liang yu - nju.edu.cn
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Algorithmic randomness theory and itsapplications
Liang Yu
Mathematical DepartmentNanjing University
March 24, 2021
Yu (Math Dept of Nanjing University) ART and its applications March 24, 2021 1 / 30
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History
What is randomness?
1 Incompressible;2 No distinguish property;3 Unpredictable.
Yu (Math Dept of Nanjing University) ART and its applications March 24, 2021 2 / 30
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History
What is randomness?1 Incompressible;
2 No distinguish property;3 Unpredictable.
Yu (Math Dept of Nanjing University) ART and its applications March 24, 2021 2 / 30
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History
What is randomness?1 Incompressible;2 No distinguish property;
3 Unpredictable.
Yu (Math Dept of Nanjing University) ART and its applications March 24, 2021 2 / 30
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History
What is randomness?1 Incompressible;2 No distinguish property;3 Unpredictable.
Yu (Math Dept of Nanjing University) ART and its applications March 24, 2021 2 / 30
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Theory aspect
Notations
We always identify a reals as its binary expansion.
Yu (Math Dept of Nanjing University) ART and its applications March 24, 2021 3 / 30
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Theory aspect
Classical Randomness
People interested in classical randomness live in a computable world.To them, randomness means random relative to the computable world.
Yu (Math Dept of Nanjing University) ART and its applications March 24, 2021 4 / 30
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Theory aspect
Kolmogorov complexity
1 Fix a Turing machine M, for each finite string σ ∈ 2<ω, defineCM(σ) = min{|τ | : M(τ) = σ}.
2 Fix a prefix free Turing machine M, for each finite stringσ ∈ 2<ω, define KM(σ) = min{|τ | : M(τ) = σ}.
If U is a universal Turing machines, both CU and KU have theminimality property.
Yu (Math Dept of Nanjing University) ART and its applications March 24, 2021 5 / 30
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Theory aspect
Kolmogorov complexity
1 Fix a Turing machine M, for each finite string σ ∈ 2<ω, defineCM(σ) = min{|τ | : M(τ) = σ}.
2 Fix a prefix free Turing machine M, for each finite stringσ ∈ 2<ω, define KM(σ) = min{|τ | : M(τ) = σ}.
If U is a universal Turing machines, both CU and KU have theminimality property.
Yu (Math Dept of Nanjing University) ART and its applications March 24, 2021 5 / 30
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Theory aspect
Kolmogorov complexity
1 Fix a Turing machine M, for each finite string σ ∈ 2<ω, defineCM(σ) = min{|τ | : M(τ) = σ}.
2 Fix a prefix free Turing machine M, for each finite stringσ ∈ 2<ω, define KM(σ) = min{|τ | : M(τ) = σ}.
If U is a universal Turing machines, both CU and KU have theminimality property.
Yu (Math Dept of Nanjing University) ART and its applications March 24, 2021 5 / 30
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Theory aspect
Basic facts
1 C(σ) ≤ |σ|+ c and so C(n) ≤ log n + c for some constant c;
2 There is some σ (or number n) so that C(σ) ≥ |σ| ( orC(n) ≥ log n).
3 C((σ, τ)) ≤ C(σ) + C(τ) + C(|σ|) + c.
Yu (Math Dept of Nanjing University) ART and its applications March 24, 2021 6 / 30
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Theory aspect
Basic facts
1 C(σ) ≤ |σ|+ c and so C(n) ≤ log n + c for some constant c;2 There is some σ (or number n) so that C(σ) ≥ |σ| ( or
C(n) ≥ log n).
3 C((σ, τ)) ≤ C(σ) + C(τ) + C(|σ|) + c.
Yu (Math Dept of Nanjing University) ART and its applications March 24, 2021 6 / 30
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Theory aspect
Basic facts
1 C(σ) ≤ |σ|+ c and so C(n) ≤ log n + c for some constant c;2 There is some σ (or number n) so that C(σ) ≥ |σ| ( or
C(n) ≥ log n).3 C((σ, τ)) ≤ C(σ) + C(τ) + C(|σ|) + c.
Yu (Math Dept of Nanjing University) ART and its applications March 24, 2021 6 / 30
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Theory aspect
Martin-Löf test
Definition (Martin-Löf)
(i) Given a real x, a Σ01 Martin-Löf test is a computable collection
{Vn : n ∈ N} of x-c.e. sets such that µ(Vn) ≤ 2−n.
(ii) Given a real x, a real y is said to pass the Σ01(x) Martin-Löf test if
y /∈∩
n∈ω Vn.
