A Generic Set That
Does Not Bound A
Minimal Pair
Mariya I. Soskova
A Generic Set That Does Not Bound A
Minimal Pair
Mariya I. Soskova
University of Leeds
Department of Pure Mathematics
17.05.2006
Background
A Generic Set That
Does Not Bound A
Minimal Pair
Mariya I. Soskova
I
Cooper, Sorbi, Lee and Yang
1. Every ∆02 e-degree bounds a minimal pair.
2. There exists a Σ02 e-degree that does not bound a
minimal pair.
I
Copestake
1. Every 2-generic e-degree bounds a minimal pair
Main Result
A Generic Set That
Does Not Bound A
Minimal Pair
Mariya I. Soskova
Theorem
There exists a 1-generic enumeration degree a, that does
not bound a minimal pair in the semi-lattice of the
enumeration degrees.
Definitions
A Generic Set That
Does Not Bound A
Minimal Pair
Mariya I. Soskova
Definition
A set A is 1-generic if for every c.e. set S the following
condition holds:
∃τ ⊆ χA (τ ∈ S ∨ ∀µ ⊇ τ (µ ∈
/ S)).
Definition
Let a and b be two enumeration degrees. We say that a
and b form a minimal pair in the semi-lattice of the
enumeration degrees if:
1. a > 0 and b > 0.
2. For every enumeration degree c
(c ≤ a ∧ c ≤ b → c = 0).
The Requirements
A Generic Set That
Does Not Bound A
Minimal Pair
Mariya I. Soskova
I
GW : ∃τ ⊆ χA (τ ∈ W ∨ ∀µ ⊇ τ (µ ∈
/ W ))
I
R Θ0 Θ1 : Θ0 (A) = X − c.e. ∨ Θ1 (A) = Y − c.e.∨
∨∃Φ0 , Φ1 ((Φ0 (X ) = Φ1 (Y ) = D)∧∀W −c.e.(W 6= D))
⇓
S W : (X − c.e ∨ Y − c.e. ∨ (Φ0 (X ) = Φ1 (Y ) = D∧
∧∃z(W (z) 6= D(z))))
The Requirements
A Generic Set That
Does Not Bound A
Minimal Pair
Mariya I. Soskova
I
GW : ∃τ ⊆ χA (τ ∈ W ∨ ∀µ ⊇ τ (µ ∈
/ W ))
I
R Θ0 Θ1 : Θ0 (A) = X − c.e. ∨ Θ1 (A) = Y − c.e.∨
∨∃Φ0 , Φ1 ((Φ0 (X ) = Φ1 (Y ) = D)∧∀W −c.e.(W 6= D))
⇓
S W : (X − c.e ∨ Y − c.e. ∨ (Φ0 (X ) = Φ1 (Y ) = D∧
∧∃z(W (z) 6= D(z))))
The Requirements
A Generic Set That
Does Not Bound A
Minimal Pair
Mariya I. Soskova
I
GW : ∃τ ⊆ χA (τ ∈ W ∨ ∀µ ⊇ τ (µ ∈
/ W ))
I
R Θ0 Θ1 : Θ0 (A) = X − c.e. ∨ Θ1 (A) = Y − c.e.∨
∨∃Φ0 , Φ1 ((Φ0 (X ) = Φ1 (Y ) = D)∧∀W −c.e.(W 6= D))
⇓
S W : (X − c.e ∨ Y − c.e. ∨ (Φ0 (X ) = Φ1 (Y ) = D∧
∧∃z(W (z) 6= D(z))))
The Requirements
A Generic Set That
Does Not Bound A
Minimal Pair
Mariya I. Soskova
I
GW : ∃τ ⊆ χA (τ ∈ W ∨ ∀µ ⊇ τ (µ ∈
/ W ))
I
R Θ0 Θ1 : Θ0 (A) = X − c.e. ∨ Θ1 (A) = Y − c.e.∨
∨∃Φ0 , Φ1 ((Φ0 (X ) = Φ1 (Y ) = D)∧∀W −c.e.(W 6= D))
⇓
S W : (X − c.e ∨ Y − c.e. ∨ (Φ0 (X ) = Φ1 (Y ) = D∧
∧∃z(W (z) 6= D(z))))
The Construction
A Generic Set That
Does Not Bound A
Minimal Pair
Mariya I. Soskova
I
Priority tree T with nodes labelled by the strategies
I
At stage s : δs , As
I
A0s = N
I
Approximations to c.e. sets.
