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Kolmogorov complexity and computably enumerable sets
, 2011
"... Abstract. We study the computably enumerable sets in terms of the: (a) Kolmogorov complexity of their initial segments; (b) Kolmogorov complexity of finite programs when they are used as oracles. We present an extended discussion of the existing research on this topic, along with recent developments ..."
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Abstract. We study the computably enumerable sets in terms of the: (a) Kolmogorov complexity of their initial segments; (b) Kolmogorov complexity of finite programs when they are used as oracles. We present an extended discussion of the existing research on this topic, along with recent developments and open problems. Besides this survey, our main original result is the following characterization of the computably enumerable sets with trivial initial segment prefixfree complexity. A computably enumerable set A is Ktrivial if and only if the family of sets with complexity bounded by the complexity of A is uniformly computable from the halting problem. 1.
RANDOMNESS AND THE LINEAR DEGREES OF COMPUTABILITY
"... Abstract. We show that there exists a real α such that, for all reals β, if α is linear reducible to β (α ≤ℓ β, previously denoted α ≤sw β) then β ≤T α. In fact, every random real satisfies this quasimaximality property. As a corollary we may conclude that there exists no ℓcomplete ∆2 real. Upon r ..."
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Abstract. We show that there exists a real α such that, for all reals β, if α is linear reducible to β (α ≤ℓ β, previously denoted α ≤sw β) then β ≤T α. In fact, every random real satisfies this quasimaximality property. As a corollary we may conclude that there exists no ℓcomplete ∆2 real. Upon realizing that quasimaximality does not characterize the random reals—there exist reals which are not random but which are of quasimaximal ℓdegree—it is then natural to ask whether maximality could provide such a characterization. Such hopes, however, are in vain since no real is of maximal ℓdegree. 1. introduction In the process of computing a real α given an oracle for β it is natural to consider the condition that for the computation of the first n bits of α we are only allowed to use the information in the first n bits of β. It is not difficult to see that this notion of oracle computation is complexity sensitive in many ways. We can then generalize this definition in a straightforward
ALGORITHMIC RANDOMNESS AND MEASURES OF COMPLEXITY
"... Abstract. We survey recent advances on the interface between computability theory and algorithmic randomness, with special attention on measures of relative complexity. We focus on (weak) reducibilities that measure (a) the initial segment complexity of reals and (b) the power of reals to compress s ..."
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Abstract. We survey recent advances on the interface between computability theory and algorithmic randomness, with special attention on measures of relative complexity. We focus on (weak) reducibilities that measure (a) the initial segment complexity of reals and (b) the power of reals to compress strings, when they are used as oracles. The results are put into context and several connections are made with various central issues in modern algorithmic randomness and computability. 1.
ANALOGUES OF CHAITIN’S OMEGA IN THE COMPUTABLY ENUMERABLE SETS
"... Abstract. We show that there are computably enumerable (c.e.) sets with maximum initial segment Kolmogorov complexity amongst all c.e. sets (with respect to both the plain and the prefixfree version of Kolmogorov complexity). These c.e. sets belong to the weak truth table degree of the halting prob ..."
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Abstract. We show that there are computably enumerable (c.e.) sets with maximum initial segment Kolmogorov complexity amongst all c.e. sets (with respect to both the plain and the prefixfree version of Kolmogorov complexity). These c.e. sets belong to the weak truth table degree of the halting problem, but not every weak truth table complete set has maximum initial segment Kolmogorov complexity. Moreover, every c.e. set with maximum initial segment prefixfree complexity is the disjoint union of two c.e. sets with the same property; and is also the disjoint union of two c.e. sets of lesser initial segment complexity. 1.