research-article

Homotopical patch theory

Authors:

Edward Morehouse,

Daniel R. Licata,

Robert HarperAuthors Info & Claims

ACM SIGPLAN Notices, Volume 49, Issue 9

Pages 243 - 256

https://doi.org/10.1145/2692915.2628158

Published: 19 August 2014 Publication History

Abstract

Homotopy type theory is an extension of Martin-Löf type theory, based on a correspondence with homotopy theory and higher category theory. In homotopy type theory, the propositional equality type becomes proof-relevant, and corresponds to paths in a space. This allows for a new class of datatypes, called higher inductive types, which are specified by constructors not only for points but also for paths. In this paper, we consider a programming application of higher inductive types. Version control systems such as Darcs are based on the notion of patches - syntactic representations of edits to a repository. We show how patch theory can be developed in homotopy type theory. Our formulation separates formal theories of patches from their interpretation as edits to repositories. A patch theory is presented as a higher inductive type. Models of a patch theory are given by maps out of that type, which, being functors, automatically preserve the structure of patches. Several standard tools of homotopy theory come into play, demonstrating the use of these methods in a practical programming context.

References

[1]

M. Abbott, T. Altenkirch, and N. Ghani. Containers: constructing strictly positive types. Theoretic Computer Science, 342(1):3--27, 2005.

Digital Library

[2]

T. Altenkirch. Containers in homotopy type theory. Talk at Mathematical Structures of Computation, Lyon, 2014.

[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]

B. Barras, T. Coquand, and S. Huber. A generalization of Takeuti-Gandy interpretation. To appear in Mathematical Structures in Computer Science, 2013.

[6]

M. Bezem, T. Coquand, and S. Huber. A model of type theory in cubical sets. Preprint, September 2013.

[7]

Camp Project. http://projects.haskell.org/camp/, 2010.

[8]

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

[9]

Coq Development Team. The Coq Proof Assistant Reference Manual, version 8.2. INRIA, 2009. Available from http://coq.inria.fr/.

[10]

J. Dagit. Type-correct changes - a safe approach to version control implementation. MS Thesis, 2009.

[11]

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

Digital Library

[12]

Ganesh Sittampalam et al. Some properties of darcs patch theory. Available from http://urchin.earth.li/darcs/ganesh/darcs-patch-theory/theory/formal.pdf, 2005.

[13]

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

Digital Library

[14]

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

[15]

R. Houston. On editing text. http://bosker.wordpress.com/2012/05/10/on-editing-text/, 2012.

[16]

J. Jacobson. A formalization of darcs patch theory using inverse semigroups. Available from ftp://ftp.math.ucla.edu/pub/camreport/cam09-83.pdf, 2009.

[17]

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

[18]

F. W. Lawvere. Functorial Semantics of Algebraic Theories and Some Algebraic Problems in the context of Functorial Semantics of Algebraic Theories. PhD thesis, Columbia University, 1963.

[19]

D. R. Licata and G. Brunerie. π_n(Sⁿ) in homotopy type theory. In Certified Programs and Proofs, 2013.

Digital Library

[20]

D. R. Licata and E. Finster. Eilenberg-MacLane spaces in homotopy type theory. Draft available from http://dlicata.web.wesleyan.edu/pubs.html, 2014.

Digital Library

[21]

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

Digital Library

[22]

D. R. Licata and R. Harper. Canonicity for 2-dimensional type theory. In ACM SIGPLAN-SIGACT Symposium on Principles of Programming Languages, 2012.

Digital Library

[23]

D. R. Licata and M. Shulman. Calculating the fundamental group of the circle in homotopy type theory. In IEEE Symposium on Logic in Computer Science, 2013.

Digital Library

[24]

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

Digital Library

[25]

P. L. Lumsdaine. Higher inductive types: a tour of the menagerie. http://homotopytypetheory.org/2011/04/24/higher-inductive-types-a-tour-of-the-menagerie/, April 2011.

