research-article

A foundation for GADTs and inductive families: dependent polynomial functor approach

Authors:

Marcelo FioreAuthors Info & Claims

WGP '11: Proceedings of the seventh ACM SIGPLAN workshop on Generic programming

Pages 59 - 70

https://doi.org/10.1145/2036918.2036927

Published: 18 September 2011 Publication History

Abstract

Every Algebraic Datatype (ADT) is characterised as the initial algebra of a polynomial functor on sets. This paper extends the characterisation to the case of more advanced datatypes: Generalised Algebraic Datatypes (GADTs) and Inductive Families. Specifically, we show that GADTs and Inductive Families are characterised as initial algebras of dependent polynomial functors. The theoretical tool we use throughout is an abstract notion of polynomial between sets together with its associated general form of polynomial functor between categories of indexed sets introduced by Gambino and Hyland.

In the context of ADTs, this fundamental result is the basis for various generic functional programming techniques. To establish the usefulness of our approach for such developments in the broader context of inductively defined dependent types, we apply the theory to construct zippers for Inductive Families.

Supplementary Material

JPG File (_talk6.jpg)

Download
16.95 KB

MP4 File (_talk6.mp4)

Download
86.41 MB

References

[1]

M. Abbott, T. Altenkirch, and N. Ghani. Representing nested inductive types using W-types. In ICALP'04, pages 59--71, 2004.

[2]

M. Abbott, T. Altenkirch, C. McBride, and N. Ghani. \partial for data: Differentiating data structures. Fundam. Inform., 65 (1-2): 1--28, 2005.

Digital Library

[3]

A. Abel, R. Matthes, and T. Uustalu. Iteration and coiteration schemes for higher-order and nested datatypes. Theor. Comput. Sci., 333 (1-2): 3--66, 2005.

Digital Library

[4]

J. Adamek. Free algebras and automata realizations in the language of categories. Comment. Math. Univ. Carolina, 15 (589602), 1974.

[5]

M. D. Adams. Scrap your zippers: A generic zipper for heterogeneous types. In WGP '10, pages 13--24. ACM, 2010.

Digital Library

[6]

T. Altenkirch and C. McBride. Generic programming within dependently typed programming. In Generic Programming, pages 1--20, 2002.

Digital Library

[7]

T. Altenkirch and P. Morris. Indexed containers. In LICS'09, pages 277--285, 2009.

Digital Library

[8]

M. Benke, P. Dybjer, and P. Jansson. Universes for generic programs and proofs in dependent type theory. Nord. J. Comput., 10 (4): 265--289, 2003.

Digital Library

[9]

R. Bird and R. Paterson. Generalised folds for nested datatypes. Form. Asp. of Comput., 11 (2): 200--222, 1999.

[10]

R. Bird and R. Paterson. De Bruijn notation as a nested datatype. J. of Funct. Program., 9 (1): 77--91, 1999.

Digital Library

[11]

A. Bove and P. Dybjer. Dependent types at work. In LerNet ALFA Summer School, pages 57--99, 2008.

Digital Library

[12]

J. Chapman, P. Dagand, C. McBride, and P. Morris. The gentle art of levitation. In Proc. of ICFP'10, pages 3--14, 2010.

Digital Library

[13]

N. A. Danielsson, J. Hughes, P. Jansson, and J. Gibbons. Fast and loose reasoning is morally correct. In Proc. of POPL'06, pages 206--217, 2006.

Digital Library

[14]

P. Dybjer. Inductive families. Formal Aspects of Computing, 6: 440--465, 1994.

[15]

P. Dybjer and A. Setzer. Indexed induction-recursion. J. Log. Algebr. Program., 66 (1): 1--49, 2006.

[16]

P. Dybjer and A. Setzer. Induction-recursion and initial algebras. Ann. Pure Appl. Logic, 124 (1-3): 1--47, 2003.

[17]

M. Fiore. Mathematical models of computational and combinatorial structures. In Proc. FOSSACS 2005, LNCS 3441, pages 25--46, 2005.

