Relative Randomness for Martin-L¨of random sets NingNing Peng

1

Mathematical Institute, Tohoku University. [email protected]

February 23, 2012

1

Join work with Kojiro Higuchi, Takeshi Yamazaki and Kazuyuki Tanaka

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Outline

1

Preliminaries

2

Γ randomness

3

Semi Γ-randomness

4

Future Study

NingNing Peng (Mathematical Institute, Tohoku Relative University. Randomness [email protected]) for Martin-L¨ of random sets

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Preliminaries

Preliminaries

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Preliminaries

Notations

B σ, τ, · · · denote the elements of 2
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Preliminaries

A c.e. open set is a set of finite string U ⊂ 2N such that: B U is computably enumerable. B if σ, τ ∈ U then σ 6⊂ τ (the basic open sets [σ], [τ ] are disjoint).

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Preliminaries

ML-randomness

ML-randomness is a central notion of algorithmic randomness for subsets of N, which defined in the following way. Definition (Martin-L¨of [1]) (i) A Martin-L¨of test, or ML-test for short, is a uniformly c.e. sequence (Gm )m∈N of open sets such that ∀m ∈ N µ(Gm ) ≤ 2−m . T (ii) A set Z ⊆ N fails the test if Z ∈ m Gm , otherwise Z passes the test. (iii) Z is ML-random if Z passes each ML-test.

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Preliminaries

weakly 2-random

Definition (Kurtz [?]) (i) A generalized ML-test is a uniformly c.e. sequence (Gm )m∈N of open T sets such that µ( m Gm ) = 0. (ii) Z is weakly 2-random if it passes every generalized ML-test.

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Preliminaries

weakly 2-random

Definition (Kurtz [?]) (i) A generalized ML-test is a uniformly c.e. sequence (Gm )m∈N of open T sets such that µ( m Gm ) = 0. (ii) Z is weakly 2-random if it passes every generalized ML-test.

Fact (i) 2-randomness ⇒ weak 2-randomness ⇒ ML-randomness. (ii) The reverse implication fails (Kurtz, Kautz).

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Γ randomness

Γ randomness

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Γ randomness

Relative Randomness

These definitions are relativised: add oracle A to tests to get A-randomness. x is A-random if x 6∈

T

UiA for universal oracle test Ui .

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Γ randomness

Relative Randomness

These definitions are relativised: add oracle A to tests to get A-randomness. x is A-random if x 6∈

T

UiA for universal oracle test Ui .

We study the colloection of these randomness notions.

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Γ randomness

Γ-randomness

We recall some notions in [3]. Definition A set Z is Γ-random if Z is ML-random relative to A for all A ∈ Γ. Any x-ML test for x ∈ Γ is called Γ-test.

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Γ randomness

Γ-randomness

We recall some notions in [3]. Definition A set Z is Γ-random if Z is ML-random relative to A for all A ∈ Γ. Any x-ML test for x ∈ Γ is called Γ-test. Γ-randomness is called L-randomness if Γ is the set of low sets. Obviously, any L-random set is ML-random. For any set Z , Z is not 1-random in Z . Thus, each low set is not L-random. Hence, L-randomness is strictly stronger than ML-randomness since there is a low ML-random set.

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Γ randomness

Usually, randomness notions stronger than ML-randomness are closed down- wards under Turing reducibility within the random sets. The notions we study here are not exception. Proposition Let X , Y be ML-random sets. If X ≤T Y and Y is L-random, then X is L-random.

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Γ randomness

Usually, randomness notions stronger than ML-randomness are closed down- wards under Turing reducibility within the random sets. The notions we study here are not exception. Proposition Let X , Y be ML-random sets. If X ≤T Y and Y is L-random, then X is L-random. In fact, it turned out that L-randomness is equivalent to ∅0 -Schnorr randomness. Theorem (Yu [4]) L-randomness is equivalent to ∅0 -Schnorr randomness.

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Γ randomness

Characterization of L-randomness A ≤LR B iff for any X , if X is B-random, then X is A-random. We would like to introduce another characterization of L-randomness. Then, the next lemma is useful. Lemma Let Γ, Γ0 ⊂ NN such that for any f ∈ Γ there is a function g ∈ Γ ∩ Γ0 with f ≤LR g . Then, Γ randomness is equivalent to Γ ∩ Γ0 randomness.

