research-article

Canonicity for 2-dimensional type theory

Authors:

Daniel R. Licata,

Robert HarperAuthors Info & Claims

ACM SIGPLAN Notices, Volume 47, Issue 1

Pages 337 - 348

https://doi.org/10.1145/2103621.2103697

Published: 25 January 2012 Publication History

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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References

[1]

Homotopy type theory Web site. www.homotopytypetheory.org, 2011.

[2]

T. Altenkirch. Extensional equality in intensional type theory. In IEEE Symposium on Logic in Computer Science, 1999.

Digital Library

[3]

T. Altenkirch, C. McBride, and W. Swierstra. Observational equality, now! In Programming Languages meets Program Verification Workshop, 2007.

Digital Library

[4]

S. Awodey and M. Warren. Homotopy theoretic models of identity types. Mathematical Proceedings of the Cambridge Philosophical Society, 2009.

[5]

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.

Digital Library

[6]

R. J. G. B. de Queiroz and A. G. de Oliveira. Propositional equality, identity types, and direct computational paths. ArXiv e-prints, July 2011.

[7]

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.

Digital Library

[8]

N. Gambino and R. Garner. The identity type weak factorisation system. Theoretical Computer Science, 409 (3): 94--109, 2008.

Digital Library

[9]

R. Garner. Two-dimensional models of type theory. Mathematical. Structures in Computer Science, 19 (4): 687--736, 2009.

Digital Library

[10]

M. Hofmann. Extensional Concepts in Intensional Type Theory. PhD thesis, University of Edinburgh, 1995.

[11]

M. Hofmann. Syntax and semantics of dependent types. In Semantics and Logics of Computation, pages 79--130. Cambridge University Press, 1997.

[12]

M. Hofmann and T. Streicher. The groupoid interpretation of type theory. In Twenty-five years of constructive type theory. Oxford University Press, 1998.

[13]

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.

[14]

D. R. Licata and R. Harper. 2-dimensional directed type theory. In Mathematical Foundations of Programming Semantics (MFPS), 2011.

Digital Library

[15]

P. L. Lumsdaine. Weak ω-categories from intensional type theory. In International Conference on Typed Lambda Calculi and Applications, 2009.

Digital Library

[16]

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.

[17]

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.

[18]

B. Nordström, K. Peterson, and J. Smith. Programming in Martin-Löf's Type Theory, an Introduction. Clarendon Press, 1990.

Digital Library

[19]

C. Paulin-Mohring. Extraction de programmes dans le Calcul des Constructions. PhD thesis, Université Paris 7, 1989.

[20]

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.

Digital Library

[21]

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/.

Digital Library

[22]

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.

[23]

V. Voevodsky. Univalent foundations of mathematics. Invited talk at WoLLIC 2011 18th Workshop on Logic, Language, Information and Computation, 2011.

Digital Library

[24]

M. A. Warren. Homotopy theoretic aspects of constructive type theory. PhD thesis, Carnegie Mellon University, 2008.

Cited By

Sojakova K(2015)Higher Inductive Types as Homotopy-Initial AlgebrasACM SIGPLAN Notices10.1145/2775051.267698350:1(31-42)Online publication date: 14-Jan-2015
https://dl.acm.org/doi/10.1145/2775051.2676983
Sojakova KRajamani SWalker D(2015)Higher Inductive Types as Homotopy-Initial AlgebrasProceedings of the 42nd Annual ACM SIGPLAN-SIGACT Symposium on Principles of Programming Languages10.1145/2676726.2676983(31-42)Online publication date: 14-Jan-2015
https://dl.acm.org/doi/10.1145/2676726.2676983
Hales T(2014)Mathematics in the age of the Turing machineTuring's Legacy10.1017/CBO9781107338579.008(253-298)Online publication date: 5-Jun-2014
https://doi.org/10.1017/CBO9781107338579.008
Show More Cited By

Index Terms

Canonicity for 2-dimensional type theory
1. Software and its engineering
  1. Software notations and tools
    1. Formal language definitions
      1. Semantics
      2. Syntax
2. Theory of computation
  1. Semantics and reasoning
    1. Program semantics
      1. Denotational semantics

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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

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
General Chair:
John Field
Google, USA
,
Program Chair:
Michael Hicks
University of Maryland, College Park, USA

Copyright © 2012 ACM.

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Association for Computing Machinery

New York, NY, United States

Publication History

Published: 25 January 2012

Published in SIGPLAN Volume 47, Issue 1

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Cited By

Sojakova K(2015)Higher Inductive Types as Homotopy-Initial AlgebrasACM SIGPLAN Notices10.1145/2775051.267698350:1(31-42)Online publication date: 14-Jan-2015
https://dl.acm.org/doi/10.1145/2775051.2676983
Sojakova KRajamani SWalker D(2015)Higher Inductive Types as Homotopy-Initial AlgebrasProceedings of the 42nd Annual ACM SIGPLAN-SIGACT Symposium on Principles of Programming Languages10.1145/2676726.2676983(31-42)Online publication date: 14-Jan-2015
https://dl.acm.org/doi/10.1145/2676726.2676983
Hales T(2014)Mathematics in the age of the Turing machineTuring's Legacy10.1017/CBO9781107338579.008(253-298)Online publication date: 5-Jun-2014
https://doi.org/10.1017/CBO9781107338579.008
Ahrens BNorth Pvan der Weide N(2023)Bicategorical type theory: semantics and syntaxMathematical Structures in Computer Science10.1017/S0960129523000312(1-45)Online publication date: 17-Oct-2023
https://doi.org/10.1017/S0960129523000312
Ahrens BNorth Pvan der Weide N(2022)Semantics for two-dimensional type theoryProceedings of the 37th Annual ACM/IEEE Symposium on Logic in Computer Science10.1145/3531130.3533334(1-14)Online publication date: 2-Aug-2022
https://dl.acm.org/doi/10.1145/3531130.3533334
Frumin DGeuvers HGondelman LWeide N(2018)Finite sets in homotopy type theoryProceedings of the 7th ACM SIGPLAN International Conference on Certified Programs and Proofs - CPP 201810.1145/3176245.3167085(201-214)Online publication date: 2018
https://doi.org/10.1145/3176245.3167085
Frumin DGeuvers HGondelman LWeide NAndronick JFelty A(2018)Finite sets in homotopy type theoryProceedings of the 7th ACM SIGPLAN International Conference on Certified Programs and Proofs10.1145/3167085(201-214)Online publication date: 8-Jan-2018
https://dl.acm.org/doi/10.1145/3167085
Angiuli CHarper R(2018)Meaning explanations at higher dimensionIndagationes Mathematicae10.1016/j.indag.2017.07.01029:1(135-149)Online publication date: Feb-2018
https://doi.org/10.1016/j.indag.2017.07.010
ANGIULI CMOREHOUSE ELICATA DHARPER R(2016)Homotopical patch theoryJournal of Functional Programming10.1017/S095679681600019826Online publication date: 13-Sep-2016
https://doi.org/10.1017/S0956796816000198
Sojakova K(2015)Higher Inductive Types as Homotopy-Initial AlgebrasACM SIGPLAN Notices10.1145/2775051.267698350:1(31-42)Online publication date: 14-Jan-2015
https://dl.acm.org/doi/10.1145/2775051.2676983
Show More Cited By

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