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Gradual typing: a new perspective

Authors:

Giuseppe Castagna,

Tommaso Petrucciani,

Jeremy G. SiekAuthors Info & Claims

Proceedings of the ACM on Programming Languages, Volume 3, Issue POPL

Article No.: 16, Pages 1 - 32

https://doi.org/10.1145/3290329

Published: 02 January 2019 Publication History

Abstract

We define a new, more semantic interpretation of gradual types and use it to ``gradualize'' two forms of polymorphism: subtyping polymorphism and implicit parametric polymorphism. In particular, we use the new interpretation to define three gradual type systems ---Hindley-Milner, with subtyping, and with union and intersection types--- in terms of two preorders, subtyping and materialization. We define these systems both declaratively ---by adding two subsumption-like rules--- which yields clearer, more intelligible, and streamlined definitions, and algorithmically by reusing existing techniques such as unification and tallying.

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References

[1]

Amal Ahmed, Robert Bruce Findler, Jeremy G. Siek, and Philip Wadler. 2011. Blame for all. ACM SIGPLAN Notices 46, 1 (2011), 201–214.

Digital Library

[2]

Amal Ahmed, Dustin Jamner, Jeremy G. Siek, and Philip Wadler. 2017. Theorems for Free for Free: Parametricity, With and Without Types. In International Conference on Functional Programming (ICFP).

[3]

Alexander Aiken and Edward L. Wimmers. 1993. Type Inclusion Constraints and Type Inference. In Proceedings of the Conference on Functional Programming Languages and Computer Architecture (FPCA ’93). ACM, New York, NY, USA, 31–41.

Digital Library

[4]

Alexander Aiken, Edward L. Wimmers, and T. K. Lakshman. 1994. Soft typing with conditional types. In POPL ’94: Proceedings of the 21st ACM SIGPLAN-SIGACT symposium on Principles of programming languages. ACM Press, New York, NY, USA, 163–173.

Digital Library

[5]

Pedro Ángelo and Mário Florido. 2018. Gradual Intersection Types. In Workshop on Intersection Types and Related Systems.

[6]

Spenser Bauman, Carl Friedrich Bolz-Tereick, Jeremy Siek, and Sam Tobin-Hochstadt. 2017. Sound Gradual Typing: Only Mostly Dead. Proc. ACM Program. Lang. 1, OOPSLA, Article 54 (Oct. 2017), 24 pages.

Digital Library

[7]

Gavin Bierman, Martín Abadi, and Mads Torgersen. 2014. Understanding TypeScript. In ECOOP 2014 – Object-Oriented Programming, Richard Jones (Ed.). Lecture Notes in Computer Science, Vol. 8586. Springer Berlin Heidelberg, 257–281.

Digital Library

[8]

John Peter Campora, Sheng Chen, Martin Erwig, and Eric Walkingshaw. 2017. Migrating Gradual Types. Proc. ACM Program. Lang. 2, POPL (Dec. 2017), 15:1–15:29.

Digital Library

[9]

Robert Cartwright and Mike Fagan. 1991. Soft typing. In Conference on Programming Language Design and Implementation (PLDI). ACM Press, 278–292.

Digital Library

[10]

Giuseppe Castagna. 2005. Semantic subtyping: challenges, perspectives, and open problems. In ICTCS 2005, Italian Conference on Theoretical Computer Science (Lecture Notes in Computer Science). Springer, 1–20.

Digital Library

[11]

Giuseppe Castagna. 2018. Covariance and Contravariance: a fresh look at an old issue (a primer in advanced type systems for learning functional programmers). (2018). First version: 02/2013, last revision: 09/2018. Unpublished manuscript.

[12]

Giuseppe Castagna and Alain Frisch. 2005. A gentle introduction to semantic subtyping. In Proceedings of PPDP ’05, the 7th ACM SIGPLAN International Symposium on Principles and Practice of Declarative Programming, pages 198-208, ACM Press (full version) and ICALP ’05, 32nd International Colloquium on Automata, Languages and Programming, Lecture Notes in Computer Science n. 3580, pages 30-34, Springer (summary). Lisboa, Portugal. Joint ICALP-PPDP keynote talk.

Digital Library

[13]

Giuseppe Castagna and Victor Lanvin. 2017. Gradual Typing with Union and Intersection Types. Proc. ACM Program. Lang. 1, ICFP ’17, Article 41 (Sept. 2017).

Digital Library

[14]

Giuseppe Castagna, Kim Nguyen, Zhiwu Xu, and Pietro Abate. 2015. Polymorphic Functions with Set-Theoretic Types. Part 2: Local Type Inference and Type Reconstruction. In Proceedings of the 42nd ACM Symposium on Principles of Programming Languages (POPL ’15). ACM, 289–302.

