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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] Presented @ SEALS, Univ. of Florida 03/06
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
Kenneth Harris Characterizing lown Degrees 17'
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low3 D B
[email protected] Presented @ SEALS, Univ. of Florida 03/06
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
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
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
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
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
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
Kenneth Harris Characterizing lown Degrees 24'
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A E F
[email protected] Presented @ SEALS, Univ. of Florida 03/06
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
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