Ershov Hierarchies and Degree Theory - Nanjing Universityyuliang/tamc.pdf · Ershov Hierarchies and...
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Ershov Hierarchies and Degree Theory
Liang YuDepartment of mathematics
National University of Singapore
Joint with F. Stephan and Y. Yang
18th May 2006
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Analytical hierarchy
DefinitionA predicate R(x ,n) on ωω × ω is recursive if there is a partialrecursive function Φ(σ,n) on ω<ω × ω so that
(i) ∀σ∀τ∀n(Φ(σ,n) ↓ ∧σ � τ =⇒ Φ(σ,n) = Φ(τ,n)).(ii) ∀x∀n∃m(R(x ,n) ⇔ Φ(x � m,n) = 0).
(iii) ∀x∀n∃m(Φ(x � m,n) ↓).
A predicate P(x ,n) is Π11 if there is a recursive predicate
R(y , x ,n) so that
P(x ,n) ⇐⇒ ∀yR(y , x ,n).
The predicate ¬P(x ,n) is called Σ11.
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Recursive ordinals and Kleene’s O
Definition
The Π11-well-ordering <o on ω is defined by transfinite induction
as follows:0 <o 1;(∀n)n <o 2n;(∀n)Φe(n) <o Φe(n + 1) =⇒ (∀n)n <o 3 · 5e.
O is the field of <o.
An ordinal α is recursive if it is isomorphic toO � n = {m|m <o n} for some n ∈ O. ωCK
1 is the leastnon-recursive ordinal.
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Some properties of O
TheoremFix an enumeration {Wn}n of r.e. sets.
1 There is a recursive function p so that for all n ∈ O,Wp(n) = O � n.
2 There is a recursive function q so that for all n ∈ O,Wq(n) = {(n0,n1)|n0 <o n1 <o n}.
The recursive function +o has the following properties for allm,n.
(i) m,n ∈ O ⇐⇒ m +o n ∈ O.(ii) m,n ∈ O =⇒ |m +o n| = |m|+ |n|.
(iii) m,n ∈ O ∧ n 6= 1 =⇒ m <o m +o n.(iv) m ∈ O ∧ k <o n ⇐⇒ m +o k <o m +o n.(v) m ∈ O ∧ n = k ∈ O ⇐⇒ m +o n = m +o k .
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Facts in higher recursion theory
Theorem (Sacks)
There is a recursive function g so that for all e, if We ⊆ O and<o on We is linear, then g(e) ∈ O and∀n(n ∈ We =⇒ n <o g(e)).
Corollary
For all m ∈ O, there is a recursive function g so that for all e, ifWe ⊆ O and <o on We is linear, then g(e) ∈ O and∀n(n ∈ We =⇒ m +o n <o g(e)). Moreover, the function g canbe found uniformly.
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continued
Theorem (Feferman and Spector)
For each n ∈ O, there is a Π11 path T ⊂ O with |T | = ωCK
1 forwhich n ∈ T .
Theorem (Spector)
Each Σ11 well ordering is strictly below ωCK
1 .
Theorem (Kleene)
Given a Π11 set R ⊆ ω × ω, there is a Π1
1 function f so that1 ∀n∃m0∃m1((n,m) ∈ R =⇒ f (n) = m1 ∧ (n,m1) ∈ R).2 ∀n∀m(f (n) = m =⇒ (n,m) ∈ R).
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Ershov Hierarchies
Definition (Ershov)
For each n ∈ O, a subset A of ω is Σ−1n if there is a recursive
function f so that1 For all i <o j <o n, Wf (i) ⊆ Wf (j).2 For all k , k ∈ A if and only if there is a notation i <o n so
thatk ∈ Wf (i).|i | 6≡ |n|( mod 2).For all j <o i , k 6∈ Wf (j).
The set ω − A is said to be Π−1n .
A set B is said to be ∆−1n if B ∈ Σ−1
n ∩ Π−1n .
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Russian school’s results
Theorem (Ershov)
1 For all n <o m, Σ−1n ∪ Π−1
n ⊂ Σ−1m ∩ Π−1
m .2 ∆0
2 =⋃
n∈O∧|n|=ω2 Σ−1n .