Digital Library

[15]

Giuseppe Castagna, Tommaso Petrucciani, and Kim Nguyen. 2016. Set-theoretic Types for Polymorphic Variants. In Proceedings of the 21st ACM SIGPLAN International Conference on Functional Programming (ICFP 2016). ACM, New York, NY, USA, 378–391.

Digital Library

[16]

Giuseppe Castagna and Zhiwu Xu. 2011. Set-theoretic Foundation of Parametric Polymorphism and Subtyping. In ICFP ’11: 16th ACM-SIGPLAN International Conference on Functional Programming. 94–106.

Digital Library

[17]

Bruno Courcelle. 1983. Fundamental properties of infinite trees. Theoretical Computer Science 25 (1983), 95–169.

[18]

Stephen Dolan and Alan Mycroft. 2017. Polymorphism, Subtyping, and Type Inference in MLsub. In Proceedings of the 44th ACM SIGPLAN Symposium on Principles of Programming Languages (POPL 2017). ACM, New York, NY, USA, 60–72.

Digital Library

[19]

Robert Bruce Findler and Matthias Felleisen. 2002. Contracts for Higher-Order Functions. Technical Report NU-CCS-02-05. Northeastern University.

[20]

Alain Frisch, Giuseppe Castagna, and Véronique Benzaken. 2008. Semantic Subtyping: dealing set-theoretically with function, union, intersection, and negation types. J. ACM 55, 4 (2008), 1–64.

Digital Library

[21]

You-Chin Fuh and Prateek Mishra. 1988. Type inference with subtypes. In ESOP ’88, H. Ganzinger (Ed.). Springer Berlin Heidelberg, Berlin, Heidelberg, 94–114.

Digital Library

[22]

Ronald Garcia. 2013. Calculating Threesomes, with Blame. In ICFP ’13: Proceedings of the International Conference on Functional Programming.

Digital Library

[23]

Ronald Garcia and Matteo Cimini. 2015. Principal Type Schemes for Gradual Programs. In Proceedings of the 42nd Annual ACM SIGPLAN-SIGACT Symposium on Principles of Programming Languages (POPL ’15). ACM, 303–315.

Digital Library

[24]

Ronald Garcia, Alison M Clark, and Éric Tanter. 2016. Abstracting gradual typing. In Proceedings of the 43rd Annual ACM SIGPLAN-SIGACT Symposium on Principles of Programming Languages (POPL ’16). ACM, 429–442.

Digital Library

[25]

Nils Gesbert, Pierre Genevès, and Nabil Layaïda. 2015. A Logical Approach to Deciding Semantic Subtyping. ACM Trans. Program. Lang. Syst. 38, 1 (2015), 3.

Digital Library

[26]

Robert Harper. 2006. Programming Languages: Theory and Practice. Carnegie Mellon University. Available on the web: http://fpl.cs.depaul.edu/jriely/547/extras/online.pdf .

[27]

Fritz Henglein. 1994. Dynamic typing: syntax and proof theory. Science of Computer Programming 22, 3 (June 1994), 197–230.

Digital Library

[28]

Haruo Hosoya, Jérôme Vouillon, and Benjamin C. Pierce. 2000. Regular Expression Types for XML. In ICFP ’00 (SIGPLAN Notices), Vol. 35(9). http://www.cis.upenn.edu/~hahosoya/papers/regsub.ps

Digital Library

[29]

Yuu Igarashi, Taro Sekiyama, and Atsushi Igarashi. 2017. On Polymorphic Gradual Typing. In International Conference on Functional Programming (ICFP). ACM.

Digital Library

[30]

Lintaro Ina and Atsushi Igarashi. 2011. Gradual typing for generics. In Proceedings of the 2011 ACM international conference on Object oriented programming systems languages and applications (OOPSLA ’11).

Digital Library

[31]

Khurram A. Jafery and Joshua Dunfield. 2017. Sums of Uncertainty: Refinements go gradual. In Symposium on Principles of Programming Languages (POPL).

Digital Library

[32]

Matthias Keil and Peter Thiemann. 2015. Blame Assignment for Higher-order Contracts with Intersection and Union. In Proceedings of the 20th ACM SIGPLAN International Conference on Functional Programming (ICFP 2015). ACM, New York, NY, USA, 375–386.

Digital Library

[33]

Assaf J. Kfoury and Joe B. Wells. 2004. Principality and type inference for intersection types using expansion variables. Theoretical Computer Science 311, 1 (2004), 1 – 70.

Digital Library

[34]

Nicolás Lehmann and Éric Tanter. 2017. Gradual Refinement Types. In Symposium on Principles of Programming Languages (POPL).

