# Bounding Nonsplitting Enumeration Degrees

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1 Bounding Nonsplitting Enumeration Degrees Thomas F. Kent Andrea Sorbi Università degli Studi di Siena Italia July 18, 2007

2 Goal: Introduce a form of Σ 0 2-permitting for the enumeration degrees. Till now, density was the only known property that held in all ideals of Σ 0 2-enumeration degrees. A is enumeration reducible to B (A e B) if we can enumerate A given any enumeration of B. Definition A e B iff there is c.e. set Φ such that A = {x : x, P Φ (P finite and P B)} = Φ B

3 Basic Facts We can embed the Turing degrees into the enumeration degrees via the embedding ι : deg T (A) deg e (A A). The image of the Turing degrees under ι is known as the total degrees. 0 e = {W : W is c.e.}. 0 e = deg e (K). Theorem (Cooper, 1984) A is Σ 0 2 iff A e K. Theorem (Cooper, 1984) The Σ 0 2-enumeration degrees are dense.

4 Partial The Global and Local Picture 0 e = ι(0 T ) Partial 0 e = deg e ( K ) Total 0 e = ι(0 T ) Σ 0 2 Total Σ e = {W : W is c.e.}

5 Nonsplitting Degrees Definition A degree a is nonsplitting if a > 0 e and for every x, y < a, x y < a. Theorem (Ahmad 1989 (c.f. Ahmad, Lachlan 1998)) There exists a nonsplitting Σ 0 2-enumeration degree. The requirements: Nontrivial N Φ : A Φ, and Nonsplitting S Ψ,Ω0,Ω 1 : A = Ψ ΩA 0 ΩA 1 Γ 0, Γ 1 ha = Γ ΩA 0 0 or A = Γ ΩA 1 1 i.

6 Bounding Nonsplitting Degrees Theorem (Kent, Sorbi 2007) Every nontrivial Σ 0 2-enumeration degree bounds a nonsplitting degree. The requirements: A e B R : A = Θ B Nontrivial N Φ : A = Φ (B = ), and Nonsplitting S Ψ,Ω0,Ω 1 : A = Ψ ΩA 0 ΩA 1 Γ 0, Γ 1 ha = Γ ΩA 0 0 or A = Γ ΩA 1 1 or Λ[B = Λ]. i

7 Some Corollaries Corollary The nonsplitting degrees are downwards dense in the 0 2-enumeration degrees. Corollary There is a properly Σ 0 2 nonsplitting enumeration degree. Corollary The c.e. Turing degrees are not elementarily equivalent to any ideal of the Σ 0 2-enumeration degrees. Question Are the nonsplitting degrees dense in the Σ 0 2 or 0 2 enumeration degrees?

8 N-Requirement - Standard N Φ : A Φ s w x Φ A x A Φ 1. Pick x and set x A. 2. If ever x Φ, set x / A.

9 N-Requirement - Bounded N Φ : A = Φ B = s 0 s 1 s 2 w x 0 Φ A x 1 Φ A x 2 Φ A... x A Φ 1. Assume x 0,... x n 1 Φ A. 2. Pick x n. 3. While 1. holds, enumerate x n, B x n Θ. 4. If ever x n Φ, stop defining x n axioms, and enumerate D n = T {D : x n, D Θ} into.

10 N-Requirement - Bounded N Φ : A = Φ B = s 0 s 1 s 2 w x 0 Φ A x 1 Φ A x 2 Φ A... x A Φ If infinitely many x i are defined and each x i A, then since D 0 D 1 B, we can conclude = B. If x i / A then x j / A for all j > i. (Conditional Dumping) While x i A, for all y S(s i ), enumerate y, B y into Θ.

11 S-Requirement - Standard S Ψ,Ω0,Ω 1 : A = Ψ ΩA 0 ΩA 1 A = Γ Ω A 0 0 or A = Γ ΩA 1 1 s γ 1 γ 0 w x Ψ ΩA 0 ΩA 1 A x A Ψ Ω A = Γ ΩA 1 A 0 ΩA 1 1 A = Γ ΩA Pick x and set x A. 2. Wait for x Ψ ΩA 0 ΩA 1 via x, F 0 F 1 Ψ. 3. Enumerate x, F i into Γ i, x into S(γ 0 ) and return to Step 1. Hopefully x A iff F 0 Ω A 0. If true for co-finitely many x, then A = Γ ΩA 0 0. Strategies below γ 0 can only use x which have this property.

12 S-Requirement - Standard S Ψ,Ω0,Ω 1 : A = Ψ ΩA 0 ΩA 1 A = Γ Ω A 0 0 or A = Γ ΩA 1 1 s γ 1 γ 0 w x Ψ ΩA 0 ΩA 1 A x A Ψ Ω A = Γ ΩA 1 A 0 ΩA 1 1 A = Γ ΩA If ever see x / A and F 0 Ω A 0 (hence x ΓΩA 0 0 A), dump S(γ 0 ) {x} into A, enumerate x into S(γ 1 ). For this x, F 0 Ω A {x} 0, killing Γ 0. Hopefully x A iff F 1 Ω A 1. If true for infinitely many x, then A = Γ ΩA 1 1. Strategies below γ 1 can only use x which have this property.

