Kenneth Harris Saturated Bounding Degrees 0 S B - - PDF document

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Kenneth Harris Saturated Bounding Degrees 0 S B - - PDF document

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Kenneth Harris Saturated Bounding Degrees 0 S B D K H Department of Computer Science University of Chicago

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SLIDE 1

Kenneth Harris Saturated Bounding Degrees

✬ ✫ ✩ ✪ S B D K H

Department of Computer Science University of Chicago

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

kaharris@uchicago.edu Master’s Presentation 12/1/04

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Kenneth Harris Saturated Bounding Degrees 1

✬ ✫ ✩ ✪

Saturated Models (Vaught) countably saturated models: Models which realize all types consistent

  • ver some finite subdomain of the model.

(Millar, Morley) A complete, decidable (CD) theory has a decidable saturated model iff there is a computable enumeration of the types of the theory. (Millar) There is a complete, decidable (CD) theory with types all computable (TAC) which has no decidable saturated model. (Millar, Goncharov/Nurtazin) Every CD theory with TAC has a 0′-decidable saturated model.

kaharris@uchicago.edu Master’s Presentation 12/1/04

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SLIDE 3

Kenneth Harris Saturated Bounding Degrees 2

✬ ✫ ✩ ✪

Saturated Bounding Degrees

  • Def. Degree d is saturated bounding if

for any CD theory T with TAC, there is a saturated A |= T with De(A) ≤T d.

  • Question. Which Turing degrees are

saturated bounding?

kaharris@uchicago.edu Master’s Presentation 12/1/04

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SLIDE 4

Kenneth Harris Saturated Bounding Degrees 3

✬ ✫ ✩ ✪

Saturated Bounding Degrees Positive Degrees Negative Degrees

0′

  • kaharris@uchicago.edu

Master’s Presentation 12/1/04

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SLIDE 5

Kenneth Harris Saturated Bounding Degrees 4

✬ ✫ ✩ ✪

PAC Π0

1 Classes

  • An extendable tree is one without terminal nodes.
  • A PAC tree is an extendable computable tree

whose paths are all computable.

  • A PAC Π0

1 class is [T ], for some PAC tree T .

  • Lemma. The types of a CD theory with TAC

form a PAC Π0

1 class.

kaharris@uchicago.edu Master’s Presentation 12/1/04

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SLIDE 6

Kenneth Harris Saturated Bounding Degrees 5

✬ ✫ ✩ ✪

Enumerating Paths in a Π0

1 Class

  • A Turing degree d has an enumeration of

a class C ⊂ 2ω if there is a d-computable function of two arguments, λnx.Fn(x) such that C = λx.Fn(x)

n∈ω

  • A Turing degree d has a subenumeration
  • f a class C ⊂ 2ω if there is a

d-computable function of two arguments, λnx.Fn(x) such that C ⊂ λx.Fn(x)

n∈ω

  • Lemma. For any extendable computable

tree, subenumerating [T ] is equivalent to enumerating [T ].

kaharris@uchicago.edu Master’s Presentation 12/1/04

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SLIDE 7

Kenneth Harris Saturated Bounding Degrees 6

✬ ✫ ✩ ✪

Saturated Bounding and Enumerations

  • Thm. (Millar, Morley) TFAE

(a) d is saturated bounding. (b) There is a d-computable enumeration

  • f the types for any CD theory with

TAC.

(c) There is a d-computable enumeration

  • f each PAC Π0

1 classes.

  • Question. Which Turing degrees can

enumerate each PAC Π0

1 class?

kaharris@uchicago.edu Master’s Presentation 12/1/04

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SLIDE 8

Kenneth Harris Saturated Bounding Degrees 7

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Upper Bound: high degrees

A degree d is high if d′ ≥ 0′′.

  • Thm. (Jockusch) Any high degree can

enumerate the computable sets.

