.
.
Lowness for uniform Kurtz randomness
Takayuki Kihara
Japan Advanced Institute of Science and Technology (JAIST)
Joint work with
Kenshi Miyabe (University of Tokyo)
4 July, 2013
Takayuki Kihara
Lowness for uniform Kurtz randomness
.
Background
.
“Lowness for Randomness” is a well-studied notion in
algorithmic randomness theory.
The highlighted theorem (Nies) is:
K -triviality = lowness for ML randomness
Downey-Griffiths-Laforte (2004) introduced the notion of
Schnorr triviality
(the triviality w.r.t. computable measure machines).
However:
Schnorr triviality , lowness for Schnorr randomness
Franklin and Stephan (2010) showed:
Schnorr triviality = tt-lowness for Schnorr randomness
.
Miyabe asked whether a characterization of tt-lowness for
Kurtz randomness exists.
Takayuki Kihara
Lowness for uniform Kurtz randomness
.
Definition
.
1. An oracle Kurtz test is a partial computable function
T :⊆ 2ω → A− (2ω ) such that:
T (X ) is null, for every X ∈ dom(T ).
.
Takayuki Kihara
Lowness for uniform Kurtz randomness
.
Definition
.
1. An oracle Kurtz test is a partial computable function
T :⊆ 2ω → A− (2ω ) such that:
T (X ) is null, for every X ∈ dom(T ).
i.e., T represents a sequence (TnX ) of clopen sets
uniformly in X ∈ 2ω and n ∈ ω such that
the measure of (TnX )n∈ω converges to 0 for every X .
∩
Then n TnX is denoted by T (X )
.
Takayuki Kihara
Lowness for uniform Kurtz randomness
.
Definition
.
1. An oracle Kurtz test is a partial computable function
T :⊆ 2ω → A− (2ω ) such that:
T (X ) is null, for every X ∈ dom(T ).
i.e., T represents a sequence (TnX ) of clopen sets
uniformly in X ∈ 2ω and n ∈ ω such that
the measure of (TnX )n∈ω converges to 0 for every X .
∩
Then n TnX is denoted by T (X )
. A is Kurtz random relative to B if
A < T (B ) for every (partial) oracle Kurtz test T .
3. A is Kurtz random tt-relative to B if
A < T (B ) for every total oracle Kurtz test T .
.4 A ∈ 2ω is (tt-)low for Kurtz randomness if every Kurtz random
2
.
sequence remains Kurtz random (tt-)relative to A .
Takayuki Kihara
Lowness for uniform Kurtz randomness
.
Example
.
1. If f : ω → ω is tt-reducible to A and strictly increasing, then
SuperSetsf = {B ⊆ ω : rng(f ) ⊆ B }
is a Kurtz test tt-relative to A .
. If V ⊆ 2ω is a null Π0 class (i.e., a Kurtz test), then
2
1
A + V = {A + B : B ∈ V}
.
is a Kurtz test tt-relative to A , where the bitwise addition
A + B is defined by (A + B )(n) ≡ A (n) + B (n) mod 2.
Takayuki Kihara
Lowness for uniform Kurtz randomness
.
Problem
.
.Find a characterization of tt-lowness for Kurtz randomness!
.
Greenberg-Miller (2009) showed that
low for Kurtz randomness = non-DNR + hyperimmune-free
The problem is that there is NO counterpart of
hyperimmune-freeness in the tt-degrees:
For every A ∈ 2ω , every f ≤tt A is majorized by a computable
function!
Hence, it seems that we cannot use the methods
.
in the original paper in Greenberg-Miller,
in a simpler proof in the book by Downey-Hirschfeldt.
Takayuki Kihara
Lowness for uniform Kurtz randomness
.
Problem
.
Find a characterization of tt-lowness for Kurtz randomness
.
Takayuki Kihara
Lowness for uniform Kurtz randomness
.
Problem
.
Find a characterization of tt-lowness for Kurtz randomness
.without the use of hyperimmune-freeness!
Takayuki Kihara
Lowness for uniform Kurtz randomness
.
Problem
.
Find a characterization of tt-lowness for Kurtz randomness
.without the use of hyperimmune-freeness!
.
Main Theorem
.
