Complete and easy bidirectional typechecking for higher-rank polymorphism

Abstract

Bidirectional typechecking, in which terms either synthesize a type or are checked against a known type, has become popular for its scalability (unlike Damas-Milner type inference, bidirectional typing remains decidable even for very expressive type systems), its error reporting, and its relative ease of implementation. Following design principles from proof theory, bidirectional typing can be applied to many type constructs. The principles underlying a bidirectional approach to polymorphism, however, are less obvious. We give a declarative, bidirectional account of higher-rank polymorphism, grounded in proof theory; this calculus enjoys many properties such as η-reduction and predictability of annotations. We give an algorithm for implementing the declarative system; our algorithm is remarkably simple and well-behaved, despite being both sound and complete.

Visible Type Application

“…Our work is also closely related to recent work on using a bidirectional type system for higher-rank polymorphism by Dunfield and Krishnaswami [10], called DK below. The closest relationship is between their declarative system (Fig.…”

“…Instead, instantiation is done only on demand, such as when a polymorphic function is applied to arguments. Lazy instantiation has been used in (non-HM) type inference before [10] and may be folklore; however this work contains the first proof that it can be used to implement the HM type system.…”

“…The grammar of System HMV appears in type schemes and monotypes; σs contain υs, which then contain τ s. 10 Thus the full form of a type scheme σ can be written as ∀{a}, b. τ , including both a set of generalized variables {a} and a list of specified variables b. Note that order never matters for generalized variables (they are in a set) while order does certainly matter for specified variables (the list specifies their order).…”

“…10 The grammar for System HMV redefines several metavariables. These metavariables then have (slightly) different meanings in different sections of this paper, but disambiguation should be clear from context.…”

“…This declarative type system itself is a novel contribution of this work. (The systems presented in related work [10,21,23] are more similar to SB than to B.) Although System B is bidirectional, it is also declarative.…”