Complete and Easy Bidirectional Typechecking for Higher-Rank Polymorphism

TL;DR

This paper presents a simple, sound, and complete bidirectional typechecking algorithm for higher-rank polymorphism, grounded in proof theory, improving scalability and error reporting in expressive type systems.

Contribution

It introduces a declarative, bidirectional approach to higher-rank polymorphism based on proof theory, with a straightforward and well-behaved implementation algorithm.

Findings

The calculus supports eta-reduction and predictable annotations.

The algorithm is both sound and complete.

The approach enhances scalability and error reporting.

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 eta-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.