476 research outputs found
Effectively Open Real Functions
A function f is continuous iff the PRE-image f^{-1}[V] of any open set V is
open again. Dual to this topological property, f is called OPEN iff the IMAGE
f[U] of any open set U is open again. Several classical Open Mapping Theorems
in Analysis provide a variety of sufficient conditions for openness.
By the Main Theorem of Recursive Analysis, computable real functions are
necessarily continuous. In fact they admit a well-known characterization in
terms of the mapping V+->f^{-1}[V] being EFFECTIVE: Given a list of open
rational balls exhausting V, a Turing Machine can generate a corresponding list
for f^{-1}[V]. Analogously, EFFECTIVE OPENNESS requires the mapping U+->f[U] on
open real subsets to be effective.
By effectivizing classical Open Mapping Theorems as well as from application
of Tarski's Quantifier Elimination, the present work reveals several rich
classes of functions to be effectively open.Comment: added section on semi-algebraic functions; to appear in Proc.
http://cca-net.de/cca200
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
Uniform test of algorithmic randomness over a general space
The algorithmic theory of randomness is well developed when the underlying
space is the set of finite or infinite sequences and the underlying probability
distribution is the uniform distribution or a computable distribution. These
restrictions seem artificial. Some progress has been made to extend the theory
to arbitrary Bernoulli distributions (by Martin-Loef), and to arbitrary
distributions (by Levin). We recall the main ideas and problems of Levin's
theory, and report further progress in the same framework.
- We allow non-compact spaces (like the space of continuous functions,
underlying the Brownian motion).
- The uniform test (deficiency of randomness) d_P(x) (depending both on the
outcome x and the measure P should be defined in a general and natural way.
- We see which of the old results survive: existence of universal tests,
conservation of randomness, expression of tests in terms of description
complexity, existence of a universal measure, expression of mutual information
as "deficiency of independence.
- The negative of the new randomness test is shown to be a generalization of
complexity in continuous spaces; we show that the addition theorem survives.
The paper's main contribution is introducing an appropriate framework for
studying these questions and related ones (like statistics for a general family
of distributions).Comment: 40 pages. Journal reference and a slight correction in the proof of
Theorem 7 adde
Computability of probability measures and Martin-Lof randomness over metric spaces
In this paper we investigate algorithmic randomness on more general spaces
than the Cantor space, namely computable metric spaces. To do this, we first
develop a unified framework allowing computations with probability measures. We
show that any computable metric space with a computable probability measure is
isomorphic to the Cantor space in a computable and measure-theoretic sense. We
show that any computable metric space admits a universal uniform randomness
test (without further assumption).Comment: 29 page
Real Hypercomputation and Continuity
By the sometimes so-called 'Main Theorem' of Recursive Analysis, every
computable real function is necessarily continuous. We wonder whether and which
kinds of HYPERcomputation allow for the effective evaluation of also
discontinuous f:R->R. More precisely the present work considers the following
three super-Turing notions of real function computability:
* relativized computation; specifically given oracle access to the Halting
Problem 0' or its jump 0'';
* encoding real input x and/or output y=f(x) in weaker ways also related to
the Arithmetic Hierarchy;
* non-deterministic computation.
It turns out that any f:R->R computable in the first or second sense is still
necessarily continuous whereas the third type of hypercomputation does provide
the required power to evaluate for instance the discontinuous sign function.Comment: previous version (extended abstract) has appeared in pp.562-571 of
"Proc. 1st Conference on Computability in Europe" (CiE'05), Springer LNCS
vol.352
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
Levels of discontinuity, limit-computability, and jump operators
We develop a general theory of jump operators, which is intended to provide
an abstraction of the notion of "limit-computability" on represented spaces.
Jump operators also provide a framework with a strong categorical flavor for
investigating degrees of discontinuity of functions and hierarchies of sets on
represented spaces. We will provide a thorough investigation within this
framework of a hierarchy of -measurable functions between arbitrary
countably based -spaces, which captures the notion of computing with
ordinal mind-change bounds. Our abstract approach not only raises new questions
but also sheds new light on previous results. For example, we introduce a
notion of "higher order" descriptive set theoretical objects, we generalize a
recent characterization of the computability theoretic notion of "lowness" in
terms of adjoint functors, and we show that our framework encompasses ordinal
quantifications of the non-constructiveness of Hilbert's finite basis theorem
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