More Web Proxy on the site http://driver.im/

research-article

Public Access

Type soundness proofs with definitional interpreters

Authors:

Tiark RompfAuthors Info & Claims

ACM SIGPLAN Notices, Volume 52, Issue 1

Pages 666 - 679

https://doi.org/10.1145/3093333.3009866

Published: 01 January 2017 Publication History

Abstract

While type soundness proofs are taught in every graduate PL class, the gap between realistic languages and what is accessible to formal proofs is large. In the case of Scala, it has been shown that its formal model, the Dependent Object Types (DOT) calculus, cannot simultaneously support key metatheoretic properties such as environment narrowing and subtyping transitivity, which are usually required for a type soundness proof. Moreover, Scala and many other realistic languages lack a general substitution property.

The first contribution of this paper is to demonstrate how type soundness proofs for advanced, polymorphic, type systems can be carried out with an operational semantics based on high-level, definitional interpreters, implemented in Coq. We present the first mechanized soundness proofs in this style for System F and several extensions, including mutable references. Our proofs use only straightforward induction, which is significant, as the combination of big-step semantics, mutable references, and polymorphism is commonly believed to require coinductive proof techniques.

The second main contribution of this paper is to show how DOT-like calculi emerge from straightforward generalizations of the operational aspects of F, exposing a rich design space of calculi with path-dependent types inbetween System F and DOT, which we dub the System D Square.

By working directly on the target language, definitional interpreters can focus the design space and expose the invariants that actually matter at runtime. Looking at such runtime invariants is an exciting new avenue for type system design.

References

[1]

A. Abel. Polarised subtyping for sized types. Mathematical Structures in Computer Science, 18:797–822, 10 2008.

[2]

M. S. Ager, D. Biernacki, O. Danvy, and J. Midtgaard. A functional correspondence between evaluators and abstract machines. In PPDP, 2003.

Digital Library

[3]

A. J. Ahmed. Semantics of types for mutable state. PhD thesis, Princeton University, 2004.

Digital Library

[4]

A. J. Ahmed. Step-indexed syntactic logical relations for recursive and quantified types. In ESOP, 2006.

Digital Library

[5]

N. Amin, S. Grütter, M. Odersky, T. Rompf, and S. Stucki. The essence of dependent object types. In WadlerFest, A List of Successes That Can Change the World: Essays Dedicated to Philip Wadler on the Occasion of His 60th Birthday, 2016.

[6]

N. Amin, A. Moors, and M. Odersky. Dependent object types. In FOOL, 2012.

[7]

N. Amin, T. Rompf, and M. Odersky. Foundations of path-dependent types. In OOPSLA, 2014.

Digital Library

[8]

A. W. Appel and D. A. McAllester. An indexed model of recursive types for foundational proof-carrying code. ACM Trans. Program. Lang. Syst., 23(5):657–683, 2001.

Digital Library

[9]

D. Aspinall and A. Compagnoni. Subtyping dependent types. Theoretical Computer Science, 266(1):273–309, 2001.

Digital Library

[10]

B. E. Aydemir, A. Bohannon, M. Fairbairn, J. N. Foster, B. C. Pierce, P. Sewell, D. Vytiniotis, G. Washburn, S. Weirich, and S. Zdancewic. Mechanized metatheory for the masses: The PoplMark Challenge. In TPHOLs, 2005.

Digital Library

[11]

H. P. Barendregt. Handbook of logic in computer science. chapter Lambda Calculi with Types. Oxford University Press, 1992.

Digital Library

[12]

N. R. Cameron, J. Noble, and T. Wrigstad. Tribal ownership. In OOPSLA, 2010.

Digital Library

[13]

L. Cardelli, S. Martini, J. C. Mitchell, and A. Scedrov. An extension of system F with subtyping. Inf. Comput., 109(1/2):4–56, 1994.

