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.
Supplemental Material
- Homotopy type theory Web site. www.homotopytypetheory.org, 2011.Google Scholar
- T. Altenkirch. Extensional equality in intensional type theory. In IEEE Symposium on Logic in Computer Science, 1999. Google Scholar
Digital Library
- T. Altenkirch, C. McBride, and W. Swierstra. Observational equality, now! In Programming Languages meets Program Verification Workshop, 2007. Google Scholar
Digital Library
- S. Awodey and M. Warren. Homotopy theoretic models of identity types. Mathematical Proceedings of the Cambridge Philosophical Society, 2009.Google Scholar
Cross Ref
- R. L. Constable, S. F. Allen, H. M. Bromley, W. R. Cleaveland, J. F. Cremer, R. W. Harper, D. J. Howe, T. B. Knoblock, N. P. Mendler, P. Panangaden, J. T. Sasaki, and S. F. Smith. Implementing Mathematics with the NuPRL Proof Development System. Prentice Hall, 1986. Google Scholar
Digital Library
- R. J. G. B. de Queiroz and A. G. de Oliveira. Propositional equality, identity types, and direct computational paths. ArXiv e-prints, July 2011.Google Scholar
- P. Dybjer and A. Filinski. Normalization and partial evaluation. In Applied Semantics: International Summer School, APPSEM 2000, volume 2395 of Lecture Notes in Computer Science, pages 137--192. Springer-Verlag, September 2000. Google Scholar
Digital Library
- N. Gambino and R. Garner. The identity type weak factorisation system. Theoretical Computer Science, 409 (3): 94--109, 2008. Google Scholar
Digital Library
- R. Garner. Two-dimensional models of type theory. Mathematical. Structures in Computer Science, 19 (4): 687--736, 2009. Google Scholar
Digital Library
- M. Hofmann. Extensional Concepts in Intensional Type Theory. PhD thesis, University of Edinburgh, 1995.Google Scholar
- M. Hofmann. Syntax and semantics of dependent types. In Semantics and Logics of Computation, pages 79--130. Cambridge University Press, 1997.Google Scholar
Cross Ref
- M. Hofmann and T. Streicher. The groupoid interpretation of type theory. In Twenty-five years of constructive type theory. Oxford University Press, 1998.Google Scholar
- D. R. Licata and R. Haprer. Canonicity for 2-dimensional type theory (extended version). Technical Report Carnegie Mellon University-CS-11--143, Carnegie Mellon University, 2011.Google Scholar
- D. R. Licata and R. Harper. 2-dimensional directed type theory. In Mathematical Foundations of Programming Semantics (MFPS), 2011.Google Scholar
Digital Library
- P. L. Lumsdaine. Weak ω-categories from intensional type theory. In International Conference on Typed Lambda Calculi and Applications, 2009. Google Scholar
Digital Library
- P. Martin-Löf. An intuitionistic theory of types: Predicative part. In H. Rose and J. Shepherdson, editors, Logic Colloquium '73, Proceedings of the Logic Colloquium, volume 80 of Studies in Logic and the Foundations of Mathematics, pages 73 -- 118. Elsevier, 1975.Google Scholar
- P. Martin-Löf. Constructive mathematics and computer programming. Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences, 312 (1522): 501--518, 1984.Google Scholar
Cross Ref
- B. Nordström, K. Peterson, and J. Smith. Programming in Martin-Löf's Type Theory, an Introduction. Clarendon Press, 1990. Google Scholar
Digital Library
- C. Paulin-Mohring. Extraction de programmes dans le Calcul des Constructions. PhD thesis, Université Paris 7, 1989.Google Scholar
- A. M. Pitts. Categorical logic. In S. Abramsky, D. M. Gabbay, and T. S. E. Maibaum, editors, Handbook of Logic in Computer Science, Volume 5. Algebraic and Logical Structures, chapter 2, pages 39--128. Oxford University Press, 2000. Google Scholar
Digital Library
- M. Sulzmann, M. M. T. Chakravarty, S. P. Jones, and K. Donnell. System f with type equality coercions. In ACM Workshop on Types in Language Design and Implementaion, 2007. Appendix at http://research.microsoft.com/ simonpj/papers/ext-f/. Google Scholar
Digital Library
- B. van den Berg and R. Garner. Types are weak ω-groupoids. Available from http://www.dpmms.cam.ac.uk/ rhgg2/Typesom/Typesom.html, 2010.Google Scholar
- V. Voevodsky. Univalent foundations of mathematics. Invited talk at WoLLIC 2011 18th Workshop on Logic, Language, Information and Computation, 2011. Google Scholar
Digital Library
- M. A. Warren. Homotopy theoretic aspects of constructive type theory. PhD thesis, Carnegie Mellon University, 2008.Google Scholar
Index Terms
Canonicity for 2-dimensional type theory
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