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


  1. ✬ ✩ Kenneth Harris Saturated Bounding Degrees 0 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

  2. ✬ ✩ Kenneth Harris Saturated Bounding Degrees 1 Saturated Models ( Vaught ) countably saturated models : Models which realize all types consistent over some finite subdomain of the model. ( Millar, Morley ) A complete , decidable (CD) theory has a decidable saturated model i ff 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

  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 D e ( A ) ≤ T d . Question . Which Turing degrees are saturated bounding? ✫ ✪ kaharris@uchicago.edu Master’s Presentation 12 / 1 / 04

  4. ✬ ✩ Kenneth Harris Saturated Bounding Degrees 3 Saturated Bounding Degrees Positive Degrees Negative Degrees ���������������������������������������������� 0 ′ ���������������������� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ���������������������� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � 0 ✫ ✪ kaharris@uchicago.edu Master’s Presentation 12 / 1 / 04

  5. ✬ ✩ Kenneth Harris Saturated Bounding Degrees 4 PAC Π 0 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 Π 0 1 class is [ T ], for some PAC tree T . Lemma . The types of a CD theory with TAC form a PAC Π 0 1 class. ✫ ✪ kaharris@uchicago.edu Master’s Presentation 12 / 1 / 04

  6. ✬ ✩ Kenneth Harris Saturated Bounding Degrees 5 Enumerating Paths in a Π 0 1 Class • A Turing degree d has an enumeration of a class C ⊂ 2 ω if there is a d -computable function of two arguments, λ nx . F n ( x ) such that C = � λ x . F n ( x ) � n ∈ ω • A Turing degree d has a subenumeration of a class C ⊂ 2 ω if there is a d -computable function of two arguments, λ nx . F n ( x ) such that C ⊂ � λ x . F n ( 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

  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 of the types for any CD theory with TAC. (c) There is a d -computable enumeration of each PAC Π 0 1 classes. Question . Which Turing degrees can enumerate each PAC Π 0 1 class? ✫ ✪ kaharris@uchicago.edu Master’s Presentation 12 / 1 / 04

  8. ✬ ✩ Kenneth Harris Saturated Bounding Degrees 7 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 Π 0 1 class. ✫ ✪ kaharris@uchicago.edu Master’s Presentation 12 / 1 / 04

  9. ✬ ✩ Kenneth Harris Saturated Bounding Degrees 8 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 Π 0 1 class. ✫ ✪ kaharris@uchicago.edu Master’s Presentation 12 / 1 / 04

  10. ✬ ✩ Kenneth Harris Saturated Bounding Degrees 9 Saturated Bounding Degrees Positive Degrees Negative Degrees ����������������������������������������������� 0 ′ ������������������������ � � � � � � � � � � � � � � � high � high high � � � � � � � � � � � � � � � � � � PA PA � � � � � � � � � � � � � � � �������������������� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � 0 ✫ ✪ kaharris@uchicago.edu Master’s Presentation 12 / 1 / 04

  11. ✬ ✩ Kenneth Harris Saturated Bounding Degrees 10 New Negative Result Thm 1 . No low c.e. degree is saturated bounding. ✫ ✪ kaharris@uchicago.edu Master’s Presentation 12 / 1 / 04

  12. ✬ ✩ Kenneth Harris Saturated Bounding Degrees 11 Saturated Bounding Degrees Positive Degrees Negative Degrees ������������������������������������������������� 0 ′ ������������������������ � � � � � � � � � � � � � � � � high high high � � � � � � � � � � � � � � � � � � PA PA � � � � � � � � � � � � � � ���������������������� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � low � � � � � � � � � � � � � � � � 0 ✫ ✪ kaharris@uchicago.edu Master’s Presentation 12 / 1 / 04

  13. ✬ ✩ Kenneth Harris Saturated Bounding Degrees 12 Millar’s Construction Against Computable Opponent Opponent: � Φ n � −→ Our Avoiding Path: −→ Φ 0 Φ 1 Φ 2 → ���� → ���� ���� ���� ���� ���� → → → → . . . . . . ↓ ↓ ↓ ✫ ✪ kaharris@uchicago.edu Master’s Presentation 12 / 1 / 04

  14. ✬ ✩ Kenneth Harris Saturated Bounding Degrees 13 Di ffi culties with Construction Against Noncomputable Opponent Opponent: � Φ n � −→ Our Avoiding Path: −→ Φ 0 → ���� ���� → . . . . . . → ↓ ✫ ✪ kaharris@uchicago.edu Master’s Presentation 12 / 1 / 04

  15. ✬ ✩ Kenneth Harris Saturated Bounding Degrees 14 Di ffi culties with Construction Against Noncomputable Opponent Opponent: � Φ n � −→ Our Avoiding Path: −→ Φ 0 Φ 0 → ���� ���� → ���� ���� → . . . . . . → → → ↓ ↓ ✫ ✪ kaharris@uchicago.edu Master’s Presentation 12 / 1 / 04

  16. ✬ ✩ Kenneth Harris Saturated Bounding Degrees 15 Di ffi culties with Construction Against Noncomputable Opponent Opponent: � Φ n � −→ Our Avoiding Path: −→ Φ 0 Φ 0 Φ 0 → ���� ���� → ���� ���� ���� ���� → → → → → → . . . . . . → → ↓ ↓ ↓ ✫ ✪ kaharris@uchicago.edu Master’s Presentation 12 / 1 / 04

  17. ✬ ✩ Kenneth Harris Saturated Bounding Degrees 16 Modified Construction Against Noncomputable Opponent Opponent: � Φ n � −→ Our Avoiding Path: −→ Φ 0 Φ 0 Φ 0 Φ 1 Φ 1 Φ 2 → ���� → ���� ���� ���� ���� ���� → → → → . . . . . . � � � � ������� � � � � ↓ ↓ ↓ � � � � � � ✫ ✪ kaharris@uchicago.edu Master’s Presentation 12 / 1 / 04

  18. ✬ ✩ Kenneth Harris Saturated Bounding Degrees 17 Modified Construction Against Noncomputable Opponent Opponent: � Φ n � −→ Our Avoiding Path: −→ Φ 0 Φ 0 Φ 0 Φ 1 Φ 1 Φ 2 → ���� ���� → ���� ���� ���� ���� → → → → → . . . . . . � � � → � → ������� � � � � ↓ ↓ ↓ � � � � � � ✫ ✪ kaharris@uchicago.edu Master’s Presentation 12 / 1 / 04

  19. ✬ ✩ Kenneth Harris Saturated Bounding Degrees 18 Modified Construction Against Noncomputable Opponent Opponent: � Φ n � −→ Our Avoiding Path: −→ Φ 0 Φ 0 Φ 0 Φ 1 Φ 1 Φ 2 → ���� ���� → ���� ���� ���� ���� → → → → → . . . . . . G � � → � � ������� � � � � ↓ ↓ ↓ � � � � � � ✫ ✪ kaharris@uchicago.edu Master’s Presentation 12 / 1 / 04

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