Introduction to Enumeration Reducibility

Charles M. Harris

Department Of Mathematics University of Leeds

Computability Theory Seminar, Autumn 2009

(University Of Leeds) Introduction to Enumeration Reducibility CTS 2009 1 / 34 Outline

1 Introducing Enumeration Reducibility The Basic Deﬁnitions Examples Alternative deﬁnitions

2 Properties of De Deﬁnition of 0e Lattice Theoretic Properties

(University Of Leeds) Introduction to Enumeration Reducibility CTS 2009 2 / 34 Outline

3 The Enumeration Jump Operator The Enumeration Jump (over sets) The Enumeration Jump Operator

4 Embedding DT into De Total Degrees Deﬁnition of the Embedding Quasi-minimal Enumeration Degrees

(University Of Leeds) Introduction to Enumeration Reducibility CTS 2009 3 / 34 Outline

5 Further Properties of De Exact Pairs and Branching Density and Minimality

Next Week

(University Of Leeds) Introduction to Enumeration Reducibility CTS 2009 4 / 34 Subsection Guide

1 Introducing Enumeration Reducibility The Basic Deﬁnitions Examples Alternative deﬁnitions

2 Properties of De Deﬁnition of 0e Lattice Theoretic Properties

(University Of Leeds) Introduction to Enumeration Reducibility CTS 2009 5 / 34 What is enumeration reducibility?

Deﬁnition (Intuitive)

A≤e B if there exists an effective procedure that, given any enumeration of B, computes an enumeration A.

Deﬁnition (Formal)

A≤e B if there exists a c.e. set W such that for all x ∈ ω

x ∈ A iff ∃u [ hx, ui ∈ W & Du ⊆ B ]

This is written A = ΦW (B).

(University Of Leeds) Introduction to Enumeration Reducibility CTS 2009 6 / 34 Subsection Guide

1 Introducing Enumeration Reducibility The Basic Deﬁnitions Examples Alternative deﬁnitions

2 Properties of De Deﬁnition of 0e Lattice Theoretic Properties

(University Of Leeds) Introduction to Enumeration Reducibility CTS 2009 7 / 34 Examples of sets W and X such that x ∈ A iff ∃D ( hx, Di ∈ W & D ⊆ X )

A≤e A via the c.e. set W = { hn, {n}i | n ∈ ω }.

If A is c.e. and B is any set, then A≤e B via the c.e. set W = { hn, ∅i | n ∈ A }. −1 If f is computable and A = f (B) (i.e. A≤m B), then A≤e B via the c.e. set W = { hn, {f (n)}i | n ∈ ω }.

(University Of Leeds) Introduction to Enumeration Reducibility CTS 2009 8 / 34 Examples of sets W and X such that x ∈ A iff ∃D ( hx, Di ∈ W & D ⊆ X )

A≤e A via the c.e. set W = { hn, {n}i | n ∈ ω }.

(University Of Leeds) Introduction to Enumeration Reducibility CTS 2009 8 / 34 Examples of sets W and X such that x ∈ A iff ∃D ( hx, Di ∈ W & D ⊆ X )

A≤e A via the c.e. set W = { hn, {n}i | n ∈ ω }.

(University Of Leeds) Introduction to Enumeration Reducibility CTS 2009 8 / 34 Examples of sets W and X such that x ∈ A iff ∃D ( hx, Di ∈ W & D ⊆ X )

A≤e A via the c.e. set W = { hn, {n}i | n ∈ ω }.

(University Of Leeds) Introduction to Enumeration Reducibility CTS 2009 8 / 34 A non trivial example

Reminder

A is regressive if there exists an enumeration {a0, a1 ...} of A and a partial computable function f such that f (0)↓ = a0 and f (an+1)↓ = an for all n ≥ 0.

Example

If A is regressive and B ⊆ A is inﬁnite then A≤e B. In effect, suppose that partial computable f regresses A. Let h(n, x) = f n(x) (so h is partial computable). Then A≤e B via

W = { hy, {x}i | ∃n [ h(n, x)↓ = y ] } .

(University Of Leeds) Introduction to Enumeration Reducibility CTS 2009 9 / 34 A non trivial example

Reminder

A is regressive if there exists an enumeration {a0, a1 ...} of A and a partial computable function f such that f (0)↓ = a0 and f (an+1)↓ = an for all n ≥ 0.

Example

If A is regressive and B ⊆ A is inﬁnite then A≤e B. In effect, suppose that partial computable f regresses A. Let h(n, x) = f n(x) (so h is partial computable). Then A≤e B via

W = { hy, {x}i | ∃n [ h(n, x)↓ = y ] } .

