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Randomness Space
 Automata, Languages and Programming, Proceedings of the 25th International Colloquium, ICALPâ€™98
, 1998
"... MartinL#of de#ned in#nite random sequences over a #nite alphabet via randomness tests which describe sets having measure zero in a constructive sense. In this paper this concept is generalized to separable topological spaces with a measure, following a suggestion of Zvonkin and Levin. After stud ..."
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MartinL#of de#ned in#nite random sequences over a #nite alphabet via randomness tests which describe sets having measure zero in a constructive sense. In this paper this concept is generalized to separable topological spaces with a measure, following a suggestion of Zvonkin and Levin. After studying basic results and constructions for such randomness spaces a general invariance result is proved which gives conditions under which a function between randomness spaces preserves randomness. This corrects and extends a result bySchnorr. Calude and J#urgensen proved that the randomness notion for real numbers obtained by considering their bary representations is independent from the base b. We use our invariance result to show that this notion is identical with the notion which one obtains by viewing the real number space directly as a randomness space. Furthermore, arithmetic properties of random real numbers are derived, for example that every computable analytic function pres...
Measure One Results In Computational Complexity Theory
 Advances in Algorithms, Languages, and Complexity
, 1997
"... This paper is dedicated to Ronald V. Book on the occasion of his 60th birthday. ..."
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Cited by 5 (3 self)
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This paper is dedicated to Ronald V. Book on the occasion of his 60th birthday.
Do the Zeros of Riemann's ZetaFunction Form a Random Sequence?
 Bull. Eur. Assoc. Theor. Comput. Sci. EATCS
, 1997
"... The aim of this note is to introduce the notion of random sequences of reals and to prove that the answer to the question in the title is negative, as anticipated by the informal discussion of Longpr#e and Kreinovich #15#. Keywords: Riemann zetafunction, random real, random sequence of reals 1 ..."
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The aim of this note is to introduce the notion of random sequences of reals and to prove that the answer to the question in the title is negative, as anticipated by the informal discussion of Longpr#e and Kreinovich #15#. Keywords: Riemann zetafunction, random real, random sequence of reals 1 Introduction Riemann's Hypothesis, a famous open problem of mathematics, states that all complex roots #zeros# s = Re#s#+iIm#s# of the Riemann's zetafunction ##s#= 1 X n=1 1 n s #i.e., the values for which ##s# = 0# are located on the straight line Re#s# = 1=2 in the complex plane #except for the known zeros, which are the negative even integers#. This hypothesis has appeared as Problem No. 8 in Hilbert's famous 1900 list of 23 open problems #see Hilbert #11#, Browder #1#, and Karatsuba and Voronin #13##. It has been proven that the real parts of the nontrivial zeros s of the Riemann's zetafunction are close to 1=2, so they form a highly organized set. In fact a large proportion ...
Separations by Random Oracles and "Almost" Classes for Generalized Reducibilities
 In Proceedings 20th Symposium on Mathematical Foundations of Computer Science
, 1995
"... . Let r and s be two binary relations on 2 N which are meant as reducibilities. Let both relations be closed under finite variation (of their set arguments) and consider the uniform distribution on 2 N , which is obtained by choosing elements of 2 N by independent tosses of a fair coin. Th ..."
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. Let r and s be two binary relations on 2 N which are meant as reducibilities. Let both relations be closed under finite variation (of their set arguments) and consider the uniform distribution on 2 N , which is obtained by choosing elements of 2 N by independent tosses of a fair coin. Then we might ask for the probability that the lower r cone of a randomly chosen set X, that is, the class of all sets A with A r X, differs from the lower scone of X. By closure under finite variation, the Kolmogorov 01 law yields immediately that this probability is either 0 or 1; in case it is 1, the relations are said to be separable by random oracles. Again by closure under finite variation, for every given set A, the probability that a randomly chosen set X is in the upper rcone of A is either 0 or 1; let Almostr be the class of sets for which the upperr cone has measure 1. In the following, results about separations by random oracles and about Almost classes are obtained ...
The Computational Complexity Column
, 1998
"... Introduction Investigation of the measuretheoretic structure of complexity classes began with the development of resourcebounded measure in 1991 [56]. Since that time, a growing body of research by more than forty scientists around the world has shown resourcebounded measure to be a powerful too ..."
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Introduction Investigation of the measuretheoretic structure of complexity classes began with the development of resourcebounded measure in 1991 [56]. Since that time, a growing body of research by more than forty scientists around the world has shown resourcebounded measure to be a powerful tool that sheds new light on many aspects of computational complexity. Recent survey papers by Lutz [60], AmbosSpies and Mayordomo [3], and Buhrman and Torenvliet [22] describe many of the achievements of this line of inquiry. In this column, we give a more recent snapshot of resourcebounded measure, focusing not so much on what has been achieved to date as on what we hope will be achieved in the near future. Section 2 below gives a brief, nontechnical overview of resourcebounded measure in terms of its motivation and principal ideas. Sections 3, 4, and 5 describe twelve specific open problems in the area. We have used the following three criteria in choosing these problems. 1. Their