NDLI: Randomness, Computability, and Density

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Randomness, Computability, and Density

Content Provider	Society for Industrial and Applied Mathematics (SIAM)
Author	Hirschfeldt, Denis R. Nies, Andr Downey, Rod G.
Copyright Year	2002
Abstract	We study effectively given positive reals (more specifically, computably enumerable reals) under a measure of relative randomness introduced by Solovay [manuscript, IBM Thomas J. Watson Research Center, Yorktown Heights, NY, 1975] and studied by Calude, Hertling, Khoussainov, and Wang [Theoret. Comput. Sci., 255 (2001), pp. 125--149], Calude [Theoret. Comput. Sci., 271 (2002), pp. 3--14], Kucera and Slaman [SIAM J. Comput., 31 (2002), pp. 199--211], and Downey, Hirschfeldt, and LaForte [Mathematical Foundations of Computer Science 2001, Springer-Verlag, Berlin, 2001, pp. 316--327], among others. This measure is called domination or Solovay reducibility and is defined by saying that $\alpha$ dominates $\beta$ if there are a constant c and a partial computable function $\varphi$ such that for all positive rationals $q < \alpha$ we have $\varphi(q)\!\downarrow < \beta$ and $\beta- \varphi(q) \leqslant c(\alpha- q)$. The intuition is that an approximating sequence for $\alpha$generates one for $\beta$ whose rate of convergence is not much slower than that of the original sequence. It is not hard to show that if $\alpha$ dominates $\beta$, then the initial segment complexity of $\alpha$ is at least that of $\beta$.In this paper we are concerned with structural properties of the degree structure generated by Solovay reducibility. We answer a natural question in this area of investigation by proving the density of the Solovay degrees. We also provide a new characterization of the random computably enumerable reals in terms of splittings in the Solovay degrees. Specifically, we show that the Solovay degrees of computably enumerable reals are dense, that any incomplete Solovay degree splits over any lesser degree, and that the join of any two incomplete Solovay degrees is incomplete, so that the complete Solovay degree does not split at all. The methodology is of some technical interest, since it includes a priority argument in which the injuries are themselves controlled by randomness considerations.
Starting Page	1169
Ending Page	1183
Page Count	15
File Format	PDF
ISSN	00975397
DOI	10.1137/S0097539700376937
e-ISSN	10957111
Journal	SIAM Journal on Computing (SMJCAT)
Issue Number	4
Volume Number	31
Language	English
Publisher	Society for Industrial and Applied Mathematics
Publisher Date	2002-01-01
Access Restriction	Subscribed
Subject Keyword	Solovay reducibility Applications of computability and recursion theory computably enumerable reals algorithmic information theory Kolmogorov complexity Other degrees and reducibilities randomness Algorithmic information theory
Content Type	Text
Resource Type	Article
Subject	Mathematics Computer Science

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