Semicomputable Points in Euclidean Spaces

We introduce the notion of a semicomputable point in R^n, defined as a point having left-c.e. projections. We study the range of such a point, which is the set of directions on which its projections are left-c.e., and is a convex cone. We provide a thorough study of these notions, proving along the way new results on the computability of convex sets. We prove realization results, by identifying computability properties of convex cones that make them ranges of semicomputable points. We give two applications of the theory. The first one provides a better understanding of the Solovay derivatives. The second one is the investigation of left-c.e. quadratic polynomials. We show that this is, in fact, a particular case of the general theory of semicomputable points

Semicomputable Geometry

Computability and semicomputability of compact subsets of the Euclidean spaces are important notions, that have been investigated for many classes of sets including fractals (Julia sets, Mandelbrot set) and objects with geometrical or topological constraints (embedding of a sphere). In this paper we investigate one of the simplest classes, namely the filled triangles in the plane. We study the properties of the parameters of semicomputable triangles, such as the coordinates of their vertices. This problem is surprisingly rich. We introduce and develop a notion of semicomputability of points of the plane which is a generalization in dimension 2 of the left-c.e. and right-c.e. numbers. We relate this notion to Solovay reducibility. We show that semicomputable triangles admit no finite parametrization, for some notion of parametrization

Computability of the Radon-Nikodym derivative

We study the computational content of the Radon-Nokodym theorem from measure theory in the framework of the representation approach to computable analysis. We define computable measurable spaces and canonical representations of the measures and the integrable functions on such spaces. For functions f,g on represented sets, f is W-reducible to g if f can be computed by applying the function g at most once. Let RN be the Radon-Nikodym operator on the space under consideration and let EC be the non-computable operator mapping every enumeration of a set of natural numbers to its characteristic function. We prove that for every computable measurable space, RN is W-reducible to EC, and we construct a computable measurable space for which EC is W-reducible to RN

Topological analysis of representations

International audienceComputable analysis is the theoretical study of the abilities of algorithms to process infinite objects. The algorithms abilities depend on the way these objects are presented to them. We survey recent results on the problem of identifying the properties of objects that are decidable or semidecidable, for several concrete classes of objects and representations of them. Topology is at the core of this study, as the decidable and semidecidable properties are closely related to the open sets induced by the representation

A Survey on Continuous Time Computations

We provide an overview of theories of continuous time computation. These theories allow us to understand both the hardness of questions related to continuous time dynamical systems and the computational power of continuous time analog models. We survey the existing models, summarizing results, and point to relevant references in the literature

Computability and dynamical systems

In this paper we explore results that establish a link between dynamical systems and computability theory (not numerical analysis). In the last few decades, computers have increasingly been used as simulation tools for gaining insight into dynamical behavior. However, due to the presence of errors inherent in such numerical simulations, with few exceptions, computers have not been used for the nobler task of proving mathematical results. Nevertheless, there have been some recent developments in the latter direction. Here we introduce some of the ideas and techniques used so far, and suggest some lines of research for further work on this fascinating topic

Systems Perspectives on the Interaction Between Human and Technological Resources

Peer reviewe

Statistical properties of dynamical systems – Simulation and abstract computation

We survey an area of recent development, relating dynamics to theoretical computer science. We discuss some aspects of the theoretical simulation and computation of the long term behavior of dynamical systems. We will focus on the statistical limiting behavior and invariant measures. We present a general method allowing the algorithmic approximation at any given accuracy of invariant measures. The method can be applied in many interesting cases, as we shall explain. On the other hand, we exhibit some examples where the algorithmic approximation of invariant measures is not possible. We also explain how it is possible to compute the speed of convergence of ergodic averages (when the system is known exactly) and how this entails the computation of arbitrarily good approximations of points of the space having typical statistical behaviour (a sort of constructive version of the pointwise ergodic theorem)