(University Of Leeds) Introduction to Enumeration Reducibility CTS 2009 9 / 34 Subsection Guide

1 Introducing Enumeration Reducibility The Basic Deﬁnitions Examples Alternative deﬁnitions

2 Properties of De Deﬁnition of 0e Lattice Theoretic Properties

(University Of Leeds) Introduction to Enumeration Reducibility CTS 2009 10 / 34 Enumeration reducibility - alternative views

Selman’s Deﬁnition X X A≤e B iff ∀X [ B ∈ Σ1 ⇒ A ∈ Σ1 ].

Scott’s Deﬁnition

A≤e B iff there exists some closed LAMBDA term u such that A = [[(u)B ]].

A Turing machine deﬁnition

A≤e B iff A is positive non deterministic Turing reducible to B.

(University Of Leeds) Introduction to Enumeration Reducibility CTS 2009 11 / 34 Enumeration reducibility - alternative views

Selman’s Deﬁnition X X A≤e B iff ∀X [ B ∈ Σ1 ⇒ A ∈ Σ1 ].

Scott’s Deﬁnition

A≤e B iff there exists some closed LAMBDA term u such that A = [[(u)B ]].

A Turing machine deﬁnition

A≤e B iff A is positive non deterministic Turing reducible to B.

(University Of Leeds) Introduction to Enumeration Reducibility CTS 2009 11 / 34 Enumeration reducibility - alternative views

Selman’s Deﬁnition X X A≤e B iff ∀X [ B ∈ Σ1 ⇒ A ∈ Σ1 ].

Scott’s Deﬁnition

A≤e B iff there exists some closed LAMBDA term u such that A = [[(u)B ]].

A Turing machine deﬁnition

A≤e B iff A is positive non deterministic Turing reducible to B.

(University Of Leeds) Introduction to Enumeration Reducibility CTS 2009 11 / 34 Enumeration reducibility - alternative views

Selman’s Deﬁnition X X A≤e B iff ∀X [ B ∈ Σ1 ⇒ A ∈ Σ1 ].

Scott’s Deﬁnition

A≤e B iff there exists some closed LAMBDA term u such that A = [[(u)B ]].

A Turing machine deﬁnition

A≤e B iff A is positive non deterministic Turing reducible to B.

(University Of Leeds) Introduction to Enumeration Reducibility CTS 2009 11 / 34 Subsection Guide

1 Introducing Enumeration Reducibility The Basic Deﬁnitions Examples Alternative deﬁnitions

2 Properties of De Deﬁnition of 0e Lattice Theoretic Properties

(University Of Leeds) Introduction to Enumeration Reducibility CTS 2009 12 / 34 The Degree of the Computably Enumerable Sets.

If A is any set and W is c.e. then W ≤e A

Hence 0e comprises the class of c.e. sets

(University Of Leeds) Introduction to Enumeration Reducibility CTS 2009 13 / 34 The Degree of the Computably Enumerable Sets.

If A is any set and W is c.e. then W ≤e A

Hence 0e comprises the class of c.e. sets

(University Of Leeds) Introduction to Enumeration Reducibility CTS 2009 13 / 34 Subsection Guide

1 Introducing Enumeration Reducibility The Basic Deﬁnitions Examples Alternative deﬁnitions

2 Properties of De Deﬁnition of 0e Lattice Theoretic Properties

(University Of Leeds) Introduction to Enumeration Reducibility CTS 2009 14 / 34 Lattice Theoretic Properties

There exists a pair of sets A,B with no g.l.b. under≤e . However every such pair A,B, always has l.u.b. A ⊕ B.

Thus De is an upper semi lattice.

(University Of Leeds) Introduction to Enumeration Reducibility CTS 2009 15 / 34 Lattice Theoretic Properties

There exists a pair of sets A,B with no g.l.b. under≤e . However every such pair A,B, always has l.u.b. A ⊕ B.

Thus De is an upper semi lattice.

(University Of Leeds) Introduction to Enumeration Reducibility CTS 2009 15 / 34 Lattice Theoretic Properties

There exists a pair of sets A,B with no g.l.b. under≤e . However every such pair A,B, always has l.u.b. A ⊕ B.

Thus De is an upper semi lattice.