(iii) Given a real x, a real y is said to be 1-x-random if it passes allΣ0
1(x) Martin-Löf tests.
There exists a universal c.e. Martin-Löf test.
Yu (Math Dept of Nanjing University) ART and its applications March 24, 2021 7 / 30
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Theory aspect
Martin-Löf test
Definition (Martin-Löf)
(i) Given a real x, a Σ01 Martin-Löf test is a computable collection
{Vn : n ∈ N} of x-c.e. sets such that µ(Vn) ≤ 2−n.
(ii) Given a real x, a real y is said to pass the Σ01(x) Martin-Löf test if
y /∈∩
n∈ω Vn.
(iii) Given a real x, a real y is said to be 1-x-random if it passes allΣ0
1(x) Martin-Löf tests.
There exists a universal c.e. Martin-Löf test.
Yu (Math Dept of Nanjing University) ART and its applications March 24, 2021 7 / 30
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Theory aspect
Betting strategy
Definition1 A martingale is a function f : 2<ω 7→ R such that for all σ ∈ 2<ω,
f(σ) = f(σ⌢0)+f(σ⌢1)2 .
2 A martingale f is said to succeed on a real y iflim supn f(y ↾ n) = ∞.
Yu (Math Dept of Nanjing University) ART and its applications March 24, 2021 8 / 30
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Theory aspect
What is randomness?
A real x is random if1 ∀nC(x ↾ n) ≥ n;
2 ∀nK(x ↾ n) ≥ n;3 x doesn’t belong to any effective Matin-Löf test;4 No effective strategy can win on x.
Theorem (Schnorr)(1) does not exist and the others are equivalent.
Yu (Math Dept of Nanjing University) ART and its applications March 24, 2021 9 / 30
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Theory aspect
What is randomness?
A real x is random if1 ∀nC(x ↾ n) ≥ n;2 ∀nK(x ↾ n) ≥ n;
3 x doesn’t belong to any effective Matin-Löf test;4 No effective strategy can win on x.
Theorem (Schnorr)(1) does not exist and the others are equivalent.
Yu (Math Dept of Nanjing University) ART and its applications March 24, 2021 9 / 30
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Theory aspect
What is randomness?
A real x is random if1 ∀nC(x ↾ n) ≥ n;2 ∀nK(x ↾ n) ≥ n;3 x doesn’t belong to any effective Matin-Löf test;
4 No effective strategy can win on x.
Theorem (Schnorr)(1) does not exist and the others are equivalent.
Yu (Math Dept of Nanjing University) ART and its applications March 24, 2021 9 / 30
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Theory aspect
What is randomness?
A real x is random if1 ∀nC(x ↾ n) ≥ n;2 ∀nK(x ↾ n) ≥ n;3 x doesn’t belong to any effective Matin-Löf test;4 No effective strategy can win on x.
Theorem (Schnorr)(1) does not exist and the others are equivalent.
Yu (Math Dept of Nanjing University) ART and its applications March 24, 2021 9 / 30
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Theory aspect
What is randomness?
A real x is random if1 ∀nC(x ↾ n) ≥ n;2 ∀nK(x ↾ n) ≥ n;3 x doesn’t belong to any effective Matin-Löf test;4 No effective strategy can win on x.
Theorem (Schnorr)(1) does not exist and the others are equivalent.
Yu (Math Dept of Nanjing University) ART and its applications March 24, 2021 9 / 30
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Theory aspect
The Kolmogorov complexity of random reals
Theorem (Miller and Y)x is 1-random iff
∑n 2n−K(x↾n) < ∞.
So if x is random, then Kx(n) ≤ K(x ↾ n)− n + c for some constant c.
Yu (Math Dept of Nanjing University) ART and its applications March 24, 2021 10 / 30
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Theory aspect
The Kolmogorov complexity of random reals
Theorem (Miller and Y)x is 1-random iff
∑n 2n−K(x↾n) < ∞.
So if x is random, then Kx(n) ≤ K(x ↾ n)− n + c for some constant c.
Yu (Math Dept of Nanjing University) ART and its applications March 24, 2021 10 / 30
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Theory aspect
Rich v.s. Power
How to compare randomness? Does higher complexity mean higherrandomness?
Obviously a random real can compress itself.But can it do more? Does richer mean more power?
Yu (Math Dept of Nanjing University) ART and its applications March 24, 2021 11 / 30
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Theory aspect
Rich v.s. Power
How to compare randomness? Does higher complexity mean higherrandomness?Obviously a random real can compress itself.
