The Computably Enumerable Sets

Themes/Review The Classic Examples Dynamic Properties Invariant Classes Index Sets

The Computably Enumerable SetsThe Past, the Present and the Future

Peter Cholak

University of Notre DameDepartment of Mathematics

[email protected]

http://www.nd.edu/~cholak/papers/

Supported by NSF Grant DMS 02-45167

May, 2006

http://www.nd.edu/~cholak/papers/


Main Goals

To understand orbits within thestructure of c.e. (r.e.) sets underinclusion.

What are the good open interestingquestions?

The story is rather complex.


Main Goals





Main Goals





The Computably Enumerable Sets, E

• We is the domain of the eth Turing machine.

• ({We : e ∈ω},⊆) are the c.e. (r.e.) sets under inclusion,E.

• These sets are the same as the Σ01 sets,

{x : (N,+,×,0,1) îϕ(x)}, where ϕ is Σ01.

• We,s is the domain of the eth Turing machine at stage s.• Dynamic Properties: How slow or fast can a c.e. set be

enumerated with respect to another c.e. set? w.r.t. thestandard enumeration of all c.e. sets?

• 0,1,∪, and ∩ are definable from ⊆ in E.

• For safety all sets are c.e., infinite, and coinfinite unlessotherwise noted.






{x : (N,+,×,0,1) îϕ(x)}, where ϕ is Σ01.










{x : (N,+,×,0,1) îϕ(x)}, where ϕ is Σ01.






Automorphisms of E

• Φ ∈ Aut(E) iff Φ(W) is a one to one, onto map takingc.e. sets to c.e. sets such that We ⊆ We′ iffΦ(We) ⊆ Φ(We′) and dually.

• For example, if p is a computable permutation of N thenΦ(W) = p(W) is an automorphism of E.

• If p is a permutation of N such that p(W) is r.e. anddually then Φ(W) = p(W) is an automorphism of E.

• Φ ∈ Aut(E) iff there is a permutation, p, of N such thatΦ(We) = Wp(e) and We ⊆ We′ iff Wp(e) ⊆ Wp(e′) anddually.

• A is automorphic to A iff there is a Φ ∈ Aut(E) such thatΦ(A) = A.


Automorphisms of E







Automorphisms of E







Automorphisms of E







Automorphisms of E







The Turing Degrees

• A ≤T B, A is Turing reducible to B, iff a Turing machinewith an oracle for B computes χA.

• ≡T is the corresponding equivalence relation.

• The Turing degrees are the corresponding equivalenceclasses on P(ω).

• A Turing degree a is c.e. iff it contains a computablyenumerable set.

• Turing Complexity: Turing reducibility, Turing jump,and the related jump classes (lown and highn classes).

• A is lown iff A(n) = ∅(n) iff ∆0n+1 = ∆An+1.

• A is highn iff A(n) = ∅(n+1) iff ∆0n+2 = ∆An+1.


The Turing Degrees

• A ≤T B, A is Turing reducible to B, iff a Turing machinewith an oracle for B computes χA.

• ≡T is the corresponding equivalence relation.

• The Turing degrees are the corresponding equivalenceclasses on P(ω).

• A Turing degree a is c.e. iff it contains a computablyenumerable set.

• Turing Complexity: Turing reducibility, Turing jump,and the related jump classes (lown and highn classes).

• A is lown iff A(n) = ∅(n) iff ∆0n+1 = ∆An+1.

• A is highn iff A(n) = ∅(n+1) iff ∆0n+2 = ∆An+1.


Core Themes

• Automorphisms and Orbits of E.

• Definability: If two sets are not in the same orbit thereis some definable difference. So definability within afixed structure; within arithmetic; (principal) types;elementary definability; within Lω1,ω, within Lω1,ω1 .

• Dynamic Properties.

• Turing Complexity.


Post’s Program

Post’s Problem. Find an incomplete, noncomputablecomputably enumerable degree. Solved by Friedberg 57 andMuchnik 56.

Post’s Program. Find a “set-theoretic” or “thinness” propertyof a computably enumerable set which guaranteesincompleteness.


Main Open Questions

Question (Completeness)Which c.e. sets are automorphic to complete sets?

Question (Cone Avoidance)Given an incomplete c.e. degree d and an incomplete c.e. setA, is there an A automorphic to A such that d 6≤T A?


The Creative Sets

Theorem (Myhill)A c.e. set is creative iff it is 1-complete.

Theorem (Harrington)Being creative is definable in E.

CorollaryThe creative sets form an orbit; the simplest automorphismof E is a computable permutation of ω, a 1-reduction.


Orbits of Complete Sets

Theorem (Harrington)The creative sets are the only orbit of E which remains anorbit when we restrict the allowable automorphisms tocomputable permutations.

Theorem (Harrington and Soare)There is another definable orbit containing just completesets.