3 For all m,n ∈ O with |m|, |n| < ω2, Σ−1m = Σ−1
n if and only if|n| = |m|.
4 For each path T ⊂ O, if |T | < ω3, then⋃
n∈T Σ−1n 6= ∆0
2.5 There is a path T ⊂ O with |T | = ω3 for which⋃
n∈T Σ−1n = ∆0
2.
Theorem (Selivanov )
For all n, there is a set A ∈ Σ−1n so that B 6∈
⋃k<on Σ−1
k for allB ≡T A.
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A basic fact
Theorem (Forklore)
For every n ∈ O and set A ⊆ ω, the following statements areequivalent:
1 A ∈ Σ−1n .
2 There is a recursive function f : ω × ω → Wp(2n) so that forall k,
1 For all i ≥ j , f (k , i) ≤o f (k , j).2 For all i , f (k , i + 1) 6= f (k , i) =⇒ |f (k , i + 1)| 6≡ |f (k , i)|(
mod 2).3 k ∈ A if and only if | lims f (k , s)| 6≡ |f (k ,0)|( mod 2).
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Paths through O
Theorem (Stephan, Yang and Yu)
1 There is a path T ⊂ O with |T | = ωCK1 so that⋃
n∈T Σ−1n = ∆0
2.2 There is a path T ⊂ O with |T | = ωCK
1 so that⋃n∈T Σ−1
n 6= ∆02.
Proof.
For (1), highly non-uniformly putting ∆02 sets into Ershov’s
hierarchy. For (2), by Feferman and Spector’s results.
Proposition (Stephan, Yang and Yu)
If T is a Π11 path so that
⋃n∈T Σ−1
n = ∆02, then T is ∆1
1.
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Lachlan’s result and it’s generalization
Theorem (Stephan, Yang and Yu)
For each notation n = 2m ∈ O with 0 < |m| < ω and setA ∈ Σ−1
n − Σ−1m , there is a non-recursive set B ∈ Σ−1
m so thatB ≤T A.
Proposition (Stephan, Yang and Yu)
If n = n0 +o n1 ∈ O, then for every set A ∈ Σ−1n − (Σ−1
n0∪ Σ−1
22n1 ),
there is a non-recursive set B ∈ Σ−1n0
so that B ≤T A.
Corollary
For each n ∈ O with n >o 2, if A ∈ Σ−12n −Σ−1
n , then A computesa non-recursive Σ−1
n set.
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A counter example to generalize Lachlan’s result
Theorem (Ding, Jin and Wang)
If n ∈ O, then there is a minimal degree a < 0′ so that A 6∈ Σ−1n
for all A ∈ a.
Proposition (Stephan, Yang and Yu)There is a notation n = n0 +o n1 ∈ O with |n1| < |n| so thatthere is a minimal degree a for which a ∩
⋃m<on Σ−1
m = ∅ buta ∩ Σ−1
n 6= ∅.
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Finite levels of Ershov hierarchy
Definition (Putnam)A set A is n-r.e. if there is a recursive functionf : ω × ω → ω so that for each m,
f (0,m) = 0.A(m) = lims f (s,m).|{s|f (s + 1,m) 6= f (s,m)}| ≤ n.
A Turing degree is n-r.e. if it contains an n-r.e. set.
A set is n-r.e. iff it is Σ−1m where m ∈ O and |m| = n + 1.
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Model theory I
The partially ordered language, L(≤), L(≤) includes variablesa,b, c, x , y , z, ... and a binary relation ≤ intended to denote apartial order. Atomic formulas are x = y , x ≤ y . Σ0 formulasare built by the following induction definition.
Each atomic formula is Σ0.¬ψ for some Σ0 formula ψ.ψ1 ∨ ψ2 for two Σ0 formula ψ1, ψ2.ψ1 ∧ ψ2 for two Σ0 formula ψ1, ψ2ψ1 =⇒ ψ2 for two Σ0 formula ψ1, ψ2.
