Roughly, a refinement type system is an extra layer of precision, enforced through subtyping, added onto an existing type system. A base type is decomposed into a set of base refinements, each of which denotes a subset of the values belonging to the base type. A subtyping relation respecting set inclusion can then be defined among the refinements of the base type. These subtyping relations can be lifted onto a subtyping relation for compound types using a standard arrow subtyping rule.
Extra type-checking precision sounds great, but what in practical terms does this precision look like? Freeman and Pfenning’s ’92 paper Refinement Types for ML proposes extending ML’s type definition language with constructs for decomposing a discriminated union type into a lattice of subtypes. For example, it allows the decomposition of a list type into a lattice including base refinements for empty lists, non-empty lists, and singletons. Those with experience in functional programming will realize this alleviates the dreaded and inescapable “non-exhaustive pattern match” warning, which tends to crop up in situations where the programmer understands that an exhaustive pattern match is not necessary.
In the late 90’s Xi and Pfenning advanced the state of refinement types by introducing a dependent refinement type system, implemented as a tool called Dependent ML. Their approach identifies a base refinement using a tuple of terms drawn from some computationally tractable constraint language called an index language. A list datatype can then be refined with a term of the linear integer arithmetic index language, denoting the subset of all lists having a specific length. One list refinement is then considered a subtype of another when a constraint solver can prove their index terms equal. Vazou et. al.’s recent project Liquid Haskell is another dependent refinement type system which decides subtyping among base types by invoking an SMT solver under a context-dependent set of constraints. It differs significantly from Dependent ML in that it refines base types with certain well-behaved program terms rather than indices.