1,241 research outputs found
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
Nondeterministic Instance Complexity and Proof Systems with Advice
Motivated by strong Karp-Lipton collapse results in bounded arithmetic, Cook and Krajíček [1] have recently introduced the notion of propositional proof systems with advice. In this paper we investigate the following question: Given a language L , do there exist polynomially bounded proof systems with advice for L ? Depending on the complexity of the underlying language L and the amount and type of the advice used by the proof system, we obtain different characterizations for this problem. In particular, we show that the above question is tightly linked with the question whether L has small nondeterministic instance complexity
Linear list-approximation for short programs (or the power of a few random bits)
A -short program for a string is a description of of length at
most , where is the Kolmogorov complexity of . We show that
there exists a randomized algorithm that constructs a list of elements that
contains a -short program for . We also show a polynomial-time
randomized construction that achieves the same list size for -short programs. These results beat the lower bounds shown by Bauwens et al.
\cite{bmvz:c:shortlist} for deterministic constructions of such lists. We also
prove tight lower bounds for the main parameters of our result. The
constructions use only ( for the polynomial-time
result) random bits . Thus using only few random bits it is possible to do
tasks that cannot be done by any deterministic algorithm regardless of its
running time
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