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23
Information Distance
, 1997
"... While Kolmogorov complexity is the accepted absolute measure of information content in an individual finite object, a similarly absolute notion is needed for the information distance between two individual objects, for example, two pictures. We give several natural definitions of a universal inf ..."
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Cited by 36 (4 self)
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While Kolmogorov complexity is the accepted absolute measure of information content in an individual finite object, a similarly absolute notion is needed for the information distance between two individual objects, for example, two pictures. We give several natural definitions of a universal information metric, based on length of shortest programs for either ordinary computations or reversible (dissipationless) computations. It turns out that these definitions are equivalent up to an additive logarithmic term. We show that the information distance is a universal cognitive similarity distance. We investigate the maximal correlation of the shortest programs involved, the maximal uncorrelation of programs (a generalization of the SlepianWolf theorem of classical information theory), and the density properties of the discrete metric spaces induced by the information distances. A related distance measures the amount of nonreversibility of a computation. Using the physical theo...
Average Case Complexity under the Universal Distribution Equals Worst Case Complexity
 Information Processing Letters
, 1992
"... The average complexity of any algorithm whatsoever (provided it always terminates) under the universal distribution is of the same order of magnitude as the worstcase complexity. This holds both for time complexity and for space complexity. To focus our discussion, we use as illustrations the pa ..."
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Cited by 25 (0 self)
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The average complexity of any algorithm whatsoever (provided it always terminates) under the universal distribution is of the same order of magnitude as the worstcase complexity. This holds both for time complexity and for space complexity. To focus our discussion, we use as illustrations the particular case of sorting algorithms, and the general case of the average case complexity of NPcomplete problems. 1
Effective randomness for computable probability measures
, 2006
"... Any notion of effective randomness that is defined with respect to arbitrary computable probability measures canonically induces an equivalence relation on such measures for which two measures are considered equivalent if their respective classes of random elements coincide. Elaborating on work of B ..."
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Cited by 4 (0 self)
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Any notion of effective randomness that is defined with respect to arbitrary computable probability measures canonically induces an equivalence relation on such measures for which two measures are considered equivalent if their respective classes of random elements coincide. Elaborating on work of Bienvenu [1], we determine all the implications that hold between the equivalence relations induced by MartinLöf randomness, computable randomness, Schnorr randomness, and weak randomness, and the equivalence and consistency relations of probability measures, except that we do not know whether two computable probability measures need to be equivalent in case their respective concepts of weakly randomness coincide. Keywords: computable probability measures, MartinLöf randomness, computable randomness, Schnorr randomness, weak randomness, equivalence of probability measures, consistency of probability measures.
Computing the speed of convergence of ergodic averages and pseudorandom points in computable dynamical systems
, 2010
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Constructive equivalence relations on computable probability measures
 International Computer Science Symposium in Russia, Lecture Notes in Computer Science
, 2006
"... Abstract. We study the equivalence relations on probability measures corresponding respectively to having the same MartinLöf random reals, having the same KolmogorovLoveland random reals, and having the same computably random reals. In particular, we show that, when restricted to the class of stro ..."
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Cited by 3 (2 self)
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Abstract. We study the equivalence relations on probability measures corresponding respectively to having the same MartinLöf random reals, having the same KolmogorovLoveland random reals, and having the same computably random reals. In particular, we show that, when restricted to the class of strongly positive generalized Bernoulli measures, they all coincide with the classical equivalence, which requires that two measures have the same nullsets. 1
Algorithmic tests and randomness with respect to a class of measures
, 2011
"... This paper offers some new results on randomness with respect to classes of measures, along with a didactical exposition of their context based on results that appeared elsewhere. We start with the reformulation of the MartinLöf definition of randomness (with respect to computable measures) in term ..."
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Cited by 2 (1 self)
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This paper offers some new results on randomness with respect to classes of measures, along with a didactical exposition of their context based on results that appeared elsewhere. We start with the reformulation of the MartinLöf definition of randomness (with respect to computable measures) in terms of randomness deficiency functions. A formula that expresses the randomness deficiency in terms of prefix complexity is given (in two forms). Some approaches that go in another direction (from deficiency to complexity) are considered. The notion of Bernoulli randomness (independent coin tosses for an asymmetric coin with some probability p of head) is defined. It is shown that a sequence is Bernoulli if it is random with respect to some Bernoulli
Characterizing the Software Development Process: A New Approach Based on Kolmogorov Complexity
 in Computer Aided Systems Theory  EUROCAST’2001, 8th International Workshop on Computer Aided Systems Theory, ser. Lecture Notes in Computer Science
, 2001
"... Our main aim is to propose a new characterization for the software development process. We suggest that software development methodology has some limits. These limits are a clue that software development process is more subjective and empirical than objective and formal. We use Kolmogorov complexity ..."
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Cited by 2 (1 self)
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Our main aim is to propose a new characterization for the software development process. We suggest that software development methodology has some limits. These limits are a clue that software development process is more subjective and empirical than objective and formal. We use Kolmogorov complexity to develop the formal argument and to outline the informal conclusions. Kolmogorov complexity is based on the size in bits of the smallest e ective description of an object and is a suitable quantitative measure of the object's information content.
Computability of the ergodic decomposition
 ANNALS OF PURE AND APPLIED LOGIC
, 2012
"... The study of ergodic theorems from the viewpoint of computable analysis is a rich field of investigation. Interactions between algorithmic randomness, computability theory and ergodic theory have recently been examined by several authors. It has been observed that ergodic measures have better comput ..."
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Cited by 1 (0 self)
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The study of ergodic theorems from the viewpoint of computable analysis is a rich field of investigation. Interactions between algorithmic randomness, computability theory and ergodic theory have recently been examined by several authors. It has been observed that ergodic measures have better computability properties than nonergodic ones. In a previous paper we studied the extent to which nonergodic measures inherit the computability properties of ergodic ones, and introduced the notion of an effectively decomposable measure. We asked the following question: if the ergodic decomposition of a stationary measure is finite, is this decomposition effective? In this paper we answer the question in the negative.
Power of fast VLSI models is insensitive to wires’ thinness
 Research Triangle Park, North Carolina
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