A ∈ 2ω is tt-low for Kurtz randomness if and only if it is
h-dimensionally
Kurtz null for every computable gauge function h.
.
Takayuki Kihara
Lowness for uniform Kurtz randomness
.
Problem
.
Find a characterization of tt-lowness for Kurtz randomness
.without the use of hyperimmune-freeness!
.
Main Theorem
.
A ∈ 2ω is tt-low for Kurtz randomness if and only if it is
h-dimensionally
Kurtz null for every computable gauge function h.
.
.
Proof Techniques
.
We have two methods to show the main theorem:
1. Modification of Greenberg-Miller’s svelte tree argument.
. Modification of Pawlikowski’s additivity characterization of
strong measure zero in set theory of the real line.
.
2
Takayuki Kihara
Lowness for uniform Kurtz randomness
We introduce a Kurtz version of effective Hausdorff dimension.
.
Definition (K.-Miyabe)
.
For an order h : ω → ω, a set E ⊆ 2ω is Kurtz h-null (K h -null) if
there is a computable sequence {Cn }n∈ω of finite sets of strings
such that
E ⊆ [Cn ] and
∑
2−h (|σ|) ≤ 2−n for all n ∈ ω.
σ∈Cn
ω
.We also say that A ∈ 2 is Kurtz h-null if {A } is Kurtz h-null.
.
Remark
.
1. Such a function h is called a gauge function or a dimension
function in geometric measure theory.
.2 If h : n 7→ s for a fixed real s ∈ (0, 1], then it is a Kurtz
version of the s-dimensional Hausdorff measure zero.
. For 1 : n 7→ 1, a real is Kurtz 1-null iff it is not Kurtz random.
.
3
Takayuki Kihara
Lowness for uniform Kurtz randomness
.
.
Fine Structure inside “Probability 0”
Non-KurR
←
K 1 -null
←
dimK ≤ 1
←
dimK ≤ s
←
dimK = 0
←
K h -null
↔
←
←
dimP ≤ s (s ∈ (0, 1])
←
dimP = 0
←
Ph -null
→
dimP ≤ 1
→
→
→
→
(h → ∞)
← (∀h) K h -null ← (∀h) Ph -null
↔
↔
(∀h) H h -null
↔
→
H h -null
→
→
dimH = 0
←
→
→
dimH ≤ s
P1 -null
→
→
dimH ≤ 1
←
→
→
H 1 -null
→
↔
Non-SchR
tt-Low(Sch, Kur) ← tt-Low(Kur) ← tt-Low(Sch)
Takayuki Kihara
Lowness for uniform Kurtz randomness
Dimension
Takayuki Kihara
tt-Low(Kur)
Lowness for uniform Kurtz randomness
Martingale
Dimension
tt-Low(Kur)
Mahine
Takayuki Kihara
Lowness for uniform Kurtz randomness
Martingale
tt-Lowtest (Kur)
Dimension
tt-Low(Kur)
Mahine
Takayuki Kihara
Lowness for uniform Kurtz randomness
Martingale
tt-Lowtest (Kur)
Dimension
tt-Low(Kur)
Mahine
Takayuki Kihara
Lowness for uniform Kurtz randomness
Martingale
tt-Lowtest (Kur)
Dimension
Mahine
tt-Low(Kur)
No KurR-Subset
Takayuki Kihara
Lowness for uniform Kurtz randomness
Martingale
tt-Lowtest (Kur)
Dimension
Mahine
K tt-trae
Takayuki Kihara
tt-Low(Kur)
No KurR-Subset
(Greenberg-Miller)
Lowness for uniform Kurtz randomness
Martingale
tt-Lowtest (Kur)
Dimension
Mahine
K tt-trae
Takayuki Kihara
tt-Low(Kur)
No KurR-Subset
(Greenberg-Miller)
Lowness for uniform Kurtz randomness
E -additive
Martingale
tt-Lowtest (Kur)
Dimension
Mahine
K tt-trae
Takayuki Kihara
tt-Low(Kur)
No KurR-Subset
(Greenberg-Miller)
Lowness for uniform Kurtz randomness
E -additive
(Pawlikowski)
Martingale
tt-Lowtest (Kur)
Dimension
Mahine
K tt-trae
Takayuki Kihara
tt-Low(Kur)
No KurR-Subset
(Greenberg-Miller)
Lowness for uniform Kurtz randomness
E -additive
(Pawlikowski)
Martingale
tt-Lowtest (Kur)
Dimension
Mahine
K tt-trae
tt-Low(Kur)
No KurR-Subset
(Greenberg-Miller)
(Galvin-Myielski-Solovay)
M-additive
Takayuki Kihara
Lowness for uniform Kurtz randomness
E -additive
(Pawlikowski)
Martingale
tt-Lowtest (Kur)
Dimension
Mahine
K tt-trae
tt-Low(Kur)
No KurR-Subset
(Greenberg-Miller)
(Galvin-Myielski-Solovay)
M-additive
tt-Low(W1G)
Takayuki Kihara
Lowness for uniform Kurtz randomness
.
