a c lown dpeople.cs.uchicago.edu/~kaharris/slides/lown-03-06.pdf · 3-uniform escape property a...

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graphics Kenneth Harris Characterizing low n Degrees 0 AC low n D K H Department of Computer Science University of Chicago http://people.cs.uchicago.edu/ kaharris [email protected] [email protected] Presented @ SEALS, Univ. of Florida 03/06

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Page 1: A C lown Dpeople.cs.uchicago.edu/~kaharris/slides/lown-03-06.pdf · 3-Uniform Escape Property A degree a is low 3 if a3 03. Definition: A degree a has the 3-Uniform Escape Property

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Kenneth Harris Characterizing lown Degrees 0'

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A C lown D

K H

Department of Computer Science

University of Chicago

http://people.cs.uchicago.edu/�kaharris

[email protected]

[email protected] Presented @ SEALS, Univ. of Florida 03/06

Page 2: A C lown Dpeople.cs.uchicago.edu/~kaharris/slides/lown-03-06.pdf · 3-Uniform Escape Property A degree a is low 3 if a3 03. Definition: A degree a has the 3-Uniform Escape Property

Kenneth Harris Characterizing lown Degrees 1'

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Dominate and Escape

Let f , g : NÑ N.

• f dominates g if

p@8xq�

f pxq ¡ gpxq�

f is a dominant function if f dominates everycomputable function.

• g escapes (domination from) f if

pD8xq�

f pxq ¤ gpxq�

f has an escape function if there is a computable gwhich escapes domination by f .

[email protected] Presented @ SEALS, Univ. of Florida 03/06

Page 3: A C lown Dpeople.cs.uchicago.edu/~kaharris/slides/lown-03-06.pdf · 3-Uniform Escape Property A degree a is low 3 if a3 03. Definition: A degree a has the 3-Uniform Escape Property

Kenneth Harris Characterizing lown Degrees 2'

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Martin’s Characterization

Theorem (Martin, 1966) Let a be a Turing degree.

• a is high�a1 ¥ 02

�iff there is an a-computable

dominant function:

pD f ¤ aqp@g ¤ 0q�

f dominates g�

• a is non-high�a1   02

�iff every a-computable

function has an escape function:

p@ f ¤ aqpDg ¤ 0q�g escapes f

[email protected] Presented @ SEALS, Univ. of Florida 03/06

Page 4: A C lown Dpeople.cs.uchicago.edu/~kaharris/slides/lown-03-06.pdf · 3-Uniform Escape Property A degree a is low 3 if a3 03. Definition: A degree a has the 3-Uniform Escape Property

Kenneth Harris Characterizing lown Degrees 3'

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Uniform Escape Property

Question: For what non-high degrees can escapefunctions be effectively produced?

Definition: A degree a has the Uniform EscapeProperty (UEP), or (1-UEP), when for any set A P a:

There is a partial computable λex.hepxq such thatwhenever ΦA

e is total, then

he total and escapes ΦAe

Recall, he escapes ΦAe if

pD8xq�ΦA

e pxq ¤ hepxq�

[email protected] Presented @ SEALS, Univ. of Florida 03/06

Page 5: A C lown Dpeople.cs.uchicago.edu/~kaharris/slides/lown-03-06.pdf · 3-Uniform Escape Property A degree a is low 3 if a3 03. Definition: A degree a has the 3-Uniform Escape Property

Kenneth Harris Characterizing lown Degrees 4'

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UEP Equivalent to low1

Theorem: For all degrees a TFAE

(A) a is low1�a1 ¤ 01

�.

(B) a has the Uniform Escape Property.

[email protected] Presented @ SEALS, Univ. of Florida 03/06

Page 6: A C lown Dpeople.cs.uchicago.edu/~kaharris/slides/lown-03-06.pdf · 3-Uniform Escape Property A degree a is low 3 if a3 03. Definition: A degree a has the 3-Uniform Escape Property

Kenneth Harris Characterizing lown Degrees 5'

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lown Degrees and Escape Functions

There is a hierarchy of properties characterized byprogressively less effective procedures, n-UniformEscape Property (n-UEP), starting with(1-UEP)=(UEP), such that

Theorem: For all degrees a and all n ¥ 1 TFAE

(A) a is lown�apnq ¤ 0pnq

�.

(B) a has (n-UEP).

[email protected] Presented @ SEALS, Univ. of Florida 03/06

Page 7: A C lown Dpeople.cs.uchicago.edu/~kaharris/slides/lown-03-06.pdf · 3-Uniform Escape Property A degree a is low 3 if a3 03. Definition: A degree a has the 3-Uniform Escape Property

Kenneth Harris Characterizing lown Degrees 6'

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Quantifiers on Steroids

p@8xq: For almost every x.Reduces to D@ and behaves like @.

pD8xq: There exists infinitely many x.Reduces to @D and behaves like D.

