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

Levels of discontinuity, limit-computability, and jump operators

We develop a general theory of jump operators, which is intended to provide an abstraction of the notion of "limit-computability" on represented spaces. Jump operators also provide a framework with a strong categorical flavor for investigating degrees of discontinuity of functions and hierarchies of sets on represented spaces. We will provide a thorough investigation within this framework of a hierarchy of $\Delta^0_2$-measurable functions between arbitrary countably based $T_0$-spaces, which captures the notion of computing with ordinal mind-change bounds. Our abstract approach not only raises new questions but also sheds new light on previous results. For example, we introduce a notion of "higher order" descriptive set theoretical objects, we generalize a recent characterization of the computability theoretic notion of "lowness" in terms of adjoint functors, and we show that our framework encompasses ordinal quantifications of the non-constructiveness of Hilbert's finite basis theorem

Rethinking the notion of oracle: A link between synthetic descriptive set theory and effective topos theory

We present three different perspectives of oracle. First, an oracle is a blackbox; second, an oracle is an endofunctor on the category of represented spaces; and third, an oracle is an operation on the object of truth-values. These three perspectives create a link between the three fields, computability theory, synthetic descriptive set theory, and effective topos theory

Realizability Toposes from Specifications

We investigate a framework of Krivine realizability with I/O effects, and present a method of associating realizability models to specifications on the I/O behavior of processes, by using adequate interpretations of the central concepts of `pole' and `proof-like term'. This method does in particular allow to associate realizability models to computable functions. Following recent work of Streicher and others we show how these models give rise to triposes and toposes

The complexity of completions in partial combinatory algebra

We discuss the complexity of completions of partial combinatory algebras, in particular of Kleene's first model. Various completions of this model exist in the literature, but all of them have high complexity. We show that although there do not exist computable completions, there exists completions of low Turing degree. We use this construction to relate completions of Kleene's first model to complete extensions of PA. We also discuss the complexity of pcas defined from nonstandard models of PA