research-article

The next 700 syntactical models of type theory

Authors:

Pierre-Marie P�drot,

Nicolas TabareauAuthors Info & Claims

CPP 2017: Proceedings of the 6th ACM SIGPLAN Conference on Certified Programs and Proofs

Pages 182 - 194

https://doi.org/10.1145/3018610.3018620

Published: 16 January 2017 Publication History

Abstract

A family of syntactic models for the calculus of construction with universes (CC_ω) is described, all of them preserving conversion of the calculus definitionally, and thus giving rise directly to a program transformation of CC_ω into itself.

Those models are based on the remark that negative type constructors (e.g. dependent product, coinductive types or universes) are underspecified in type theory-which leaves some freedom on extra intensional specifications.

The model construction can be seen as a compilation phase from a complex type theory into a simpler type theory.

Such models can be used to derive (the negative part of) independence results with respect to CC_ω, such as functional extensionality, propositional extensionality, univalence or the fact that bisimulation on a coinductive type may not coincide with equality.

They can also be used to add new principles to the theory, which we illustrate by defining a version of CC_ω with ad-hoc polymorphism that shows in particular that parametricity is not an implicit requirement of type theory.

The correctness of some of the models/program transformations have been checked in the Coq proof assistant and have been instrumented as a Coq plugin.

References

[1]

T. Altenkirch. Extensional equality in intensional type theory. In Proceedings of LICS, Trento, Italy, July 1999.

Digital Library

[2]

T. Altenkirch and A. Kaposi. Type theory in type theory using quotient inductive types. In Proceedings of POPL, 2016.

Digital Library

[3]

J.-P. Bernardy and M. Lasson. Realizability and Parametricity in Pure Type Systems. In Foundations of Software Science and Computational Structures, volume 6604, pages 108–122, Saarbrücken, Germany, Mar. 2011.

Digital Library

[4]

J.-P. Bernardy and G. Moulin. A computational interpretation of parametricity. In Proceedings of LICS, Dubrovnik, Croatia, June 2012.

Digital Library

[5]

J.-P. Bernardy, P. Jansson, and R. Paterson. Parametricity and dependent types. In Proceedings of ICFP, Baltimore, Maryland, USA, 2010.

Digital Library

[6]

J. Chapman. Type theory should eat itself. Electron. Notes Theor. Comput. Sci., 228:21–36, Jan. 2009. ISSN 1571-0661.

Digital Library

[7]

P. Dybjer. Internal type theory. In S. Berardi and M. Coppo, editors, Types for Proofs and Programs, volume 1158 of LNCS, pages 120–134. Springer Berlin Heidelberg, 1996. ISBN 978-3-540- 61780-8. P. Dybjer. A general formulation of simultaneous inductiverecursive definitions in type theory. J. Symb. Log., 65(2):525– 549, 2000.

Digital Library

[8]

M. Hoffman. Syntax and semantics of dependent types. In Semantics and Logics of Computation, pages 241–298, 1997.

[9]

M. Hofmann. Extensional constructs in intensional type theory. CPHC/BCS distinguished dissertations. Springer, 1997. ISBN 978-3-540-76121-1. G. Jaber, N. Tabareau, and M. Sozeau. Extending Type Theory with Forcing. In Proceedings of LICS, Dubrovnik, Croatia, June 2012.

Digital Library

[10]

G. Jaber, G. Lewertoski, P.-M. Pédrot, N. Tabareau, and M. Sozeau. The Definitional Side of the Forcing. In Proceedings of LICS, New-York, USA, July 2016.

Digital Library

[11]

B. Jacobs. Comprehension categories and the semantics of type dependency. Theoretical Computer Science, 107(2):169 – 207, 1993.

Digital Library

[12]

C. Kapulkin, P. L. Lumsdaine, and V. Voevodsky. The simplicial model of univalent foundations. arXiv preprint arXiv:1211.2851, 2012.

[13]

Y. Lafont, B. Reus, and T. Streicher. Continuation semantics or expressing implication by negation. Univ. München, Inst. für Informatik, 1993.

[14]

Z. Luo. ECC, an extended calculus of constructions. In Proceedings of LICS, Pacific Grove, CA, USA, June 1989.

Digital Library

[15]

C. Paulin-Mohring. Extraction de programmes dans le Calcul des Constructions. Thèse d’université, Paris 7, Jan. 1989.

[16]

The A GDA Development Team. Agda. 2015.

Cited By

Kov�cs A(2024)Closure-Free Functional Programming in a Two-Level Type TheoryProceedings of the ACM on Programming Languages10.1145/36746488:ICFP(659-692)Online publication date: 15-Aug-2024
https://dl.acm.org/doi/10.1145/3674648
Winterhalter T(2024)Dependent Ghosts Have a Reflection for FreeProceedings of the ACM on Programming Languages10.1145/36746478:ICFP(630-658)Online publication date: 15-Aug-2024
https://dl.acm.org/doi/10.1145/3674647
Altenkirch TChamoun YKaposi AShulman M(2024)Internal Parametricity, without an IntervalProceedings of the ACM on Programming Languages10.1145/36329208:POPL(2340-2369)Online publication date: 5-Jan-2024
https://dl.acm.org/doi/10.1145/3632920
Show More Cited By

Index Terms

The next 700 syntactical models of type theory
1. Theory of computation
  1. Logic
  2. Models of computation
    1. Computability

