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Local Realizability Toposes and a Modal Logic for Computability (Extended Abstracts)
 Presented at Tutorial Workshop on Realizability Semantics, FLoC'99
, 1999
"... ) Steven Awodey 1 Lars Birkedal 2y Dana S. Scott 2z 1 Department of Philosophy, Carnegie Mellon University 2 School of Computer Science, Carnegie Mellon University April 15, 1999 Abstract This work is a step toward developing a logic for types and computation that includes both the usual ..."
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) Steven Awodey 1 Lars Birkedal 2y Dana S. Scott 2z 1 Department of Philosophy, Carnegie Mellon University 2 School of Computer Science, Carnegie Mellon University April 15, 1999 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 configuration 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 first 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 [SAB + ]. The general goal of this research program is to develop a logical fra...
Relative and Modified Relative Realizability
 Annals of Pure and Applied Logic
, 2001
"... this paper) and it was described by means of tripos theory right from the beginnings of that theory, see, e.g., [17, Section 1.5, item (ii)]. Recently there has been a renewed interest in Relative Realizability, both in Thomas Streicher's "Topos for Computable Analysis" [18] and in [2 ..."
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this paper) and it was described by means of tripos theory right from the beginnings of that theory, see, e.g., [17, Section 1.5, item (ii)]. Recently there has been a renewed interest in Relative Realizability, both in Thomas Streicher's "Topos for Computable Analysis" [18] and in [2, 1, 4]. The idea is, that instead of doing realizability with one partial combinatory algebra A one uses an inclusion of partial combinatory algebras A ] ` A (such that there are combinators k; s 2 A ] which also serve as combinators for A), the principal point being that "(A ] ) computable" functions may also act on data (in A) that need not be computable
QUANTUM GAUGE FIELD THEORY IN COHESIVE HOMOTOPY TYPE THEORY
"... Abstract. We implement in the formal language of homotopy type theory a new set of axioms called cohesion. Then we indicate how the resulting cohesive homotopy type theory naturally serves as a formal foundation for central concepts in quantum gauge field theory. This is a brief survey of work by th ..."
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Abstract. We implement in the formal language of homotopy type theory a new set of axioms called cohesion. Then we indicate how the resulting cohesive homotopy type theory naturally serves as a formal foundation for central concepts in quantum gauge field theory. This is a brief survey of work by the authors developed in detail elsewhere [48, 45]. Contents
A New Category for Semantics
"... now is to use this idea as a unifying platform for semantics and reasoning. Our small group of faculty and students at Carnegie Mellon, namely Steven Awodey, Andrej Bauer, Lars Birkedal, and Jesse Hughes, has begun work on this program. A selection of our theses and papers is listed in the bibliogra ..."
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now is to use this idea as a unifying platform for semantics and reasoning. Our small group of faculty and students at Carnegie Mellon, namely Steven Awodey, Andrej Bauer, Lars Birkedal, and Jesse Hughes, has begun work on this program. A selection of our theses and papers is listed in the bibliography. Equilogical Spaces Note: We build on the known cartesian closed category CLat of continuous lattices and continuous maps. Types: Pairs X = (hXi; X ), where hXi is a continuous lattice and is a partial equivalence relation on hXi. We set jXj = fx j x xg. A type is also called an equilogical space. Equivalence Classes: [x] = fx j x g, and kXk = f[x] X j x 2 jXjg, which we regard as a topological space with the quotient topology inherited from the subspace jXj of hXi. Theorem: Each continuous lattice can be regarded as an equilogical space X where jXj = hXi and each [x] = fxg. Some Flat Lattices: We set L n = f?; 0; 1; : : : ; n 1; >g and 1 = f?; 0; 1;