Motivations Framework Applications Questions
Canonical notions of forcing in computability theory
Ludovic PATEY Joint work with Denis Hirschfeldt
1 / 37
November 27, 2019 Motivations Framework Applications Questions
Have we found the right techniques?
Would martians come up with the same proof?
Do we loose in generality with our constructions?
2 / 37 Motivations Framework Applications Questions
Example : weak 1-genericity
A set D ⊆ 2<ω is dense if for every σ ∈ 2<ω there is a τ ⪰ σ in D.
A real R meets D if σ ∈ D for some σ ≺ R.
A real R is weakly 1-generic if it meets 0 every dense Σ1 set.
3 / 37 Motivations Framework Applications Questions
Example : weak 1-genericity
0 · · · ⊆ <ω List all the Σ1 sets W0, W1, W2, 2
Build a real with the finite extension method σ0 ≺ σ1 ≺ σ2 ≺ ...
Let f : ω → ω be an increasing time function.
Search for an extension σs+1 of σs in some unsatisfied We such that σs has no extension in Wi[f(|σs+1|)] for any unsatisfied Wi with i < e
Thm (Kurtz) Every weakly 1-generic real computes a function f which makes this construction produce a weakly 1-generic real.
4 / 37 Motivations Framework Applications Questions
Example : minimal degree
A Turing degree d > 0 is minimal if there is no degree in between 0 and d.
Two strings τ0, τ1 are a Ψ-splitting if Ψτ0 and Ψτ1 are incompatible.
T ⊆ 2<ω is a delayed Ψ-splitting tree if T is a tree and whenever σ0, σ1 ∈ T are incompatible, any τ0, τ1 ∈ T properly extending σ0 and σ1 respectively are a Ψ-splitting.
5 / 37 Motivations Framework Applications Questions
Example : minimal degree
A set T ⊆ 2<ω is a c.e. tree if there is a computable enumeration of finite sets {Tn}n≥0 such that |T0| = 1 and if σ ∈ Ts+1 \ Ts, then σ extends a leaf of Ts.
Thm (Lewis) A set G is of minimal degree iff G is incomputable, and whenever ΨG is total and incomputable, then ΨG lies on a delayed Ψ-splitting c.e. tree.
6 / 37 Motivations Framework Applications Questions
Both constructions are without loss of generality
The constructions are natural
The resulting objects carry their own construction
7 / 37 Motivations Framework Applications Questions
Consider mathematical problems
Intermediate value theorem For every continuous function f over an a · interval [a, b] such that f(a) f(b) < 0, there b is a real x ∈ [a, b] such that f(x) = 0.
König’s lemma Every infinite, finitely branching tree admits an infinite path.
8 / 37 Motivations Framework Applications Questions
Computable reduction
P solver
Computable Computable Q solver transformation transformation
P ≤c Q
Every P-instance I computes a Q-instance J such that for every solution X to J, X ⊕ I computes a solution to I.
9 / 37 Motivations Framework Applications Questions
Observations
When proving that P ̸≤c Q, we usually
construct a computable instance of P with complex solutions construct for every computable instance of Q a simple solution use a notion of forcing to build solutions to Q-instances
10 / 37 Motivations Framework Applications Questions
Observations
The notion of forcing for Q does not depend on P
Q seems to have a canonical notion of forcing
Separation proofs can be obtained without loss of generality using this notion of forcing
11 / 37 Motivations Framework Applications Questions
Examples
0 For WKL, forcing with Π1 classes : ACA ̸≤c WKL (cone avoidance) 2 ̸≤ RT2 c WKL (ω hyperimmunities) ...
For ADS, forcing with split pairs :
ACA ̸≤c ADS (cone avoidance)
CAC ̸≤c ADS (Towsner) 2 ̸≤ RT2 c ADS (dep. hyperimmunity) DNC ̸≤c ADS (non-DNC degree)
For DNC, forcing with bushy trees
12 / 37 Motivations Framework Applications Questions
Towards a framework
13 / 37 Motivations Framework Applications Questions
Weakness property
A weakness property is a class W ⊆ 2ω which is closed downward under Turing reducibility.
