximum:

e Ecemax e′ ⇐⇒ max(We) = max(We′)

Both are reducible to =ce, by saturating We upwards for Eminand downwards for Emax.

These saturation reductions are computable selectors,choosing a representative of each class.



Max not reducible to Min

Theorem

Ecemax is not computably reducible to Ecemin. Consequently, Ecemin

lies properly below =ce.

Proof.

This holds because Ecemin is ∆02, while E

cemax is Π02 complete.

Even the Emax class INF = {e | |We| = ℵ0 } is Π02 complete.

In fact, we show next that Ecemax and Ecemin are incomparable, andboth lie properly below =ce.



Monotonicity lemma

f well-defined: We = We′ =⇒ Wf (e) = Wf (e′).f monotone: We ⊆We′ =⇒ Wf (e) ⊆Wf (e′).

Monotonicity Lemma

Every computable well-defined f ... N→ N is monotone.

Proof.

Suppose We ⊆We′ and x ∈Wf (e). Using the recursiontheorem, let Wp look like We until x appears in Wf (p), and thenWp looks like We′ . Note that x must appear in Wf (p), sinceotherwise Wp = We and so Wf (p) = Wf (e). But now Wp = We′and so x ∈Wf (p) = Wf (e′), as desired.



Min not reducible to Max

Theorem

Ecemin does not reduce to Ecemax. Consequently, Ecemax


Same median relation, Ecemed

e Ecemed e′ ⇐⇒ median(We) = median(We′)

Theorem

Ecemed lies between Ecemax and Ece0 in the reducibility hierarchy.

Proof.

Ecemax ≤c Ecemed by saturating downwards.

Ecemed ≤c Ece0 via f , where as long as rs = median(We,s) doesn’t

change, Wf (e) enumerates multiples of rs into Wf (e). When rsdoes change, Wf (e) saturates below current max and nowenumerates multiples of new rs. This reduces, since if We hasmedian r , then this stabilizes, so Wf (e) is eventually rN.



Median incomparable with Min and equalityTheorem

Ecemed is incomparable with both Ecemin and =

ce.

Proof.

Suppose f : Ecemed ≤c=ce. So f well-defined, hence monotone.Consider We1 = {1}, We2 = {1,2}, We3 = {0,1,2}. By monotonicity,Wf (e1) ( Wf (e2) ( Wf (e3), but med(We1 ) = med(We3 ), contradiction.

Suppose f : Ecemin ≤c Ecemed. Fix We0 nonempty with Wf (e0) finite medianr0. Find min(We0 ) < min(We1 ) < min(We2 ) < · · · < min(WeN ),N = 2r0 + 2, with ri = med(Wf (ei )) finite. Consider program e, whichenumerates min(WeN ) into We until med(Wf (e),sN ) = rN . Nowenumerate min(WeN−1 ) into We, and so on. After N iterations,med(Wf (e)) = med(Wf (e0)) = r0, but Wf (e) has ≥ N = 2r0 + 2 elements,impossible.



Reducibility diagram

=ce

Ecemin Ecemax

Ecemed

Ece0

This diagram is complete for these relations.



Generalization to other ordersFor computable linear ordering L and W ⊆ L, considerDedekind cut

cutL(W ) = { l ∈ L | ∃w ∈W (l


Complete embeddingsLet L be the c.e. cuts of L. Define computably embeddableL1 ↪→c L2, if there is computable α ... N→ N such that

cutL1(We) < cutL1(We′) ⇐⇒ cutL2(Wα(e)) < cutL2(Wα(e′)).

Theorem

If L1,L2 computable linear orders, then EL1 ≤ EL2 iff L1 ↪→c L2.

Proof.

If f : EL1 ≤c EL2 , wlog f is well-defined. By monotonicy, fpreserves cut order, so f witnesses L1 ↪→c L2.

If α : L1 ↪→c L2, let Wf (e) = cutL2(Wα(e)), which is a reduction ofEL1 to EL2 .

The result generalizes to partial orders.Equivalence relations on N, CUNY MAMLS 2012 Joel David Hamkins, New York


Cuts and Hulls in computable ordinals

Hω

EωEmax ∼ Eω∗∼ Emin

EQ

· ··

Eα

· · ·

· · ·Eα∗

· ··

···

Hα

···

∼ =ce

Figure: Diagram showing the cut and hull relations for computableordinals α and their reverse orderings α∗. In 2017 this will be the firstdiagram to land on Gliese 581g.



Enumerable relations and orbit equivalence relations

Enumerable relations

A major focus for Borel theory: countable Borel equivalencerelations.

The Lusin/Novikov theorem: every countable Borel equivalencerelation has a uniform Borel enumeration of each class.

The Feldman/Moore argument uses this to show everycountable Borel equivalence relation is the orbit relation of aBorel action of a countable group.

