Computability and realizability for interactive computations

Computability structures, simulations and realizability

We generalize the standard construction of realizability models (specifically, of categories of assemblies) to a wide class of computability structures, broad enough to embrace models of computation such as labelled transition systems and process algebras. We consider a general notion of simulation between such computability structures, and show how these simulations correspond precisely to cer...

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...

Local Realizability Toposes and a Modal Logic for Computability

Developing Theories of Types and Computability via Realizability

We investigate the development of theories of types and computability via realizability. In the rst part of the thesis, we suggest a general notion of realizability, based on weakly closed partial cartesian categories, which generalizes the usual notion of realizability over a partial combinatory algebra. We show how to construct categories of so-called assemblies and modest sets over any weakl...

A Monadic Framework for Interactive Realizability

We give a new presentation of interactive realizability with a more explicit syntax. Interactive realizability is a realizability semantics that extends the Curry-Howard correspondence to (sub-)classical logic, more precisely to first-order intuitionistic arithmetic (Heyting Arithmetic) extended by the law of the excluded middle restricted to simply existential formulas, a system motivated by i...