Main Theorem
.
The following are equivalent for a real A ∈ 2ω :
1. A is tt-low for Kurt randomness (tests).
. A is Kurtz h-null for every computable order h.
2
.
Takayuki Kihara
Lowness for uniform Kurtz randomness
.
Main Theorem
.
The following are equivalent for a real A ∈ 2ω :
1. A is tt-low for Kurt randomness (tests).
. A is Kurtz h-null for every computable order h.
3. There is no f ≤
tt A such that
2
rng(f ) has no infinite subset of a Kurtz random set.
. A is Kurtz tt-traceable.
4
.
Takayuki Kihara
Lowness for uniform Kurtz randomness
.
Main Theorem
.
The following are equivalent for a real A ∈ 2ω :
1. A is tt-low for Kurt randomness (tests).
. A is Kurtz h-null for every computable order h.
3. There is no f ≤
tt A such that
2
rng(f ) has no infinite subset of a Kurtz random set.
. A is Kurtz tt-traceable.
5. ∀comp. order h ∃comp. order g ∃comp. martingale d s.t.
4
∀n∃k ∈ [g (n), g (n + 1)) d (A ↾ k ) ≥ 2n 2n−h (k ) .
. ∀comp. order h ∃comp. order g ∃c.m.m. M s.t.
6
∀n∃k ∈ [g (n), g (n + 1)) KM (A ↾ k ) ≥ h (k ) − n.
.
Takayuki Kihara
Lowness for uniform Kurtz randomness
.
Main Theorem
.
The following are equivalent for a real A ∈ 2ω :
1. A is tt-low for Kurt randomness (tests).
. A is Kurtz h-null for every computable order h.
3. There is no f ≤
tt A such that
2
rng(f ) has no infinite subset of a Kurtz random set.
. A is Kurtz tt-traceable.
5. ∀comp. order h ∃comp. order g ∃comp. martingale d s.t.
4
∀n∃k ∈ [g (n), g (n + 1)) d (A ↾ k ) ≥ 2n 2n−h (k ) .
. ∀comp. order h ∃comp. order g ∃c.m.m. M s.t.
6
∀n∃k ∈ [g (n), g (n + 1)) KM (A ↾ k ) ≥ h (k ) − n.
. A + B is Kurtz random iff B is Kurtz random.
.
. 8 A + B is weakly 1-generic iff B is weakly 1-generic.
7
Takayuki Kihara
Lowness for uniform Kurtz randomness
.
Lemma
.
Assume that V ⊆ 2ω is Kurtz h-null for every computable order h,
and E : 2ω → A− (2ω ) is a total Kurtz test.
∪
.Then, E [V] = A ∈V E (A ) is covered by a Kurtz test.
.
Z ↾ u (n )
u: En
is determined for every Z ∈ 2ω .
h: 2−h (u(n)) ≥
1
.
n +1
Since V is Kurtz h-null, ∃Vn : seq. of clopen s.t.
∩
∑
1
V ⊆ n Vn and σ∈Vn 2−h (|σ|) < n+
.
1
Every Vn has at most k strings length ≤ u(n + k ), since
∑
.
σ∈Vn
2−h (|σ|) <
1
n +1
≤
k +1
n +k +1
≤ (k + 1)2−h (u(n+k )) .
Vn can be think of as a seq. (σk )k of strings with
|σk | = u(n + k ).
Takayuki Kihara
Lowness for uniform Kurtz randomness
E
u(n + 4)
u(n + 3)
u(n + 2)
u(n + 1)
.