Orbits of Complete Sets

Theorem (Harrington)The creative sets are the only orbit of E which remains anorbit when we restrict the allowable automorphisms tocomputable permutations.

Theorem (Harrington and Soare)There is another definable orbit containing just completesets.


The Maximal Sets

DefinitionM is maximal if for all W either W ⊆∗ M or M ∪W =∗ ω.

Theorem (Friedberg)Maximal sets exist.

Proof.Make M cohesive. Use markers Γe to mark the eth element ofMs . Let σ(e,x, s) = {i ≤ e : x ∈ Wi,s}; the eth state of x atstage s. Allow Γe to pull at any stage to increase itse-state.

CorollaryLet M be maximal. If M ∪W =∗ ω then there is a computableset RW such that RW ⊆∗ M and W ∪ RW =ω. SoW = RW t (W ∩ RW ).


Building Automorphisms

Theorem (Soare)If A and A are both noncomputable then there is anisomorphism, Φ, between the c.e. subsets of A, E(A), to thec.e. subsets of A sending computable subsets of A tocomputable subsets of A and dually.

False Proof.Let p be a computable 1-1 map from A to A. LetΦ(W) = p(W).


Orbits of Maximal Sets

Theorem (Soare 74)The maximal sets form an orbit.

Proof.Define if W ⊆∗ M then let Ψ(W) = Φ(W). If M ∪W =∗ ω thenlet Ψ(W) = Φ(RW )t Φ(RW ∩W).

Theorem (Martin)A c.e. degree h is high iff there is a maximal set M such thatM ∈ h.


Orbits of Maximal Sets

Theorem (Soare 74)The maximal sets form an orbit.

Proof.Define if W ⊆∗ M then let Ψ(W) = Φ(W). If M ∪W =∗ ω thenlet Ψ(W) = Φ(RW )t Φ(RW ∩W).

Theorem (Martin)A c.e. degree h is high iff there is a maximal set M such thatM ∈ h.


Friedberg Splits of Maximal Sets

DefinitionS1 t S2 = M is a Friedberg split of M, for all W , if W ↘ M isinfinite then W ↘ Si is also.

Theorem (Downey and Stob)The Friedberg splits of the maximal sets form an definableorbit.

Theorem (Downey and Stob)All high degrees are hemimaximal degrees. All promptdegrees are hemimaximal. There is a low degree which is nota hemimaximal degree.


Friedberg Splits of Maximal Sets

DefinitionS1 t S2 = M is a Friedberg split of M, for all W , if W ↘ M isinfinite then W ↘ Si is also.

Theorem (Downey and Stob)The Friedberg splits of the maximal sets form an definableorbit.

Theorem (Downey and Stob)All high degrees are hemimaximal degrees. All promptdegrees are hemimaximal. There is a low degree which is nota hemimaximal degree.


Almost Prompt Sets

DefinitionLet Xne be the eth n-r.e. set. A is almost prompt iff there is acomputable nondecreasing function p(s) such that for all eand n if Xne = A then (∃x)(∃s)[x ∈ Xne,s and x ∈ Ap(s)].

Theorem (Maass, Shore, Stob)There is a definable property P(A) which implies A is promptand furthermore for all prompt degree, d, there is set A suchthat P(A) and A ∈ d.

Theorem (Harrington and Soare)All almost prompt sets are automorphic to a complete set.


Tardy Sets

DefinitionD is 2-tardy iff for every computable nondecreasing functionp(s) there is an e such that X2

e = D and (∀x)(∀s)[if x ∈ X2e,s

then x 6∈ Dp(s)].

DefinitionD is codable iff for all A there is an A in the orbit of A suchthat D ≤T A.

Theorem (Harrington and Soare)There are E definable properties Q(D) and P(D,C) such that

• Q(D) implies that D is 2-tardy,

• if there is an C such that P(D,C) and D is 2-tardy thenQ(D) (and D is high),

• X is codable iff there is a D such that X ≤T D and Q(D).


Tardy Sets

DefinitionD is 2-tardy iff for every computable nondecreasing functionp(s) there is an e such that X2

e = D and (∀x)(∀s)[if x ∈ X2e,s

then x 6∈ Dp(s)].

DefinitionD is codable iff for all A there is an A in the orbit of A suchthat D ≤T A.

Theorem (Harrington and Soare)There are E definable properties Q(D) and P(D,C) such that

• Q(D) implies that D is 2-tardy,

• if there is an C such that P(D,C) and D is 2-tardy thenQ(D) (and D is high),

• X is codable iff there is a D such that X ≤T D and Q(D).


Questions about Tardiness

QuestionHow do the following sets of degrees compare:

• the hemimaximal degrees,

• the tardy degrees,

• for each n, {d : there is a n-tardy D such that d ≤T D},• {d : there is a 2-tardy D such that Q(D) and d ≤T D},• {d : there is a A ∈ d which is not automorphic to a

complete set}.

