Multi-stage Programming with Functors and Monads: Eliminating Abstraction Overhead from Generic Code

Abstract

We use multi-stage programming, monads and Ocaml's advanced module system to demonstrate how to eliminate all abstraction overhead from generic programs while avoiding any inspection of the resulting code. We demonstrate this clearly with Gaussian Elimination as a representative family of symbolic and numeric algorithms. We parameterize our code to a great extent -over domain, input and permutation matrix representations, determinant and rank tracking, pivoting policies, result types, etc. -at no run-time cost. Because the resulting code is generated just right and not changed afterward, MetaOCaml guarantees that the generated code is well-typed. We further demonstrate that various abstraction parameters (aspects) can be made orthogonal and compositional, even in the presence of name-generation for temporaries, and "interleaving" of aspects. We also show how to encode some domain-specific knowledge so that "clearly wrong" compositions can be rejected at or before generation time, rather than during the compilation or running of the generated code.

Partially-static data as free extension of algebras

“…Combining the techniques in this paper with Carette and Kiselyov's [2005] modular approach by instantiating numeric signatures with free extensions is a promising avenue for future exploration. Yallop [2017] uses several partially-static structures in the staging of an implementation of the Scrap Your Boilerplate generic programming library, including a partially-static structure for monoids that reassociates subexpressions similarly to the free extension presented here.…”

Partial evaluation of Maple

“…The authors were convinced that partial evaluation (and metaprogramming in general [1,2]), should be a very powerful tool when combined with a Computer Algebra System (CAS). Certainly code generation from a CAS had already shown itself [3,4] to be a successful technique.…”