Computability and Randomness

Nies, André

doi:10.1093/acprof:oso/9780199230761.001.0001

Cited by 426 publications

(99 citation statements)

References 0 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…It is convenient here to use Schnorr's martingale characterization [22,21,23,13,19,6] of the algorithmic randomness notion introduced by Martin-Löf [18]. If ν is a probability measure on Σ ∞ , then a ν-martingale is a function d : Σ * → [0, ∞) satisfying d(w)ν(w) = a∈Σ d(wa)ν(wa) for all w ∈ Σ * .…”

Section: For Everymentioning

confidence: 99%

Mutual Dimension and Random Sequences

Case

Lutz

2015

Mathematical Foundations of Computer Science 2015

View full text Add to dashboard Cite

If S and T are infinite sequences over a finite alphabet, then the lower and upper mutual dimensions mdim(S : T ) and M dim(S : T ) are the upper and lower densities of the algorithmic information that is shared by S and T . In this paper we investigate the relationships between mutual dimension and coupled randomness, which is the algorithmic randomness of two sequences R 1 and R 2 with respect to probability measures that may be dependent on one another. For a restricted but interesting class of coupled probability measures we prove an explicit formula for the mutual dimensions mdim(R 1 : R 2 ) and M dim(R 1 : R 2 ), and we show that the condition M dim(R 1 : R 2 ) = 0 is necessary but not sufficient for R 1 and R 2 to be independently random.We also identify conditions under which Billingsley generalizations of the mutual dimensions mdim(S : T ) and M dim(S : T ) can be meaningfully defined; we show that under these conditions these generalized mutual dimensions have the "correct" relationships with the Billingsley generalizations of dim(S), Dim(S), dim(T ), and Dim(T ) that were developed and applied by Lutz and Mayordomo; and we prove a divergence formula for the values of these generalized mutual dimensions.

show abstract

Section: For Everymentioning

confidence: 99%

Mutual Dimension and Random Sequences

Case

Lutz

2015

Mathematical Foundations of Computer Science 2015

View full text Add to dashboard Cite

show abstract

“…(Nies [93]). A good account of this material can be found in Nies [94,95], but things are constantly changing, with perhaps seventeen characterizations of this class at present. We also refer to [46] for the situation up to mid-2010.…”

Section: Computability and Randomnessmentioning

confidence: 99%

“…In particular, we have seen significant clarification as to the mathematical relationship between algorithmic computational power of infinite random sources and level algorithmic randomness. Much of this material has been reported in the short surveys Downey [41,42], Nies [95] and long surveys [40,47] and long monographs Downey and Hirschfeldt [46] and Nies [94]. Also the book edited by Hector Zenil [152] has a lot of discussion of randomness of varying levels of technicality, many aimed at a general audience.…”

mentioning

confidence: 99%