Generic bidirectional typing for dependent type theories

TL;DR

This paper presents a generic framework for bidirectional typing applicable to a wide range of dependent type theories, providing formal definitions, equivalence proofs, and a practical type-checking algorithm.

Contribution

It introduces a theory-independent approach to bidirectional typing for dependent type theories, including formal definitions, equivalence proofs, and a practical implementation.

Findings

Proved equivalence of declarative and bidirectional systems across theories

Established decidability of bidirectional typing for normalizing theories

Developed and implemented a generic type-checking algorithm

Abstract

Bidirectional typing is a discipline in which the typing judgment is decomposed explicitly into inference and checking modes, allowing to control the flow of type information in typing rules and to specify algorithmically how they should be used. Bidirectional typing has been fruitfully studied and bidirectional systems have been developed for many type theories. However, the formal development of bidirectional typing has until now been kept confined to specific theories, with general guidelines remaining informal. In this work, we give a generic account of bidirectional typing for a general class of dependent type theories. This is done by first giving a general definition of type theories (or equivalently, a logical framework), for which we define declarative and bidirectional type systems. We then show, in a theory-independent fashion, that the two systems are equivalent. Finally, we establish the decidability of bidirectional typing for normalizing theories, yielding a generic type-checking algorithm that has been implemented in a prototype and used in practice with many theories.