964 research outputs found
Random strings and tt-degrees of Turing complete C.E. sets
We investigate the truth-table degrees of (co-)c.e.\ sets, in particular,
sets of random strings. It is known that the set of random strings with respect
to any universal prefix-free machine is Turing complete, but that truth-table
completeness depends on the choice of universal machine. We show that for such
sets of random strings, any finite set of their truth-table degrees do not meet
to the degree~0, even within the c.e. truth-table degrees, but when taking the
meet over all such truth-table degrees, the infinite meet is indeed~0. The
latter result proves a conjecture of Allender, Friedman and Gasarch. We also
show that there are two Turing complete c.e. sets whose truth-table degrees
form a minimal pair.Comment: 25 page
Algorithmic statistics: forty years later
Algorithmic statistics has two different (and almost orthogonal) motivations.
From the philosophical point of view, it tries to formalize how the statistics
works and why some statistical models are better than others. After this notion
of a "good model" is introduced, a natural question arises: it is possible that
for some piece of data there is no good model? If yes, how often these bad
("non-stochastic") data appear "in real life"?
Another, more technical motivation comes from algorithmic information theory.
In this theory a notion of complexity of a finite object (=amount of
information in this object) is introduced; it assigns to every object some
number, called its algorithmic complexity (or Kolmogorov complexity).
Algorithmic statistic provides a more fine-grained classification: for each
finite object some curve is defined that characterizes its behavior. It turns
out that several different definitions give (approximately) the same curve.
In this survey we try to provide an exposition of the main results in the
field (including full proofs for the most important ones), as well as some
historical comments. We assume that the reader is familiar with the main
notions of algorithmic information (Kolmogorov complexity) theory.Comment: Missing proofs adde
Coding-theorem Like Behaviour and Emergence of the Universal Distribution from Resource-bounded Algorithmic Probability
Previously referred to as `miraculous' in the scientific literature because
of its powerful properties and its wide application as optimal solution to the
problem of induction/inference, (approximations to) Algorithmic Probability
(AP) and the associated Universal Distribution are (or should be) of the
greatest importance in science. Here we investigate the emergence, the rates of
emergence and convergence, and the Coding-theorem like behaviour of AP in
Turing-subuniversal models of computation. We investigate empirical
distributions of computing models in the Chomsky hierarchy. We introduce
measures of algorithmic probability and algorithmic complexity based upon
resource-bounded computation, in contrast to previously thoroughly investigated
distributions produced from the output distribution of Turing machines. This
approach allows for numerical approximations to algorithmic
(Kolmogorov-Chaitin) complexity-based estimations at each of the levels of a
computational hierarchy. We demonstrate that all these estimations are
correlated in rank and that they converge both in rank and values as a function
of computational power, despite fundamental differences between computational
models. In the context of natural processes that operate below the Turing
universal level because of finite resources and physical degradation, the
investigation of natural biases stemming from algorithmic rules may shed light
on the distribution of outcomes. We show that up to 60\% of the
simplicity/complexity bias in distributions produced even by the weakest of the
computational models can be accounted for by Algorithmic Probability in its
approximation to the Universal Distribution.Comment: 27 pages main text, 39 pages including supplement. Online complexity
calculator: http://complexitycalculator.com
On the information carried by programs about the objects they compute
In computability theory and computable analysis, finite programs can compute
infinite objects. Presenting a computable object via any program for it,
provides at least as much information as presenting the object itself, written
on an infinite tape. What additional information do programs provide? We
characterize this additional information to be any upper bound on the
Kolmogorov complexity of the object. Hence we identify the exact relationship
between Markov-computability and Type-2-computability. We then use this
relationship to obtain several results characterizing the computational and
topological structure of Markov-semidecidable sets
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
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
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