A formula ϕ is Σ1 if it is of the form ∃x1∃x2...∃xnψ(x1, x2, ..., xn)for some Σ0 formula ψ.A formula ϕ is Πn if it is the form ¬ψ for some Σn formula ψ anda formula ϕ is Σn+1 if it is the form ∃x1∃x2...∃xmψ(x1, x2, ..., xm)for some Πn formula ψ. A sentence is a formula without freevariables.
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Model theory II
Given two structures A(A,≤A) and B(B,≤B) for L(≤), we saythat A(A,≤A) is a substructure of B(B,≤B), writeA(A,≤A) ⊆ B(B,≤B), if A ⊆ B and the interpretation ≤A is arestriction to A of ≤B.Examples:D(≤ 0′) = (D(≤ 0′),≤). The structures of n-r.e. degreesDn = (Dn,≤).
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Definition(i) We say that A(A,≤A) is a Σn substructure of B(B,≤B),
write A(A,≤A) �Σn B(B,≤B), if A(A,≤A) ⊆ B(B,≤B) andfor all Σn formulas ϕ(
−→x ) and any −→a ⊆ A,
A(A,≤A) |= ϕ(−→a ) if and only if B(B,≤B) |= ϕ(
−→a ).
(ii) We say that A(A,≤A) is Σn-elementary-equivalent toB(B,≤B), write A(A,≤A) ≡Σn B(B,≤B), if for all Σnsentences ϕ,
A(A,≤A) |= ϕ if and only if B(B,≤B) |= ϕ.
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Elementary difference among Ershov hierarchies
Theorem (Folklore)
For all n ∈ ω, Dn ≡Σ1 D(≤ 0′).
Theorem (Arslanov)For each natural number n > 1, D1 6≡Σ3 Dn.
Theorem (An accumulation of lots of results)
Dn 6≡Σ2 D(≤ 0′).For each natural number n > 1, D1 6≡Σ2 Dn.
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Downey’s conjecture and its solution
Conjecture (Downey )
For each n > 1 and k ≥ 0, Dn ≡Σk Dn+m.
Theorem (Arslanov, Kalimullin, Lempp )
D2 6≡Σ2 D3.
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Σ1-substructures of D(≤ 0′)
Theorem (Slaman)
(i) There are r.e. sets A,B and C and a ∆02 set E such that
∅ <T E ≤T A;C 6≤T B ⊕ E;For all r.e. set W (∅ <T W ≤T A ⇒ C ≤T W ⊕ B).
(ii) For each natural number n ≥ 1, Dn 6�Σ1 D(≤ 0′).
Proof.Take a Σ1 formula
ϕ(x1, x2, x3) ≡ ∃e∃y∃z(e ≤ x1 ∧ e ≥ y ∧ e 6= y ∧ z ≥ x2 ∧ z ≥ e ∧ z 6≥ x3).
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Slaman’s conjecture and its generalization
Conjecture (Slaman)
For each n > 1, D1 �Σ1 Dn?
Conjecture (Arslanov and Lempp)For all n > m, Dm �Σ1 Dn?
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The solution
Theorem (Yang and Yu )
There are r.e. sets A,B,C and E and a d.r.e. set D such that1 D ≤T A and D 6≤T E;2 C 6≤T B ⊕ D;3 For all r.e. sets W (W ≤T A ⇒ either C ≤T W ⊕ B or
W ≤T E).
Theorem (Yang and Yu)For all n > 1, D1 6�Σ1 Dn.
Proof.
ϕ(x1, x2, x3, x4) ≡ ∃d∃g(d ≤ x1∧d 6≤ x4∧g ≥ x2∧g ≥ d∧x3 6≤ g).
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Some questions
Question (Khoussainov)For n > 1, is there a function f : D1 → Dn so that for anyΣ1-formula ϕ(x1, ..., xm),
D1 |= ϕ(x1,x2, ...,xm) iff Dn |= ϕ(f (x1), f (x2), ..., f (xm)),
where x1,x2, ...,xm range over D1?
QuestionFinding out a proper Σ1-substructure of D(≤ 0′)
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Thank you