Some Results on Effective Randomness

Merkle, Wolfgang; Mihailovic, Nenad; Slaman, Theodore A.

doi:10.1007/s00224-005-1212-8

Cited by 10 publications

(24 citation statements)

References 21 publications

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“…In the terminology of this paper, a martingale process is basically a computable betting strategy on 2 ω with the fair‐coin measure which bets on decidable sets (i.e., finite unions of basic open sets). Merkle, Mihailović and Slaman showed that Martin‐Löf randomness is equivalent to the randomness characterized by martingale processes . The proof of this next theorem is basically their proof…”

Section: Betting Strategies and Kolmogorov‐loveland Randomnessmentioning

confidence: 89%

“…The first is due to Downey, Griffiths, and LaForte . However, we shall use the other due to Merkle, Mihailović, and Slaman . Definition On

(2^{ω}, λ)

a Martin‐Löf test

(U_{n})

is called a bounded Martin‐Löf test if there is a computable measure

ν : 2^{< ω} \to [0, \infty)

such that for all

n \in N

and

σ \in 2^{< ω}

λ (U_{n} \cap {[σ]}^{≺}) \leq 2^{- n} ν (σ) .

We say that the test

(U_{n})

is bounded by the measure ν.…”

Section: Computable Randomness With Respect To Computable Probabilitymentioning

confidence: 99%

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Computable randomness and betting for computable probability spaces

Rute

2016

Mathematical Logic Qtrly

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Abstract. Unlike Martin-Löf randomness and Schnorr randomness, computable randomness has not been defined, except for a few ad hoc cases, outside of Cantor space. This paper offers such a definition (actually, several equivalent definitions), and further, provides a general method for abstracting "bit-wise" definitions of randomness from Cantor space to arbitrary computable probability spaces. This same method is also applied to give machine characterizations of computable and Schnorr randomness for computable probability spaces, extending the previously known results. The paper contains a new type of randomness-endomorphism randomness-which the author hopes will shed light on the open question of whether Kolmogorov-Loveland randomness is equivalent to Martin-Löf randomness. The last section contains ideas for future research.

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Section: Betting Strategies and Kolmogorov‐loveland Randomnessmentioning

confidence: 89%

“…The first is due to Downey, Griffiths, and LaForte . However, we shall use the other due to Merkle, Mihailović, and Slaman . Definition On

(2^{ω}, λ)

a Martin‐Löf test

(U_{n})

is called a bounded Martin‐Löf test if there is a computable measure

ν : 2^{< ω} \to [0, \infty)

such that for all

n \in N

and

σ \in 2^{< ω}

λ (U_{n} \cap {[σ]}^{≺}) \leq 2^{- n} ν (σ) .

We say that the test

(U_{n})

is bounded by the measure ν.…”

Section: Computable Randomness With Respect To Computable Probabilitymentioning

confidence: 99%

Computable randomness and betting for computable probability spaces

Rute

2016

Mathematical Logic Qtrly

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“…Definition 2.12 (Merkle, Mihailovic, and Slaman 15) A computable rational probability distribution (or measure) is a computable function \documentclass{article}\usepackage{amssymb}\pagestyle{empty}\begin{document}$\nu :2^{< \omega }\longrightarrow \mathbb {Q}$\end{document}, with ν(λ) = 1 and ν(σ) = ν(σ0) + ν(σ1). A bounded Martin‐Löf test is a Martin‐Löf test for which there exists a computable rational probability distribution ν with for all \documentclass{article}\usepackage{amssymb}\pagestyle{empty}\begin{document}$n\in \mathbb {N}$\end{document} and σ ∈ 2 <ω .…”

Section: Preliminariesmentioning

confidence: 99%

“…Theorem 2.14 (Merkle, Mihailovic, and Slaman 15, Downey, Griffiths, and LaForte 3) A real is computably random iff it withstands all bounded Martin‐Löf tests (iff it withstands all computably graded tests).…”

Section: Preliminariesmentioning

confidence: 99%

Truth-table Schnorr randomness and truth-table reducible randomness

Miyabe

2011

Mathematical Logic Quarterly

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MSC (2010) 68Q30, 03D15, 03D25Schnorr randomness and computable randomness are natural concepts of random sequences. However van Lambalgen's Theorem fails for both randomnesses. In this paper we define truth-table Schnorr randomness (defined in [6] too only by martingales) and truth-table reducible randomness, for which we prove that van Lambalgen's Theorem holds. We also show that the classes of truth-table Schnorr random reals relative to a high set contain reals Turing equivalent to the high set. It follows that each high Schnorr random real is half of a real for which van Lambalgen's Theorem fails. Moreover we establish the coincidence between triviality and lowness notions for truth-table Schnorr randomness.

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“…Miyabe [19] obtains a compelling computational difference between optimal ML-tests and universal ML-tests: by a result of Merkle, Mihailović, and Slaman [18], there is a universal ML-test U and a left-c.e. real α such that ∀n(λ(U n ) = 2 −n α).…”

Section: Introductionmentioning

confidence: 99%

Universality, optimality, and randomness deficiency

Hölzl

Shafer

2015

Annals of Pure and Applied Logic

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Abstract. A Martin-Löf test U is universal if it captures all non-Martin-Löf random sequences, and it is optimal if for every ML-test V there is a c ∈ ω such that ∀n(Vn+c ⊆ Un). We study the computational differences between universal and optimal ML-tests as well as the effects that these differences have on both the notion of layerwise computability and the Weihrauch degree of LAY, the function that produces a bound for a given Martin-Löf random sequence's randomness deficiency. We prove several robustness and idempotence results concerning the Weihrauch degree of LAY, and we show that layerwise computability is more restrictive than Weihrauch reducibility to LAY. Along similar lines we also study the principle RD, a variant of LAY outputting the precise randomness deficiency of sequences instead of only an upper bound as LAY.

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Some Results on Effective Randomness

Cited by 10 publications

References 21 publications

Computable randomness and betting for computable probability spaces

Computable randomness and betting for computable probability spaces

Truth-table Schnorr randomness and truth-table reducible randomness

Universality, optimality, and randomness deficiency

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