Kenneth Harris Characterizing low n Degrees 0 A C - - PDF document

Kenneth Harris Characterizing lown Degrees

✬ ✫ ✩ ✪ A C   lown D K H

Department of Computer Science University of Chicago

http://people.cs.uchicago.edu/✒kaharris kaharris@uchicago.edu

kaharris@uchicago.edu Presented @ SEALS, Univ. of Florida 03/06

Kenneth Harris Characterizing lown Degrees 1

✬ ✫ ✩ ✪

Dominate and Escape

Let f, g : N Ñ N.

• f dominates g if

♣❅✽xq ✏ f♣xq → g♣xq ✘ f is a dominant function if f dominates every computable function.

• g escapes (domination from) f if

♣❉✽xq ✏ f♣xq ↕ g♣xq ✘ f has an escape function if there is a computable g which escapes domination by f.

kaharris@uchicago.edu Presented @ SEALS, Univ. of Florida 03/06

Kenneth Harris Characterizing lown Degrees 2

✬ ✫ ✩ ✪

Martin’s Characterization

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

• a is high
• a✶ ➙ 0✷✟

iff there is an a-computable dominant function: ♣❉ f ↕ aq♣❅g ↕ 0q ✏ f dominates g ✘

• a is non-high
• a✶ ➔ 0✷✟

iff every a-computable function has an escape function: ♣❅ f ↕ aq♣❉g ↕ 0q ✏ g escapes f ✘

kaharris@uchicago.edu Presented @ SEALS, Univ. of Florida 03/06

Kenneth Harris Characterizing lown Degrees 3

✬ ✫ ✩ ✪

Uniform Escape Property

Question: For what non-high degrees can escape functions be effectively produced? Definition: A degree a has the Uniform Escape Property (UEP), or (1-UEP), when for any set A P a: There is a partial computable λex.he♣xq such that whenever ΦA

e is total, then

he total and escapes ΦA

e

Recall, he escapes ΦA

e if

♣❉✽xq ✏ ΦA

e ♣xq ↕ he♣xq

✘

kaharris@uchicago.edu Presented @ SEALS, Univ. of Florida 03/06

Kenneth Harris Characterizing lown Degrees 4

✬ ✫ ✩ ✪

UEP Equivalent to low1

Theorem: For all degrees a TFAE (A) a is low1

• a✶ ↕ 0✶✟

. (B) a has the Uniform Escape Property.

kaharris@uchicago.edu Presented @ SEALS, Univ. of Florida 03/06

Kenneth Harris Characterizing lown Degrees 5

✬ ✫ ✩ ✪

lown Degrees and Escape Functions

There is a hierarchy of properties characterized by progressively less effective procedures, n-Uniform Escape Property (n-UEP), starting with (1-UEP)=(UEP), such that Theorem: For all degrees a and all n ➙ 1 TFAE (A) a is lown

• a♣nq ↕ 0♣nq✟

. (B) a has (n-UEP).

kaharris@uchicago.edu Presented @ SEALS, Univ. of Florida 03/06

Kenneth Harris Characterizing lown Degrees 6

✬ ✫ ✩ ✪

Quantifiers on Steroids

♣❅✽xq: For almost every x. Reduces to ❉❅ and behaves like ❅. ♣❉✽xq: There exists infinitely many x. Reduces to ❅❉ and behaves like ❉. Fundamental Relations ❉✽ P ðñ ✥❅✽✥ P ❅ ùñ ❅✽ ùñ ❉✽ ùñ ❉ Theorem (Strong Normal Form): The arithmetic hierarchy is characterized by alternations of the two strongest quantifiers, ❅ and ❅✽.

kaharris@uchicago.edu Presented @ SEALS, Univ. of Florida 03/06

Kenneth Harris Characterizing lown Degrees 7

✬ ✫ ✩ ✪ low1 D   1-U E P

kaharris@uchicago.edu Presented @ SEALS, Univ. of Florida 03/06

Kenneth Harris Characterizing lown Degrees 8

✬ ✫ ✩ ✪

low1 implies Uniform Escape Property

Theorem: All low1 sets A have (1-UEP): There is a partial computable function λex.he♣xq such that whenever ΦA

e is total, then

he total and escapes ΦA

e kaharris@uchicago.edu Presented @ SEALS, Univ. of Florida 03/06

Kenneth Harris Characterizing lown Degrees 9

✬ ✫ ✩ ✪

The Key Idea

Let A be low1, so ΠA

2 ⑨ Π2.

