Computability structures, simulations and realizability

Local Realizability Toposes and a Modal Logic for Computability

Interactive Realizability for second-order Heyting arithmetic with EM1 and SK1

We introduce a classical realizability semantics based on interactive learning for full second-order Heyting Arithmetic with excluded middle and Skolem axioms over Σ1-formulas. Realizers are written in a classical version of Girard’s System F. Since the usual computability semantics does not apply to such a system, we introduce a constructive forcing/computability semantics: though realizers ar...

RZ: A Tool for Bringing Constructive and Computable Mathematics Closer to Programming Practice

Realizability theory is not just a fundamental tool in logic and computability. It also has direct application to the design and implementation of programs, since it can produce code interfaces for the data structure corresponding to a mathematical theory. Our tool, called RZ, serves as a bridge between the worlds of constructive mathematics and programming. By using the realizability interpret...

Workshop on Realizability Preliminary Version Local Realizability Toposes and a Modal Logic for Computability (extended Abstract)

This work is a step toward developing a logic for types and computation that includes both the usual spaces of mathematics and constructions and spaces from logic and domain theory. Using realizability, we investigate a connguration of three toposes, which we regard as describing a notion of relative computability. Attention is focussed on a certain local map of toposes, which we study rst axio...

Realizability as the Connection between Computable and Constructive Mathematics

These are lecture notes for a tutorial seminar which I gave at a satellite seminar of “Computability and Complexity in Analysis 2004” in Kyoto. The main message of the notes is that computable mathematics is the realizability interpretation of constructive mathematics. The presentation is targeted at an audience which is familiar with computable mathematics but less so with constructive mathema...