Solovay functions and their applications in algorithmic randomness

Bienvenu, Laurent; Downey, Rodney G.; Nies, André; Merkle, Wolfgang

doi:10.1016/j.jcss.2015.04.004

Cited by 13 publications

(13 citation statements)

References 37 publications

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“…To finish the proof, we appeal to the theory of Solovay functions. When h is a computable positive function, the sum n 2 -h(n) is not random if and only if h(n) -K (n) → ∞ [4,5]. This is the case here as = n 2 -h(n) is Solovay-incomplete hence not random.…”

Section: Claim 4 α Is Non-computablementioning

confidence: 99%

Some Questions of Uniformity in Algorithmic Randomness

Bienvenu¹,

Csima²,

Harrison-Trainor³

2021

J. symb. log.

Self Cite

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The $\Omega $ numbers—the halting probabilities of universal prefix-free machines—are known to be exactly the Martin-Löf random left-c.e. reals. We show that one cannot uniformly produce, from a Martin-Löf random left-c.e. real $\alpha $ , a universal prefix-free machine U whose halting probability is $\alpha $ . We also answer a question of Barmpalias and Lewis-Pye by showing that given a left-c.e. real $\alpha $ , one cannot uniformly produce a left-c.e. real $\beta $ such that $\alpha - \beta $ is neither left-c.e. nor right-c.e.

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Section: Claim 4 α Is Non-computablementioning

confidence: 99%

Some Questions of Uniformity in Algorithmic Randomness

Bienvenu¹,

Csima²,

Harrison-Trainor³

2021

J. symb. log.

Self Cite

View full text Add to dashboard Cite

show abstract

“…To finish the proof, we appeal to the theory of Solovay functions. When h is a computable positive function, the sum ∑ n 2 −h(n) is not random if and only if h(n) − K(n) → ∞ [BD09,BDNM15]. This is the case here as δ = ∑ n 2 −h(n) is Solovayincomplete hence not random.…”

Section: End Constructionmentioning

confidence: 99%

Some Questions of Uniformity in Algorithmic Randomness

Bienvenu,

Csima,

Harrison-Trainor

2021

Preprint

Self Cite

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The Ω numbers-the halting probabilities of universal prefix-free machines-are known to be exactly the Martin-Löf random left-c.e. reals. We show that one cannot uniformly produce, from a Martin-Löf random left-c.e. real α, a universal prefix-free machine U whose halting probability is α. We also answer a question of Barmpalias and Lewis-Pye by showing that given a left-c.e. real α, one cannot uniformly produce a left-c.e. real β such that α − β is neither left-c.e. nor right-c.e.

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“…There are also other relevant and promising approaches to measures motivated by algorithmic complexity that should be explored and supported by the community as alternatives to statistical compression. Some of these include Solovay functions [ 68 , 69 ], minimal computer languages [ 70 ] and, approaches similar to ours [ 42 ], model proposals involving restricted levels of computational power (but beyond statistical compression algorithms) such as finite-state and transducer complexity [ 71 , 72 , 73 ]. We ourselves have produced a variant of a transducer complexity model in a CTM fashion [ 42 ], while also introducing other CTM approaches based on different models and computational power [ 42 ].…”

Section: Alternatives To Lossless Compressionmentioning

confidence: 99%

A Review of Methods for Estimating Algorithmic Complexity: Options, Challenges, and New Directions

Zenil

2020

Entropy

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Some established and also novel techniques in the field of applications of algorithmic (Kolmogorov) complexity currently co-exist for the first time and are here reviewed, ranging from dominant ones such as statistical lossless compression to newer approaches that advance, complement and also pose new challenges and may exhibit their own limitations. Evidence suggesting that these different methods complement each other for different regimes is presented and despite their many challenges, some of these methods can be better motivated by and better grounded in the principles of algorithmic information theory. It will be explained how different approaches to algorithmic complexity can explore the relaxation of different necessary and sufficient conditions in their pursuit of numerical applicability, with some of these approaches entailing greater risks than others in exchange for greater relevance. We conclude with a discussion of possible directions that may or should be taken into consideration to advance the field and encourage methodological innovation, but more importantly, to contribute to scientific discovery. This paper also serves as a rebuttal of claims made in a previously published minireview by another author, and offers an alternative account.

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Solovay functions and their applications in algorithmic randomness

Cited by 13 publications

References 37 publications

Some Questions of Uniformity in Algorithmic Randomness

Some Questions of Uniformity in Algorithmic Randomness

Some Questions of Uniformity in Algorithmic Randomness

A Review of Methods for Estimating Algorithmic Complexity: Options, Challenges, and New Directions

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