A problem P computably satisfies a weakness property W if every computable instance of P has a solution in W.
Example : Given a set A, let WA = {X : X ̸≥T A}. Then WKL computably satisfies WA for every A ̸≤T ∅.
14 / 37 Motivations Framework Applications Questions
Weakness property
If Q computably satisfies W but P does not, then
P ̸≤c Q
15 / 37 Motivations Framework Applications Questions
P-forcing
Fix a problem P.
A P-forcing is a forcing family P = (PI : I ∈ dom P) such that for every P-instance I, every sufficiently generic filter yields a solution to I.
A P-forcing P computably satisfies a weakness property W if every computable I ∈ dom(P), every sufficiently generic filter yields an element in W.
Example : Given a set A, let WA = {X : X ̸≥T A}. 0 W Forcing with Π1 classes computably satisfies A for every A ̸≤T ∅.
16 / 37 Motivations Framework Applications Questions
Fix a class W of weakness properties.
Defi A P-forcing P is canonical for W if for every W ∈ W such that P computably satisfies W, then so does P.
What class W to consider?
17 / 37 Motivations Framework Applications Questions
Weakness properties
Effectiveness properties: Genericity properties:
′ ′ Lowness (W = {X : X ≤T ∅ }) Cone avoidance (WA = {X : X ̸≥T A} for A ̸≤T ∅) Arithmetical hierarchy (W = {X : X is arithmetical }) Preservation of hyperimmunity (Wf = {X : f is X-hyperimmune})
0 Preservation of non-Σ1 definitions W { ̸∈ 0,X} ̸∈ 0 ( A = X : A Σ1 for A Σ1)
18 / 37 Motivations Framework Applications Questions
Closed set avoidance
A closed set avoidance property is a property of the form
WC = {X : C has no X-computable member}
for some closed set C ⊆ ωω in the Baire space.
Cone avoidance: CA = {A} ω Preservation of hyperimmunity: Cf = {g ∈ ω : g ≥ f} ω Non-DNC degree C = {g ∈ ω : ∃n(g(n) = Φn(n))}
19 / 37 Motivations Framework Applications Questions
What problems admit a canonical forcing?
20 / 37 Motivations Framework Applications Questions
Cohen genericity
Lem (Folklore) If C ⊆ ωω is a closed set with no computable member, then C has no G-computable member for every sufficiently Cohen generic.
Proof: Given a Cohen condition σ ∈ 2<ω forcing totality of a ⪰ τ ∩ C ∅ functional Φe, there is a τ σ such that [Φe ] = .
The Atomic Model Theorem (AMT) admits a canonical notion of forcing for closed set avoidance properties.
21 / 37 Motivations Framework Applications Questions
Highness
Lem (Folklore) If C ⊆ ωω is a closed set with no computable member and A ∈ 2ω, then C has no G-computable member for some G such ′ that G ≥T A.
⊆ 2 → 0 Proof: Use forcing conditions (h, n), where h ω 2 is a finite ∆2 approximation, and n fixes the first n columns to A.
Cohesiveness (COH) and highness admit a canonical notion of forcing for closed set avoidance properties.
22 / 37 Motivations Framework Applications Questions
Weak König’s lemma
WKL: Every infinite binary tree has an infinite path
Let C be the closed set of all completions of PA.
Then WKL does not computably preserve WC.
Thm 0 The WKL-forcing with non-empty Π1 classes is canonical for closed set avoidance properties.
23 / 37 Motivations Framework Applications Questions
Weak König’s lemma
Thm 0 The WKL-forcing with non-empty Π1 classes is canonical for closed set avoidance properties.
ω Fix a closed set C ⊆ ω and a functional Φe. 0 G ̸∈ C Try to prove that the set of Π1 classes forcing Φe is dense.
If it fails, show that WKL does not computably preserve WC.
24 / 37 Motivations Framework Applications Questions
Weak König’s lemma
C ⊆ ω 0 D Fix a closed set ω , a non-empty Π1 class and Φe.