(University Of Leeds) Introduction to Enumeration Reducibility CTS 2009 15 / 34 Subsection Guide

3 The Enumeration Jump Operator The Enumeration Jump (over sets) The Enumeration Jump Operator

4 Embedding DT into De Total Degrees Deﬁnition of the Embedding Quasi-minimal Enumeration Degrees

(University Of Leeds) Introduction to Enumeration Reducibility CTS 2009 16 / 34 The Enumeration Jump (over sets)

Notation For any set A,

KA =def { x | x ∈ Φx (A) } ,

JA =def KA ⊕ KA , (0) (k+1) JA =def JA = A and JA =def J (k) . JA

Lemma For any sets A and B,

KA is 1-complete for Enum(A),

A≤e B iff A≤1 KB iff KA ≤1 KB .

Forward

(University Of Leeds) Introduction to Enumeration Reducibility CTS 2009 17 / 34 The Enumeration Jump (over sets)

Notation For any set A,

KA =def { x | x ∈ Φx (A) } ,

JA =def KA ⊕ KA , (0) (k+1) JA =def JA = A and JA =def J (k) . JA

Lemma For any sets A and B,

KA is 1-complete for Enum(A),

A≤e B iff A≤1 KB iff KA ≤1 KB .

Forward

(University Of Leeds) Introduction to Enumeration Reducibility CTS 2009 17 / 34 The Enumeration Jump (over sets)

Notation For any set A,

KA =def { x | x ∈ Φx (A) } ,

JA =def KA ⊕ KA , (0) (k+1) JA =def JA = A and JA =def J (k) . JA

Lemma For any sets A and B,

KA is 1-complete for Enum(A),

A≤e B iff A≤1 KB iff KA ≤1 KB .

Forward

(University Of Leeds) Introduction to Enumeration Reducibility CTS 2009 17 / 34 The Enumeration Jump (over sets)

Notation For any set A,

KA =def { x | x ∈ Φx (A) } ,

JA =def KA ⊕ KA , (0) (k+1) JA =def JA = A and JA =def J (k) . JA

Lemma For any sets A and B,

KA is 1-complete for Enum(A),

A≤e B iff A≤1 KB iff KA ≤1 KB .

Forward

(University Of Leeds) Introduction to Enumeration Reducibility CTS 2009 17 / 34 The Enumeration Jump (over sets)

Notation For any set A,

KA =def { x | x ∈ Φx (A) } ,

JA =def KA ⊕ KA , (0) (k+1) JA =def JA = A and JA =def J (k) . JA

Lemma For any sets A and B,

KA is 1-complete for Enum(A),

A≤e B iff A≤1 KB iff KA ≤1 KB .

Forward

(University Of Leeds) Introduction to Enumeration Reducibility CTS 2009 17 / 34 Subsection Guide

3 The Enumeration Jump Operator The Enumeration Jump (over sets) The Enumeration Jump Operator

4 Embedding DT into De Total Degrees Deﬁnition of the Embedding Quasi-minimal Enumeration Degrees

(University Of Leeds) Introduction to Enumeration Reducibility CTS 2009 18 / 34 The Enumeration Jump Operator (over degrees)

Deﬁnition

For any ae and A ∈ ae, 0 ae =def dege(JA) , (0) (k+1) (k) 0 ae = ae and ae = (ae ) .

Remark

Choose any Turing degree aT and any set A ∈ aT. Then, by results Lemma 0 below , KA ≡1 KA⊕A (and so KA⊕A ∈ aT ). Let ae = dege(A ⊕ A). Then

0 0 ι( aT ) = ι( degT(KA⊕A) ) = dege( KA⊕A ⊕ KA⊕A ) = ae .

Proposition

(University Of Leeds) Introduction to Enumeration Reducibility CTS 2009 19 / 34 The Enumeration Jump Operator (over degrees)

Deﬁnition

For any ae and A ∈ ae, 0 ae =def dege(JA) , (0) (k+1) (k) 0 ae = ae and ae = (ae ) .

Remark

Choose any Turing degree aT and any set A ∈ aT. Then, by results Lemma 0 below , KA ≡1 KA⊕A (and so KA⊕A ∈ aT ). Let ae = dege(A ⊕ A). Then

0 0 ι( aT ) = ι( degT(KA⊕A) ) = dege( KA⊕A ⊕ KA⊕A ) = ae .

Proposition

(University Of Leeds) Introduction to Enumeration Reducibility CTS 2009 19 / 34 The Enumeration Jump Operator (over degrees)

Deﬁnition

For any ae and A ∈ ae, 0 ae =def dege(JA) , (0) (k+1) (k) 0 ae = ae and ae = (ae ) .

