823 research outputs found
Computable de Finetti measures
We prove a computable version of de Finetti's theorem on exchangeable
sequences of real random variables. As a consequence, exchangeable stochastic
processes expressed in probabilistic functional programming languages can be
automatically rewritten as procedures that do not modify non-local state. Along
the way, we prove that a distribution on the unit interval is computable if and
only if its moments are uniformly computable.Comment: 32 pages. Final journal version; expanded somewhat, with minor
corrections. To appear in Annals of Pure and Applied Logic. Extended abstract
appeared in Proceedings of CiE '09, LNCS 5635, pp. 218-23
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
Computability and analysis: the legacy of Alan Turing
We discuss the legacy of Alan Turing and his impact on computability and
analysis.Comment: 49 page
Independence, Relative Randomness, and PA Degrees
We study pairs of reals that are mutually Martin-L\"{o}f random with respect
to a common, not necessarily computable probability measure. We show that a
generalized version of van Lambalgen's Theorem holds for non-computable
probability measures, too. We study, for a given real , the
\emph{independence spectrum} of , the set of all so that there exists a
probability measure so that and is
-random. We prove that if is r.e., then no set
is in the independence spectrum of . We obtain applications of this fact to
PA degrees. In particular, we show that if is r.e.\ and is of PA degree
so that , then
- …