Theorem (Harrington and Soare)There is a maximal 2-tardy set.

QuestionIs there a nonhigh 2-tardy set which is automorphic to acomplete set?


Invariant Classes

DefinitionA class D of degrees is invariant if there is a class S of (c.e.)sets such that

1. d ∈ D implies there is a W in S and d.

2. W ∈ S implies deg(W) ∈ D and

3. S is closed under automorphic images (but need not beone orbit).

CorollaryThe high degrees are invariant by a single orbit.

CorollaryThe prompt degrees are invariant by a single orbit.


More Invariant degree classes

Theorem (Shoenfield 76)The nonlow2 degrees are invariant.

Theorem (Harrington and Soare)

• The nonlow degrees are not invariant.

• There is properly low2 degree d such that if A ∈ d thenA is automorphic to a low set.

• There is a low2 set which is not automorphic to a low set.

Theorem (Cholak and Harrington 02)For n ≥ 2, nonlown and highn degrees are invariant.


Questions on Jump Classes and Degrees

Question (Can single jumps be coded into E?)Let J be C.E.A. in 0′ but not of degree 0′′. Is there a degree asuch that a′ ≡T J and, for all A ∈ a, there is an A with Aautomorphic to A and A′ <T a′ or A′|Ta′?

Question (Can a single Turing degree be coded into E?)Is there a degree d and an incomplete set A such that, for allA automorphic to A, d ≤ A? A ∈ d?


More Single Orbit Invariant Classes?

Theorem (Downey and Harrington – No fat orbit)There is a property S(A), a prompt low degree d1, a prompthigh2 degree d2 greater than d1, and tardy high2 degree esuch that for all E ≤T e, ¬S(E) and if d1 ≤T D ≤T d2 thenS(D).


Orbits Containing Prompt (Tardy) High Sets

QuestionLet A be incomplete. If the orbit of A contains a set of highprompt degree must the orbit of A contain a set from all highprompt degrees?

QuestionIf the orbit of A contains a set of high tardy degree must theorbit of A contain a set from all high tardy degrees?

A positive answer to both questions would answer the coneavoidance question. But not the completeness question.

QuestionFor every degree a there is a set A ∈ a whose orbit containsevery high degree.


Questions on Invariant Degree Classes

QuestionAre the 2-tardy degrees invariant? The degrees containingsets not automorphic to a complete set?

QuestionThere are the array non-computable degrees invariant?

QuestionIs there is an anc degree d such that if A ∈ d then A isautomorphic an non-anc set?


On the Complexity of the Orbits

Look at the index set of all A in the orbit of A with the hopesof finding some answers. The index set of such A is in Σ1

1.

Theorem (Cholak and Harrington)If A is hhsimple then {e : We is automorphic to A} is Σ0

5.

Theorem (Cholak and Harrington)If A is simple then {e : We is automorphic to A} is Σ0

8.

QuestionIs every low2 simple set automorphic to a complete set? Atleast this is an arithmetic question.




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Σ11-completeness

Theorem (Cholak, Downey, and Harrington)There is an set A such that the index set{i : Wi is automorphic to A} is Σ1

1-complete.

Corollary

• Not all orbits are elementarily definable.

• No arithmetic description of all orbits.

• Scott rank of E is ωCK1 + 1.

• (of the proof) For all α ≥ 8, there is a properly ∆0α orbit.


Σ11-completeness

Theorem (Cholak, Downey, and Harrington)There is an set A such that the index set{i : Wi is automorphic to A} is Σ1

1-complete.

Corollary

• Not all orbits are elementarily definable.

• No arithmetic description of all orbits.

• Scott rank of E is ωCK1 + 1.

• (of the proof) For all α ≥ 8, there is a properly ∆0α orbit.


The Game Plan

The isomorphism problem for computable infinitelybranching trees which is known to be Σ1

1-complete. Embedthis problem into orbits.

Given a tree T build AT using ideas from maximal sets,Friedberg splits, and cohesiveness arguments and a jazzedup version of Soare’s theorem on isomorphisms to code atree T into the orbit of AT .


Last Questions

T is an invariant which determines the orbit of AT . Onlyworks for some A; those in the orbit of some AT .

QuestionCan we find an “invariant” (in terms of trees?) which worksfor all A?

ConjectureThe degrees of the possible AT are the hemimaximal degrees.


Last Questions

T is an invariant which determines the orbit of AT . Onlyworks for some A; those in the orbit of some AT .

QuestionCan we find an “invariant” (in terms of trees?) which worksfor all A?

ConjectureThe degrees of the possible AT are the hemimaximal degrees.


Main Open Questions, Again



QuestionAre these arithmetical questions?


Main Open Questions, Again



QuestionAre these arithmetical questions?

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The Computably Enumerable Sets