Remark

Choose any Turing degree aT and any set A ∈ aT. Then, by results Lemma 0 below , KA ≡1 KA⊕A (and so KA⊕A ∈ aT ). Let ae = dege(A ⊕ A). Then

0 0 ι( aT ) = ι( degT(KA⊕A) ) = dege( KA⊕A ⊕ KA⊕A ) = ae .

Proposition

(University Of Leeds) Introduction to Enumeration Reducibility CTS 2009 19 / 34 Subsection Guide

3 The Enumeration Jump Operator The Enumeration Jump (over sets) The Enumeration Jump Operator

4 Embedding DT into De Total Degrees Deﬁnition of the Embedding Quasi-minimal Enumeration Degrees

(University Of Leeds) Introduction to Enumeration Reducibility CTS 2009 20 / 34 Total Functions and Characteristic Sets

Deﬁnition An e-degree is total if it contains the graph of a total function (or, equivalently, the graph of a characteristic function, or a characteristic set).

Lemma For any set A :

CA ≡e CA ≡e CA ≡e A ⊕ A

(University Of Leeds) Introduction to Enumeration Reducibility CTS 2009 21 / 34 Total Functions and Characteristic Sets

Deﬁnition An e-degree is total if it contains the graph of a total function (or, equivalently, the graph of a characteristic function, or a characteristic set).

Lemma For any set A :

CA ≡e CA ≡e CA ≡e A ⊕ A

(University Of Leeds) Introduction to Enumeration Reducibility CTS 2009 21 / 34 Subsection Guide

3 The Enumeration Jump Operator The Enumeration Jump (over sets) The Enumeration Jump Operator

4 Embedding DT into De Total Degrees Deﬁnition of the Embedding Quasi-minimal Enumeration Degrees

(University Of Leeds) Introduction to Enumeration Reducibility CTS 2009 22 / 34 Enumeration Reducibility and Turing Reducibility

Lemma

For any sets A and B, A c.e. in B iff A≤e CB .

Remark Proof B Suppose that A = Wi for some i ≥ 0. Then the c.e. set

0 D + Wei = { hx, D ⊕ D i | ϕi (x) ↓ & Q (ϕi , x, D) = D − 0 & Q (ϕi , x, D) = D }

witnesses A≤e B ⊕ B.

(University Of Leeds) Introduction to Enumeration Reducibility CTS 2009 23 / 34 Enumeration Reducibility and Turing Reducibility

Lemma

For any sets A and B, A c.e. in B iff A≤e CB .

Remark Proof B Suppose that A = Wi for some i ≥ 0. Then the c.e. set

0 D + Wei = { hx, D ⊕ D i | ϕi (x) ↓ & Q (ϕi , x, D) = D − 0 & Q (ϕi , x, D) = D }

witnesses A≤e B ⊕ B.

(University Of Leeds) Introduction to Enumeration Reducibility CTS 2009 23 / 34 Enumeration Reducibility and Turing Reducibility

Corollary

For any sets A and B, A≤T B iff A ⊕ A≤e B ⊕ B iff CA ≤e CB .

(University Of Leeds) Introduction to Enumeration Reducibility CTS 2009 24 / 34 The Embedded Turing Degrees

Proposition

The embedding ιe of the Turing degrees into the e-degrees induced by the map X 7→ CX is structure preserving and also preserves suprema, least element, and the jump. Remark

Remark The embedding does not preserve inﬁma. This is witnessed by Cooper 0 and McEvoy’s result that there exists a (high) Σ1 minimal pair (in DT) 0 which embeds to a Π1 pair (in De) that is not minimal and below which 0 all the non zero e-degrees are properly Σ2.

(University Of Leeds) Introduction to Enumeration Reducibility CTS 2009 25 / 34 The Embedded Turing Degrees

Proposition

The embedding ιe of the Turing degrees into the e-degrees induced by the map X 7→ CX is structure preserving and also preserves suprema, least element, and the jump. Remark

(University Of Leeds) Introduction to Enumeration Reducibility CTS 2009 25 / 34 Subsection Guide

3 The Enumeration Jump Operator The Enumeration Jump (over sets) The Enumeration Jump Operator

4 Embedding DT into De Total Degrees Deﬁnition of the Embedding Quasi-minimal Enumeration Degrees

(University Of Leeds) Introduction to Enumeration Reducibility CTS 2009 26 / 34 Gaps in the Embedding

Theorem (Medvedev)

There exist non total enumeration degrees. In fact there exist enumeration degrees below which no non zero degree is total.

