9 research outputs found
Dimension Extractors and Optimal Decompression
A *dimension extractor* is an algorithm designed to increase the effective
dimension -- i.e., the amount of computational randomness -- of an infinite
binary sequence, in order to turn a "partially random" sequence into a "more
random" sequence. Extractors are exhibited for various effective dimensions,
including constructive, computable, space-bounded, time-bounded, and
finite-state dimension. Using similar techniques, the Kucera-Gacs theorem is
examined from the perspective of decompression, by showing that every infinite
sequence S is Turing reducible to a Martin-Loef random sequence R such that the
asymptotic number of bits of R needed to compute n bits of S, divided by n, is
precisely the constructive dimension of S, which is shown to be the optimal
ratio of query bits to computed bits achievable with Turing reductions. The
extractors and decompressors that are developed lead directly to new
characterizations of some effective dimensions in terms of optimal
decompression by Turing reductions.Comment: This report was combined with a different conference paper "Every
Sequence is Decompressible from a Random One" (cs.IT/0511074, at
http://dx.doi.org/10.1007/11780342_17), and both titles were changed, with
the conference paper incorporated as section 5 of this new combined paper.
The combined paper was accepted to the journal Theory of Computing Systems,
as part of a special issue of invited papers from the second conference on
Computability in Europe, 200
Constructive Dimension and Turing Degrees
This paper examines the constructive Hausdorff and packing dimensions of
Turing degrees. The main result is that every infinite sequence S with
constructive Hausdorff dimension dim_H(S) and constructive packing dimension
dim_P(S) is Turing equivalent to a sequence R with dim_H(R) <= (dim_H(S) /
dim_P(S)) - epsilon, for arbitrary epsilon > 0. Furthermore, if dim_P(S) > 0,
then dim_P(R) >= 1 - epsilon. The reduction thus serves as a *randomness
extractor* that increases the algorithmic randomness of S, as measured by
constructive dimension.
A number of applications of this result shed new light on the constructive
dimensions of Turing degrees. A lower bound of dim_H(S) / dim_P(S) is shown to
hold for the Turing degree of any sequence S. A new proof is given of a
previously-known zero-one law for the constructive packing dimension of Turing
degrees. It is also shown that, for any regular sequence S (that is, dim_H(S) =
dim_P(S)) such that dim_H(S) > 0, the Turing degree of S has constructive
Hausdorff and packing dimension equal to 1.
Finally, it is shown that no single Turing reduction can be a universal
constructive Hausdorff dimension extractor, and that bounded Turing reductions
cannot extract constructive Hausdorff dimension. We also exhibit sequences on
which weak truth-table and bounded Turing reductions differ in their ability to
extract dimension.Comment: The version of this paper appearing in Theory of Computing Systems,
45(4):740-755, 2009, had an error in the proof of Theorem 2.4, due to
insufficient care with the choice of delta. This version modifies that proof
to fix the error
Constructive dimension and weak truth-table degrees
submitted to Theory of Computing SystemsThis paper examines the constructive Hausdorff and packing dimensions of weak truth-table degrees. The main result is that every infinite sequence with constructive Hausdorff dimension and constructive packing dimension \Dim(S) is weak truth-table equivalent to a sequence with \dim(R) \geq \dim(S) / \Dim(S) - \epsilon, for arbitrary . Furthermore, if \Dim(S) > 0, then \Dim(R) \geq 1 - \epsilon. The reduction thus serves as a \emph{randomness extractor} that increases the algorithmic randomness of , as measured by constructive dimension. A number of applications of this result shed new light on the constructive dimensions of wtt degrees (and, by extension, Turing degrees). A lower bound of \dim(S) / \Dim(S) is shown to hold for the wtt degree of any sequence . A new proof is given of a previously-known zero-one law for the constructive packing dimension of wtt degrees. It is also shown that, for any \emph{regular} sequence (that is, \dim(S) = \Dim(S)) such that , the wtt degree of has constructive Hausdorff and packing dimension equal to 1. Finally, it is shown that no single Turing reduction can be a \emph{universal} constructive Hausdorff dimension extractor, and that \emph{bounded} Turing reductions cannot extract constructive Hausdorff dimension. We also exhibit sequences on which weak truth-table and bounded Turing reductions differ in their ability to extract dimension
Compression of data streams down to their information content
According to the Kolmogorov complexity, every finite binary string is compressible to a shortest code-its information content-from which it is effectively recoverable. We investigate the extent to which this holds for the infinite binary sequences (streams). We devise a new coding method that uniformly codes every stream X into an algorithmically random stream Y , in such a way that the first n bits of X are recoverable from the first I(X \upharpoonright -{n}) bits of Y , where I is any partial computable information content measure that is defined on all prefixes of X , and where X \upharpoonright -{n} is the initial segment of X of length n. As a consequence, if g is any computable upper bound on the initial segment prefix-free complexity of X , then X is computable from an algorithmically random Y with oracle-use at most g. Alternatively (making no use of such a computable bound g ), one can achieve an the oracle-use bounded above by K(X \upharpoonright -{n})+\log n. This provides a strong analogue of Shannon's source coding theorem for the algorithmic information theory