Digital Library

[35]

André Murbach Maidl, Fabio Mascarenhas, and Roberto Ierusalimschy. 2014. Typed Lua: An Optional Type System for Lua. In Proceedings of the Workshop on Dynamic Languages and Applications (Dyla’14). ACM, New York, NY, USA, Article 3, 10 pages.

Digital Library

[36]

Alberto Martelli and Ugo Montanari. 1982. An Efficient Unification Algorithm. ACM Trans. Program. Lang. Syst. 4, 2 (1982), 258–282.

Digital Library

[37]

John C. Mitchell. 1991. Type inference with simple subtypes. Journal of Functional Programming 1, 3 (1991), 245–285.

[38]

Francisco Ortin and Miguel García. 2011. Union and intersection types to support both dynamic and static typing. Inform. Process. Lett. 111, 6 (2011), 278 – 286.

Digital Library

[39]

Tommaso Petrucciani, Giuseppe Castagna, Davide Ancona, and Elena Zucca. 2018. Semantic subtyping for non-strict languages. Technical Report. https://arxiv.org/abs/1810.05555 .

[40]

François Pottier. 2001. Simplifying subtyping constraints: a theory. Inf. Comput. 170, 2 (2001), 153–183.

Digital Library

[41]

François Pottier and Didier Rémy. 2005. The essence of ML type inference. In Advanced Topics in Types and Programming Languages, Benjamin C. Pierce (Ed.). MIT Press, Chapter 10, 389–489.

[42]

Aseem Rastogi, Avik Chaudhuri, and Basil Hosmer. 2012. The ins and outs of gradual type inference. In Symposium on Principles of Programming Languages (POPL). 481–494.

Digital Library

[43]

Simona Ronchi Della Rocca. 1988. Principal type scheme and unification for intersection type discipline. Theor. Comput. Sci. 59, 1-2 (1988), 181–209.

Digital Library

[44]

Taro Sekiyama, Atsushi Igarashi, and Michael Greenberg. 2017. Polymorphic Manifest Contracts, Revised and Resolved. ACM Trans. Program. Lang. Syst. 39, 1 (Feb. 2017), 3:1–3:36.

Digital Library

[45]

Jeremy G. Siek and Walid Taha. 2006. Gradual Typing for Functional Languages. In Proceedings of Scheme and Functional Programming Workshop. ACM, 81–92.

[46]

Jeremy G. Siek and Walid Taha. 2007. Gradual Typing for Objects. In European Conference on Object-Oriented Programming (LCNS), Vol. 4609. 2–27.

Digital Library

[47]

Jeremy G. Siek, Peter Thiemann, and Philip Wadler. 2015a. Blame and coercion: together again for the first time. In ACM SIGPLAN Notices, Vol. 50. ACM, 425–435.

Digital Library

[48]

Jeremy G. Siek and Manish Vachharajani. 2008a. Gradual typing with unification-based inference. In Proceedings of the 2008 Symposium on Dynamic languages. ACM, 7.

Digital Library

[49]

Jeremy G. Siek and Manish Vachharajani. 2008b. Gradual Typing with Unification-based Inference. Technical Report CU-CS-1039-08. University of Colorado at Boulder.

[50]

Jeremy G. Siek, Michael M. Vitousek, Matteo Cimini, and John Tang Boyland. 2015b. Refined criteria for gradual typing. In LIPIcs-Leibniz International Proceedings in Informatics, Vol. 32. Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik.

[51]

Nikhil Swamy, Cédric Fournet, Aseem Rastogi, Karthikeyan Bhargavan, Juan Chen, Pierre-Yves Strub, and Gavin Bierman. 2014. Gradual Typing Embedded Securely in JavaScript. In ACM Conference on Principles of Programming Languages (POPL).

Digital Library

[52]

Asumu Takikawa, Daniel Feltey, Ben Greenman, Max S. New, Jan Vitek, and Matthias Felleisen. 2016. Is Sound Gradual Typing Dead?. In Proceedings of the 43rd Annual ACM SIGPLAN-SIGACT Symposium on Principles of Programming Languages (POPL ’16). ACM, 456–468.

Digital Library

[53]

Matías Toro and Éric Tanter. 2017. A Gradual Interpretation of Union Types. In Proceedings of the 24th Static Analysis Symposium (SAS 2017) (Lecture Notes in Computer Science), Vol. 10422. Springer-Verlag, New York City, NY, USA, 382–404.

[54]

Philip Wadler and Robert Bruce Findler. 2009. Well-typed programs can’t be blamed. In European Symposium on Programming. Springer, 1–16.