[26]

P. L. Lumsdaine and M. Shulman. Higher inductive types. In preparation, 2013.

[27]

C. McBride. Dependently Typed Functional Programs and Their Proofs. PhD thesis, University of Edinburgh, 2000.

[28]

S. Mimram and C. Di Giusto. A categorical theory of patches. Electronic Notes in Theoretic Computer Science, 298:283--307, 2013.

Digital Library

[29]

U. Norell. Towards a practical programming language based on dependent type theory. PhD thesis, Chalmers University of Technology, 2007.

[30]

J. C. Reynolds. Separation logic: A logic for shared mutable data structures. In IEEE Symposium on Logic in Computer Science, 2002.

Digital Library

[31]

D. Roundy. Darcs: Distributed version management in haskell. In ACM SIGPLAN Workshop on Haskell. ACM, 2005.

Digital Library

[32]

M. Shulman. Homotopy type theory VI: higher inductive types. http://golem.ph.utexas.edu/category/2011/04/homotopy_type_theory_vi.html, April 2011.

[33]

M. Shulman. Univalence for inverse diagrams, oplax limits, and gluing, and homotopy canonicity. arXiv:1203.3253, 2013.

[34]

W. Swierstra and A. Löh. The semantics of version control. Available from http://www.staff.science.uu.nl/~swier004/, 2014.

[35]

The Univalent Foundations Program, Institute for Advanced Study. Homotopy Type Theory: Univalent Foundations Of Mathematics. Available from homotopytypetheory.org/book, 2013.

[36]

B. van den Berg and R. Garner. Types are weak ω-groupoids. Proceedings of the London Mathematical Society, 102(2):370--394, 2011.

[37]

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

Digital Library

[38]

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

Cited By

Clark CBohrer RMilano MLeo J(2023)Homotopy Type Theory for Sewn QuiltsProceedings of the 11th ACM SIGPLAN International Workshop on Functional Art, Music, Modelling, and Design10.1145/3609023.3609803(32-43)Online publication date: 30-Aug-2023
https://dl.acm.org/doi/10.1145/3609023.3609803
Lewowski TMadeyski L(2019)Mutants as Patches: Towards a formal approach to Mutation TestingFoundations of Computing and Decision Sciences10.2478/fcds-2019-001944:4(379-405)Online publication date: 25-Nov-2019
https://doi.org/10.2478/fcds-2019-0019
Angiuli CHarper RWilson T(2017)Computational higher-dimensional type theoryACM SIGPLAN Notices10.1145/3093333.300986152:1(680-693)Online publication date: 1-Jan-2017
https://dl.acm.org/doi/10.1145/3093333.3009861
Show More Cited By

Index Terms

Homotopical patch theory
1. Theory of computation
  1. Semantics and reasoning
    1. Program constructs
      1. Type structures

Recommendations

Homotopical patch theory
ICFP '14: Proceedings of the 19th ACM SIGPLAN international conference on Functional programming

Homotopy type theory is an extension of Martin-Löf type theory, based on a correspondence with homotopy theory and higher category theory. In homotopy type theory, the propositional equality type becomes proof-relevant, and corresponds to paths in a ...
Self-Representation in Girard's System U
POPL '15: Proceedings of the 42nd Annual ACM SIGPLAN-SIGACT Symposium on Principles of Programming Languages

In 1991, Pfenning and Lee studied whether System F could support a typed self-interpreter. They concluded that typed self-representation for System F "seems to be impossible", but were able to represent System F in F_ω. Further, they found that the ...
Type inference with non-structural subtyping
Abstract
We present an O(n³) time type inference algorithm for a type system with a largest typeΤ, a smallest type ⊥, and the usual ordering between function types. The algorithm infers type annotations of least shape, and it works equally well for ...

Comments

Information & Contributors

Information

Published In

cover image ACM SIGPLAN Notices

ACM SIGPLAN Notices Volume 49, Issue 9

ICFP '14

September 2014

361 pages

ISSN:0362-1340

EISSN:1558-1160

DOI:10.1145/2692915

Editors:
Mark W. Bailey
Hamilton College, Clinton, NY
,
Rajeev Balasubramonian
University of Utah
,
Al Davis
University of Utah
,
Sarita Adve
University of Illinois at Urbana-Champ

Issue’s Table of Contents

ICFP '14: Proceedings of the 19th ACM SIGPLAN international conference on Functional programming
August 2014
390 pages
ISBN:9781450328739
DOI:10.1145/2628136
General Chair:
Johan Jeuring
Universiteit Utrecht, The Netherlands
,
Program Chair:
Manuel M.T. Chakravarty
University of New South Wales, Australia

Copyright © 2014 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].

