Schnorr Randomness

Downey, Rodney G.; Griffiths, Evan J.

doi:10.1016/s1571-0661(04)80376-1

Cited by 7 publications

(8 citation statements)

References 11 publications

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“…In order to prove the main result, it is also convenient to use the notion of computable measure machine, which was introduced by Downey and Griffiths [6] in 2004 for the purpose of characterizing the notion of Schnorr randomness of a real in terms of program-size complexity.…”

Section: The Proof Of the Main Resultsmentioning

confidence: 99%

A statistical mechanical interpretation of algorithmic information theory III: Composite systems and fixed points

Tadaki

2009

2009 IEEE Information Theory Workshop

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Abstract-The statistical mechanical interpretation of algorithmic information theory (AIT, for short) was introduced and developed by our former works [K. Tadaki, Local Proceedings of CiE 2008, pp. 425-434, 2008 and [K. Tadaki, Proceedings of LFCS'09, Springer's LNCS, vol. 5407, pp. 422-440, 2009], where we introduced the notion of thermodynamic quantities, such as partition function Z(T ), free energy F (T ), energy E(T ), and statistical mechanical entropy S(T ), into AIT. We then discovered that, in the interpretation, the temperature T equals to the partial randomness of the values of all these thermodynamic quantities, where the notion of partial randomness is a stronger representation of the compression rate by means of program-size complexity. Furthermore, we showed that this situation holds for the temperature itself as a thermodynamic quantity, namely, for each of all the thermodynamic quantities above, the computability of its value at temperature T gives a sufficient condition for T 2 (0; 1) to be a fixed point on partial randomness. In this paper, we develop the statistical mechanical interpretation of AIT further and pursue its formal correspondence to normal statistical mechanics. The thermodynamic quantities in AIT are defined based on the halting set of an optimal computer, which is a universal decoding algorithm used to define the notion of program-size complexity. We show that there are infinitely many optimal computers which give completely different sufficient conditions in each of the thermodynamic quantities in AIT. We do this by introducing the notion of composition of computers to AIT, which corresponds to the notion of composition of systems in normal statistical mechanics.

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Section: The Proof Of the Main Resultsmentioning

confidence: 99%

A statistical mechanical interpretation of algorithmic information theory III: Composite systems and fixed points

Tadaki

2009

2009 IEEE Information Theory Workshop

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“…The intuition can not be taken literally because every real number belongs to the singleton set containing it, which has measure zero. To prevent the property being void one can restrict to computably defined sets, as done by Martin-Löf in [21] and Schnorr in [23,12].…”

Section: Randomnessmentioning

confidence: 99%

Randomness and uniform distribution modulo one

Becher¹,

Grigorieff²

2021

Preprint

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We elaborate the notions of Martin-Löf and Schnorr randomness for real numbers in terms of uniform distribution of sequences. We give a necessary condition for a real number to be Schnorr random expressed in terms of classical uniform distribution of sequences. This extends the result proved by Avigad for sequences of linear functions with integer coefficients to the wider classical class of Koksma sequences of functions. And, by requiring equidistribution with respect to every computably enumerable open set (respectively, computably enumerable open set with computable measure) in the unit interval, we give a sufficient condition for Martin-Löf (respectively Schnorr) randomness.

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“…Schnorr triviality. Franklin and Stephan [11] characterise the Schnorr trivial sets (defined by Downey and Griffiths in [5]) as those sets whose truth-table degree is recursively traceable, that is, there is some order function h which bounds traces for all functions f truth-table reducible to the degree a, but where the trace T n is required to be given recursively (as a sequence of finite sets) rather than merely uniformly recursively enumerably. In other words, there is a recursive function g such that for all n, g(n) is the canonical index for the finite set T n (in Soare's [31] notation, T n = D g(n) ).…”

Section: 1mentioning

confidence: 99%

Anti-Complex Sets and Reducibilities with Tiny Use

Franklin¹,

Greenberg

Stephan³

et al. 2013

J. symb. log.

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Abstract. In contrast with the notion of complexity, a set A is called anticomplex if the Kolmogorov complexity of the initial segments of A chosen by a recursive function is always bounded by the identity function. We show that, as for complexity, the natural arena for examining anti-complexity is the weak-truth table degrees. In this context, we show the equivalence of anticomplexity and other lowness notions such as r.e. traceability or being weak truth-table reducible to a Schnorr trivial set. A set A is anti-complex if and only if it is reducible to another set B with tiny use, whereby we mean that the use function for reducing A to B can be made to grow arbitrarily slowly, as gauged by unbounded nondecreasing recursive functions. This notion of reducibility is then studied in its own right, and we also investigate its range and the range of its uniform counterpart.

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Schnorr Randomness

Cited by 7 publications

References 11 publications

A statistical mechanical interpretation of algorithmic information theory III: Composite systems and fixed points

A statistical mechanical interpretation of algorithmic information theory III: Composite systems and fixed points

Randomness and uniform distribution modulo one

Anti-Complex Sets and Reducibilities with Tiny Use

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