OutsideIn(X)Modular type inference with local assumptions

Abstract

Advanced type system features, such as GADTs, type classes, and type families have proven to be invaluable language extensions for ensuring data invariants and program correctness among others. Unfortunately, they pose a tough problem for type inference, because they introduce local type assumptions.In this article we present a novel constraint-based type inference approach for local type assumptions. Our system, called OutsideIn(X), is parameterised over the particular underlying constraint domain X, in the same way as HM(X). This stratification allows us to use a common metatheory and inference algorithm.Going beyond the general framework, we also give a particular constraint solver for X = type classes + GADTs + type families, a non-trivial challenge in its own right.

Skalpel: A constraint-based type error slicer for Standard ML

“…Variants of this system have been proposed over the years, for example in [55,56]. Similarly, in order to deal with Haskell's rich type system, Vytiniotis et al developed the parameterized OutsideIn(X) constraint-based type inference algorithm [59]. Using these let-constraints Pottier and Rémy "achieve the desired separation between constraint generation, on the one hand, and constraint solving and simplification, on the other hand, without compromising efficiency" [48].…”

Visible Type Application

“…The advantage of studying a syntax-directed system such as System C is that doing so separates concerns: System C fixes the structure of the typing derivation (and of any implementation) while leaving monotype-guessing as a separate problem. Algorithm W deduces the monotypes via unification, but a constraint-based approach [25,27] would also work.…”

Type Class Instances for Type-Level Lambdas in Haskell

“…In the this section, we will give a short description of the type checking and constraint solving for Haskell that is documented in (Vytiniotis et al 2011). Type checking is divided into two phases: first type inference, where constraints are generated, then these constraints are solved.…”

“…Using the notation from In (Vytiniotis et al 2011), → is a typing judgment with an input tuple ᾱ, ϕ, Qg, Qw and an output tuple ᾱ , ϕ , Q g , Q w . Here, Q is the top-level environment (containing, for example, all defined class instances), Qw are the wanted constraints and Qg are the given constraints.ᾱ is a set of touchable variables (the variables which may be substituted) and ϕ is a set of substitutions.…”

“…The → judgment is applied until a fixed-point is found. In (Vytiniotis et al 2011) a number of cases for → are described, Figure 20 adds a new case to deal with typelevel lambdas in instances. We have also removed a case which is not explicitly given in (Vytiniotis et al 2011), namely the rule that would automatically decompose f a ∼ g b into f ∼ g and a ∼ b.…”