Digital Library

[18]

M. Fiore, G. Plotkin, and D. Turi. Abstract syntax and variable binding. In Proc. of 14th Annual Symposium on Logic in Computer Science, pages 193--202, 1999.

Digital Library

[19]

N. Gambino and M. Hyland. Wellfounded trees and dependent polynomial functors. In TYPES'03, pages 210--225, 2003.

[20]

N. Gambino and J. Kock. Polynomial functors and polynomial monads. ArXiv:0906.4931, 2010.

[21]

N. Ghani, P. Johann, T. Uustalu, and V. Vene. Monadic augment and generalised short cut fusion. In Proc. of ICFP'05, pages 294--305, 2005.

Digital Library

[22]

T. Hagino. A typed lambda calculus with categorical type constructors. In Pitt et al., editor, Category Theory in Computer Science'87, LNCS 283, pages 140--157. Springer-Verlag, 1987.

Digital Library

[23]

R. Hinze. Fun with phantom types. In The Fun of Programming, pages 245--262. Palgrave Macmillan, 2003.

[24]

R. Hinze. Adjoint folds and unfolds. In Proc. of MPC'10, pages 195--228, 2010.

Digital Library

[25]

R. Hinze and J. Jeuring. Generic Haskell: Practice and theory. In Generic Programming, LNCS 2793, pages 1--56, 2003.

[26]

Z. Hu, H. Iwasaki, and M. Takeichi. Deriving structural hylomorphisms from recursive definitions. In Proc. of ICFP'96, pages 73--82, 1996.

Digital Library

[27]

G. Huet. Functional pearl: The zipper. Journal of Functional Programming, 6 (5): 549--554, 1997.

Digital Library

[28]

B. Jacobs. Categorical Logic and Type Theory. Studies in Logic and the Foundations of Mathematics 141. North Holland, Elsevier, 1999.

[29]

P. Jansson and J. Jeuring. Polyp - a polytypic programming language. In POPL, pages 470--482, 1997.

Digital Library

[30]

P. Johann and N. Ghani. Foundations for structured programming with GADTs. In Proc. of POPL'08, pages 297--308, 2008.

Digital Library

[31]

S. Katsumata and S. Nishimura. Algebraic fusion of functions with an accumulating parameter and its improvement. J. Funct. Program., 18 (5-6): 781--819, 2008.

Digital Library

[32]

J. Kock. Notes on polynomial functors. Manuscript, version 2009-08-05, 2009.

[33]

J. Launchbury and T. Sheard. Warm fusion: Deriving build-cata's from recursive definitions. In Proc. of FPCA'95, pages 314--323, 1995.

Digital Library

[34]

F.W. Lawvere. Adjointness in foundations. Dialectica, pages 281--296, 1969.

[35]

S. Mac Lane. Categories for the Working Mathematician, volume 5 of Graduate Texts in Mathematics. Springer-Verlag, 1971.

[36]

S. Mac Lane and I. Moerdijk. Sheaves in Geometry and Logic: A First Introduction to Topos Theory. Springer-Verlag, 1994.

[37]

C. McBride. Epigram: Practical programming with dependent types. In Advanced Functional Programming, pages 130--170, 2004.

Digital Library

[38]

C. McBride. The derivative of a regular type is its type of one-hole contexts. Manuscript, 2001.

[39]

C. McBride. Dependently Typed Programs and their Proofs. PhD thesis, University of Edinburgh, 1999.

[40]

E. Meijer, M. M. Fokkinga, and R. Paterson. Functional programming with bananas, lenses, envelopes and barbed wire. In Proc of FPCA'91, pages 124--144, 1991.

Digital Library

[41]

A. Morihata, K. Matsuzaki, Z. Hu, and M. Takeichi. The third homomorphism theorem on trees: downward & upward lead to divide-and-conquer. In POPL'09, pages 177--185, 2009.