But can it do more? Does richer mean more power?
Yu (Math Dept of Nanjing University) ART and its applications March 24, 2021 11 / 30
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Theory aspect
Rich v.s. Power
How to compare randomness? Does higher complexity mean higherrandomness?Obviously a random real can compress itself.But can it do more? Does richer mean more power?
Yu (Math Dept of Nanjing University) ART and its applications March 24, 2021 11 / 30
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Theory aspect
Some evidence from computability theory
Theorem (de Leeuw, Moore, Shannon, Shapiro; Sacks)If x is noncomputable, then µ({y : y ≥T x}) = 0.
Theorem (Stephan)A random real x is PA-complete iff x can compute the halting problem.
Theorem (Merkle and Y)Let E(n) =
∫Kx(n)dx, then E(n) =∗ K(n).
So a random real cannot be very powerful.
Yu (Math Dept of Nanjing University) ART and its applications March 24, 2021 12 / 30
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Theory aspect
Some evidence from computability theory
Theorem (de Leeuw, Moore, Shannon, Shapiro; Sacks)If x is noncomputable, then µ({y : y ≥T x}) = 0.
Theorem (Stephan)A random real x is PA-complete iff x can compute the halting problem.
Theorem (Merkle and Y)Let E(n) =
∫Kx(n)dx, then E(n) =∗ K(n).
So a random real cannot be very powerful.
Yu (Math Dept of Nanjing University) ART and its applications March 24, 2021 12 / 30
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Theory aspect
Some evidence from computability theory
Theorem (de Leeuw, Moore, Shannon, Shapiro; Sacks)If x is noncomputable, then µ({y : y ≥T x}) = 0.
Theorem (Stephan)A random real x is PA-complete iff x can compute the halting problem.
Theorem (Merkle and Y)Let E(n) =
∫Kx(n)dx, then E(n) =∗ K(n).
So a random real cannot be very powerful.
Yu (Math Dept of Nanjing University) ART and its applications March 24, 2021 12 / 30
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Theory aspect
K-degrees, vL-degrees, LR-degrees and LK-degrees
Definition1 x ≤K y if ∀n(K(x ↾ n) ≤ K(y ↾ n));2 x ≤vL y if for all z, x ⊕ z is random implies y ⊕ z is random;3 x ≤LR y if for all z, z is y-random implies z is x-random;4 x ≤LK y if ∃c∀n(Ky(n) ≤ Kx(n) + c).
Note that x ≤vL y implies x ≥LR y for x, y random. This gives anexplicit description for comparing randomness with power.
Yu (Math Dept of Nanjing University) ART and its applications March 24, 2021 13 / 30
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Theory aspect
K-degrees, vL-degrees, LR-degrees and LK-degrees
Definition1 x ≤K y if ∀n(K(x ↾ n) ≤ K(y ↾ n));2 x ≤vL y if for all z, x ⊕ z is random implies y ⊕ z is random;3 x ≤LR y if for all z, z is y-random implies z is x-random;4 x ≤LK y if ∃c∀n(Ky(n) ≤ Kx(n) + c).
Note that x ≤vL y implies x ≥LR y for x, y random. This gives anexplicit description for comparing randomness with power.
Yu (Math Dept of Nanjing University) ART and its applications March 24, 2021 13 / 30
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Theory aspect
Relationships of the reductions
Theorem (Miller, Yu)x ≤K y implies x ≤vL y.
So more complicated means more random.
Theorem (Kjos-Hanssen, Miller, Solomon)x ≤LR y iff x ≤LK y.
So compressing random exactly means compressing everything. Puttwo results together, we can say that more random means less power.
Yu (Math Dept of Nanjing University) ART and its applications March 24, 2021 14 / 30
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Theory aspect
Relationships of the reductions
Theorem (Miller, Yu)x ≤K y implies x ≤vL y.
So more complicated means more random.
Theorem (Kjos-Hanssen, Miller, Solomon)x ≤LR y iff x ≤LK y.
So compressing random exactly means compressing everything.
Puttwo results together, we can say that more random means less power.
Yu (Math Dept of Nanjing University) ART and its applications March 24, 2021 14 / 30
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Theory aspect
Relationships of the reductions
Theorem (Miller, Yu)x ≤K y implies x ≤vL y.
So more complicated means more random.
Theorem (Kjos-Hanssen, Miller, Solomon)x ≤LR y iff x ≤LK y.
So compressing random exactly means compressing everything. Puttwo results together, we can say that more random means less power.
