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Canonicity for 2-dimensional type theory

Published:25 January 2012Publication History
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Abstract

Higher-dimensional dependent type theory enriches conventional one-dimensional dependent type theory with additional structure expressing equivalence of elements of a type. This structure may be employed in a variety of ways to capture rather coarse identifications of elements, such as a universe of sets considered modulo isomorphism. Equivalence must be respected by all families of types and terms, as witnessed computationally by a type-generic program. Higher-dimensional type theory has applications to code reuse for dependently typed programming, and to the formalization of mathematics. In this paper, we develop a novel judgemental formulation of a two-dimensional type theory, which enjoys a canonicity property: a closed term of boolean type is definitionally equal to true or false. Canonicity is a necessary condition for a computational interpretation of type theory as a programming language, and does not hold for existing axiomatic presentations of higher-dimensional type theory. The method of proof is a generalization of the NuPRL semantics, interpreting types as syntactic groupoids rather than equivalence relations.

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        • Published in

          cover image ACM SIGPLAN Notices
          ACM SIGPLAN Notices  Volume 47, Issue 1
          POPL '12
          January 2012
          569 pages
          ISSN:0362-1340
          EISSN:1558-1160
          DOI:10.1145/2103621
          Issue’s Table of Contents
          • cover image ACM Conferences
            POPL '12: Proceedings of the 39th annual ACM SIGPLAN-SIGACT symposium on Principles of programming languages
            January 2012
            602 pages
            ISBN:9781450310833
            DOI:10.1145/2103656

          Copyright © 2012 ACM

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          Association for Computing Machinery

          New York, NY, United States

          Publication History

          • Published: 25 January 2012

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