Digital Library

[14]

A. Charguéraud. The locally nameless representation. J. Autom. Reasoning, 49(3):363–408, 2012.

[15]

D. Clarke, S. Drossopoulou, J. Noble, and T. Wrigstad. Tribe: a simple virtual class calculus. In AOSD, 2007.

Digital Library

[16]

V. Cremet, F. Garillot, S. Lenglet, and M. Odersky. A core calculus for Scala type checking. In MFCS, 2006.

Digital Library

[17]

N. A. Danielsson. Operational semantics using the partiality monad. In ICFP, 2012.

Digital Library

[18]

O. Danvy and J. Johannsen. Inter-deriving semantic artifacts for object-oriented programming. J. Comput. Syst. Sci., 76(5):302–323, 2010.

Digital Library

[19]

O. Danvy, K. Millikin, J. Munk, and I. Zerny. On inter-deriving smallstep and big-step semantics: A case study for storeless call-by-need evaluation. Theor. Comput. Sci., 435:21–42, 2012.

Digital Library

[20]

D. Dreyer and A. Rossberg. Mixin’ up the ML module system. In ICFP, 2008.

Digital Library

[21]

E. Ernst. Family polymorphism. In ECOOP, 2001.

Digital Library

[22]

E. Ernst. Higher-order hierarchies. In ECOOP, 2003.

[23]

E. Ernst, K. Ostermann, and W. R. Cook. A virtual class calculus. In POPL, 2006.

Digital Library

[24]

M. Flatt. Binding as sets of scopes. In POPL, 2016.

Digital Library

[25]

V. Gasiunas, M. Mezini, and K. Ostermann. Dependent classes. In OOPSLA, 2007.

Digital Library

[26]

C. A. Gunter and D. Rémy. A proof-theoretic assesment of runtime type errors. Technical Report Technical Memo 11261-921230-43TM, AT&T Bell Laboratories, 1993.

[27]

R. Harper. A simplified account of polymorphic references. Information Processing Letters, 51(4):201 – 206, 1994.

Digital Library

[28]

R. Harper. A simplified account of polymorphic references – followup. https://www.cs.cmu.edu/~rwh/papers/refs/ipl-followup.pdf, 1995.

[29]

R. Harper and M. Lillibridge. A type-theoretic approach to higherorder modules with sharing. In POPL, 1994.

Digital Library

[30]

G. Kahn. Natural semantics. In STACS, 1987.

Digital Library

[31]

D. K. Lee, K. Crary, and R. Harper. Towards a mechanized metatheory of standard ML. In POPL, pages 173–184. ACM, 2007.

Digital Library

[32]

X. Leroy. Manifest types, modules and separate compilation. In POPL, 1994.

Digital Library

[33]

X. Leroy and H. Grall. Coinductive big-step operational semantics. Inf. Comput., 207(2):284–304, 2009.

Digital Library

[34]

D. Macqueen. Using dependent types to express modular structure. In POPL, 1986.

Digital Library

[35]

J. Midtgaard, N. Ramsey, and B. Larsen. Engineering definitional interpreters. In PPDP, 2013.

Digital Library

[36]

E. Moggi. Notions of computation and monads. Inf. Comput., 93(1):55–92, 1991.

Digital Library

[37]

A. Moors, F. Piessens, and M. Odersky. Safe type-level abstraction in Scala. In FOOL, 2008.

[38]

P. Neron, A. P. Tolmach, E. Visser, and G. Wachsmuth. A theory of name resolution. In ESOP, 2015.

[39]

N. Nystrom, S. Chong, and A. C. Myers. Scalable extensibility via nested inheritance. In OOPSLA, 2004.

Digital Library

[40]

M. Odersky, V. Cremet, C. Röckl, and M. Zenger. A nominal theory of objects with dependent types. In ECOOP, 2003.

[41]

S. Owens, M. O. Myreen, R. Kumar, and Y. K. Tan. Functional bigstep semantics. In ESOP, 2016.

[42]

B. C. Pierce. Programming with Intersection Types and Bounded Polymorphism. PhD thesis, Carnegie Mellon University, December 1991.

Digital Library

[43]

G. D. Plotkin. A structural approach to operational semantics. J. Log. Algebr. Program., 60-61:17–139, 2004.

[44]

J. C. Reynolds. Definitional interpreters for higher-order programming languages. Higher-Order and Symbolic Computation, 11(4):363–397, 1998.