Yu (Math Dept of Nanjing University) ART and its applications March 24, 2021 14 / 30
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Theory aspect
Higher Randomness Theory
It is arguable that the real world is computable.
At least mathematicians do not live in the computable world.
But, really...?
Yu (Math Dept of Nanjing University) ART and its applications March 24, 2021 15 / 30
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Theory aspect
Higher Randomness Theory
It is arguable that the real world is computable.
At least mathematicians do not live in the computable world.
But, really...?
Yu (Math Dept of Nanjing University) ART and its applications March 24, 2021 15 / 30
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Theory aspect
Higher Randomness Theory
It is arguable that the real world is computable.
At least mathematicians do not live in the computable world.
But, really...?
Yu (Math Dept of Nanjing University) ART and its applications March 24, 2021 15 / 30
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Theory aspect
Constructibility
We may perform recursive operators over sets just like numbers.
L0 = ∅,
Lα+1 is the closure of recursive operators over Lα,
Lα =∪β<α
Lβ,when α is limit.
L =∪α
Lα
Yu (Math Dept of Nanjing University) ART and its applications March 24, 2021 16 / 30
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Theory aspect
Constructibility
We may perform recursive operators over sets just like numbers.
L0 = ∅,
Lα+1 is the closure of recursive operators over Lα,
Lα =∪β<α
Lβ,when α is limit.
L =∪α
Lα
Yu (Math Dept of Nanjing University) ART and its applications March 24, 2021 16 / 30
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Theory aspect
Gödel’s Theorem
TheoremL satisfies ZFC.
So it is safe to say that mathematicians live in a “computable world”.
Yu (Math Dept of Nanjing University) ART and its applications March 24, 2021 17 / 30
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Theory aspect
Gödel’s Theorem
TheoremL satisfies ZFC.
So it is safe to say that mathematicians live in a “computable world”.
Yu (Math Dept of Nanjing University) ART and its applications March 24, 2021 17 / 30
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Theory aspect
The real world for recursion theorists
Not every ordinal exists.
Only computable ordinals exist.The lease upper bound of computable ordinal is ωCK
1 .
LωCK1
is the world for recursion theorists.
Yu (Math Dept of Nanjing University) ART and its applications March 24, 2021 18 / 30
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Theory aspect
The real world for recursion theorists
Not every ordinal exists.
Only computable ordinals exist.The lease upper bound of computable ordinal is ωCK
1 .
LωCK1
is the world for recursion theorists.
Yu (Math Dept of Nanjing University) ART and its applications March 24, 2021 18 / 30
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Theory aspect
The real world for recursion theorists
Not every ordinal exists.
Only computable ordinals exist.The lease upper bound of computable ordinal is ωCK
1 .
LωCK1
is the world for recursion theorists.
Yu (Math Dept of Nanjing University) ART and its applications March 24, 2021 18 / 30
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Theory aspect
Computation in LωCK1
For logicians, a computation is a Σ1-procedure. So a computation inLωCK
1is a searching procedure over the elements of LωCK
1.
The time in LωCK1
is longer than the space. This results in that sometechniques in computable world do not work in LωCK
1.
Yu (Math Dept of Nanjing University) ART and its applications March 24, 2021 19 / 30
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Theory aspect
Computation in LωCK1
For logicians, a computation is a Σ1-procedure. So a computation inLωCK
1is a searching procedure over the elements of LωCK
1.
The time in LωCK1
is longer than the space. This results in that sometechniques in computable world do not work in LωCK
1.
Yu (Math Dept of Nanjing University) ART and its applications March 24, 2021 19 / 30
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Theory aspect
Π11-randomness
DefinitionA real x is Π1
1-random if it does not belong to any Π11-null set.
Theorem (Sacks)If x is Π1
1-random, then ωx1 = ωCK
1 .
Yu (Math Dept of Nanjing University) ART and its applications March 24, 2021 20 / 30
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Theory aspect
Beyond ZFC
But Gödel’s L is not the right model to push randomness theoryfurther.
We need a “right” way to generalize “computation”.
This relates to some deep results in set theory.
Yu (Math Dept of Nanjing University) ART and its applications March 24, 2021 21 / 30
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Mathematical aspect
Euclid’s theorem
Theorem (Euclid)There are infinitely many prime numbers.
Proof.Suppose that there are only m-many prime numbers. So for anynumber n, n = pn1
1 · · · pnmm . Thus
C(n) ≤∑i≤m
2 log ni ≤ 2m max{log ni | i ≤ m} ≤ 2m log log n.
But if m is random, then C(n) ≥ log n, a contradiction.
