Efficient type inference using ranked type variables

Abstract

The Damas-Milner type inference algorithm (commonly known as algorithm W) is at the core of all ML type checkers. Although the algorithm is known to have poor worst-case behavior [8], in practice well-engineered type checkers will run in approximately linear time. To achieve this efficiency, implementations need to improve on algorithm W's method of scanning the complete type environment to determine whether a type variable can be generalized at a let binding. Following a suggestion of Damas, most ML type checkers use an alternative method based on ranking unification variables to track their position in the type environment.Here we formalize two such ranking systems, one based on lambda depth (used in the SML/NJ compiler), and the other based on let depth (used in OCaml, for instance). Each of these systems is formalized both with and without the value restriction, and they are proved correct relative to the classic algorithm W. Our formalizations of the various algorithms use simple abstract machines that are similar to those derived from small-step evaluation semantics.

Defunctionalized interpreters for programming languages

“…(If the initial program is block-structured, just lambda-lift it [39].) This combination of transformations can also be used for deriving or relating programs [24,26,60] and is used today, e.g., for web programming [32] and for type inference [43].…”

Hindley-milner elaboration in applicative style

“…Let us just emphasize that it is is modular: (a) at the lowest layer lies Tarjan's efficient union-find algorithm [23]; (b) above it, one finds Huet's first-order unification algorithm [9]; (c) then comes the treatment of generalization and instantiation, which exploits Rémy's integer ranks [20,12,11] to efficiently determine which type variables must be generalized; (d) the last layer interprets the syntax of constraints. The solver meets McAllester asymptotic time bound [12]: under the assumption that all of the type schemes that are ever constructed have bounded size, its time complexity is O(nk), where n is the size of the constraint and k is the left-nesting depth of CLet nodes.…”

Hindley-milner elaboration in applicative style

“…We lack space to describe the implementation of the low-level solver, and it is, anyway, beside the point of the paper. Let us just emphasize that it is is modular: (a) at the lowest layer lies Tarjan's efficient union-find algorithm [23]; (b) above it, one finds Huet's first-order unification algorithm [9]; (c) then comes the treatment of generalization and instantiation, which exploits Rémy's integer ranks [20,12,11] to efficiently determine which type variables must be generalized; (d) the last layer interprets the syntax of constraints. The solver meets McAllester asymptotic time bound [12]: under the assumption that all of the type schemes that are ever constructed have bounded size, its time complexity is O(nk), where n is the size of the constraint and k is the left-nesting depth of CLet nodes.…”