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Γ randomness

Characterization of L-randomness A ≤LR B iff for any X , if X is B-random, then X is A-random. We would like to introduce another characterization of L-randomness. Then, the next lemma is useful. Lemma Let Γ, Γ0 ⊂ NN such that for any f ∈ Γ there is a function g ∈ Γ ∩ Γ0 with f ≤LR g . Then, Γ randomness is equivalent to Γ ∩ Γ0 randomness.

Proof. g f 0 For T any fΓ-test T {Un }gn∈N , there exists Γ ∩ Γ test {V0 n }n∈N such that n∈N Un ⊆ n∈N Vn , since f ≤LR g . Then, Γ ∩ Γ randomness implies Γ randomness.

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Γ randomness

1-generic

Definition An r.e. set W ⊂ 2
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Γ randomness

1-generic

Definition An r.e. set W ⊂ 2
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Γ randomness

By previous Lemma, L randomness can be also given by subsets of L as follows. PA denotes the set of reals of PA degrees. Proposition The following are equivalent: (i) X is L randomness, (ii) X is L ∩ G randomness, (iii) X is L ∩ PA randomness.

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Γ randomness

By previous Lemma, L randomness can be also given by subsets of L as follows. PA denotes the set of reals of PA degrees. Proposition The following are equivalent: (i) X is L randomness, (ii) X is L ∩ G randomness, (iii) X is L ∩ PA randomness. Proof. (ii) ⇒ (i) is proved from previous Lemma since there exsits a low g such that g is 1-generic relative to f and f ≤T g , for any low f . (iii) ⇒ (i) is the same.

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Γ randomness

Let MLR denote the class of ML-random reals. Conjecture L randomness is equivalent to L ∩ MLR randomness, In other word, L randomness ⇔ {x | ∀ y : low & ML-random, x is y-random }

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Semi Γ-randomness

semi Γ-randomness

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Semi Γ-randomness

Motivation

The present paper is concerned with the algorithmic notion of randomness such as originally introduced by P. Martin-L¨of [1] in 1966. One approach is to generalize the Martin-L¨ of-test by giving the m-th component (a c.e. set of measure at most 2−m ) via a function in some function class Γ. A main purpose of this paper is to give a general framework for such randomness notions. To do this, we introduce the notion of semi Γ-randomness.

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Semi Γ-randomness

Semi Γ-random In this section, we investigate a new randomness notion weaker than Γ-random. We concentrate on index function for the componets of a test. Definition Let Γ ⊆ NN : 1

2 3

We say that a sequence {Gn }n∈N of c.e open sets is a Γ-indexed test if and only if there exists f ∈ Γ such that Gn = Wf (n) for all n ∈ N and µ(Gn ) ≤ 2−n . T A set Z ⊆ N fails the test if Z ∈ n Gn , otherwise Z passes the test. A is semi Γ-random if it passes every Γ-indexed test.

Note that, there is no semi NN -random set. It is straightforward from the definition that ML-randomness is equivalent to semi ∆01 -randomness.

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Semi Γ-randomness

Theorem A set is semi ∆02 -random if and only if it is weakly 2-random.

Proposition Every L-random real is semi L-random.

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Semi Γ-randomness

Theorem A set is semi ∆02 -random if and only if it is weakly 2-random.

Proposition Every L-random real is semi L-random.

Theorem For any Γ ⊂ NN , there is no universal Γ indexed Martin-L¨of test unless any Martin-L¨of random set is semi Γ-random.

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Semi Γ-randomness

Separation between random notions

The following lemma is for separation between weak randomness and semi Γ-randomness. Lemma For any Γ, Γ0 ⊂ NN such that Γ randomness is not equivalent to Martin-L¨of randomness and Γ0 is countable, there exists a semi Γ0 random real which is not Γ random.

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Semi Γ-randomness

Proof of Lemma

T Choose a Martin-L¨of random f which is in i∈N Vi , where {Vi }i∈N is a universal g -Martin-L¨of test for some g ∈ Γ. Let {{Ugi (j) }j∈N }i∈N be a sequence of all Γ0 indexed test.

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Semi Γ-randomness

Proof of Lemma

T Choose a Martin-L¨of random f which is in i∈N Vi , where {Vi }i∈N is a universal g -Martin-L¨of test for some g ∈ Γ. Let {{Ugi (j) }j∈N }i∈N be a sequence of all Γ0 indexed test. We construct a function h and a ⊂-increasing sequence {σi }i∈N such that [σi ] ⊂ Vi and (limj→∞ σj ) 6∈ Ugi (h(i)) for any i ∈ N. Then a semi Γ0 -random

S

σi is not Γ-random.