Yu (Math Dept of Nanjing University) ART and its applications March 24, 2021 22 / 30
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Mathematical aspect
pn ≤ n(log n)2
Theorempn ≤ n(log n)2.
Proof.Given any number m, let pn be the largest prime dividing m. Then
C(m) ≤ C(n, mpn
) ≤ log n + log mpn
+ 2 log log n
If m is random, then log m ≤ C(m) and so
log m ≤ log n + log m − log pn + 2 log log n.
So pn ≤ n(log n)2.
Yu (Math Dept of Nanjing University) ART and its applications March 24, 2021 23 / 30
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Mathematical aspect
Banach’s theorem
Theorem (Banach)If f is a continuous function and has property (N), then it has propertyT2. I.e. for almost every real r, f−1(r) is countable.
Theorem (Russian school; Louveau)If f is has property (N) and measurable, then it has property T2.
Yu (Math Dept of Nanjing University) ART and its applications March 24, 2021 24 / 30
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Mathematical aspect
Banach’s theorem
Theorem (Banach)If f is a continuous function and has property (N), then it has propertyT2. I.e. for almost every real r, f−1(r) is countable.
Theorem (Russian school; Louveau)If f is has property (N) and measurable, then it has property T2.
Yu (Math Dept of Nanjing University) ART and its applications March 24, 2021 24 / 30
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Mathematical aspect
A recursion theoretical proof of Banach’s theorem
Let f be a recursive function having property N. Let r be a∆1
1(O)-random real. Then f−1(r) only contains ∆11(O)-random. So
f(x) = r implies ωx1 = ωCK
1 . Note that f−1(r) is a ∆11(r) set and only
contains reals Turing computing r. By Martin’s theorem, f−1(r) iscountable.
To prove the second theorem, note that a measurable function fequals to a Borel function almost everywhere. Since f hasproperty-(N), we may ignore the null part. Then apply the methodabove to the Borel function.
Yu (Math Dept of Nanjing University) ART and its applications March 24, 2021 25 / 30
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Mathematical aspect
A recursion theoretical proof of Banach’s theorem
Let f be a recursive function having property N. Let r be a∆1
1(O)-random real. Then f−1(r) only contains ∆11(O)-random. So
f(x) = r implies ωx1 = ωCK
1 . Note that f−1(r) is a ∆11(r) set and only
contains reals Turing computing r. By Martin’s theorem, f−1(r) iscountable.
To prove the second theorem, note that a measurable function fequals to a Borel function almost everywhere. Since f hasproperty-(N), we may ignore the null part. Then apply the methodabove to the Borel function.
Yu (Math Dept of Nanjing University) ART and its applications March 24, 2021 25 / 30
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Mathematical aspect
Some more results (1)
DefinitionA function f has Luzin-(M)-Property if it maps a null set to acountable set.
Theorem (Pauly, Westrick, Y)If f is a continuous function and has property (M), then f is aconstant function.If f is has property (M) and measurable, then the range of f iscountable.
Yu (Math Dept of Nanjing University) ART and its applications March 24, 2021 26 / 30
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Mathematical aspect
Some more results (2)
Proof.We only prove (1) for a recursive function f. For any x, let g be∆1
1(Ox)-generic. Then x and g forma a minimal pair of ∆11-degrees.
Moreover, x belongs to a ∆11(g)-null set A. So f(A) is a Σ1
1(g) andcountable set. Thus f(x) ≤∆1
1g, x and so must be ∆1
1. In other words,the range of f is countable and so constant.
Peng and I also proved that the general conclusion is independent ofZFC.
Yu (Math Dept of Nanjing University) ART and its applications March 24, 2021 27 / 30
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Mathematical aspect
Some questions
Question(1) Is there a function having property-(N) but not having property
T2?(2) Is there an uncountable ideal I of Turing degrees so that for any
real x and almost every real r, r is random relative to the idealgenerated by I ∪ {x}?
Both questions have positive answers under certain set theoryassumptions.
Yu (Math Dept of Nanjing University) ART and its applications March 24, 2021 28 / 30
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Mathematical aspect
Further readings
An introduction to Kolmogorov complexity, Li and Vitany, 2008.
Computability and randomness, Nies, 2012.
Algorithmic randomness and complexity, Downey and Hirschfeldt,2010.
Recursion theory, Chong and Yu, 2015.
Yu (Math Dept of Nanjing University) ART and its applications March 24, 2021 29 / 30
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Thanks!
Yu (Math Dept of Nanjing University) ART and its applications March 24, 2021 30 / 30