Multi-Level Languages are Generalized Arrows

Multi-level languages and Arrows both facilitate metaprogramming, the act of writing a program which generates a program. The arr function required of all Arrows turns arbitrary host language expressions into guest language expressions; because of this, Arrows may be used for metaprogramming only when the guest language is a superset of the host language. This restriction is also present in multi-level languages which offer unlimited cross-level persistence. The converse restriction, that the host language is a subset of the guest language, is imposed for multi-stage languages – those multi-level languages with a run construct. This paper introduces generalized arrows and proves that they generalize Arrows in the following sense: every Arrow in a programming language arises from a generalized arrow with that language’s term category as its codomain (Theorem 4.5.4). Generalized arrows impose no containment relationship between the guest language and host language; they facilitate heterogeneous metaprogramming. The category having all generalized arrows as its morphisms and the category having all multi-level languages as its morphisms are isomorphic categories (Theorem 4.7.8). This is proven formally in Coq, and the proof is offered as justification for the assertion that multi-level languages are generalized arrows. Combined with the existence of a particular kind of retraction (Definition 4.8.3) in the host language, this proof can be used to define an invertible translation from two-level terms to one-level terms parameterized by a generalized arrow instance. This is ergonomically significant: it lets guest language providers write generalized arrow instances while the users of those guest languages write multi-level terms. This is beneficial because implementing a generalized arrow instance is easier than modifying a compiler, whereas writing two-level terms is easier than manipulating generalized arrow terms. Haskell is one example of a host language with the necessary kind of retraction. A modified version of GHC with multi-level terms is offered as a proof of concept; the Haskell extraction of the Coq proofs mentioned above has been compiled into this modified GHC, and is made available as a new flattening pass.