Recommendations

Decidability of conversion for type theory in type theory

Type theory should be able to handle its own meta-theory, both to justify its foundational claims and to obtain a verified implementation. At the core of a type checker for intensional type theory lies an algorithm to check equality of types, or in other ...
A relationally parametric model of dependent type theory
POPL '14: Proceedings of the 41st ACM SIGPLAN-SIGACT Symposium on Principles of Programming Languages

Reynolds' theory of relational parametricity captures the invariance of polymorphically typed programs under change of data representation. Reynolds' original work exploited the typing discipline of the polymorphically typed lambda-calculus System F, ...
A relationally parametric model of dependent type theory
POPL '14

Reynolds' theory of relational parametricity captures the invariance of polymorphically typed programs under change of data representation. Reynolds' original work exploited the typing discipline of the polymorphically typed lambda-calculus System F, ...

Comments

Information & Contributors

Information

Published In

cover image ACM Other conferences

CPP 2017: Proceedings of the 6th ACM SIGPLAN Conference on Certified Programs and Proofs

January 2017

234 pages

ISBN:9781450347051

DOI:10.1145/3018610

Program Chairs:
Yves Bertot
Inria, France / Universit� Cote d'Azur, France
,
Viktor Vafeiadis
MPI-SWS, Germany

Copyright � 2017 ACM.

Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than the author(s) must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from [email protected].

In-Cooperation

Publisher

Association for Computing Machinery

New York, NY, United States

Publication History

Published: 16 January 2017

Permissions

Request permissions for this article.

Request Permissions

Check for updates

Author Tags

Qualifiers

Research-article

Funding Sources

European Research Council

Conference

CPP '17

CPP '17: Certified Proofs and Programs

January 16 - 17, 2017

Paris, France

Acceptance Rates

Overall Acceptance Rate 18 of 26 submissions, 69%

Contributors

Other Metrics

View Article Metrics

Bibliometrics & Citations

Bibliometrics

Article Metrics

40
Total Citations
View Citations
393
Total Downloads

Downloads (Last 12 months)29
Downloads (Last 6 weeks)2

Reflects downloads up to 22 Oct 2024

Other Metrics

View Author Metrics

Citations

Cited By

Kov�cs A(2024)Closure-Free Functional Programming in a Two-Level Type TheoryProceedings of the ACM on Programming Languages10.1145/36746488:ICFP(659-692)Online publication date: 15-Aug-2024
https://dl.acm.org/doi/10.1145/3674648
Winterhalter T(2024)Dependent Ghosts Have a Reflection for FreeProceedings of the ACM on Programming Languages10.1145/36746478:ICFP(630-658)Online publication date: 15-Aug-2024
https://dl.acm.org/doi/10.1145/3674647
Altenkirch TChamoun YKaposi AShulman M(2024)Internal Parametricity, without an IntervalProceedings of the ACM on Programming Languages10.1145/36329208:POPL(2340-2369)Online publication date: 5-Jan-2024
https://dl.acm.org/doi/10.1145/3632920
Van Muylder ANuyts ADevriese D(2024)Internal and Observational Parametricity for Cubical AgdaProceedings of the ACM on Programming Languages10.1145/36328508:POPL(209-240)Online publication date: 5-Jan-2024
https://dl.acm.org/doi/10.1145/3632850
Cohen CCrance EMahboubi A(2024)Trocq: Proof Transfer for Free, With or Without UnivalenceProgramming Languages and Systems10.1007/978-3-031-57262-3_10(239-268)Online publication date: 6-Apr-2024
https://dl.acm.org/doi/10.1007/978-3-031-57262-3_10
Castro F(2023)An Interpretation of E-HA^w inside HA^wElectronic Proceedings in Theoretical Computer Science10.4204/EPTCS.396.5396(52-66)Online publication date: 17-Nov-2023
https://doi.org/10.4204/EPTCS.396.5
Huang YYallop J(2023)Defunctionalization with Dependent TypesProceedings of the ACM on Programming Languages10.1145/35912417:PLDI(516-538)Online publication date: 6-Jun-2023
https://dl.acm.org/doi/10.1145/3591241
Maillard KLennon-Bertrand MTabareau NTanter �(2022)A reasonably gradual type theoryProceedings of the ACM on Programming Languages10.1145/35476556:ICFP(931-959)Online publication date: 31-Aug-2022
https://dl.acm.org/doi/10.1145/3547655
Vazou NGreenberg MPolikarpova N(2022)How to safely use extensionality in Liquid HaskellProceedings of the 15th ACM SIGPLAN International Haskell Symposium10.1145/3546189.3549919(13-26)Online publication date: 6-Sep-2022
https://dl.acm.org/doi/10.1145/3546189.3549919
Lennon-Bertrand MMaillard KTabareau NTanter �(2022)Gradualizing the Calculus of Inductive ConstructionsACM Transactions on Programming Languages and Systems10.1145/349552844:2(1-82)Online publication date: 6-Apr-2022
https://dl.acm.org/doi/10.1145/3495528
Show More Cited By

View Options

Get Access

Login options

Check if you have access through your login credentials or your institution to get full access on this article.

Full Access

Get this Publication

View options

PDF

View or Download as a PDF file.

eReader

View online with eReader.

Media

Figures

Other

Tables

View Table of Contents