Article

Numbering matters: first-order canonical forms for second-order recursive types

Authors:

Nadji Gauthier,

François PottierAuthors Info & Claims

ICFP '04: Proceedings of the ninth ACM SIGPLAN international conference on Functional programming

Pages 150 - 161

https://doi.org/10.1145/1016850.1016872

Published: 19 September 2004 Publication History

Abstract

We study a type system equipped with universal types and equire-cursive types, which we refer to as F_͘. We show that type equality may be decided in time O(nlog n), an improvement over the previous known bound of O(n² ). In fact, we show that two more general problems, namely entailment of type equations and type unification, may be decided in time O(nlog n), a new result. To achieve this bound, we associate, with every F_͘ type, a first-order canonical form, which may be computed in time O(nlogn). By exploiting this notion, we reduce all three problems to equality and unification of first-order recursive terms, for which efficient algorithms are known.

References

[1]

Martín Abadi and Marcelo P. Fiore. Syntactic considerations on recursive types. In IEEE Symposium on Logic in Computer Science (LICS), pages 242--252, July 1996.

Digital Library

[2]

Michael Brandt and Fritz Henglein. Coinductive axiomatization of recursive type equality and subtyping. Fundamenta Informaticæ, 33:309--338, 1998.

Digital Library

[3]

Kim B. Bruce, Luca Cardelli, and Benjamin C. Pierce. Comparing object encodings. Information and Computation, 155(1/2):108--133, November 1999.

Digital Library

[4]

Dario Colazzo and Giorgio Ghelli. Subtyping recursive types in Kernel Fun. In IEEE Symposium on Logic in Computer Science (LICS), pages 137--146, July 1999.

Digital Library

[5]

Gregory D. Collins and Zhong Shao. Intensional analysis of higher-kinded recursive types. Technical Report YALEU/DCS/TR-1240, Yale University, 2002.

[6]

Jeffrey Considine. Efficient hash-consing of recursive types. Technical Report 2000-006, Boston University, January 2000.

Digital Library

[7]

Karl Crary. Simple, efficient object encoding using intersection types. Technical Report CMU-CS-99-100, Carnegie Mellon University, 1999.

[8]

Gilles Dowek. Higher-order unification and matching. In J. Alan Robinson and Andrei Voronkov, editors, Handbook of Automated Reasoning, pages 1009--1062. Elsevier Science, 2001.

Digital Library

[9]

Murdoch J. Gabbay and Andrew M. Pitts. A new approach to abstract syntax with variable binding. Formal Aspects of Computing, 13(3-5):341--363, July 2002.

Digital Library

[10]

Vladimir Gapeyev, Michael Levin, and Benjamin Pierce. Recursive subtyping revealed. Journal of Functional Programming, 12(6):511--548, 2003.

Digital Library

[11]

Nadji Gauthier. Implementation of N. http://caml.inria.fr/~gauthier/naming.tar.gz, April 2004.

[12]

Neal Glew. An efficient class and object encoding. In ACM Conference on Object-Oriented Programming, Systems, Languages, and Applications (OOPSLA), pages 311--324, October 2000.

Digital Library

[13]

Neal Glew. A theory of second-order trees. In European Symposium on Programming (ESOP), volume 2305 of Lecture Notes in Computer Science, pages 147--161. Springer Verlag, April 2002.

Digital Library

[14]

Gérard Huet. Résolution d'équations dans des langages d'ordre 1, 2, .., w . PhD thesis, Université Paris 7, September 1976.

[15]

Christopher League, Zhong Shao, and Valery Trifonov. Representing Java classes in a typed intermediate language. In ACM International Conference on Functional Programming (ICFP),pages 183--196, September 1999.

Digital Library

[16]

Christopher League, Zhong Shao, and Valery Trifonov. Type-preserving compilation of Featherweight Java. ACM Transactions on Programming Languages and Systems, 24(2):112--152, March 2002.

Digital Library

[17]

Benjamin C. Pierce. Types and Programming Languages. MIT Press, 2002.

Digital Library

[18]

François Pottier and Nadji Gauthier. Polymorphic typed defunctionalization. In ACMSymposium on Principles of Programming Languages (POPL), pages 89--98, January 2004.

Digital Library

[19]

Didier Rémy. Projective ML. In ACM Symposium on Lisp and Functional Programming (LFP), pages 66--75, 1992.

Digital Library

[20]

Didier Rémy. Type inference for records in a natural extension of ML. In Carl A. Gunter and John C. Mitchell, editors, Theoretical Aspects Of Object-Oriented Programming. Types, Semantics and Language Design. MIT Press, 1994.

Digital Library

[21]

Christian Urban, Andrew Pitts, and Murdoch Gabbay. Nominal unification. In Computer Science Logic, volume 2803 of Lecture Notes in Computer Science, pages 513--527. Springer Verlag, August 2003.

[22]

Hongwei Xi, Chiyan Chen, and Gang Chen. Guarded recursive datatype constructors. In ACM Symposium on Principles of Programming Languages (POPL), January 2003.

