Local realizability toposes and a modal logic for computability

Authors:
Steven Awodey;Lars Birkedal;Dana S. Scott
Affiliations:
Department of Philosophy, Carnegie Mellon University, Pittsburgh, PA 15213, USA;The IT University of Copenhagen, Glentevej 67, DK–2400 Copenhagen NV, Denmark;School of Computer Science, Carnegie Mellon University, Pittsburgh, PA 15213, USA
Venue:
Mathematical Structures in Computer Science
Year:
2002

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Abstract

This work is a step toward the development of a logic for types and computation that includes not only the usual spaces of mathematics and constructions, but also spaces from logic and domain theory. Using realizability, we investigate a configuration of three toposes that we regard as describing a notion of relative computability. Attention is focussed on a certain local map of toposes, which we first study axiomatically, and then by deriving a modal calculus as its internal logic. The resulting framework is intended as a setting for the logical and categorical study of relative computability.