We develop the computable analogue of this theory.




Computable analogue of Lusin/Novikov

An equivalence relation E on c.e. sets is enumerable in theindices if there is computable α ... N× N→ N with

We E We′ ⇐⇒ ∃n Wα(n,e) = We′ .

This is the computable analogue of the Lusin/Novikov propertyfor countable Borel equivalence relations.

For example, Ece0 has this property, since let Wα(n,e) be Wemodified to have nth finite set at bottom. As n varies, Wα(e,n)enumerates all finite modifications of We, and so α witnessesthat Ece0 is enumerable in the indices.




Enumerable relations reduce to Eset

Theorem

If Ece is enumerable in the indices, then Ece ≤ Eceset.

Proof.

Simply map program e to a program for a subset of N× Nwhich puts Wα(n,e) on the nth column.

Of course Eceset is not itself enumerable, since enumerablerelations are easily seen to be Σ03, whereas E

ceset is Π

03 complete.




Computable group actions

The action of a computable group Γ acting on the c.e. sets iscomputable in the indices if there is computable α ... N× N→ Nwith Wα(γ,e) = γWe.

For example, left translation action of Γ on c.e. subsets of Γ.

Induced orbit equivalence relation

e EceΓ e′ ⇐⇒ ∃γ ∈ Γ We′ = γWe.

One would prefer an analogue of Feldman/Moore, saying thatevery enumerable relation is the orbit relation of an actioncomputable in the indices.

Unfortunately, this is not the case.




Counterexamples to orbit equivalence relations

Theorem

If E is an equivalence relation on c.e. sets and |[We]E | ≥ 2 forsome e with We ⊆We′ for all e′ Ece e, then E is not an orbitequivalence of any action computable in the indices.

Proof.

Suppose E is orbit equivalence of computable action of Γ,witnessed by α. For each γ, the map e 7→ α(γ,e) iswell-defined on c.e. sets, hence monotone. Choose e′ Ece ewith We ( We′ and γWe′ = We. So γWe ⊆ γWe′ = We, so byhypothesis, γWe = We also, a contradiction.




Ece0 is not an orbit equivalence

Corollary

Ece0 is not induced by any action which is computable in theindices.

Proof.

The empty set We = ∅ is minimal in its E0 equivalence class,which consists of all the finite sets.

Nevertheless, we don’t know:

Question

Is Ece0 computably bireducible with an orbit relation induced byan action which is computable in the indices?




Universal orbit equivalence

Like their Borel counterparts, the class of orbit relationsinduced by actions computable in the indices exhibits auniversal element, which gives hope that the structure of theorbit equivalence relations on c.e. sets will mirror the countableBorel equivalence relations.

Theorem

There is Ece∞ induced by an action computable in the indices,and EΓ ≤ Ece∞ whenever EΓ arises from an action computable inthe indices.

The proof uses the free group on ω generators to simulate allother actions.




enumerable

orbit

=ce

Ece0EceΓ

UFω

Eceset

The enumerable relations, with orbit equivalence relations.



Isomorphism of c.e. structures

Definition

Let ∼=cebin denote the isomorphism relation on the codes for c.e.binary relations. That is, let e ∼=cebin e

′ if and only if We and We′ ,thought of as binary relations on N, are isomorphic.

We remark that in order to analyze the isomorphism onarbitrary L-structures, it is enough to consider just the binaryrelations, since if L is a computable language then theisomorphism relation ∼=ceL on the c.e. L-structures is computablyreducible to ∼=cebin.



What is isomorphism on c.e. graphs?Two different ways to treat isomorphism of c.e. structures.

Could restrict ∼=cebin to indices for We that are the right kindof structure. (Problem: these relations are not total)Or, identify structures as c.e. substructures of fixeduniversal structure.

For example, fix a computable copy Γ of the countable randomgraph, and we define ∼=cegraph to be the isomorphism relation onthe c.e. subsets of Γ.

Similarly, we consider c.e. linear orders as subsets of a copy ofQ, and c.e. trees as the downward closure of c.e. subsets ofN


Isomorphisms of c.e. graphs, linear orders, trees

Theorem

Isomorphism of binary relations ∼=cebin is computably bireduciblewith each of isomorphisms of graphs, linear orders and trees.

∼=cegraph ∼=celo

∼=cetree

The classical reductions go through easily.

Part of the reason it works is that these classical reductions useonly the positive information about the structures; they need toknow when two elements are related, but not instances ofnon-relation.



Same set is strictly below ∼=binThe isomorphism relation of binary relations ∼=cebin is very high inour hierarchy. For example,

Theorem

Eceset lies properly below ∼=cebin.

Proof.

To reduce Eceset to ∼=cebin, let Wf (e) be a code for We as ahereditarily countable set. In other words, Wf (e) is awell-founded tree coding the transitive closure of {We}.