The GW - Strategy
A Generic Set That
Does Not Bound A
Minimal Pair
Mariya I. Soskova
γ
γˆ0
I
Choose a finite string λγ
I
Ask ∃µ ⊃ λγ (µ ∈ W ) ?
I
Yes : outcome 0
I
No : outcome 1
γˆ1
The GW - Strategy
A Generic Set That
Does Not Bound A
Minimal Pair
Mariya I. Soskova
γ
γˆ0
I
Choose a finite string λγ
I
Ask ∃µ ⊃ λγ (µ ∈ W ) ?
I
Yes : outcome 0
I
No : outcome 1
γˆ1
The GW - Strategy
A Generic Set That
Does Not Bound A
Minimal Pair
Mariya I. Soskova
γ
γˆ0
I
Choose a finite string λγ
I
Ask ∃µ ⊃ λγ (µ ∈ W ) ?
I
Yes : outcome 0
I
No : outcome 1
γˆ1
A Generic Set That
Does Not Bound A
Minimal Pair
The R Strategy
Mariya I. Soskova
α
αˆ0
I
Check all S W substrategies.
I
At this level operators Φ0 and Φ1 are built
I
Outcome 0
The S W Strategy – o− = ∞X
A Generic Set That
Does Not Bound A
Minimal Pair
Mariya I. Soskova
β
βˆ∞X
I
I
...
βˆh∞Y , ki
βˆhl, ki
βˆd0
Build a set U, aiming to prove that X is c.e.
Scan U for errors.
I
I
No errors found - outcome ∞X
An error found at point k - build agitator set Ek for k ,
such that k ∈ X ⇔ Ek ⊂ A,
Outcome h∞Y , k i
The S W Strategy – o− = ∞X
A Generic Set That
Does Not Bound A
Minimal Pair
Mariya I. Soskova
β
βˆ∞X
I
I
...
βˆh∞Y , ki
βˆhl, ki
βˆd0
Build a set U, aiming to prove that X is c.e.
Scan U for errors.
I
I
No errors found - outcome ∞X
An error found at point k - build agitator set Ek for k ,
such that k ∈ X ⇔ Ek ⊂ A,
Outcome h∞Y , k i
The S W Strategy – o− = ∞X
A Generic Set That
Does Not Bound A
Minimal Pair
Mariya I. Soskova
β
βˆ∞X
I
I
...
βˆh∞Y , ki
βˆhl, ki
βˆd0
Build a set U, aiming to prove that X is c.e.
Scan U for errors.
I
I
No errors found - outcome ∞X
An error found at point k - build agitator set Ek for k ,
such that k ∈ X ⇔ Ek ⊂ A,
Outcome h∞Y , k i
The S W Strategy – o− = h∞Y , k i
A Generic Set That
Does Not Bound A
Minimal Pair
Mariya I. Soskova
β
βˆ∞X
βˆh∞Y , k i
βˆhl, ki
...
βˆd0
I
Check if the agitator for k is still valid. If not the
outcome is ∞X
I
Build a set Vk , aiming to prove that Y is c.e.
Scan Vk for errors.
I
I
I
No errors - outcome h∞Y , k i
An error found at element l, then build an agitator for
l, enumerate a witness z in D: hz, {k }i & Φ0 and
hz, {l}i & Φ1 , outcome d0
The S W Strategy – o− = h∞Y , k i
A Generic Set That
Does Not Bound A
Minimal Pair
Mariya I. Soskova
β
βˆ∞X
βˆh∞Y , k i
βˆhl, ki
...