Success if ∈ <ω ∩ D ̸ ∅ σ ∩ C ∅ there is a σ 2 such that [σ] = and [Φe ] = . { ∈ D X ↑} ̸ ∅ or X :Φe (n) = for some n.
{ X ∈ D} C Otherwise Φe : X is an effectively compact subset of . Every PA degree computes a member of C.
25 / 37 Motivations Framework Applications Questions
Weak König’s lemma
Thm WKL computably preserves WC iff C has no non-empty effectively compact subset.
Cone avoidance : C = {A} if A ̸≤T ∅ ω Preservation of hyperimmunity: Cf = {g ∈ ω : g ≥ f} 0 { } DNC : The Π1 class of 0, 1 -valued DNC is a non-empty effectively compact subset
26 / 37 Motivations Framework Applications Questions
Ascending Descending Sequence
SADS: Every linear order of type ω + ω∗ has an infinite ascending or descending sequence.
Let L = (ω, Forcing conditions : (σ0, σ1) such that <ω σ0, σ1 ∈ ω are σ0 ⊆ U is σ1 ⊆ V is 27 / 37 Motivations Framework Applications Questions Ascending Descending Sequence Thm The SADS-forcing is canonical for closed set avoidance properties. A split pair is a pair (τ0, τ1) such that <ω τ0, τ1 ∈ ω are τ0 is maxL τ0 28 / 37 Motivations Framework Applications Questions Closed set jump avoidance A closed set jump avoidance property is a property of the form ′ JC = {X : C has no X -computable member} for some closed set C ⊆ ωω in the Baire space. Let C be the closed set of all completions of PA relative to 0’. Then COH does not computably preserve JC. 29 / 37 Motivations Framework Applications Questions Cohesiveness Let R0, R1, R2, ... be an instance of COH ∩ ∩ Let Rσ = σ(i)=1 Ri σ(i)=0 Ri Forcing conditions: (F, σ, D) such that F is a finite set, σ ∈ 2<ω ′ D 0,∅ is a non-empty Π1 subclass of [σ] (E, τ, E) ≤ (F, σ, D) if (E, Rτ ) Mathias extends (F, Rσ). σ ≺ τ and E ⊆ D. 30 / 37 Motivations Framework Applications Questions Cohesiveness Thm The COH-forcing is canonical for closed set jump avoidance properties. Thm COH computably preserves JC iff C has no non-empty ∅′-effectively compact subset. 0 If A is not ∆2, every computable instance of COH admits a 0 solution G such that A is not ∆2(G). 31 / 37 Motivations Framework Applications Questions Open questions 32 / 37 Motivations Framework Applications Questions DNC functions A function f is DNC if for every e, f(e) ≠ Φe(e). A tree T ⊆ ω<ω is k-bushy above σ ∈ ω<ω if every element of T is comparable with T, and for every τ ∈ T which extends σ and is not a leaf, τ has at least k immediate extensions in T. A set B ⊆ ω<ω is k-small above σ if there is no finite tree k-bushy above σ whose leaves belong to B. 33 / 37 Motivations Framework Applications Questions DNC functions Bushy tree forcing : (σ, B) where σ ∈ ω<ω B is k-small above σ for some k. Question Is bushy tree DNC-forcing canonical for closed set avoidance properties? 34 / 37 Motivations Framework Applications Questions Intuition Lem Let X be a set. TFAE X computes a DNC function X computes a function g such that if |We| ≤ n, then g(e, n) ̸∈ We. Lem Suppose B is a k-small c.e. set above σ. Then the set {n : B is not k-small above σn} is c.e. of size at most k − 1. 35 / 37 Motivations Framework Applications Questions Conclusion Natural combinatorial problems seem to have canonical notions of forcing. The proofs of canonicity yield forcing-free criteria of preservations. The right notion of forcing for DNC functions is not fully understood. 36 / 37 Motivations Framework Applications Questions References Mushfeq Khan and Joseph S Miller. Forcing with bushy trees. Bulletin of Symbolic Logic, 23(2):160–180, 2017. Publisher: Cambridge University Press. Ludovic Patey. Controlling iterated jumps of solutions to combinatorial problems. Computability, 6(1):47–78, 2017. 37 / 37