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Gradual type theory

Part of: Gradual Typing

Published online by Cambridge University Press:  14 October 2021

MAX S. NEW Open the ORCID record for MAX S. NEW [Opens in a new window] ,
DANIEL R. LICATA  and
AMAL AHMED
MAX S. NEW
Affiliation:
Khoury College of Computer Sciences, Northeastern University, Boston, MA 02115, USA (e-mail: maxsnew@umich.edu)
DANIEL R. LICATA
Affiliation:
Mathematics and Computer Science, Wesleyan University, Middletown, CT 06459, USA (e-mail: dlicata@wesleyan.edu)
AMAL AHMED
Affiliation:
Khoury College of Computer Sciences, Northeastern University, Boston, MA 02115, USA (e-mail: amal@ccs.neu.edu)
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Abstract

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Gradually typed languages are designed to support both dynamically typed and statically typed programming styles while preserving the benefits of each. Sound gradually typed languages dynamically check types at runtime at the boundary between statically typed and dynamically typed modules. However, there is much disagreement in the gradual typing literature over how to enforce complex types such as tuples, lists, functions and objects. In this paper, we propose a new perspective on the design of runtime gradual type enforcement: runtime type casts exist precisely to ensure the correctness of certain type-based refactorings and optimizations. For instance, for simple types, a language designer might desire that beta-eta equality is valid. We show that this perspective is useful by demonstrating that a cast semantics can be derived from beta-eta equality. We do this by providing an axiomatic account program equivalence in a gradual cast calculus in a logic we call gradual type theory (GTT). Based on Levy’s call-by-push-value, GTT allows us to axiomatize both call-by-value and call-by-name gradual languages. We then show that we can derive the behavior of casts for simple types from the corresponding eta equality principle and the assumption that the language satisfies a property called graduality, also known as the dynamic gradual guarantee. Since we can derive the semantics from the assumption of eta equality, we also receive a useful contrapositive: any observably different cast semantics that satisfies graduality must violate the eta equality. We show the consistency and applicability of our axiomatic theory by proving that a contract-based implementation using the lazy cast semantics gives a logical relations model of our type theory, where equivalence in GTT implies contextual equivalence of the programs. Since GTT also axiomatizes the dynamic gradual guarantee, our model also establishes this central theorem of gradual typing. The model is parameterized by the implementation of the dynamic types, and so gives a family of implementations that validate type-based optimization and the gradual guarantee.

Type
Research Article
Information
Journal of Functional Programming , Volume 31 , 2021 , e21
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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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