  Liquid typing provides a decidable refinement inference mechanism that is
convenient but subject to two major issues: (1) inference is global and
requires top-level annotations, making it unsuitable for inference of modular
code components and prohibiting its applicability to library code, and (2)
inference failure results in obscure error messages. These difficulties
seriously hamper the migration of existing code to use refinements. This paper
shows that gradual liquid type inference---a novel combination of liquid
inference and gradual refinement types---addresses both issues. Gradual
refinement types, which support imprecise predicates that are optimistically
interpreted, can be used in argument positions to constrain liquid inference so
that the global inference process e effectively infers modular specifications
usable for library components. Dually, when gradual refinements appear as the
result of inference, they signal an inconsistency in the use of static
refinements. Because liquid refinements are drawn from a nite set of
predicates, in gradual liquid type inference we can enumerate the safe
concretizations of each imprecise refinement, i.e. the static refinements that
justify why a program is gradually well-typed. This enumeration is useful for
static liquid type error explanation, since the safe concretizations exhibit
all the potential inconsistencies that lead to static type errors. We develop
the theory of gradual liquid type inference and explore its pragmatics in the
setting of Liquid Haskell.