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Semi Γ-randomness

How to construct h and σi ? S Stage s:SLet σ = i 0.

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Semi Γ-randomness

How to construct h and σi ? S Stage s:SLet σ = i 0. Define h(s) by the least number x such that [ µ([σ] \ ( Ugi (h(i)) ∪ Ugs (x) )) > 0. i
Choose a tail f 0 of f such that σf 0 ∈ [σ] \

[

Ugi (h(i))

i≤s

Note that σf 0 ∈ and [σs ] ⊂ Vs .

T

i∈N Vi .

Thus, we can choose σs ) σ such that σs ⊂ σf 0

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Semi Γ-randomness

Theorem If n > 2, then weakly n randomness is strictly stronger than semi ∆0n -randomness.

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Semi Γ-randomness

Theorem If n > 2, then weakly n randomness is strictly stronger than semi ∆0n -randomness.

Proof. It is clear that weakly n randomness is stronger than semi ∆0n randomness. This relation is strict since weakly n randomness is stronger that n − 1 randomness which contains some Martin-L¨ of random set so that we can apply to the previous Lemma.

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Semi Γ-randomness

Finally, we consider the case around n = 2. Lemma There exists a Π01 (∅0 ) null set P containing a semi L random element.

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Semi Γ-randomness

Proof sketch:

Let f be a strictly increasing ∅0 -computable function such that no low function dominates f and 2f (x) ≤ 2f (x)−f (x−1) . Let F refer to the set {σ ∈ N
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Semi Γ-randomness

Let B(σ) be an element τ ∈ 2f (lh(σ)) such that [ µ([τ ] \ ( [Wσ(i) ])) ≥ 2−lh(στ )−1 i∈A(σ)

for all σ ∈ F . Define a Π01 (∅0 ) null set P by P = [B(F )]. Let {{Whi (x) }x∈N }i∈N be a sequence of all L indexed test. (Indeed, we gives some technical assumption to hi ’s.) Then we can construct a ⊂-increasing sequence {σi }i∈N of elements Sof F and a function h : N → N such that limi→∞ B(σi ) 6∈ [Whj (h(j)) ]. So B(σi ) ∈ P is semi L random.

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Semi Γ-randomness

The following is straightfoward from Lemma 13. Theorem Weakly 2 randomness is strictly stronger than semi L-randomness

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Semi Γ-randomness

The following is straightfoward from Lemma 13. Theorem Weakly 2 randomness is strictly stronger than semi L-randomness

Proof. This is because for any Π01 (∅0 ) null set PTthere exists a generalized Martin-L¨of test {Ui }i∈N such that P = i∈N Ui .

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Semi Γ-randomness

Proposition 2-randomness does not imply semi ∆03 -randomness.

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Semi Γ-randomness

Proposition 2-randomness does not imply semi ∆03 -randomness.

Proof. Since there is a ∆03 2 random set, we show that there is no ∆03 semi ∆03 -random set. Let A be a ∆03 set, then there is a ∆03 function f such that Wf (n) = {Akn}. Note that Wf (n) is a singleton set where only element isTAkn and µ(Wf (n) ) ≤ 2−n . So, {Wf (n) }n∈N is a semi ∆03 -test. But A ∈ n∈N Wf (n) . Hence, A is not semi ∆03 -random.

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Future Study

Future Study

Conjecture L randomness is equivalent to L ∩ MLR randomness, Conjecture smei L-randomness and Demuth randomness are incompareble. Question Is the a characterization of ∅0 -computable randomness via ML randomness?

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Future Study

References

Per. Martin-L¨of: The definition of random sequences. Information and Control, vol. 9, no. 6, pp. 602-619 (1966) Andr´e Nies: Computability and Randomness. Oxford University Press (2009) NingNing Peng: The notions between Martin L¨ of randomness and 2-randomness. RIMS Kˆ o kyˆ u roku, No. 1792, pp. 117-122 (2010) Liang Yu: Characterizing strong randomness via Martin-L¨of randomness. Annals of Pure and Applied Logic, vol. 163, no. 3, pp. 214-224 (2012)

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Future Study

Thank you very much!

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