Dynamic Type Inference for Gradual Hindley--Milner Typing

TL;DR

This paper introduces a dynamic type inference approach for gradually typed languages, ensuring soundness, completeness, and divergence avoidance by deferring type variable instantiation to runtime, and proves the gradual guarantee.

Contribution

It formalizes a new blame calculus with deferred type variable instantiation and integrates dynamic type inference into its semantics, improving correctness and divergence handling.

Findings

The DTI-based semantics is sound and complete with respect to fully instantiated terms.

The approach avoids divergence caused by static type variable instantiation.

The paper proves the gradual guarantee for the ITGL language.

Abstract

Garcia and Cimini study a type inference problem for the ITGL, an implicitly and gradually typed language with let-polymorphism, and develop a sound and complete inference algorithm for it. Soundness and completeness mean that, if the algorithm succeeds, the input term can be translated to a well-typed term of an explicitly typed blame calculus by cast insertion and vice versa. However, in general, there are many possible translations depending on how type variables that were left undecided by static type inference are instantiated with concrete static types. Worse, the translated terms may behave differently---some evaluate to values but others raise blame. In this paper, we propose and formalize a new blame calculus $\lambda^{\textsf{DTI}}_{\textsf{B}}$ that avoids such divergence as an intermediate language for the ITGL. A main idea is to allow a term to contain type variables (that have not been instantiated during static type inference) and defer instantiation of these type variables to run time. We introduce dynamic type inference (DTI) into the semantics of $\lambda^{\textsf{DTI}}_{\textsf{B}}$ so that type variables are instantiated along reduction. The DTI-based semantics not only avoids the divergence described above but also is sound and complete with respect to the semantics of fully instantiated terms in the following sense: if the evaluation of a term succeeds (i.e., terminates with a value) in the DTI-based semantics, then there is a fully instantiated version of the term that also succeeds in the explicitly typed blame calculus and vice versa. Finally, we prove the gradual guarantee, which is an important correctness criterion of a gradually typed language, for the ITGL.