1,750 research outputs found
Probabilistic Computability and Choice
We study the computational power of randomized computations on infinite
objects, such as real numbers. In particular, we introduce the concept of a Las
Vegas computable multi-valued function, which is a function that can be
computed on a probabilistic Turing machine that receives a random binary
sequence as auxiliary input. The machine can take advantage of this random
sequence, but it always has to produce a correct result or to stop the
computation after finite time if the random advice is not successful. With
positive probability the random advice has to be successful. We characterize
the class of Las Vegas computable functions in the Weihrauch lattice with the
help of probabilistic choice principles and Weak Weak K\H{o}nig's Lemma. Among
other things we prove an Independent Choice Theorem that implies that Las Vegas
computable functions are closed under composition. In a case study we show that
Nash equilibria are Las Vegas computable, while zeros of continuous functions
with sign changes cannot be computed on Las Vegas machines. However, we show
that the latter problem admits randomized algorithms with weaker failure
recognition mechanisms. The last mentioned results can be interpreted such that
the Intermediate Value Theorem is reducible to the jump of Weak Weak
K\H{o}nig's Lemma, but not to Weak Weak K\H{o}nig's Lemma itself. These
examples also demonstrate that Las Vegas computable functions form a proper
superclass of the class of computable functions and a proper subclass of the
class of non-deterministically computable functions. We also study the impact
of specific lower bounds on the success probabilities, which leads to a strict
hierarchy of classes. In particular, the classical technique of probability
amplification fails for computations on infinite objects. We also investigate
the dependency on the underlying probability space.Comment: Information and Computation (accepted for publication
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
Representing Probability Measures using Probabilistic Processes
In the Type-2 Theory of Effectivity, one considers representations of topological spaces in which infinite words are used as “names ” for the elements they represent. Given such a representation, we show that probabilistic processes on infinite words generate Borel probability measures on the represented space. Conversely, for several well-behaved types of space, every Borel probability measure is represented by a corresponding probabilistic process. Accordingly, we consider probabilistic processes as providing “probabilistic names ” for Borel probability measures. We show that integration is computable with respect to the induced representation of measures.
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
A Convenient Category of Domains
We motivate and define a category of "topological domains",
whose objects are certain topological spaces, generalising
the usual -continuous dcppos of domain theory.
Our category supports all the standard constructions of domain theory,
including the solution of recursive domain equations. It also
supports the construction of free algebras for (in)equational
theories, provides a model of parametric polymorphism,
and can be used as the basis for a theory of computability.
This answers a question of Gordon Plotkin, who asked
whether it was possible to construct a category of domains
combining such properties
Towards a Convenient Category of Topological Domains
We propose a category of topological spaces that promises to be convenient for the purposes of domain theory as a mathematical theory for modelling computation. Our notion of convenience presupposes the usual properties of domain theory, e.g. modelling the basic type constructors, fixed points, recursive types, etc. In addition, we seek to model parametric polymorphism, and also to provide a flexible toolkit for modelling computational effects as free algebras for algebraic theories. Our convenient category is obtained as an application of recent work on the remarkable closure conditions of the category of quotients of countably-based topological spaces. Its convenience is a consequence of a connection with realizability models
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