Fundamental Relations

D8 P ðñ @8 P

@ ùñ @8 ùñ D8 ùñ D

Theorem (Strong Normal Form):The arithmetic hierarchy is characterized byalternations of the two strongest quantifiers, @ and @8.

[email protected] Presented @ SEALS, Univ. of Florida 03/06

Page 8: A C lown Dpeople.cs.uchicago.edu/~kaharris/slides/lown-03-06.pdf · 3-Uniform Escape Property A degree a is low 3 if a3 03. Definition: A degree a has the 3-Uniform Escape Property

Kenneth Harris Characterizing lown Degrees 7'

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low1 D 1-U E P

[email protected] Presented @ SEALS, Univ. of Florida 03/06

Page 9: A C lown Dpeople.cs.uchicago.edu/~kaharris/slides/lown-03-06.pdf · 3-Uniform Escape Property A degree a is low 3 if a3 03. Definition: A degree a has the 3-Uniform Escape Property

Kenneth Harris Characterizing lown Degrees 8'

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low1 implies Uniform Escape Property

Theorem: All low1 sets A have (1-UEP):

There is a partial computable function λex.hepxqsuch that whenever ΦA

e is total, then

he total and escapes ΦAe

[email protected] Presented @ SEALS, Univ. of Florida 03/06

Page 10: A C lown Dpeople.cs.uchicago.edu/~kaharris/slides/lown-03-06.pdf · 3-Uniform Escape Property A degree a is low 3 if a3 03. Definition: A degree a has the 3-Uniform Escape Property

Kenneth Harris Characterizing lown Degrees 9'

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The Key Idea

Let A be low1, so ΠA2 � Π2.

Want: Computable g such that for each total ΦAe ,

Wgpeq satisfies

pescapeq pD8xqpDsq�ΦA

e,spxqÓ¤ s & x < Wgpeq,s�

ptotalq Wgpeq � ω

Define:hepxq � pµsq

�x P Wgpeq,s

Problem: How to match (escape) with (total)?

[email protected] Presented @ SEALS, Univ. of Florida 03/06

Page 11: A C lown Dpeople.cs.uchicago.edu/~kaharris/slides/lown-03-06.pdf · 3-Uniform Escape Property A degree a is low 3 if a3 03. Definition: A degree a has the 3-Uniform Escape Property

Kenneth Harris Characterizing lown Degrees 10'

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Strong Normal Form: Π2, Σ2

Normal Form: For V P Π2, there is some v (the Π2

index for V) with

Vpeq ðñ p@yq�xe, yy P Wv

Strong Normal Form (SNF): There is a computableg, such that for any V P Π2 with index e

Vpeq ùñ�Wgpv,eq � ω

� Vpeq ùñ

�Wgpv,eq finite

[email protected] Presented @ SEALS, Univ. of Florida 03/06

Page 12: A C lown Dpeople.cs.uchicago.edu/~kaharris/slides/lown-03-06.pdf · 3-Uniform Escape Property A degree a is low 3 if a3 03. Definition: A degree a has the 3-Uniform Escape Property

Kenneth Harris Characterizing lown Degrees 11'

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Implementation of Key Idea

Let A be low1 (thus ΠA2 � Π2).

Define ΠA2 predicate (escape)

pD8xqpDsq�ΦA

e,spxqÓ¤ s & x < Wgpv,eq,s�

where g is the computable function given by (SNF)from a Π2 index v for pescapeq:

pescapeq ùñ�Wgpv,eq � ω

� pescapeq ùñ

�Wgpv,eq finite

then define

hepxq � pµsq�x P Wgpv,eq,s

[email protected] Presented @ SEALS, Univ. of Florida 03/06

Page 13: A C lown Dpeople.cs.uchicago.edu/~kaharris/slides/lown-03-06.pdf · 3-Uniform Escape Property A degree a is low 3 if a3 03. Definition: A degree a has the 3-Uniform Escape Property

Kenneth Harris Characterizing lown Degrees 12'

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low2 D 2-U E P

[email protected] Presented @ SEALS, Univ. of Florida 03/06

Page 14: A C lown Dpeople.cs.uchicago.edu/~kaharris/slides/lown-03-06.pdf · 3-Uniform Escape Property A degree a is low 3 if a3 03. Definition: A degree a has the 3-Uniform Escape Property

Kenneth Harris Characterizing lown Degrees 13'

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2-Uniform Escape Property: First Change

Definition: A set A is low2 if A2 ¤ 02.

With low2 we add one jump class and one layer ofquantifier complexity.

