Multi-Stage Programs are Generalized Arrows

The lambda calculus, subject to typing restrictions, provides a syntax for the internal language of cartesian closed categories. This paper establishes a parallel result: staging annotations [? ], subject to named level restrictions, provide a syntax for the internal language of Freyd categories, which are known to be in bijective correspondence with Arrows. The connection is made by interpreting multistage type systems as indexed functors from polynomial categories to their reindexings (Definitions 16 and 17). This result applies only to multi-stage languages which are (1) homogeneous, (2) allow cross-stage persistence and (3) place no restrictions on the use of structural rules in typing derivations. Removing these restrictions and repeating the construction yields generalized arrows, of which Arrows are a particular case. A translation from well-typed multi-stage programs to single-stage GArrow terms is provided. The translation is defined by induction on the structure of the proof that the multi-stage program is well-typed, relying on information encoded in the proof’s use of structural rules (weakening, contraction, exchange, and context associativity). Metalanguage designers can now factor out the syntactic machinery of metaprogramming by providing a single translation from staging syntax into expressions of generalized arrow type. Object language providers need only implement the functions of the generalized arrow type class in point-free style. Object language users may write metaprograms over these object languages in a point-ful style, using the same binding, scoping, abstraction, and application mechanisms in both the object language and metalanguage. This paper’s principal contributions are the GArrow definition of Figures 2 and 3, the translation in Figure 5 and the category-theoretic semantics of Definition 16. An accompanying Coq proof formalizes the type system, translation procedure, and key theorems. [Copyright notice will appear here once ’preprint’ option is removed.]