Digital Library

Cited By

Costa DMordido APoças DVasconcelos V(2024)Polymorphic higher-order context-free session typesTheoretical Computer Science10.1016/j.tcs.2024.1145821001:COnline publication date: 27-Jun-2024
https://dl.acm.org/doi/10.1016/j.tcs.2024.114582
Poças DCosta DMordido AVasconcelos V(2023)System with Context-free Session TypesProgramming Languages and Systems10.1007/978-3-031-30044-8_15(392-420)Online publication date: 22-Apr-2023
https://dl.acm.org/doi/10.1007/978-3-031-30044-8_15
Cai YGiarrusso POstermann K(2016)System f-omega with equirecursive types for datatype-generic programmingACM SIGPLAN Notices10.1145/2914770.283766051:1(30-43)Online publication date: 11-Jan-2016
https://dl.acm.org/doi/10.1145/2914770.2837660
Show More Cited By

Index Terms

Numbering matters: first-order canonical forms for second-order recursive types
1. Software and its engineering
  1. Software notations and tools
    1. General programming languages
      1. Language features
        Polymorphism
        Recursion

Recommendations

Numbering matters: first-order canonical forms for second-order recursive types
ICFP '04

We study a type system equipped with universal types and equire-cursive types, which we refer to as F_͘. We show that type equality may be decided in time O(nlog n), an improvement over the previous known bound of O(n² ). In fact, we show that two more ...
System F with coercion constraints
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)

We present a second-order λ-calculus with coercion constraints that generalizes a previous extension of System F with parametric coercion abstractions by allowing multiple but simultaneous type and coercion abstractions, as well as recursive coercions ...
Embedding polymorphic dynamic typing
WGP '11: Proceedings of the seventh ACM SIGPLAN workshop on Generic programming

Dynamic typing in a statically typed functional language allows us to defer type unification until run time. This is typically useful when interacting with the 'outside' world where the type of values involved may not be known statically. Haskell has ...

Comments

Information & Contributors

Information

Published In

cover image ACM Conferences

ICFP '04: Proceedings of the ninth ACM SIGPLAN international conference on Functional programming

September 2004

264 pages

ISBN:1581139055

DOI:10.1145/1016850

General Chair:
Chris Okasaki
United States Military Academy, West Point
,
Program Chair:
Kathleen Fisher
AT&T Labs Research

ACM SIGPLAN Notices Volume 39, Issue 9
ICFP '04
September 2004
254 pages
ISSN:0362-1340
EISSN:1558-1160
DOI:10.1145/1016848
Issue’s Table of Contents

Copyright © 2004 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

Publisher

Association for Computing Machinery

New York, NY, United States

Publication History

Published: 19 September 2004

Permissions

Request permissions for this article.

Request Permissions

Check for updates

Author Tags

Qualifiers

Article

Conference

ICFP04

Sponsor:

ICFP04: ACM SIGPLAN International Conference on Functional Programming

September 19 - 21, 2004

UT, Snow Bird, USA

Acceptance Rates

Overall Acceptance Rate 333 of 1,064 submissions, 31%

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

16
Total Citations
View Citations
264
Total Downloads

Downloads (Last 12 months)11
Downloads (Last 6 weeks)0

Reflects downloads up to 20 Jan 2025

Other Metrics

View Author Metrics

Citations

Cited By

Costa DMordido APoças DVasconcelos V(2024)Polymorphic higher-order context-free session typesTheoretical Computer Science10.1016/j.tcs.2024.1145821001:COnline publication date: 27-Jun-2024
https://dl.acm.org/doi/10.1016/j.tcs.2024.114582
Poças DCosta DMordido AVasconcelos V(2023)System with Context-free Session TypesProgramming Languages and Systems10.1007/978-3-031-30044-8_15(392-420)Online publication date: 22-Apr-2023
https://dl.acm.org/doi/10.1007/978-3-031-30044-8_15
Cai YGiarrusso POstermann K(2016)System f-omega with equirecursive types for datatype-generic programmingACM SIGPLAN Notices10.1145/2914770.283766051:1(30-43)Online publication date: 11-Jan-2016
https://dl.acm.org/doi/10.1145/2914770.2837660
Cai YGiarrusso POstermann KBodik RMajumdar R(2016)System f-omega with equirecursive types for datatype-generic programmingProceedings of the 43rd Annual ACM SIGPLAN-SIGACT Symposium on Principles of Programming Languages10.1145/2837614.2837660(30-43)Online publication date: 11-Jan-2016
https://dl.acm.org/doi/10.1145/2837614.2837660
Kilpatrick SDreyer DPeyton Jones SMarlow S(2014)BackpackACM SIGPLAN Notices10.1145/2578855.253588449:1(19-31)Online publication date: 8-Jan-2014
https://dl.acm.org/doi/10.1145/2578855.2535884
Kilpatrick SDreyer DPeyton Jones SMarlow SJagannathan SSewell P(2014)BackpackProceedings of the 41st ACM SIGPLAN-SIGACT Symposium on Principles of Programming Languages10.1145/2535838.2535884(19-31)Online publication date: 11-Jan-2014
https://dl.acm.org/doi/10.1145/2535838.2535884
Ke WLiu ZWang SZhao L(2013)A graph-based generic type system for object-oriented programsFrontiers of Computer Science: Selected Publications from Chinese Universities10.1007/s11704-012-1307-87:1(109-134)Online publication date: 1-Feb-2013
https://dl.acm.org/doi/10.1007/s11704-012-1307-8
Glew N(2012)Subtyping for f-bounded quantifiers and equirecursive typesLogic and Program Semantics10.5555/2340820.2340827(66-82)Online publication date: 1-Jan-2012
https://dl.acm.org/doi/10.5555/2340820.2340827
Rémy DYakobowski BPottier FNecula G(2007)A graphical presentation of MLF types with a linear-time unification algorithmProceedings of the 2007 ACM SIGPLAN international workshop on Types in languages design and implementation10.1145/1190315.1190321(27-38)Online publication date: 16-Jan-2007
https://dl.acm.org/doi/10.1145/1190315.1190321
Calvès CFernández M(2007)Implementing Nominal UnificationElectronic Notes in Theoretical Computer Science (ENTCS)10.1016/j.entcs.2006.09.027176:1(25-37)Online publication date: 1-May-2007
https://dl.acm.org/doi/10.1016/j.entcs.2006.09.027
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 Table of Contents