Our first change in defining (2-UEP):

There are uniformly enumerable (u.e. ) families ofpartial computable functions λe.

he,y

(yPω such

that whenever ΦAe is total, then

p@8yq�

he,y total and escapes ΦAe

[email protected] Presented @ SEALS, Univ. of Florida 03/06

Page 15: A C lown Dpeople.cs.uchicago.edu/~kaharris/slides/lown-03-06.pdf · 3-Uniform Escape Property A degree a is low 3 if a3 03. Definition: A degree a has the 3-Uniform Escape Property

Kenneth Harris Characterizing lown Degrees 14'

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2-Uniform Escape Property

Definition: A degree a has the 2-Uniform EscapeProperty (2-UEP), when for any set A P a:

There are uniformly enumerable (u.e. ) families ofpartial computable functions λe.

he,y

(yPω such

that for any u.e. family of functions ΦA

e,y

(yPω

satisfyingp@8yq

�ΦA

e,y total�

then

p@8yq�

he,y total and escapes ΦAe,y

[email protected] Presented @ SEALS, Univ. of Florida 03/06

Page 16: A C lown Dpeople.cs.uchicago.edu/~kaharris/slides/lown-03-06.pdf · 3-Uniform Escape Property A degree a is low 3 if a3 03. Definition: A degree a has the 3-Uniform Escape Property

Kenneth Harris Characterizing lown Degrees 15'

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low2 Equivalent to 2-UEP

For all degrees a TFAE

(A) a is low2�a2 ¤ 02

�.

(B) a has the 2-Uniform Escape Property.

[email protected] Presented @ SEALS, Univ. of Florida 03/06

Page 17: A C lown Dpeople.cs.uchicago.edu/~kaharris/slides/lown-03-06.pdf · 3-Uniform Escape Property A degree a is low 3 if a3 03. Definition: A degree a has the 3-Uniform Escape Property

Kenneth Harris Characterizing lown Degrees 16'

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Strong Normal Form: Σ3, Π3

If A is low2 then ΣA3 � Σ3.

Strategy of Proof: Pump (escape) property (ΠA2 ) with

strong quantifiers to ΣA3 and exploit weakness of A.

p@8yqpD8xqpDsq�ΦA

e,y,spxqÓ¤ s & x < Wgpv,e,yq,s�

Strong Normal Form (SNF): There is a computableg, such that for any V P Σ3 with index e

pescapeq ùñ p@8yq�Wgpv,e,yq � ω

� pescapeq ùñ p@yq

�Wgpv,e,yq finite

[email protected] Presented @ SEALS, Univ. of Florida 03/06

Page 18: A C lown Dpeople.cs.uchicago.edu/~kaharris/slides/lown-03-06.pdf · 3-Uniform Escape Property A degree a is low 3 if a3 03. Definition: A degree a has the 3-Uniform Escape Property

Kenneth Harris Characterizing lown Degrees 17'

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low3 D B

[email protected] Presented @ SEALS, Univ. of Florida 03/06

Page 19: A C lown Dpeople.cs.uchicago.edu/~kaharris/slides/lown-03-06.pdf · 3-Uniform Escape Property A degree a is low 3 if a3 03. Definition: A degree a has the 3-Uniform Escape Property

Kenneth Harris Characterizing lown Degrees 18'

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3-Uniform Escape Property

A degree a is low3 if a3 � 03.

Definition: A degree a has the 3-Uniform EscapeProperty (3-UEP) when for any set A P a:

There are uniformly enumerable (u.e. ) families ofpartial computable functions λe.

he,y1,y2

(y1,y2Pω

such that for any u.e. family of functions ΦA

e,y1,y2

(y1,y2Pω

satisfying

pD8y2qp@8y1q

�ΦA

e,y1,y2total

then

pD8y2qp@8y1q

�he,y1,y2 total and escapes ΦA

e,y1,y2

[email protected] Presented @ SEALS, Univ. of Florida 03/06

Page 20: A C lown Dpeople.cs.uchicago.edu/~kaharris/slides/lown-03-06.pdf · 3-Uniform Escape Property A degree a is low 3 if a3 03. Definition: A degree a has the 3-Uniform Escape Property

Kenneth Harris Characterizing lown Degrees 19'

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low3 Equivalent to 3-UEP

Theorem: For all degrees a TFAE

(A) a is low3�a3 ¤ 03

�.

(B) a has the 3-Uniform Escape Property.

[email protected] Presented @ SEALS, Univ. of Florida 03/06

Page 21: A C lown Dpeople.cs.uchicago.edu/~kaharris/slides/lown-03-06.pdf · 3-Uniform Escape Property A degree a is low 3 if a3 03. Definition: A degree a has the 3-Uniform Escape Property

Kenneth Harris Characterizing lown Degrees 20'

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Strong Normal Form: Π4, Σ4

If A is low3 then ΠA4 � Π4.

