7 research outputs found
Limit complexities revisited [once more]
The main goal of this article is to put some known results in a common
perspective and to simplify their proofs.
We start with a simple proof of a result of Vereshchagin saying that
equals . Then we use the same argument to prove
similar results for prefix complexity, a priori probability on binary tree, to
prove Conidis' theorem about limits of effectively open sets, and also to
improve the results of Muchnik about limit frequencies. As a by-product, we get
a criterion of 2-randomness proved by Miller: a sequence is 2-random if and
only if there exists such that any prefix of is a prefix of some
string such that . (In the 1960ies this property was
suggested in Kolmogorov as one of possible randomness definitions.) We also get
another 2-randomness criterion by Miller and Nies: is 2-random if and only
if for some and infinitely many prefixes of .
This is a modified version of our old paper that contained a weaker (and
cumbersome) version of Conidis' result, and the proof used low basis theorem
(in quite a strange way). The full version was formulated there as a
conjecture. This conjecture was later proved by Conidis. Bruno Bauwens
(personal communication) noted that the proof can be obtained also by a simple
modification of our original argument, and we reproduce Bauwens' argument with
his permission.Comment: See http://arxiv.org/abs/0802.2833 for the old pape
Limit complexities revisited
The main goal of this paper is to put some known results in a common
perspective and to simplify their proofs. We start with a simple proof of a
result from (Vereshchagin, 2002) saying that \limsup_n\KS(x|n) (here
\KS(x|n) is conditional (plain) Kolmogorov complexity of when is
known) equals \KS^{\mathbf{0'}(x), the plain Kolmogorov complexity with
\mathbf{0'-oracle. Then we use the same argument to prove similar results for
prefix complexity (and also improve results of (Muchnik, 1987) about limit
frequencies), a priori probability on binary tree and measure of effectively
open sets. As a by-product, we get a criterion of Martin-L\"of
randomness (called also 2-randomness) proved in (Miller, 2004): a sequence
is 2-random if and only if there exists such that any prefix
of is a prefix of some string such that \KS(y)\ge |y|-c. (In the
1960ies this property was suggested in (Kolmogorov, 1968) as one of possible
randomness definitions; its equivalence to 2-randomness was shown in (Miller,
2004) while proving another 2-randomness criterion (see also (Nies et al.
2005)): is 2-random if and only if \KS(x)\ge |x|-c for some and
infinitely many prefixes of . Finally, we show that the low-basis
theorem can be used to get alternative proofs for these results and to improve
the result about effectively open sets; this stronger version implies the
2-randomness criterion mentioned in the previous sentence
Generic algorithms for halting problem and optimal machines revisited
The halting problem is undecidable --- but can it be solved for "most"
inputs? This natural question was considered in a number of papers, in
different settings. We revisit their results and show that most of them can be
easily proven in a natural framework of optimal machines (considered in
algorithmic information theory) using the notion of Kolmogorov complexity. We
also consider some related questions about this framework and about asymptotic
properties of the halting problem. In particular, we show that the fraction of
terminating programs cannot have a limit, and all limit points are Martin-L\"of
random reals. We then consider mass problems of finding an approximate solution
of halting problem and probabilistic algorithms for them, proving both positive
and negative results. We consider the fraction of terminating programs that
require a long time for termination, and describe this fraction using the busy
beaver function. We also consider approximate versions of separation problems,
and revisit Schnorr's results about optimal numberings showing how they can be
generalized.Comment: a preliminary version was presented at the ICALP 2015 conferenc