Z ↾ u (n )
µ(En
.
2 n 2
2 n 1
) ≤ 2−n .
Vn is a seq. (σk )k of strings with |σk | = u(n + k ).
∪ σ
Then E [Vn ] ⊆ k E k has measure 2−n+1
k
Takayuki Kihara
Lowness for uniform Kurtz randomness
.
h
.(∀comp. h) V is K -null ⇐⇒ (∀Kurtz test E) V + E is Kurtz null.
.
Lemma (K.-Miyabe [3])
.
Assume that V is K h -null for every computable h. Then,
∪
ω
. y ∈V E (y ) is Kurtz null for every uniform Kurtz test E : 2 → A− .
Takayuki Kihara
Lowness for uniform Kurtz randomness
.
h
.(∀comp. h) V is K -null ⇐⇒ (∀Kurtz test E) V + E is Kurtz null.
.
Lemma
.
Assume that V + E is Kurtz null for every Kurtz null set E.
h
.Then, V is K -null for every computable h.
.
1. h: given. g (n ) = g (n − 1) + h (n ) + 2h (n ) .
. Ek ⊆ 2g(k ) : strings of the form τ⌢ σi ⌢ ρ s.t.
2
|τ| = g (k − 1), |σi | = h (k ), |ρ| = 2h (k ) , and ρ(i ) = 0,
where {σi : i < 2h (k ) } is an enumeration of 2h (k ) .
. By assumption, V + E is covered by a Kurtz test D = ∩ Dn .
n
g (k −1)
4. D
: µ(D
|τ) < 1/8 for any τ ∈ 2
.
3
.
e (k )
e (k )
Takayuki Kihara
Lowness for uniform Kurtz randomness
.
(. ∀comp. h) V is K h -null ⇐ (∀Kurtz test E) V + E is Kurtz null.
.
1. d (k ) = e (k ) if E
⊆ 2≤g(k −1) ; d (k ) = d (k − 1) o.w.
d (k −1)
. Given τ ∈ 2g(k −1) , σ ∈ 2h (k )+2h (k ) gets k -closer to D /τ if
2
(1 − 2−k −1 ) µ(Dd (k ) |τσ) > µ(Dd (k ) |τ).
Dτ [k ]: all σ which get k -closer to D /τ.
. (Remark) µ(Dτ [k ]) ≤ 1 − 2−k −1 .
4. V [k ] = {σ ∈ 2h (k ) : (∃σ′ ⪰ σ) σ′ + E ∈ D [k ]}.
τ
τ
5. V [k ] = {τσ : σ ∈ D [k ]}.
τ
3
.
Takayuki Kihara
Lowness for uniform Kurtz randomness
.
Claim
.
h (k )
#
.
. V [k ] ≤ (k + 1) · 2
.
1. Note that V [k ] + E
τ
k ⊆ Dτ [k ].
.2 By probability independence, µ(Vτ [k ] + Ek ) = 1 − 2−|Vτ [k ]| .
3. However, µ(D [k ]) ≤ 1 − 2−k −1 .
τ
.
. Hence, |Vτ [k ]| ≤ k + 1.
4
Takayuki Kihara
Lowness for uniform Kurtz randomness
.
Claim
.
Let {k (l )}l ∈ω be the list of all k s.t. d (k ) , d (k − 1).
∩ ∪k (l )−1
Then, V ⊆ l
V [j ].
j =k (l −1)
.
.
∪k (l )−1
1. Otherwise, there is x ∈ V s.t. x <
j =k (l −1)
V [j ] for some l.
. a := k (l − 1), b = k (l ) − 1
g (a −1)
3. By def., µ(D
.
d (a ) |x + τ) < 1/8 for any τ ∈ 2
4. µ(D
d (a ) |x + τ) = 1 for ∀τ ∈ E ↾ g (b ), since V + E ⊆ Dd (a ) .
∪b
5. This is impossible, since x <
V [j ] implies we can find
2
j =a
.
sequence τa , τa +1 , · · · ∈ E s.t.
∏
µ(Dd (a ) |x + τa ) ≥ b (1 − 2−j −1 ) · µ(Dd (a ) |x + τb ) > 1/8.
j =a
Takayuki Kihara
Lowness for uniform Kurtz randomness