Digital Library

[42]

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

[43]

U. Norell. Dependently typed programming in Agda. In Advanced Functional Programming, pages 230--266, 2008.

Digital Library

[44]

N. Oury and W. Swierstra. The power of Pi. In Proc. of ICFP'08, pages 39--50, 2008.

Digital Library

[45]

T. Schrijvers, S. L. Peyton Jones, M. Sulzmann, and D. Vytiniotis. Complete and decidable type inference for GADTs. In ICFP'09, pages 341--352, 2009.

Digital Library

[46]

T. Sheard and L. Fegaras. A fold for all seasons. In Proc. of FPCA'93, pages 233--242, 1993.

Digital Library

[47]

M. B. Smyth and G. D. Plotkin. The category-theoretic solution of recursive domain equations. SIAM J. Comput, 11 (4): 763--783, 1982.

[48]

A. Takano and E. Meijer. Shortcut deforestation in calculational form. In Proc. of FPCA'95, pages 306--313, 1995.

Digital Library

[49]

N. Yoneda. On Ext and exact sequences. Jour. Fac. Sci. Univ. Tokyo, pages 507--576, 1960.

Cited By

Cagne PGhiorzi EJohann P(2024)GADTs are not (Even partial) functorsMathematical Structures in Computer Science10.1017/S0960129524000161(1-24)Online publication date: 27-Aug-2024
https://doi.org/10.1017/S0960129524000161
Johann PGhiorzi EJeffries D(2022)GADTs, Functoriality, Parametricity: Pick TwoElectronic Proceedings in Theoretical Computer Science10.4204/EPTCS.357.6357(77-92)Online publication date: 7-Apr-2022
https://doi.org/10.4204/EPTCS.357.6
Patterson ELynch OFairbanks J(2022)Categorical Data Structures for Technical ComputingCompositionality10.32408/compositionality-4-54(5)Online publication date: 28-Dec-2022
https://doi.org/10.32408/compositionality-4-5
Show More Cited By

Index Terms

A foundation for GADTs and inductive families: dependent polynomial functor approach
1. Software and its engineering
  1. Software notations and tools
    1. General programming languages
      1. Language features
        Data types and structures
        Frameworks
2. Theory of computation
  1. Semantics and reasoning
    1. Program constructs
      1. Type structures

Recommendations

Revisiting the categorical interpretation of dependent type theory

We show that Hofmann's and Curien's interpretations of Martin-Lof's type theory, which were both designed to cure a mismatch between syntax and semantics in Seely's original interpretation in locally cartesian closed categories, are related via a ...
Type Theory based on Dependent Inductive and Coinductive Types
LICS '16: Proceedings of the 31st Annual ACM/IEEE Symposium on Logic in Computer Science

We develop a dependent type theory that is based purely on inductive and coinductive types, and the corresponding recursion and corecursion principles. This results in a type theory with a small set of rules, while still being fairly expressive. For ...
A type theory for productive coprogramming via guarded recursion
CSL-LICS '14: Proceedings of the Joint Meeting of the Twenty-Third EACSL Annual Conference on Computer Science Logic (CSL) and the Twenty-Ninth Annual ACM/IEEE Symposium on Logic in Computer Science (LICS)

To ensure consistency and decidability of type checking, proof assistants impose a requirement of productivity on corecursive definitions. In this paper we investigate a type-based alternative to the existing syntactic productivity checks of Coq and ...

Comments

Information & Contributors

Information

Published In

cover image ACM Conferences

WGP '11: Proceedings of the seventh ACM SIGPLAN workshop on Generic programming

September 2011

102 pages

ISBN:9781450308618

DOI:10.1145/2036918

General Chairs:
Jaakko Järvi
Texas A&M University, USA
,
Shin-Cheng Mu
Academia Sinica, Taiwan

Copyright © 2011 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 ACM 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]

Sponsors

SIGPLAN: ACM Special Interest Group on Programming Languages

Publisher

Association for Computing Machinery

New York, NY, United States

Publication History

Published: 18 September 2011

Permissions

Request permissions for this article.

