3,749 research outputs found
Exhaustible sets in higher-type computation
We say that a set is exhaustible if it admits algorithmic universal
quantification for continuous predicates in finite time, and searchable if
there is an algorithm that, given any continuous predicate, either selects an
element for which the predicate holds or else tells there is no example. The
Cantor space of infinite sequences of binary digits is known to be searchable.
Searchable sets are exhaustible, and we show that the converse also holds for
sets of hereditarily total elements in the hierarchy of continuous functionals;
moreover, a selection functional can be constructed uniformly from a
quantification functional. We prove that searchable sets are closed under
intersections with decidable sets, and under the formation of computable images
and of finite and countably infinite products. This is related to the fact,
established here, that exhaustible sets are topologically compact. We obtain a
complete description of exhaustible total sets by developing a computational
version of a topological Arzela--Ascoli type characterization of compact
subsets of function spaces. We also show that, in the non-empty case, they are
precisely the computable images of the Cantor space. The emphasis of this paper
is on the theory of exhaustible and searchable sets, but we also briefly sketch
applications
The Rice-Shapiro theorem in Computable Topology
We provide requirements on effectively enumerable topological spaces which
guarantee that the Rice-Shapiro theorem holds for the computable elements of
these spaces. We show that the relaxation of these requirements leads to the
classes of effectively enumerable topological spaces where the Rice-Shapiro
theorem does not hold. We propose two constructions that generate effectively
enumerable topological spaces with particular properties from wn--families and
computable trees without computable infinite paths. Using them we propose
examples that give a flavor of this class
A Local to Global Principle for the Complexity of Riemann Mappings (Extended Abstract)
We show that the computational complexity of Riemann mappings can be bounded
by the complexity needed to compute conformal mappings locally at boundary
points. As a consequence we get first formally proven upper bounds for
Schwarz-Christoffel mappings and, more generally, Riemann mappings of domains
with piecewise analytic boundaries
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
Computability of Julia sets
In this paper we settle most of the open questions on algorithmic
computability of Julia sets. In particular, we present an algorithm for
constructing quadratics whose Julia sets are uncomputable. We also show that a
filled Julia set of a polynomial is always computable.Comment: Revised. To appear in Moscow Math. Journa
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