Want: Computable g such that for each total ΦA

e ,

Wg♣eq satisfies ♣escapeq ♣❉✽xq♣❉sq ✏ ΦA

e,s♣xqÓ↕ s & x Wg♣eq,s

✘ ♣totalq Wg♣eq ✏ ω Define: he♣xq ✏ ♣µsq ✏ x P Wg♣eq,s ✘ Problem: How to match (escape) with (total)?

kaharris@uchicago.edu Presented @ SEALS, Univ. of Florida 03/06

Kenneth Harris Characterizing lown Degrees 10

✬ ✫ ✩ ✪

Strong Normal Form: Π2, Σ2

Normal Form: For V P Π2, there is some v (the Π2 index for V) with V♣eq ðñ ♣❅yq ✏ ①e, y② P Wv ✘ Strong Normal Form (SNF): There is a computable g, such that for any V P Π2 with index e V♣eq ùñ ✏ Wg♣v,eq ✏ ω ✘ ✥V♣eq ùñ ✏ Wg♣v,eq finite ✘

kaharris@uchicago.edu Presented @ SEALS, Univ. of Florida 03/06

Kenneth Harris Characterizing lown Degrees 11

✬ ✫ ✩ ✪

Implementation of Key Idea

Let A be low1 (thus ΠA

2 ⑨ Π2).

Define ΠA

2 predicate (escape)

♣❉✽xq♣❉sq ✏ ΦA

e,s♣xqÓ↕ s & x Wg♣v,eq,s

✘ where g is the computable function given by (SNF) from a Π2 index v for ♣escapeq: ♣escapeq ùñ ✏ Wg♣v,eq ✏ ω ✘ ✥♣escapeq ùñ ✏ Wg♣v,eq finite ✘ then define he♣xq ✏ ♣µsq ✏ x P Wg♣v,eq,s ✘

kaharris@uchicago.edu Presented @ SEALS, Univ. of Florida 03/06

Kenneth Harris Characterizing lown Degrees 12

✬ ✫ ✩ ✪ low2 D   2-U E P

kaharris@uchicago.edu Presented @ SEALS, Univ. of Florida 03/06

Kenneth Harris Characterizing lown Degrees 13

✬ ✫ ✩ ✪

2-Uniform Escape Property: First Change

Definition: A set A is low2 if A✷ ↕ 0✷. With low2 we add one jump class and one layer of quantifier complexity. Our first change in defining (2-UEP): There are uniformly enumerable (u.e. ) families of partial computable functions λe. ✥ he,y ✭

yPω such

that whenever ΦA

e is total, then

♣❅✽yq ✒ he,y total and escapes ΦA

e

✚

kaharris@uchicago.edu Presented @ SEALS, Univ. of Florida 03/06

Kenneth Harris Characterizing lown Degrees 14

✬ ✫ ✩ ✪

2-Uniform Escape Property

Definition: A degree a has the 2-Uniform Escape Property (2-UEP), when for any set A P a: There are uniformly enumerable (u.e. ) families of partial computable functions λe. ✥ he,y ✭

yPω such

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

e,y

✭

yPω

satisfying ♣❅✽yq ✏ ΦA

e,y total

✘ then ♣❅✽yq ✒ he,y total and escapes ΦA

e,y

✚

kaharris@uchicago.edu Presented @ SEALS, Univ. of Florida 03/06

Kenneth Harris Characterizing lown Degrees 15

✬ ✫ ✩ ✪

low2 Equivalent to 2-UEP

For all degrees a TFAE (A) a is low2

• a✷ ↕ 0✷✟

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

kaharris@uchicago.edu Presented @ SEALS, Univ. of Florida 03/06

Kenneth Harris Characterizing lown Degrees 16

✬ ✫ ✩ ✪

Strong Normal Form: Σ3, Π3

If A is low2 then ΣA

3 ❸ Σ3.

Strategy of Proof: Pump (escape) property (ΠA

2) with

strong quantifiers to ΣA

3 and exploit weakness of A.

♣❅✽yq♣❉✽xq♣❉sq ✏ ΦA

e,y,s♣xqÓ↕ s & x Wg♣v,e,yq,s

✘ Strong Normal Form (SNF): There is a computable g, such that for any V P Σ3 with index e ♣escapeq ùñ ♣❅✽yq ✏ Wg♣v,e,yq ✏ ω ✘ ✥♣escapeq ùñ ♣❅yq ✏ Wg♣v,e,yq finite ✘

kaharris@uchicago.edu Presented @ SEALS, Univ. of Florida 03/06

Kenneth Harris Characterizing lown Degrees 17

✬ ✫ ✩ ✪ low3 D  B

kaharris@uchicago.edu Presented @ SEALS, Univ. of Florida 03/06

Kenneth Harris Characterizing lown Degrees 18

✬ ✫ ✩ ✪

3-Uniform Escape Property

A degree a is low3 if a✸ ✏ 0✸. Definition: A degree a has the 3-Uniform Escape Property (3-UEP) when for any set A P a: There are uniformly enumerable (u.e. ) families of partial computable functions λe. ✥ he,y1,y2 ✭

y1,y2Pω

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

e,y1,y2

✭

y1,y2Pω satisfying

♣❉✽y2q♣❅✽y1q ✏ ΦA

e,y1,y2 total

✘ then ♣❉✽y2q♣❅✽y1q ✒ he,y1,y2 total and escapes ΦA

e,y1,y2

✚

kaharris@uchicago.edu Presented @ SEALS, Univ. of Florida 03/06

Kenneth Harris Characterizing lown Degrees 19

✬ ✫ ✩ ✪

low3 Equivalent to 3-UEP

Theorem: For all degrees a TFAE (A) a is low3

• a✸ ↕ 0✸✟

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

kaharris@uchicago.edu Presented @ SEALS, Univ. of Florida 03/06

Kenneth Harris Characterizing lown Degrees 20

✬ ✫ ✩ ✪

Strong Normal Form: Π4, Σ4

If A is low3 then ΠA

4 ❸ Π4.