Request Permissions

Check for updates

Author Tags

Qualifiers

Research-article

Conference

ICFP '11

Sponsor:

SIGPLAN

ICFP '11: ACM SIGPLAN International Conference on Functional Programming

September 18, 2011

Tokyo, Japan

Acceptance Rates

WGP '11 Paper Acceptance Rate 8 of 10 submissions, 80%;

Overall Acceptance Rate 30 of 43 submissions, 70%

Upcoming Conference

ICFP '25

Sponsor:
sigplan

ACM SIGPLAN International Conference on Functional Programming

October 12 - 18, 2025

Singapore , Singapore

Contributors

Other Metrics

View Article Metrics

Bibliometrics & Citations

Bibliometrics

Article Metrics

10
Total Citations
View Citations
210
Total Downloads

Downloads (Last 12 months)7
Downloads (Last 6 weeks)1

Reflects downloads up to 05 Mar 2025

Other Metrics

View Author Metrics

Citations

Cited By

Cagne PGhiorzi EJohann P(2024)GADTs are not (Even partial) functorsMathematical Structures in Computer Science10.1017/S0960129524000161(1-24)Online publication date: 27-Aug-2024
https://doi.org/10.1017/S0960129524000161
Johann PGhiorzi EJeffries D(2022)GADTs, Functoriality, Parametricity: Pick TwoElectronic Proceedings in Theoretical Computer Science10.4204/EPTCS.357.6357(77-92)Online publication date: 7-Apr-2022
https://doi.org/10.4204/EPTCS.357.6
Patterson ELynch OFairbanks J(2022)Categorical Data Structures for Technical ComputingCompositionality10.32408/compositionality-4-54(5)Online publication date: 28-Dec-2022
https://doi.org/10.32408/compositionality-4-5
Ahrens BMatthes RMörtberg APopescu AZdancewic S(2022)Implementing a category-theoretic framework for typed abstract syntaxProceedings of the 11th ACM SIGPLAN International Conference on Certified Programs and Proofs10.1145/3497775.3503678(307-323)Online publication date: 17-Jan-2022
https://dl.acm.org/doi/10.1145/3497775.3503678
Johann PCagne P(2022)Characterizing Functions Mappable over GADTsProgramming Languages and Systems10.1007/978-3-031-21037-2_7(135-154)Online publication date: 25-Nov-2022
https://doi.org/10.1007/978-3-031-21037-2_7
Johann PPolonsky ABouyer P(2019)Higher-kinded data typesProceedings of the 34th Annual ACM/IEEE Symposium on Logic in Computer Science10.5555/3470152.3470155(1-13)Online publication date: 24-Jun-2019
https://dl.acm.org/doi/10.5555/3470152.3470155
Keidel SPoulsen CErdweg S(2018)Compositional soundness proofs of abstract interpretersProceedings of the ACM on Programming Languages10.1145/32367672:ICFP(1-26)Online publication date: 30-Jul-2018
https://dl.acm.org/doi/10.1145/3236767
Basold HGeuvers HKoskinen EGrohe MShankar N(2016)Type Theory based on Dependent Inductive and Coinductive TypesProceedings of the 31st Annual ACM/IEEE Symposium on Logic in Computer Science10.1145/2933575.2934514(327-336)Online publication date: 5-Jul-2016
https://dl.acm.org/doi/10.1145/2933575.2934514
Dagand PMcBride C(2013)A Categorical Treatment of OrnamentsProceedings of the 2013 28th Annual ACM/IEEE Symposium on Logic in Computer Science10.5555/2591370.2591396(530-539)Online publication date: 25-Jun-2013
https://dl.acm.org/doi/10.5555/2591370.2591396
Dagand PMcBride C(2013)A Categorical Treatment of Ornaments2013 28th Annual ACM/IEEE Symposium on Logic in Computer Science10.1109/LICS.2013.60(530-539)Online publication date: Jun-2013
https://doi.org/10.1109/LICS.2013.60

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.

Figures

Tables

Media

View Table of Conten