Digital Library

[45]

T. Rompf and N. Amin. Type soundness for dependent object types. In OOPSLA, 2016.

Digital Library

[46]

A. Rossberg. 1ML - core and modules united (f-ing first-class modules). In ICFP, 2015.

Digital Library

[47]

A. Rossberg, C. V. Russo, and D. Dreyer. F-ing modules. J. Funct. Program., 24(5):529–607, 2014.

[48]

D. S. Scott. Domains for denotational semantics. In Automata, Languages and Programming, 1982.

Digital Library

[49]

J. Siek. Type safety in three easy lemmas. http://siek.blogspot. com/2013/05/type-safety-in-three-easy-lemmas.html, 2013.

[50]

M. Steffen. Polarized higher-order subtyping. PhD thesis, University of Erlangen-Nuremberg, 1997.

[51]

M. Tofte. Operational Semantics and Polymorphic Type Inference. PhD thesis, 1988.

[52]

M. VanInwegen. The Machine-Assisted Proof of Programming Language Properties. PhD thesis, May 1996.

Digital Library

[53]

D. von Oheimb. Re: Subject reduction fails in java. http://www.seas.upenn.edu/~sweirich/types/archive/1997-98/ msg00452.html, 1998.

[54]

P. Wadler. The essence of functional programming. In POPL, pages 1–14. ACM Press, 1992.

Digital Library

[55]

P. Wadler. Propositions as types. Presentation at Strange Loop, 2015.

[56]

A. K. Wright and M. Felleisen. A syntactic approach to type soundness. Inf. Comput., 115(1):38–94, 1994.

Digital Library

Cited By

Rajić GSruk V(2024)Definitions and Computational Properties of OCL: A Systematic ReviewIEEE Access10.1109/ACCESS.2024.342886512(99704-99738)Online publication date: 2024
https://doi.org/10.1109/ACCESS.2024.3428865
Zhou LZhou YOliveira B(2023)Recursive Subtyping for AllProceedings of the ACM on Programming Languages10.1145/35712417:POPL(1396-1425)Online publication date: 11-Jan-2023
https://dl.acm.org/doi/10.1145/3571241
Blaudeau CLiu F(2022)A conceptual framework for safe object initialization: a principled and mechanized soundness proof of the Celsius modelProceedings of the ACM on Programming Languages10.1145/35633146:OOPSLA2(729-757)Online publication date: 31-Oct-2022
https://dl.acm.org/doi/10.1145/3563314
Show More Cited By

Index Terms

Type soundness proofs with definitional interpreters
1. Software and its engineering
  1. Software notations and tools
    1. General programming languages
      1. Language features

Recommendations

Type soundness proofs with definitional interpreters
POPL '17: Proceedings of the 44th ACM SIGPLAN Symposium on Principles of Programming Languages

While type soundness proofs are taught in every graduate PL class, the gap between realistic languages and what is accessible to formal proofs is large. In the case of Scala, it has been shown that its formal model, the Dependent Object Types (DOT) ...
Intrinsically-typed definitional interpreters for imperative languages

A definitional interpreter defines the semantics of an object language in terms of the (well-known) semantics of a host language, enabling understanding and validation of the semantics through execution. Combining a definitional interpreter with a ...
Type soundness for dependent object types (DOT)
OOPSLA '16

Scala’s type system unifies aspects of ML modules, object- oriented, and functional programming. The Dependent Object Types (DOT) family of calculi has been proposed as a new theoretic foundation for Scala and similar expressive languages. ...

Comments

Please enable JavaScript to view thecomments powered by Disqus.

Information & Contributors

Information

Published In

cover image ACM SIGPLAN Notices

ACM SIGPLAN Notices Volume 52, Issue 1

POPL '17

January 2017

901 pages

ISSN:0362-1340

EISSN:1558-1160

DOI:10.1145/3093333

Editor:
Matthew Fluet

Issue’s Table of Contents

POPL '17: Proceedings of the 44th ACM SIGPLAN Symposium on Principles of Programming Languages
January 2017
901 pages
ISBN:9781450346603
DOI:10.1145/3009837
General Chair:
Giuseppe Castagna
Paris Diderot University, France / CNRS, France
,
Program Chair:
Andrew D. Gordon
Microsoft Research, UK / University of Edinburgh, UK

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].

