On partial randomness

Calude, Cristian S.; Staiger, Ludwig; Terwijn, Sebastiaan A.

doi:10.1016/j.apal.2005.06.004

Cited by 46 publications

(49 citation statements)

References 9 publications

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“…In the works [11], [12], we generalized the notion of the randomness of a real so that the degree of the randomness, which is often referred to as the partial randomness recently [2], [9], [3], can be characterized by a real T with 0 ≤ T ≤ 1 as follows.…”

Section: Partial Randomnessmentioning

confidence: 99%

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

Tadaki¹

2012

Math. Struct. Comp. Sci.

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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: Partial Randomnessmentioning

confidence: 99%

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

Tadaki¹

2012

Math. Struct. Comp. Sci.

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“…The known bound in Equation 6 is taken from [1]. Again, let F ⊆ X ω an ω-language having L α (F ) > 0 and c > 0 an arbitrary constant.…”

Section: Prefix Complexitymentioning

confidence: 99%

“…The concepts of fractal geometry are described at full length in Falconer's book [2]. In the papers [8] and [1] have been proved lower bounds for simple and prefix complexity depending on the Hausdorff dimension. We use the general Hausdorff measures to improve these bounds and to find a lower bound on a priori complexity for ω-languages.…”

Section: Introductionmentioning

confidence: 99%

Refined Bounds on Kolmogorov Complexity for ω-Languages

Mielke

2008

Electronic Notes in Theoretical Computer Science

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The paper investigates bounds on various notions of complexity for ω-languages. We understand the complexity of an ω-languages as the complexity of the most complex strings contained in it. There have been shown bounds on simple and prefix complexity using fractal Hausdorff dimension. Here these bounds are refined by using general Hausdorff measure originally introduced by Felix Hausdorff. Furthermore a lower bound for a priori complexity is shown.

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“…Alternative characterizations of constructive dimension in terms of variations of Martin-Löf tests and effectivizations of Hausdorff measure have been given by Reimann and Stephan [54] and Calude et al [4]. Doty has considered Turing reduction compression ratio in [8].…”

Section: Constructive Dimensionmentioning

confidence: 99%

Effective Fractal Dimension in Algorithmic Information Theory

Mayordomo

2008

New Computational Paradigms

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On partial randomness

Cited by 46 publications

References 9 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

Refined Bounds on Kolmogorov Complexity for ω-Languages

Effective Fractal Dimension in Algorithmic Information Theory

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