Publisher

Association for Computing Machinery

New York, NY, United States

Publication History

Published: 19 August 2014

Published in SIGPLAN Volume 49, Issue 9

Check for updates

Author Tags

Qualifiers

Research-article

Funding Sources

Division of Computing and Communication Foundations

Contributors

Other Metrics

View Article Metrics

Bibliometrics & Citations

Bibliometrics

Article Metrics

23
Total Citations
View Citations
436
Total Downloads

Downloads (Last 12 months)17
Downloads (Last 6 weeks)2

Reflects downloads up to 23 Dec 2024

Other Metrics

View Author Metrics

Citations

Cited By

Clark CBohrer RMilano MLeo J(2023)Homotopy Type Theory for Sewn QuiltsProceedings of the 11th ACM SIGPLAN International Workshop on Functional Art, Music, Modelling, and Design10.1145/3609023.3609803(32-43)Online publication date: 30-Aug-2023
https://dl.acm.org/doi/10.1145/3609023.3609803
Lewowski TMadeyski L(2019)Mutants as Patches: Towards a formal approach to Mutation TestingFoundations of Computing and Decision Sciences10.2478/fcds-2019-001944:4(379-405)Online publication date: 25-Nov-2019
https://doi.org/10.2478/fcds-2019-0019
Angiuli CHarper RWilson T(2017)Computational higher-dimensional type theoryACM SIGPLAN Notices10.1145/3093333.300986152:1(680-693)Online publication date: 1-Jan-2017
https://dl.acm.org/doi/10.1145/3093333.3009861
Angiuli CHarper RWilson TCastagna GGordon A(2017)Computational higher-dimensional type theoryProceedings of the 44th ACM SIGPLAN Symposium on Principles of Programming Languages10.1145/3009837.3009861(680-693)Online publication date: 1-Jan-2017
https://dl.acm.org/doi/10.1145/3009837.3009861
Kotiuga PLahtinen V(2024)An electrical engineering perspective on naturality in computational physicsAdvances in Computational Mathematics10.1007/s10444-024-10197-650:5Online publication date: 1-Oct-2024
https://dl.acm.org/doi/10.1007/s10444-024-10197-6
Erdweg SSzabó TPacak AFreund SYahav E(2021)Concise, type-safe, and efficient structural diffingProceedings of the 42nd ACM SIGPLAN International Conference on Programming Language Design and Implementation10.1145/3453483.3454052(406-419)Online publication date: 19-Jun-2021
https://dl.acm.org/doi/10.1145/3453483.3454052
Hartel JLammel R(2020)Incremental Map-Reduce on Repository History2020 IEEE 27th International Conference on Software Analysis, Evolution and Reengineering (SANER)10.1109/SANER48275.2020.9054811(320-331)Online publication date: Feb-2020
https://doi.org/10.1109/SANER48275.2020.9054811
Yallop JWhite L(2019)Lambda: the ultimate sublanguage (experience report)Proceedings of the ACM on Programming Languages10.1145/33427133:ICFP(1-17)Online publication date: 26-Jul-2019
https://dl.acm.org/doi/10.1145/3342713
Hameer APientka B(2019)Teaching the art of functional programming using automated grading (experience report)Proceedings of the ACM on Programming Languages10.1145/33417193:ICFP(1-15)Online publication date: 26-Jul-2019
https://dl.acm.org/doi/10.1145/3341719
Hackett JHutton G(2019)Call-by-need is clairvoyant call-by-valueProceedings of the ACM on Programming Languages10.1145/33417183:ICFP(1-23)Online publication date: 26-Jul-2019
https://dl.acm.org/doi/10.1145/3341718
Show More Cited By

View Options

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 Issue’s Table of Contents