Type Inference for Nakano’s Modality

Around fifteen years ago, Nakano introduced an extension to a system of recursive types for the Lambda Calculus consisting of a unary type constructor, or modality, guarding the occurrences of recursive references. This modest extension afforded a powerful result: the guarantee of a particular kind of termination (specifically, head normalisation). That is, programs typeable in this system are guaranteed to produce output. Since then, much research has been done to understand the semantics of this modality, and utilise it in more sophisticated type systems. Notable contributions to this effort include: the work of Birkedal et al. and Benton and Krishnaswami, who study semantic models induced by the modality; research by Appel at al. and Pottier who employ the modality in typed intermediate representations of programs; and Atkey and McBride’s work on co-programming for infinite data. While some of this work explicitly addresses the possibility of type checking (e.g., that of Pottier), to the best of our knowledge type inference is still an unsolved problem. The lack of a type inference procedure presents a significant barrier to the wider adoption of Nakano’s technique in practice. In this paper, we describe a (conservative) extension to Nakano’s system which allows us to develop a unification-based algorithm for deciding whether a term is typeable by constructing a suitably general type.