Publisher

Association for Computing Machinery

New York, NY, United States

Publication History

Published: 01 January 2017

Published in SIGPLAN Volume 52, Issue 1

Check for updates

Author Tags

Qualifiers

Research-article

Funding Sources

National Science Foundation

Contributors

Other Metrics

View Article Metrics

Bibliometrics & Citations

Bibliometrics

Article Metrics

39
Total Citations
View Citations
1,097
Total Downloads

Downloads (Last 12 months)165
Downloads (Last 6 weeks)24

Reflects downloads up to 05 Jan 2025

Other Metrics

View Author Metrics

Citations

Cited By

Rajić GSruk V(2024)Definitions and Computational Properties of OCL: A Systematic ReviewIEEE Access10.1109/ACCESS.2024.342886512(99704-99738)Online publication date: 2024
https://doi.org/10.1109/ACCESS.2024.3428865
Zhou LZhou YOliveira B(2023)Recursive Subtyping for AllProceedings of the ACM on Programming Languages10.1145/35712417:POPL(1396-1425)Online publication date: 11-Jan-2023
https://dl.acm.org/doi/10.1145/3571241
Blaudeau CLiu F(2022)A conceptual framework for safe object initialization: a principled and mechanized soundness proof of the Celsius modelProceedings of the ACM on Programming Languages10.1145/35633146:OOPSLA2(729-757)Online publication date: 31-Oct-2022
https://dl.acm.org/doi/10.1145/3563314
Liu FLhoták OXing EPham NRichard-Foy JDoeraene S(2021)Safe object initialization, abstractlyProceedings of the 12th ACM SIGPLAN International Symposium on Scala10.1145/3486610.3486895(33-43)Online publication date: 17-Oct-2021
https://dl.acm.org/doi/10.1145/3486610.3486895
Bao YWei GBračevac OJiang YHe QRompf T(2021)Reachability types: tracking aliasing and separation in higher-order functional programsProceedings of the ACM on Programming Languages10.1145/34855165:OOPSLA(1-32)Online publication date: 15-Oct-2021
https://dl.acm.org/doi/10.1145/3485516
Sasdelli FAmaro MCardoso EFeitosa SRibeiro RCavalcante EDantas FBatista T(2020)Syntax vs SemanticsProceedings of the 24th Brazilian Symposium on Context-Oriented Programming and Advanced Modularity10.1145/3427081.3427083(9-16)Online publication date: 19-Oct-2020
https://dl.acm.org/doi/10.1145/3427081.3427083
Feitosa SRibeiro RCavalcante EDantas FBatista T(2020)An Intrinsically-Typed Solution for the List-Machine BenchmarkProceedings of the 24th Brazilian Symposium on Context-Oriented Programming and Advanced Modularity10.1145/3427081.3427082(1-8)Online publication date: 19-Oct-2020
https://dl.acm.org/doi/10.1145/3427081.3427082
Sergey INagaraj VJohannsen JKumar ATrunov AHao K(2019)Safer smart contract programming with ScillaProceedings of the ACM on Programming Languages10.1145/33606113:OOPSLA(1-30)Online publication date: 10-Oct-2019
https://dl.acm.org/doi/10.1145/3360611
Essertel GWei GRompf T(2019)Precise reasoning with structured time, structured heaps, and collective operationsProceedings of the ACM on Programming Languages10.1145/33605833:OOPSLA(1-30)Online publication date: 10-Oct-2019
https://dl.acm.org/doi/10.1145/3360583
Feitosa SMena ARibeiro RBois A(2019)An Inherently-Typed Formalization for Featherweight JavaProceedings of the XXIII Brazilian Symposium on Programming Languages10.1145/3355378.3355385(11-18)Online publication date: 23-Sep-2019
https://dl.acm.org/doi/10.1145/3355378.3355385
Show More Cited By

View Options

View options

PDF

View or Download as a PDF file.

eReader

View online with eReader.

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

Media

Figures

Other

Tables

View Issue’s Table of Contents