Corollary Proof Construct a set A such that (for all e ≥ 0) A satisﬁes

R2e : A 6= We

R2e+1 :Φe(A) characteristic ⇒ Φe(A) c.e.

(University Of Leeds) Introduction to Enumeration Reducibility CTS 2009 27 / 34 Gaps in the Embedding

Theorem (Medvedev)

There exist non total enumeration degrees. In fact there exist enumeration degrees below which no non zero degree is total.

Corollary Proof Construct a set A such that (for all e ≥ 0) A satisﬁes

R2e : A 6= We

R2e+1 :Φe(A) characteristic ⇒ Φe(A) c.e.

(University Of Leeds) Introduction to Enumeration Reducibility CTS 2009 27 / 34 Quasi Minimal Degrees

Deﬁnition An e-degree a is said to be quasi-minimal if a > 0 and ∀d( d < a & d total ⇒ d = 0 ).

Corollary (To the previous Theorem)

There exists a quasi-minimal e-degree.

Last Theorem

(University Of Leeds) Introduction to Enumeration Reducibility CTS 2009 28 / 34 Quasi Minimal Degrees

Deﬁnition An e-degree a is said to be quasi-minimal if a > 0 and ∀d( d < a & d total ⇒ d = 0 ).

Corollary (To the previous Theorem)

There exists a quasi-minimal e-degree.

Last Theorem

(University Of Leeds) Introduction to Enumeration Reducibility CTS 2009 28 / 34 Subsection Guide

5 Further Properties of De Exact Pairs and Branching Density and Minimality

(University Of Leeds) Introduction to Enumeration Reducibility CTS 2009 29 / 34 Exact Pairs and Branching

Proposition (Case) There exists an exact pair of enumeration degrees.

Theorem (Case, Jockusch) There exists a minimal pair of enumeration degrees. In fact every non zero degree is part of a minimal pair.

Theorem (Rozinas) Every enumeration degree is branching.

(University Of Leeds) Introduction to Enumeration Reducibility CTS 2009 30 / 34 Exact Pairs and Branching

Proposition (Case) There exists an exact pair of enumeration degrees.

Theorem (Case, Jockusch) There exists a minimal pair of enumeration degrees. In fact every non zero degree is part of a minimal pair.

Theorem (Rozinas) Every enumeration degree is branching.

(University Of Leeds) Introduction to Enumeration Reducibility CTS 2009 30 / 34 Exact Pairs and Branching

Proposition (Case) There exists an exact pair of enumeration degrees.

Theorem (Case, Jockusch) There exists a minimal pair of enumeration degrees. In fact every non zero degree is part of a minimal pair.

Theorem (Rozinas) Every enumeration degree is branching.

(University Of Leeds) Introduction to Enumeration Reducibility CTS 2009 30 / 34 Subsection Guide

5 Further Properties of De Exact Pairs and Branching Density and Minimality

(University Of Leeds) Introduction to Enumeration Reducibility CTS 2009 31 / 34 Density and Minimality

Proposition (Cooper, Calhoun and Slaman)

De is not dense.

Lemma (Gutteridge) 0 If ae is minimal then it is ∆2.

Theorem (Gutteridge) There is no minimal enumeration degree.

(University Of Leeds) Introduction to Enumeration Reducibility CTS 2009 32 / 34 Density and Minimality

Proposition (Cooper, Calhoun and Slaman)

De is not dense.

Lemma (Gutteridge) 0 If ae is minimal then it is ∆2.

Theorem (Gutteridge) There is no minimal enumeration degree.

(University Of Leeds) Introduction to Enumeration Reducibility CTS 2009 32 / 34 Density and Minimality

Proposition (Cooper, Calhoun and Slaman)

De is not dense.

Lemma (Gutteridge) 0 If ae is minimal then it is ∆2.

Theorem (Gutteridge) There is no minimal enumeration degree.

(University Of Leeds) Introduction to Enumeration Reducibility CTS 2009 32 / 34 The End

(University Of Leeds) Introduction to Enumeration Reducibility CTS 2009 33 / 34 Next Week: The Local Structure of the Enumeration Degrees

0 0 0 Π1, ∆2, and properly Σ2 degrees. High and low classes. 0 Good Σ2 Enumerations Density. Embeddings of the local Turing degrees. Branching and Splitting.

Return to Outline End

(University Of Leeds) Introduction to Enumeration Reducibility CTS 2009 34 / 34