  • Thm. Any high degree can enumerate

every PAC Π0

1 class.

kaharris@uchicago.edu Master’s Presentation 12/1/04

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Kenneth Harris Saturated Bounding Degrees 8

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Upper Bound: PA degrees

A degree d is a PA degree if d computes a completion of Peano Arithmetic.

  • Thm. (Jockusch) Any PA degree can

subenumerate the computable sets.

  • Thm. Any PA degree can enumerate

every PAC Π0

1 class.

kaharris@uchicago.edu Master’s Presentation 12/1/04

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Kenneth Harris Saturated Bounding Degrees 9

✬ ✫ ✩ ✪

Saturated Bounding Degrees Positive Degrees Negative Degrees

0′ high high high PA PA

  • kaharris@uchicago.edu

Master’s Presentation 12/1/04

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SLIDE 11

Kenneth Harris Saturated Bounding Degrees 10

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New Negative Result Thm 1. No low c.e. degree is saturated bounding.

kaharris@uchicago.edu Master’s Presentation 12/1/04

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SLIDE 12

Kenneth Harris Saturated Bounding Degrees 11

✬ ✫ ✩ ✪

Saturated Bounding Degrees Positive Degrees Negative Degrees

0′ high high high PA PA

  • low
  • kaharris@uchicago.edu

Master’s Presentation 12/1/04

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SLIDE 13

Kenneth Harris Saturated Bounding Degrees 12

✬ ✫ ✩ ✪

Millar’s Construction Against Computable Opponent Opponent: Φn

  • −→

Our Avoiding Path: −→

Φ0 Φ1 Φ2

→ →

  • ↓

→ →

  • ↓

→

  • ↓

→

. . . . . .

kaharris@uchicago.edu Master’s Presentation 12/1/04

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SLIDE 14

Kenneth Harris Saturated Bounding Degrees 13

✬ ✫ ✩ ✪

Difficulties with Construction Against Noncomputable Opponent Opponent: Φn

  • −→

Our Avoiding Path: −→

Φ0

→ →

  • ↓

→ . . . . . . kaharris@uchicago.edu Master’s Presentation 12/1/04

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SLIDE 15

Kenneth Harris Saturated Bounding Degrees 14

✬ ✫ ✩ ✪

Difficulties with Construction Against Noncomputable Opponent Opponent: Φn

  • −→

Our Avoiding Path: −→

Φ0 Φ0

→ →

  • ↓

→ → →

  • ↓

→ . . . . . . kaharris@uchicago.edu Master’s Presentation 12/1/04

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SLIDE 16

Kenneth Harris Saturated Bounding Degrees 15

✬ ✫ ✩ ✪

Difficulties with Construction Against Noncomputable Opponent Opponent: Φn

  • −→

Our Avoiding Path: −→

Φ0 Φ0 Φ0

→ →

  • ↓

→ → →

  • ↓

→ → →

  • ↓

→ →

. . . . . .

kaharris@uchicago.edu Master’s Presentation 12/1/04

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SLIDE 17

Kenneth Harris Saturated Bounding Degrees 16

✬ ✫ ✩ ✪

Modified Construction Against Noncomputable Opponent Opponent: Φn

  • −→

Our Avoiding Path: −→

Φ0 Φ0 Φ1 Φ0 Φ1 Φ2

→ →

  • ↓

→ →

  • ↓
  • →
  • ↓
  • →

. . . . . .

kaharris@uchicago.edu Master’s Presentation 12/1/04

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SLIDE 18

Kenneth Harris Saturated Bounding Degrees 17

✬ ✫ ✩ ✪

Modified Construction Against Noncomputable Opponent Opponent: Φn

  • −→

Our Avoiding Path: −→

Φ0 Φ0 Φ1 Φ0 Φ1 Φ2

→ →

  • ↓

→ → →

  • ↓
  • →

→

  • ↓
  • →

→

. . . . . .