13 S-Requirement - Standard S Ψ,Ω0,Ω 1 : A = Ψ ΩA 0 ΩA 1 A = Γ Ω A 0 0 or A = Γ ΩA 1 1 s γ 1 γ 0 w x Ψ ΩA 0 ΩA 1 A x A Ψ Ω A = Γ ΩA 1 A 0 ΩA 1 1 A = Γ ΩA If ever see x / A and F 1 Ω A 1 (hence x ΓΩA 1 1 A), dump S(γ 1 ) S(γ 0 ) x into A, and set x / A. For this x, F 1 Ω A {x} 1, killing Γ 1. Not a problem since now x Ψ ΩA {x} 0 Ω A {x} 1.

14 S-Requirement - Bounded (v. 1.0) S Ψ,Ω0,Ω 1 : A = Ψ ΩA 0 ΩA 1 A = Γ Ω A 0 0 or A = Γ ΩA 1 1 or B = Λ s i γ 1 γ 0 w x Ψ ΩA 0 ΩA 1 A A = Γ Ω A 1 1 A = Γ ΩA 0 0 x A Ψ ΩA 0 ΩA 1 As with the N-strategy, expand the outcome s to s 0, s 1,.... If we choose x 0, x 1,... as possible diagonalization witness, and for all i, x i Ψ ΩA 0 ΩA 1 A, then B = Λ. (Conditional Dumping) While x i A, for all y S(s i ), enumerate y, B y into Θ.

15 S-Requirement - Potential Problem S Ψ,Ω0,Ω 1 : A = Ψ ΩA 0 ΩA 1 A = Γ Ω A 0 0 or A = Γ ΩA 1 1 or B = Λ s i γ 1 γ 0 w x Ψ ΩA 0 ΩA 1 A A = Γ Ω A 1 1 A = Γ ΩA 0 0 x A Ψ ΩA 0 ΩA 1 Γ 0 assumes all elements of S(γ 1 ) have settled down. Possibly there is x S(γ 1 ) and y S(γ 0 ) such that while x A, y A iff y Γ ΩA 0 0, but while x / A, y / Γ ΩA 0 0. lim s A(x) does not exist, i.e. A is Σ 0 2.

16 S-Requirement - Solution S Ψ,Ω0,Ω 1 : A = Ψ ΩA 0 ΩA 1 A = Γ Ω A 0 0 or A = Γ ΩA 1 1 or B = Λ s i γ 1 γ 0,0 γ 0,1 w x Ψ ΩA 0 ΩA 1 A A = Γ Ω A 1 1 A = Γ ΩA 0,0 0,0 A = Γ ΩA 0,1 x A Ψ ΩA 0 ΩA 1 0,1 Assume S(γ 1 ) = {x}. Construct two enumeration operators: Γ 0,0 and Γ 0,1. Γ 0,0 assumes x / A and Γ 0,1 assumes x A. Accounts for Σ 0 2 nature of A. In general, if S(γ 1 ) = n, then we construct 2 n enumeration operators.

17 Quasi-Lexicographical Ordering Definition Define the quasi-lexicographical ordering < b on 2 <ω by σ < b τ if 1. σ < τ, or 2. σ = τ and there is a k < σ such that σ(k) = 0, τ(k) = 1, and for all i < k, σ(i) = τ(i). 0 < b 1 < b 00 < b 01 < b 10 < b 11 < b 000 < b 001 < b 010 < b 011 < b 100 < b 101 < b

18 S-Requirement - Bounded S Ψ,Ω0,Ω 1 : A = Ψ ΩA 0 ΩA 1 A = Γ Ω A 0 0 or A = Γ ΩA 1 1 or B = Λ s i γ 1 γ 0,σ w σ x Ψ ΩA 0 ΩA 1 A x A Ψ Ω A = Γ ΩA 1 A 0 ΩA 1 1 A = Γ ΩA 0 0,σ The strategy proceeds basically the same as before. We have added infinitely many outcomes to the tree to account for all possible states of elements to the left of the current outcome. When an element is moved left through the streams, it must make sure that the assumptions of the new stream are consistent with the assumptions of the previous stream.

19 S-Requirement - Bounded S Ψ,Ω0,Ω 1 : A = Ψ ΩA 0 ΩA 1 A = Γ Ω A 0 0 or A = Γ ΩA 1 1 or B = Λ s i γ 1 γ 0,σ w σ x Ψ ΩA 0 ΩA 1 A x A Ψ Ω A = Γ ΩA 1 A 0 ΩA 1 1 A = Γ ΩA 0 0,σ Other technical concerns not covered here include: Other uses of conditional dumping. Local approximations to B. etc.

20 The End! Kent, Sorbi Bounding Nonsplitting Enumeration Degrees, to appear in The Journal of Symbolic Logic.

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