βˆd0
I
Check if the agitator for k is still valid. If not the
outcome is ∞X
I
Build a set Vk , aiming to prove that Y is c.e.
Scan Vk for errors.
I
I
I
No errors - outcome h∞Y , k i
An error found at element l, then build an agitator for
l, enumerate a witness z in D: hz, {k }i & Φ0 and
hz, {l}i & Φ1 , outcome d0
The S W Strategy – o− = h∞Y , k i
A Generic Set That
Does Not Bound A
Minimal Pair
Mariya I. Soskova
β
βˆ∞X
βˆh∞Y , k i
βˆhl, ki
...
βˆd0
I
Check if the agitator for k is still valid. If not the
outcome is ∞X
I
Build a set Vk , aiming to prove that Y is c.e.
Scan Vk for errors.
I
I
I
No errors - outcome h∞Y , k i
An error found at element l, then build an agitator for
l, enumerate a witness z in D: hz, {k }i & Φ0 and
hz, {l}i & Φ1 , outcome d0
The S W Strategy – o− = h∞Y , k i
A Generic Set That
Does Not Bound A
Minimal Pair
Mariya I. Soskova
β
βˆ∞X
βˆh∞Y , k i
βˆhl, ki
...
βˆd0
I
Check if the agitator for k is still valid. If not the
outcome is ∞X
I
Build a set Vk , aiming to prove that Y is c.e.
Scan Vk for errors.
I
I
I
No errors - outcome h∞Y , k i
An error found at element l, then build an agitator for
l, enumerate a witness z in D: hz, {k }i & Φ0 and
hz, {l}i & Φ1 , outcome d0
The S W Strategy – o− = h∞Y , k i
A Generic Set That
Does Not Bound A
Minimal Pair
Mariya I. Soskova
β
βˆ∞X
βˆh∞Y , k i
βˆhl, ki
...
βˆd0
I
Check if the agitator for k is still valid. If not the
outcome is ∞X
I
Build a set Vk , aiming to prove that Y is c.e.
Scan Vk for errors.
I
I
I
No errors - outcome h∞Y , ki
An error found at element l, then build an agitator for
l, enumerate a witness z in D: hz, {k }i & Φ0 and
hz, {l}i & Φ1 , outcome d0
The S W Strategy – o− = d0
A Generic Set That
Does Not Bound A
Minimal Pair
Mariya I. Soskova
β
βˆ∞X
βˆh∞Y , ki
...
βˆhl, k i
βˆd0
I
Check if z ∈ W .
I
If not outcome d0
I
If yes, then extract z from D, outcome hl, ki
The S W Strategy – o− = d0
A Generic Set That
Does Not Bound A
Minimal Pair
Mariya I. Soskova
β
βˆ∞X
βˆh∞Y , ki
...
βˆhl, k i
βˆd0
I
Check if z ∈ W .
I
If not outcome d0
I
If yes, then extract z from D, outcome hl, ki
The S W Strategy – o− = d0
A Generic Set That
Does Not Bound A
Minimal Pair
Mariya I. Soskova
β
βˆ∞X
βˆh∞Y , ki
...
βˆhl, k i
βˆd0
I
Check if z ∈ W .
I
If not outcome d0
I
If yes, then extract z from D, outcome hl, ki
The S W Strategy – o− = hl, k i
A Generic Set That
Does Not Bound A
Minimal Pair
Mariya I. Soskova
β
βˆ∞X
βˆh∞Y , ki
...
βˆhl, k i
βˆd0
I
Check if the agitator for k is valid - if not outcome ∞X
I
Check if the agitator for l is valid - if not outcome
h∞Y , k i
I
Otherwise - outcome hl, ki
The S W Strategy – o− = hl, k i
A Generic Set That
Does Not Bound A
Minimal Pair
Mariya I. Soskova
β
βˆ∞X
βˆh∞Y , ki
...