kaharris@uchicago.edu Master’s Presentation 12/1/04

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SLIDE 19

Kenneth Harris Saturated Bounding Degrees 18

✬ ✫ ✩ ✪

Modified Construction Against Noncomputable Opponent Opponent: Φn

  • −→

Our Avoiding Path: −→

Φ0 Φ0 Φ1 Φ0 Φ1 Φ2

→ →

  • ↓

→ → →

  • ↓
  • →

→

  • G
  • ↓
  • →

. . . . . .

kaharris@uchicago.edu Master’s Presentation 12/1/04

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Kenneth Harris Saturated Bounding Degrees 19

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Escape Property

  • Def. A degree d has the Escape Property if

(∀Φ ≤T d)(∃ f ≤T 0)(∃∞x)Φ(x) ≤ f(x)

The computable functions escape domination from any d-computable function.

  • Thm. (Martin) All nonhigh degrees have the

Escape Property.

kaharris@uchicago.edu Master’s Presentation 12/1/04

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Kenneth Harris Saturated Bounding Degrees 20

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Escape Property Escaping Values

. . .

  • 6
  • 5
  • 4
  • 3
  • 2
  • 1
  • kaharris@uchicago.edu

Master’s Presentation 12/1/04

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Kenneth Harris Saturated Bounding Degrees 21

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Review: Modified Construction Needed Nodes on Path Opponent: Φn

  • −→

Our Avoiding Path: −→

Φ0 Φ0 Φ1 Φ0 Φ1 Φ2

→ →

  • 1

↓ → → →

  • 3

↓

  • →

→

  • 5
  • ↓
  • →

→

. . . . . .

kaharris@uchicago.edu Master’s Presentation 12/1/04

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Kenneth Harris Saturated Bounding Degrees 22

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Escape Property Escaping Values Values Needed by construction

. . .

  • 6
  • 5
  • 4
  • 3
  • 2
  • 1
  • kaharris@uchicago.edu

Master’s Presentation 12/1/04

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Kenneth Harris Saturated Bounding Degrees 23

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Uniform Escape Property

  • Def. A degree d has the Uniform Escape

Property if there is an h ≤T 0 such that

(∀e)(∃∞x)Φd

e(x) ≤ Φh(e)(x)

Escape functions for d-computable functions can be found effectively.

Thm 2. For c.e. degrees d, the following are equivalent

(a) d is low. (b) d has the Uniform Escape Property

kaharris@uchicago.edu Master’s Presentation 12/1/04

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Kenneth Harris Saturated Bounding Degrees 24

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Aligned Escape Property

  • Def. A degree d has the Aligned Escape

Property if for any g ≤T 0 where ϕg(e) is total provided ϕe is total, (∀Φ ≤T d)(∃e)(∃∞x)Φ(ϕg(e)(x)) ≤ ϕe(x)

  • Lemma. No degree with the Aligned Escape

Property is saturated bounding.

kaharris@uchicago.edu Master’s Presentation 12/1/04

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Kenneth Harris Saturated Bounding Degrees 25

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Aligned Escape Property Escaping Values Values Needed by construction Aligned Values

. . .

  • 6
  • 5
  • 4
  • 3
  • 2
  • 1
  • kaharris@uchicago.edu

Master’s Presentation 12/1/04

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Kenneth Harris Saturated Bounding Degrees 26

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Extending Negative Results

A degree d is lown if d(n) = 0(n), for some n ∈ ω.

Thm 3. All lown c.e. degrees have the Aligned Escape Property.

  • Cor. No lown c.e. degree is saturated

bounding.

kaharris@uchicago.edu Master’s Presentation 12/1/04

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Kenneth Harris Saturated Bounding Degrees 27

✬ ✫ ✩ ✪

Saturated Bounding Degrees Positive Degrees Negative Degrees

0′ high high high PA PA

  • lown

low

  • kaharris@uchicago.edu

Master’s Presentation 12/1/04