Digital Library

[55]

Mitchell Wand. 1987. A simple algorithm and proof for type inference. Fundamenta Informaticae 10 (1987), 115–122.

[56]

Ningning Xie, Xuan Bi, and Bruno C. d. S. Oliveira. 2018. Consistent Subtyping for All. In Programming Languages and Systems, Amal Ahmed (Ed.). Springer International Publishing, Cham, 3–30.

Cited By

SU CCHEN LYANHUI LZHOU Y(2024)Static Blame for gradual typingJournal of Functional Programming10.1017/S095679682400002934Online publication date: 25-Mar-2024
https://doi.org/10.1017/S0956796824000029
Khan MChen S(2024)Gradual Typing Performance, Micro Configurations and Macro PerspectivesTheoretical Aspects of Software Engineering10.1007/978-3-031-64626-3_15(261-278)Online publication date: 29-Jul-2024
https://dl.acm.org/doi/10.1007/978-3-031-64626-3_15
New MGiovannini ELicata D(2023)Gradual Typing for Effect HandlersProceedings of the ACM on Programming Languages10.1145/36228607:OOPSLA2(1758-1786)Online publication date: 16-Oct-2023
https://dl.acm.org/doi/10.1145/3622860
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Index Terms

Gradual typing: a new perspective
1. Software and its engineering
  1. Software notations and tools
    1. General programming languages
      1. Language features
        Polymorphism

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cover image Proceedings of the ACM on Programming Languages

Proceedings of the ACM on Programming Languages Volume 3, Issue POPL

January 2019

2275 pages

EISSN:2475-1421

DOI:10.1145/3302515

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Copyright © 2019 Owner/Author.

This work is licensed under a Creative Commons Attribution International 4.0 License.

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Association for Computing Machinery

New York, NY, United States

Publication History

Published: 02 January 2019

Published in PACMPL Volume 3, Issue POPL

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Cited By

SU CCHEN LYANHUI LZHOU Y(2024)Static Blame for gradual typingJournal of Functional Programming10.1017/S095679682400002934Online publication date: 25-Mar-2024
https://doi.org/10.1017/S0956796824000029
Khan MChen S(2024)Gradual Typing Performance, Micro Configurations and Macro PerspectivesTheoretical Aspects of Software Engineering10.1007/978-3-031-64626-3_15(261-278)Online publication date: 29-Jul-2024
https://dl.acm.org/doi/10.1007/978-3-031-64626-3_15
New MGiovannini ELicata D(2023)Gradual Typing for Effect HandlersProceedings of the ACM on Programming Languages10.1145/36228607:OOPSLA2(1758-1786)Online publication date: 16-Oct-2023
https://dl.acm.org/doi/10.1145/3622860
Greenman B(2023)GTP Benchmarks for Gradual Typing PerformanceProceedings of the 2023 ACM Conference on Reproducibility and Replicability10.1145/3589806.3600034(102-114)Online publication date: 27-Jun-2023
https://dl.acm.org/doi/10.1145/3589806.3600034
Parvin SMandal MGopi PSingh SKhan MPandya PIslam MGazi H(2023)A comparative study on DNA and protein binding properties of thymol and thymoquinoneJournal of Biomolecular Structure and Dynamics10.1080/07391102.2023.218066541:20(10944-10956)Online publication date: 25-Feb-2023
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Castagna G(2023)Programming with Union, Intersection, and Negation TypesThe French School of Programming10.1007/978-3-031-34518-0_12(309-378)Online publication date: 11-Oct-2023
https://doi.org/10.1007/978-3-031-34518-0_12
Schimpf AWehr SBieniusa A(2022)Set-theoretic Types for ErlangProceedings of the 34th Symposium on Implementation and Application of Functional Languages10.1145/3587216.3587220(1-14)Online publication date: 31-Aug-2022
https://dl.acm.org/doi/10.1145/3587216.3587220
Ângelo PFlorido M(2022)A Typed Lambda Calculus with Gradual Intersection TypesProceedings of the 24th International Symposium on Principles and Practice of Declarative Programming10.1145/3551357.3551382(1-13)Online publication date: 20-Sep-2022
https://dl.acm.org/doi/10.1145/3551357.3551382
Castagna GLaurent MNguyễn KLutze M(2022)On type-cases, union elimination, and occurrence typingProceedings of the ACM on Programming Languages10.1145/34986746:POPL(1-31)Online publication date: 12-Jan-2022
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Lennon-Bertrand MMaillard KTabareau NTanter É(2022)Gradualizing the Calculus of Inductive ConstructionsACM Transactions on Programming Languages and Systems10.1145/349552844:2(1-82)Online publication date: 6-Apr-2022
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