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Trivial Reals
"... Solovay showed that there are noncomputable reals ff such that H(ff _ n) 6 H(1n) + O(1), where H is prefixfree Kolmogorov complexity. Such Htrivial reals are interesting due to the connection between algorithmic complexity and effective randomness. We give a new, easier construction of an Htrivi ..."
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Cited by 56 (31 self)
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Solovay showed that there are noncomputable reals ff such that H(ff _ n) 6 H(1n) + O(1), where H is prefixfree Kolmogorov complexity. Such Htrivial reals are interesting due to the connection between algorithmic complexity and effective randomness. We give a new, easier construction of an Htrivial real. We also analyze various computabilitytheoretic properties of the Htrivial reals, showing for example that no Htrivial real can compute the halting problem. Therefore, our construction of an Htrivial computably enumerable set is an easy, injuryfree construction of an incomplete computably enumerable set. Finally, we relate the Htrivials to other classes of "highly nonrandom " reals that have been previously studied.
Randomness, relativization, and Turing degrees
 J. Symbolic Logic
, 2005
"... We compare various notions of algorithmic randomness. First we consider relativized randomness. A set is nrandom if it is MartinLof random relative to . We show that a set is 2random if and only if there is a constant c such that infinitely many initial segments x of the set are cincompre ..."
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Cited by 39 (17 self)
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We compare various notions of algorithmic randomness. First we consider relativized randomness. A set is nrandom if it is MartinLof random relative to . We show that a set is 2random if and only if there is a constant c such that infinitely many initial segments x of the set are cincompressible: C(x) c. The `only if' direction was obtained independently by Joseph Miller. This characterization can be extended to the case of timebounded Ccomplexity.
Lowness for the Class of Random Sets
, 1998
"... A positive answer to a question of M. van Lambalgen and D. Zambella whether there exist nonrecursive sets that are low for the class of random sets is obtained. Here a set A is low for the class RAND of random sets if RAND = RAND A . 1 Introduction The present paper is concerned with the noti ..."
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Cited by 30 (3 self)
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A positive answer to a question of M. van Lambalgen and D. Zambella whether there exist nonrecursive sets that are low for the class of random sets is obtained. Here a set A is low for the class RAND of random sets if RAND = RAND A . 1 Introduction The present paper is concerned with the notion of randomness as originally defined by P. MartinLof in [8]. A set is MartinLofrandom, or 1random for short, if it cannot be approximated in measure by recursive means. These sets have played a central role in the study of algorithmic randomness. One can relativize the definition of randomness to an arbitrary oracle. Relativized randomness has been studied by several authors. The intuitive meaning of "A is 1random relative to B" is that A is independent of B. A justification for this interpretation is given by M. van Lambalgen [7]. In this introduction we review some of the basic properties of sets which are 1random and we state the main problem. We work in the Cantor space 1 The fi...
On the Quantitative Structure of ...
, 2000
"... We analyze the quantitative structure of 0 2 . Among other things, we prove that a set is Turing complete if and only if its lower cone is nonnegligible, and that the sets of r.e.degree form a small subset of 0 2 . Mathematical Subject Classification: 03D15, 03D30, 28E15 Keywords: Comput ..."
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We analyze the quantitative structure of 0 2 . Among other things, we prove that a set is Turing complete if and only if its lower cone is nonnegligible, and that the sets of r.e.degree form a small subset of 0 2 . Mathematical Subject Classification: 03D15, 03D30, 28E15 Keywords: Computable measure theory, Turing degrees, completeness. 1 Introduction We study an eective measure theory suited for the study of 0 2 , the second level of the arithmetical hierarchy (alternatively, the sets computable relative to the halting problem K). This work may be seen as part of the constructivist tradition in mathematics as documented in [6]. The framework for eectivizing measure theory that we employ uses martingales. Martingales were rst applied to the study of random sequences by J. Ville [22]. Recursive martingales were studied in Schnorr [19], and became popular in complexity theory in more recent years through the work of Lutz [14, 15]. Lutz Research supported by a Ma...