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Blame and coercion: Together again for the first time

Published online by Cambridge University Press:  13 October 2021

JEREMY G. SIEK
Affiliation:
Department of Computer Science, Bloomington, Indiana (e-mail: jsiek@indiana.edu)
PETER THIEMANN
Affiliation:
Faculty of Engineering, Freiburg, Germany (e-mail: thiemann@informatik.uni-freiburg.de)
PHILIP WADLER
Affiliation:
School of Informatics, Edinburgh (e-mail: wadler@inf.ed.ac.uk)
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Abstract

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C#, Dart, Pyret, Racket, TypeScript, VB: many recent languages integrate dynamic and static types via gradual typing. We systematically develop four calculi for gradual typing and the relations between them, building on and strengthening previous work. The calculi are as follows: $\lambda{B}$ , based on the blame calculus of Wadler and Findler (2009); $\lambda{C}$ , inspired by the coercion calculus of Henglein (1994); $\lambda{S}$ inspired by the space-efficient calculus of Herman, Tomb, and Flanagan (2006); and $\lambda{T}$ based on the threesome calculus of Siek and Wadler (2010). While $\lambda{B}$ and $\lambda{T}$ are little changed from previous work, $\lambda{C}$ and $\lambda{S}$ are new. Together, $\lambda{B}$ , $\lambda{C}$ , $\lambda{S}$ , and $\lambda{T}$ provide a coherent foundation for design, implementation, and optimization of gradual types. We define translations from $\lambda{B}$ to $\lambda{C}$ , from $\lambda{C}$ to $\lambda{S}$ , and from $\lambda{S}$ to $\lambda{T}$ . Much previous work lacked proofs of correctness or had weak correctness criteria; here we demonstrate the strongest correctness criterion one could hope for, that each of the translations is fully abstract. Each of the calculi reinforces the design of the others: $\lambda{C}$ has a particularly simple definition, and the subtle definition of blame safety for $\lambda{B}$ is justified by the simple definition of blame safety for $\lambda{C}$ . Our calculus $\lambda{S}$ is implementation-ready: the first space-efficient calculus that is both straightforward to implement and easy to understand. We give two applications: first, using full abstraction from $\lambda{C}$ to $\lambda{S}$ to establish an equational theory of coercions; and second, using full abstraction from $\lambda{B}$ to $\lambda{S}$ to easily establish the Fundamental Property of Casts, which required a custom bisimulation and six lemmas in earlier work.

Type
Research Article
Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author(s), 2021. Published by Cambridge University Press

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