βˆhl, k i
βˆd0
I
Check if the agitator for k is valid - if not outcome ∞X
I
Check if the agitator for l is valid - if not outcome
h∞Y , k i
I
Otherwise - outcome hl, ki
The S W Strategy – o− = hl, k i
A Generic Set That
Does Not Bound A
Minimal Pair
Mariya I. Soskova
β
βˆ∞X
βˆh∞Y , ki
...
βˆhl, k i
βˆd0
I
Check if the agitator for k is valid - if not outcome ∞X
I
Check if the agitator for l is valid - if not outcome
h∞Y , k i
I
Otherwise - outcome hl, ki
The S W Strategy – o− = hl, k i
A Generic Set That
Does Not Bound A
Minimal Pair
Mariya I. Soskova
β
βˆ∞X
βˆh∞Y , ki
...
βˆhl, k i
βˆd0
I
Check if the agitator for k is valid - if not outcome ∞X
I
Check if the agitator for l is valid - if not outcome
h∞Y , k i
I
Otherwise - outcome hl, ki
A Generic Set That
Does Not Bound A
Minimal Pair
Local priority
Mariya I. Soskova
β
βˆ∞X
...
h∞Y , k i
...
γ
I
An element k enters U, relying on axiom hk, Ei ∈ Θ0
I
γ extracts k from X by extracting some e ∈ E from A
I
β builds agitator Ek ⊃ E for k and moves to the right.
A Generic Set That
Does Not Bound A
Minimal Pair
Local priority
Mariya I. Soskova
β
βˆ∞X
...
h∞Y , k i
...
γ
γ0
I
A new axiom hk, E 0 i makes Ek invalid and fixes the
error in U
I
A new γ 0 extracts k from X again by extracting some
e ∈ E 0 from A
A Generic Set That
Does Not Bound A
Minimal Pair
Local priority
Mariya I. Soskova
β
βˆ∞X
...
h∞Y , k i
...
γ
I
A computable bijection σ : Γ → N assigns local
priority to GW - strategies below a S W strategy.
I
If an element k enters U and σ(γ) > k, then γ is
initialized.
A Generic Set That
Does Not Bound A
Minimal Pair
The Structure Watched
α1
Mariya I. Soskova
α2
β1
β1ˆ∞X
...
β1ˆh∞Y , ki
...
β2
I
I
I
β2 chooses agitators E2 and F2 , modifying (Φ0 , Φ1 )α2
β1 chooses E1 s.t E2 ⊂ E1 and F2 * E1 .
Agitators are chosen more carefully to include all
agitators of nodes with lower priority in the
corresponding subtree
A Generic Set That
Does Not Bound A
Minimal Pair
The Structure Watched
Mariya I. Soskova
α1
α2
β1
β1ˆ∞X
...
β1ˆh∞Y , ki
...
β2
I
Now (E2 ∪ F2 ) ⊂ E1 , but E2 becomes invalid.
I
A list Watchedα is kept by all R W strategies. It fixes
mistakes in the operators Φ0 and Φ1
Bibliography
A Generic Set That
Does Not Bound A
Minimal Pair
Mariya I. Soskova
R. I. Soare, Recursively enumerable sets and
degrees, Springer-Verlag, Heidelberg, 1987.
S. B. Cooper, Computability Theory, Chapman &
Hall/CRC Mathematics, Boca Raton, FL, 2004.
P. G. Odifreddi, Classical Recursion Theory, Volume
II, North-Holland/Elsevier, Amsterdam, Lausanne,
New York, Oxford, Shannon, Singapore, Tokyo 1999.
K. Copestake, 1-genericity in the enumeration
degrees, J. Symbolic Logic 53 (1988), 878–887.
S. B. Cooper, Andrea Sorbi, Angsheng Li and Yue
Yang, Bounding and nonbounding minimal pairs in
the enumeration degrees, Journal of Symbolic Logic
70 (2005), 741-766.