11 research outputs found

    Calibrating the complexity of Delta 2 sets via their changes

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    The computational complexity of a Delta 2 set will be calibrated by the amount of changes needed for any of its computable approximations. Firstly, we study Martin-Loef random sets, where we quantify the changes of initial segments. Secondly, we look at c.e. sets, where we quantify the overall amount of changes by obedience to cost functions. Finally, we combine the two settings. The discussions lead to three basic principles on how complexity and changes relate

    Lowness and Π nullsets

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    We prove that there exists a noncomputable c.e. real which is low for weak 2-randomness, a definition of randomness due to Kurtz, and that all reals which are low for weak 2-randomness are low for Martin-Lof randomness

    Characterizing the strongly jump-traceable sets via randomness

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    We show that if a set AA is computable from every superlow 1-random set, then AA is strongly jump-traceable. This theorem shows that the computably enumerable (c.e.) strongly jump-traceable sets are exactly the c.e.\ sets computable from every superlow 1-random set. We also prove the analogous result for superhighness: a c.e.\ set is strongly jump-traceable if and only if it is computable from every superhigh 1-random set. Finally, we show that for each cost function cc with the limit condition there is a 1-random Δ20\Delta^0_2 set YY such that every c.e.\ set A≤TYA \le_T Y obeys cc. To do so, we connect cost function strength and the strength of randomness notions. This result gives a full correspondence between obedience of cost functions and being computable from Δ20\Delta^0_2 1-random sets.Comment: 41 page

    K-Triviality and Computable Measures

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    Algorithmic randomness is primarily concerned with quantifying the degree of randomness of infinite binary strings, and is usually carried out in the setting of Cantor space. One characterization of randomness involves prefixes ``being as hard as possible to describe . Also of interest are the infinite binary strings whose prefixes are as easy as possible to describe i.e., the KK-trivial strings. We will study these strings in the setting of computable metric spaces, and investigate several definitions which attempt to correctly generalize KK-triviality. We describe some of the difficulties inherent in a natural-seeming approach, and offer partial results where new definitions relate to a more established definition of KK-triviality under the right conditions

    Degrees of Computability and Randomness

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