Strategy of Proof: Pump (escape) property (ΠA

2) with

strong quantifiers to ΠA

4 and exploit weakness of A.

♣❉✽y2q♣❅✽y1q♣❉✽xq♣❉sq ✏ ΦA

e,y1,y2,s♣xqÓ↕ s

& x Wg♣v,e,y1,y2q,s ✘ Strong Normal Form (SNF): There is a computable g, such that for any V P Π4 with index e ♣escapeq ùñ ♣❅y2q♣❅✽y1q ✏ Wg♣v,e,y1,y2q ✏ ω ✘ ✥♣escapeq ùñ ♣❅✽y2q♣❅y1q ✏ Wg♣v,e,y1,y2q finite ✘

kaharris@uchicago.edu Presented @ SEALS, Univ. of Florida 03/06

Kenneth Harris Characterizing lown Degrees 21

✬ ✫ ✩ ✪

n-Uniform Escape Property

A degree a is lown if a♣nq ✏ 0♣nq. Definition: A degree a has the n-Uniform Escape Property (n-UEP) when for any set A P a: There are uniformly enumerable (u.e. ) families of partial computable functions λe. ✥ he,y ✭

yPω such

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

e,y

✭

yPω

satisfying ♣Q1yn✁1q♣Q2yn✁2q . . . ✏ ΦA

e,y total

✘ then ♣Q1yn✁1q♣Q2yn✁2q . . . ✒ he,y total and escapes ΦA

e,y

✚ where Q1, Q2 P ✥ ❉✽, ❅✽✭ by

• For odd n: alternate ❉✽❅✽
• For even n: alternate ❅✽❉✽

kaharris@uchicago.edu Presented @ SEALS, Univ. of Florida 03/06

Kenneth Harris Characterizing lown Degrees 22

✬ ✫ ✩ ✪

lown Equivalent to n-UEP

Theorem: For all degrees a TFAE (A) a is lown

• a♣nq ↕ 0♣nq✟

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

kaharris@uchicago.edu Presented @ SEALS, Univ. of Florida 03/06

Kenneth Harris Characterizing lown Degrees 23

✬ ✫ ✩ ✪

Strong Normal Form Theorem

Strong Normal Form Theorem (SNF) (with n ➙ 1) All arithmetic formulas equivalent to formulas using

• nly the beefiest quantifiers

✥ ❅, ❅✽✭ : For any V P Σ2n1 with index v there is a computable g, such that V♣eq ùñ ♣❅✽y2n✁1q♣❅y2n✁2q . . . ✏ Wg♣v,e,yq ✏ ω ✘ ✥V♣eq ùñ ♣❅y2n✁1q♣❅✽yy2n✁2q . . . ✏ Wg♣v,e,yq finite ✘ For any U P Π2n with index u there is a computable g, such that U♣eq ùñ ♣❅y2n✁2q♣❅✽y2n✁3q . . . ✏ Wg♣u,e,yq ✏ ω ✘ ✥U♣eq ùñ ♣❅✽y2n✁3q♣❅y2n✁3q . . . ✏ Wg♣u,e,yq finite ✘

kaharris@uchicago.edu Presented @ SEALS, Univ. of Florida 03/06

Kenneth Harris Characterizing lown Degrees 24

✬ ✫ ✩ ✪ A  E F

kaharris@uchicago.edu Presented @ SEALS, Univ. of Florida 03/06

Kenneth Harris Characterizing lown Degrees 25

✬ ✫ ✩ ✪

Bounding Saturated Models

Theorem: There is a complete decidable theory T whose types are all computable, which has no saturated model of lown c.e. degree for any n.

kaharris@uchicago.edu Presented @ SEALS, Univ. of Florida 03/06

Kenneth Harris Characterizing lown Degrees 26

✬ ✫ ✩ ✪

Bibliography

My work: Kenneth Harris, ”A Characterization of the lown Degrees using Escape Functions, preprint at

people.cs.uchicago.edu/✒kaharris/papers/lown.pdf

Kenneth Harris, ”On Bounding Saturated Models”, preprint at

people.cs.uchicago.edu/✒kaharris/papers/sat.pdf

kaharris@uchicago.edu Presented @ SEALS, Univ. of Florida 03/06