A long-standing shortcoming of statically typed functional languages is that type checking does not rule out pattern-matching failures (run-time match exceptions). Refinement types distinguish different values of datatypes; if a program annotated with refinements passes type checking, pattern-matching failures become impossible. Unfortunately, refinement is a monolithic property of a type, exacerbating the difficulty of adding refinement types to nontrivial programs.Gradual typing has explored how to incrementally move between static typing and dynamic typing. We develop a type system of gradual sums that combines refinement with imprecision. Then, we develop a bidirectional version of the type system, which rules out excessive imprecision, and give a type-directed translation to a target language with explicit casts. We prove that the static sublanguage cannot have match failures, that a well-typed program remains well-typed if its type annotations are made less precise, and that making annotations less precise causes target programs to fail later. Several of these results correspond to criteria for gradual typing given by Siek et al. (2015).

Abstracting Gradual Typing (AGT) is a systematic approach to designing gradually-typed languages. Languages developed using AGT automatically satisfy the formal semantic criteria for gradual languages identified by Siek et al. Nonetheless, vanilla AGT semantics can still have important shortcomings. First, a gradual language's runtime checks should preserve the space-efficiency guarantees inherent to the underlying static and dynamic languages. To the contrary, the default operational semantics of AGT break proper tail calls. Second, a gradual language's runtime checks should enforce basic modular type-based invariants expected from the static type discipline. To the contrary, the default operational semantics of AGT may fail to enforce some invariants in surprising ways. We demonstrate this in the GTFL ≲ language of Garcia et al. This paper addresses both problems at once by refining the theory underlying AGT's dynamic checks. Garcia et al. observe that AGT involves two abstractions of static types: one for the static semantics and one for the dynamic semantics. We recast the latter as an abstract interpretation of subtyping itself, while gradual types still abstract static types. Then we show how forward-completeness (Giacobazzi and Quintarelli) is key to supporting both space-efficient execution and reliable runtime type enforcement.

A long-standing shortcoming of statically typed functional languages is that type checking does not rule out pattern-matching failures (run-time match exceptions). Refinement types distinguish different values of datatypes; if a program annotated with refinements passes type checking, pattern-matching failures become impossible. Unfortunately, refinement is a monolithic property of a type, exacerbating the difficulty of adding refinement types to nontrivial programs.Gradual typing has explored how to incrementally move between static typing and dynamic typing. We develop a type system of gradual sums that combines refinement with imprecision. Then, we develop a bidirectional version of the type system, which rules out excessive imprecision, and give a type-directed translation to a target language with explicit casts. We prove that the static sublanguage cannot have match failures, that a well-typed program remains well-typed if its type annotations are made less precise, and that making annotations less precise causes target programs to fail later. Several of these results correspond to criteria for gradual typing given by Siek et al. (2015).

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