 
Summary: Under consideration for publication in Math. Struct. in Comp. Science
Local Realizability Toposes and a Modal
Logic for Computability
STEVEN AWODEY 2 and LARS B I RKEDAL 1 and DANA S. SCOTT 1
1
School of Computer Science, Carnegie Mellon University, Pittsburgh, PA 15213, USA.
2
Department of Philosophy, Carnegie Mellon University, Pittsburgh, PA 15213, USA.
Received 17 January 2000
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 conguration 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 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.
1. Introduction
We report here on the current status of research on the Logic of Types and Computation
at Carnegie Mellon University (Scott et al., 1998). The general goal of this research
program is to develop a logical framework for the theories of types and computability that