Strategy of Proof: Pump (escape) property (ΠA2 ) with

strong quantifiers to ΠA4 and exploit weakness of A.

pD8y2qp@8y1qpD

8xqpDsq�ΦA

e,y1,y2,spxqÓ¤ s

& x < Wgpv,e,y1,y2q,s�

Strong Normal Form (SNF): There is a computableg, such that for any V P Π4 with index e

pescapeq ùñ p@y2qp@8y1q

�Wgpv,e,y1,y2q � ω

� pescapeq ùñ p@8y2qp@y1q

�Wgpv,e,y1,y2q finite

[email protected] Presented @ SEALS, Univ. of Florida 03/06

Page 22: A C lown Dpeople.cs.uchicago.edu/~kaharris/slides/lown-03-06.pdf · 3-Uniform Escape Property A degree a is low 3 if a3 03. Definition: A degree a has the 3-Uniform Escape Property

Kenneth Harris Characterizing lown Degrees 21'

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n-Uniform Escape Property

A degree a is lown if apnq � 0pnq.

Definition: A degree a has the n-Uniform EscapeProperty (n-UEP) when for any set A P a:

There are uniformly enumerable (u.e. ) families ofpartial computable functions λe.

he,y

(yPω such

that for any u.e. family of functions ΦA

e,y

(yPω

satisfying

pQ1yn�1qpQ2yn�2q . . .�ΦA

e,y total�

then

pQ1yn�1qpQ2yn�2q . . .

�he,y total and escapes ΦA

e,y

where Q1,Q2 P D8,@8

(by

• For odd n: alternate D8@8

• For even n: alternate @8D8

[email protected] Presented @ SEALS, Univ. of Florida 03/06

Page 23: A C lown Dpeople.cs.uchicago.edu/~kaharris/slides/lown-03-06.pdf · 3-Uniform Escape Property A degree a is low 3 if a3 03. Definition: A degree a has the 3-Uniform Escape Property

Kenneth Harris Characterizing lown Degrees 22'

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lown Equivalent to n-UEP

Theorem: For all degrees a TFAE

(A) a is lown�apnq ¤ 0pnq

�.

(B) a has the n-Uniform Escape Property.

[email protected] Presented @ SEALS, Univ. of Florida 03/06

Page 24: A C lown Dpeople.cs.uchicago.edu/~kaharris/slides/lown-03-06.pdf · 3-Uniform Escape Property A degree a is low 3 if a3 03. Definition: A degree a has the 3-Uniform Escape Property

Kenneth Harris Characterizing lown Degrees 23'

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Strong Normal Form Theorem

Strong Normal Form Theorem (SNF) (with n ¥ 1)All arithmetic formulas equivalent to formulas usingonly the beefiest quantifiers

@,@8

(:

For any V P Σ2n�1 with index v there is a computableg, such that

Vpeq ùñ p@8y2n�1qp@y2n�2q . . .�Wgpv,e,yq � ω

� Vpeq ùñ p@y2n�1qp@

8yy2n�2q . . .�Wgpv,e,yq finite

For any U P Π2n with index u there is a computable g,such that

Upeq ùñ p@y2n�2qp@8y2n�3q . . .

�Wgpu,e,yq � ω

� Upeq ùñ p@8y2n�3qp@y2n�3q . . .

�Wgpu,e,yq finite

[email protected] Presented @ SEALS, Univ. of Florida 03/06

Page 25: A C lown Dpeople.cs.uchicago.edu/~kaharris/slides/lown-03-06.pdf · 3-Uniform Escape Property A degree a is low 3 if a3 03. Definition: A degree a has the 3-Uniform Escape Property

Kenneth Harris Characterizing lown Degrees 24'

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A E F

[email protected] Presented @ SEALS, Univ. of Florida 03/06

Page 26: A C lown Dpeople.cs.uchicago.edu/~kaharris/slides/lown-03-06.pdf · 3-Uniform Escape Property A degree a is low 3 if a3 03. Definition: A degree a has the 3-Uniform Escape Property

Kenneth Harris Characterizing lown Degrees 25'

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Bounding Saturated Models

Theorem: There is a complete decidable theory Twhose types are all computable, which has nosaturated model of lown c.e. degree for any n.

[email protected] Presented @ SEALS, Univ. of Florida 03/06

Page 27: A C lown Dpeople.cs.uchicago.edu/~kaharris/slides/lown-03-06.pdf · 3-Uniform Escape Property A degree a is low 3 if a3 03. Definition: A degree a has the 3-Uniform Escape Property

Kenneth Harris Characterizing lown Degrees 26'

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Bibliography

My work:

Kenneth Harris, ”A Characterization of the lown

Degrees using Escape Functions, preprint atpeople.cs.uchicago.edu/�kaharris/papers/lown.pdf

Kenneth Harris, ”On Bounding SaturatedModels”, preprint atpeople.cs.uchicago.edu/�kaharris/papers/sat.pdf

[email protected] Presented @ SEALS, Univ. of Florida 03/06