The absence of any reverse reduction follows from complexity,since Eceset is Π

04.



Isomorphism of c.e. groupsIsomorphism of c.e. groups admits several coding methods.

We could consider indices e such that We, as a subset ofN× N× N, is the graph of a group operation, and then define∼=cegroup to be the restriction of the isomorphism relation ∼=cetern onthe c.e. ternary relations to this set.

Alternatively we can code a group by a presentation, a set ofwords in Fω, thinking of the group as the correspondingquotient. Thus, e ∼=cepres e′ iff We and We′ , as sets of relations inFω, determine isomorphic quotient groups.

Thus, the classification problem for groups splits into twoseparate problems: for computable group operations, and forcomputably presented groups.



Two versions of group isomorphism

Theorem∼=cegroup ≤ ∼=cebin ≤ ∼=

cepres.

We suspect that neither reduction is reversible.



Computable isomorphisms

Consider now the case of computable isomorphisms, ratherthan arbitrary isomorphisms.

e 'cebin e′ iff We,We′ , as binary relations, are isomorphic by a

computable bijection.

It happens that most of the previous reductions, such as frombinary relations to graphs, go through for computableisomorphisms.

Thus, the computable isomorphism relation on the class of c.e.binary relations, graphs, linear orders and trees are allcomputably bireducible to each other.



Computable isomorphisms

But they are lower in the hierarchy:

Theorem

'cebin lies properly below Eceset.

Proof.

For the reduction, the basic idea is to reduce to Eset by makinga subset of N× N whose slices enumerate all computablecopies of the original structure (plus all finite sets).

Conversely, Eceset is not reducible to 'cebin because 'cebin is Σ

03, but

Eceset has a Π03-complete equivalence class.



Isomorphisms and computable isomorphisms

=ce

'cebin

Eceset

∼=ceL

∼=cebin ∼ ∼=cegraph ∼ ∼=

celo ∼ ∼=

cetree

∼=cegroup

∼=cepres



Relations from computability theory

Consider now the principal equivalence relations ofcomputability theory, viewed as equivalence relations on thec.e. sets.

Turing equivalence, =TMany-one equivalence, =mone-equivalence, =1



Theorem

=ce lies properly below each of =ceT , =ce1 , and =

cem in the

computable reducibility hierarchy.

Proof.

Reduce to all at once. Fix strong antichain: c.e. sets Ai withAi 6≤T

⊕j 6=i Aj . Let Wf (e) ⊆ N× N have k th column Ank , when

nk is k th element enumerated into We.

If We = We′ , then Wf (e) ≡1 Wf (e′), by suitable permutation ofcolumns. Conversely, if We 6= We′ , then Wf (e) 6≡T Wf (e′), for ifi ∈We r We′ , then Ai ≤T Wf (e) but not Wf (e′).

None of the relations reduces to =ce, because they areΣ03-complete, whilst =

ce is Π02.



Theorem

=cem is computably reducible to =ce1 .

Proof.

Simply let Wf (e) = We × N. If φ ... N→ N many-one reduces Weto We′ , then (m,n) 7→ (φ(m), 〈m,n〉) one-one reduces Wf (e) toWf (e′). Converse is similar.

Theorem

=ce1 is computably reducible to 'cebin.

Proof.

This is because =ce1 is the computable isomorphism relation onthe set of c.e. unary relations. So =ce1 ≤ '

cebin via f where Wf (e)

codes the graph on N ∪ {?} where ?→ n iff n ∈We.Equivalence relations on N, CUNY MAMLS 2012 Joel David Hamkins, New York


Degree relations

=ce

=cem

=ce1

'cebin

=T



Difference hierarchy

We have extended the analysis to the difference hierarchy.

For e1, . . . ,en, the corresponding n-c.e. set is

W(ei ) = (We1 r We2) ∪ (We3 r We4) ∪ · · ·r ∪Wen .

Every equivalence relation E on n-c.e. sets has correspondingrelation

(ei) En-ce (fi) ⇐⇒ W(ei ) E W(fi )

It is easy to see that En-ce ≤ En+1-ce. Just fix e with We = ∅,and use (e1, . . . ,en) 7→ (e1, . . . ,en,e).



Relations in the difference hierarchy

=ce

=dce

=n-ce

=


Thank you.

Joel David HamkinsThe City University of New York

http://jdh.hamkins.org

http://cantorsattic.info


Introduction, Equivalences of c.e. setsEquivalences of c.e. sets

Below equality, =ceEnumerable relations and orbit equivalence relations

Classifications of c.e. structuresComputability relations

Documents

The hierarchy of equivalence relations on N under …jdh.hamkins.org/wp-content/uploads/2012/03/Comp-equiv...Borel equivalence relations Many naturally arising equivalence relations,