7,412 research outputs found
On the Weak Computability of Continuous Real Functions
In computable analysis, sequences of rational numbers which effectively
converge to a real number x are used as the (rho-) names of x. A real number x
is computable if it has a computable name, and a real function f is computable
if there is a Turing machine M which computes f in the sense that, M accepts
any rho-name of x as input and outputs a rho-name of f(x) for any x in the
domain of f. By weakening the effectiveness requirement of the convergence and
classifying the converging speeds of rational sequences, several interesting
classes of real numbers of weak computability have been introduced in
literature, e.g., in addition to the class of computable real numbers (EC), we
have the classes of semi-computable (SC), weakly computable (WC), divergence
bounded computable (DBC) and computably approximable real numbers (CA). In this
paper, we are interested in the weak computability of continuous real functions
and try to introduce an analogous classification of weakly computable real
functions. We present definitions of these functions by Turing machines as well
as by sequences of rational polygons and prove these two definitions are not
equivalent. Furthermore, we explore the properties of these functions, and
among others, show their closure properties under arithmetic operations and
composition
The Tsallis entropy and the Shannon entropy of a universal probability
We study the properties of Tsallis entropy and Shannon entropy from the point
of view of algorithmic randomness. In algorithmic information theory, there are
two equivalent ways to define the program-size complexity K(s) of a given
finite binary string s. In the standard way, K(s) is defined as the length of
the shortest input string for the universal self-delimiting Turing machine to
output s. In the other way, the so-called universal probability m is introduced
first, and then K(s) is defined as -log_2 m(s) without reference to the concept
of program-size. In this paper, we investigate the properties of the Shannon
entropy, the power sum, and the Tsallis entropy of a universal probability by
means of the notion of program-size complexity. We determine the convergence or
divergence of each of these three quantities, and evaluate its degree of
randomness if it converges.Comment: 5 pages, to appear in the Proceedings of the 2008 IEEE International
Symposium on Information Theory, Toronto, ON, Canada, July 6 - 11, 200
The Bolzano-Weierstrass Theorem is the Jump of Weak K\"onig's Lemma
We classify the computational content of the Bolzano-Weierstrass Theorem and
variants thereof in the Weihrauch lattice. For this purpose we first introduce
the concept of a derivative or jump in this lattice and we show that it has
some properties similar to the Turing jump. Using this concept we prove that
the derivative of closed choice of a computable metric space is the cluster
point problem of that space. By specialization to sequences with a relatively
compact range we obtain a characterization of the Bolzano-Weierstrass Theorem
as the derivative of compact choice. In particular, this shows that the
Bolzano-Weierstrass Theorem on real numbers is the jump of Weak K\"onig's
Lemma. Likewise, the Bolzano-Weierstrass Theorem on the binary space is the
jump of the lesser limited principle of omniscience LLPO and the
Bolzano-Weierstrass Theorem on natural numbers can be characterized as the jump
of the idempotent closure of LLPO. We also introduce the compositional product
of two Weihrauch degrees f and g as the supremum of the composition of any two
functions below f and g, respectively. We can express the main result such that
the Bolzano-Weierstrass Theorem is the compositional product of Weak K\"onig's
Lemma and the Monotone Convergence Theorem. We also study the class of weakly
limit computable functions, which are functions that can be obtained by
composition of weakly computable functions with limit computable functions. We
prove that the Bolzano-Weierstrass Theorem on real numbers is complete for this
class. Likewise, the unique cluster point problem on real numbers is complete
for the class of functions that are limit computable with finitely many mind
changes. We also prove that the Bolzano-Weierstrass Theorem on real numbers
and, more generally, the unbounded cluster point problem on real numbers is
uniformly low limit computable. Finally, we also discuss separation techniques.Comment: This version includes an addendum by Andrea Cettolo, Matthias
Schr\"oder, and the authors of the original paper. The addendum closes a gap
in the proof of Theorem 11.2, which characterizes the computational content
of the Bolzano-Weierstra\ss{} Theorem for arbitrary computable metric space
Closed Choice and a Uniform Low Basis Theorem
We study closed choice principles for different spaces. Given information
about what does not constitute a solution, closed choice determines a solution.
We show that with closed choice one can characterize several models of
hypercomputation in a uniform framework using Weihrauch reducibility. The
classes of functions which are reducible to closed choice of the singleton
space, of the natural numbers, of Cantor space and of Baire space correspond to
the class of computable functions, of functions computable with finitely many
mind changes, of weakly computable functions and of effectively Borel
measurable functions, respectively. We also prove that all these classes
correspond to classes of non-deterministically computable functions with the
respective spaces as advice spaces. Moreover, we prove that closed choice on
Euclidean space can be considered as "locally compact choice" and it is
obtained as product of closed choice on the natural numbers and on Cantor
space. We also prove a Quotient Theorem for compact choice which shows that
single-valued functions can be "divided" by compact choice in a certain sense.
Another result is the Independent Choice Theorem, which provides a uniform
proof that many choice principles are closed under composition. Finally, we
also study the related class of low computable functions, which contains the
class of weakly computable functions as well as the class of functions
computable with finitely many mind changes. As one main result we prove a
uniform version of the Low Basis Theorem that states that closed choice on
Cantor space (and the Euclidean space) is low computable. We close with some
related observations on the Turing jump operation and its initial topology
Effective Choice and Boundedness Principles in Computable Analysis
In this paper we study a new approach to classify mathematical theorems
according to their computational content. Basically, we are asking the question
which theorems can be continuously or computably transferred into each other?
For this purpose theorems are considered via their realizers which are
operations with certain input and output data. The technical tool to express
continuous or computable relations between such operations is Weihrauch
reducibility and the partially ordered degree structure induced by it. We have
identified certain choice principles which are cornerstones among Weihrauch
degrees and it turns out that certain core theorems in analysis can be
classified naturally in this structure. In particular, we study theorems such
as the Intermediate Value Theorem, the Baire Category Theorem, the Banach
Inverse Mapping Theorem and others. We also explore how existing
classifications of the Hahn-Banach Theorem and Weak K"onig's Lemma fit into
this picture. We compare the results of our classification with existing
classifications in constructive and reverse mathematics and we claim that in a
certain sense our classification is finer and sheds some new light on the
computational content of the respective theorems. We develop a number of
separation techniques based on a new parallelization principle, on certain
invariance properties of Weihrauch reducibility, on the Low Basis Theorem of
Jockusch and Soare and based on the Baire Category Theorem. Finally, we present
a number of metatheorems that allow to derive upper bounds for the
classification of the Weihrauch degree of many theorems and we discuss the
Brouwer Fixed Point Theorem as an example
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