1,986 research outputs found
A Galois connection between Turing jumps and limits
Limit computable functions can be characterized by Turing jumps on the input
side or limits on the output side. As a monad of this pair of adjoint
operations we obtain a problem that characterizes the low functions and dually
to this another problem that characterizes the functions that are computable
relative to the halting problem. Correspondingly, these two classes are the
largest classes of functions that can be pre or post composed to limit
computable functions without leaving the class of limit computable functions.
We transfer these observations to the lattice of represented spaces where it
leads to a formal Galois connection. We also formulate a version of this result
for computable metric spaces. Limit computability and computability relative to
the halting problem are notions that coincide for points and sequences, but
even restricted to continuous functions the former class is strictly larger
than the latter. On computable metric spaces we can characterize the functions
that are computable relative to the halting problem as those functions that are
limit computable with a modulus of continuity that is computable relative to
the halting problem. As a consequence of this result we obtain, for instance,
that Lipschitz continuous functions that are limit computable are automatically
computable relative to the halting problem. We also discuss 1-generic points as
the canonical points of continuity of limit computable functions, and we prove
that restricted to these points limit computable functions are computable
relative to the halting problem. Finally, we demonstrate how these results can
be applied in computable analysis
Bounded time computation on metric spaces and Banach spaces
We extend the framework by Kawamura and Cook for investigating computational
complexity for operators occurring in analysis. This model is based on
second-order complexity theory for functions on the Baire space, which is
lifted to metric spaces by means of representations. Time is measured in terms
of the length of the input encodings and the required output precision. We
propose the notions of a complete representation and of a regular
representation. We show that complete representations ensure that any
computable function has a time bound. Regular representations generalize
Kawamura and Cook's more restrictive notion of a second-order representation,
while still guaranteeing fast computability of the length of the encodings.
Applying these notions, we investigate the relationship between purely metric
properties of a metric space and the existence of a representation such that
the metric is computable within bounded time. We show that a bound on the
running time of the metric can be straightforwardly translated into size bounds
of compact subsets of the metric space. Conversely, for compact spaces and for
Banach spaces we construct a family of admissible, complete, regular
representations that allow for fast computation of the metric and provide short
encodings. Here it is necessary to trade the time bound off against the length
of encodings
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
Dynamics and abstract computability: computing invariant measures
We consider the question of computing invariant measures from an abstract
point of view. We work in a general framework (computable metric spaces,
computable measures and functions) where this problem can be posed precisely.
We consider invariant measures as fixed points of the transfer operator and
give general conditions under which the transfer operator is (sufficiently)
computable. In this case, a general result ensures the computability of
isolated fixed points and hence invariant measures (in given classes of
"regular" measures). This implies the computability of many SRB measures.
On the other hand, not all computable dynamical systems have a computable
invariant measure. We exhibit two interesting examples of computable dynamics,
one having an SRB measure which is not computable and another having no
computable invariant measure at all, showing some subtlety in this kind of
problems
Approximation systems for functions in topological and in metric spaces
A notable feature of the TTE approach to computability is the representation
of the argument values and the corresponding function values by means of
infinitistic names. Two ways to eliminate the using of such names in certain
cases are indicated in the paper. The first one is intended for the case of
topological spaces with selected indexed denumerable bases. Suppose a partial
function is given from one such space into another one whose selected base has
a recursively enumerable index set, and suppose that the intersection of base
open sets in the first space is computable in the sense of Weihrauch-Grubba.
Then the ordinary TTE computability of the function is characterized by the
existence of an appropriate recursively enumerable relation between indices of
base sets containing the argument value and indices of base sets containing the
corresponding function value.This result can be regarded as an improvement of a
result of Korovina and Kudinov. The second way is applicable to metric spaces
with selected indexed denumerable dense subsets. If a partial function is given
from one such space into another one, then, under a semi-computability
assumption concerning these spaces, the ordinary TTE computability of the
function is characterized by the existence of an appropriate recursively
enumerable set of quadruples. Any of them consists of an index of element from
the selected dense subset in the first space, a natural number encoding a
rational bound for the distance between this element and the argument value, an
index of element from the selected dense subset in the second space and a
natural number encoding a rational bound for the distance between this element
and the function value. One of the examples in the paper indicates that the
computability of real functions can be characterized in a simple way by using
the first way of elimination of the infinitistic names.Comment: 21 pages, published in Logical Methods in Computer Scienc
The descriptive theory of represented spaces
This is a survey on the ongoing development of a descriptive theory of
represented spaces, which is intended as an extension of both classical and
effective descriptive set theory to deal with both sets and functions between
represented spaces. Most material is from work-in-progress, and thus there may
be a stronger focus on projects involving the author than an objective survey
would merit.Comment: survey of work-in-progres
Computing the speed of convergence of ergodic averages and pseudorandom points in computable dynamical systems
A pseudorandom point in an ergodic dynamical system over a computable metric
space is a point which is computable but its dynamics has the same statistical
behavior as a typical point of the system.
It was proved in [Avigad et al. 2010, Local stability of ergodic averages]
that in a system whose dynamics is computable the ergodic averages of
computable observables converge effectively. We give an alternative, simpler
proof of this result.
This implies that if also the invariant measure is computable then the
pseudorandom points are a set which is dense (hence nonempty) on the support of
the invariant measure
- …