The prospects for mathematical logic in the twenty-first century

The four authors present their speculations about the future developments of mathematical logic in the twenty-first century. The areas of recursion theory, proof theory and logic for computer science, model theory, and set theory are discussed independently.Comment: Association for Symbolic Logi

Computable geometric complex analysis and complex dynamics

We discuss computability and computational complexity of conformal mappings and their boundary extensions. As applications, we review the state of the art regarding computability and complexity of Julia sets, their invariant measures and external rays impressions

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

Real-number Computability from the Perspective of Computer Assisted Proofs in Analysis

Inspired by computer assisted proofs in analysis, we present an interval approach to real-number computations

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

Hypercomputing the Mandelbrot Set?

The Mandelbrot set is an extremely well-known mathematical object that can be described in a quite simple way but has very interesting and non-trivial properties. This paper surveys some results that are known concerning the (non-)computability of the set. It considers two models of decidability over the reals (which have been treated much more thoroughly and technically by Hertling (2005), Blum, Shub and Smale, Brattka (2003) and Weihrauch (1999 and 2003) among others), two over the computable reals (the Russian school and hypercomputation) and a model over the rationals.Comment: 11 pages, 2 figure

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

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

On the Complexity of Real Functions

We develop a notion of computability and complexity of functions over the reals, which seems to be very natural when one tries to determine just how "difficult" a certain function is. This notion can be viewed as an extension of both BSS computability [Blum, Cucker, Shub, Smale 1998], and bit computability in the tradition of computable analysis [Weihrauch 2000] as it relies on the latter but allows some discontinuities and multiple values

Computational Problems in Metric Fixed Point Theory and their Weihrauch Degrees

We study the computational difficulty of the problem of finding fixed points of nonexpansive mappings in uniformly convex Banach spaces. We show that the fixed point sets of computable nonexpansive self-maps of a nonempty, computably weakly closed, convex and bounded subset of a computable real Hilbert space are precisely the nonempty, co-r.e. weakly closed, convex subsets of the domain. A uniform version of this result allows us to determine the Weihrauch degree of the Browder-Goehde-Kirk theorem in computable real Hilbert space: it is equivalent to a closed choice principle, which receives as input a closed, convex and bounded set via negative information in the weak topology and outputs a point in the set, represented in the strong topology. While in finite dimensional uniformly convex Banach spaces, computable nonexpansive mappings always have computable fixed points, on the unit ball in infinite-dimensional separable Hilbert space the Browder-Goehde-Kirk theorem becomes Weihrauch-equivalent to the limit operator, and on the Hilbert cube it is equivalent to Weak Koenig's Lemma. In particular, computable nonexpansive mappings may not have any computable fixed points in infinite dimension. We also study the computational difficulty of the problem of finding rates of convergence for a large class of fixed point iterations, which generalise both Halpern- and Mann-iterations, and prove that the problem of finding rates of convergence already on the unit interval is equivalent to the limit operator.Comment: 44 page