A Low-Level Look at A-Normal Form

A-normal form (ANF) is a widely studied intermediate form in which local control and data flow is made explicit in syntax, and a normal form in which many programs with equivalent control-flow graphs have a single normal syntactic representation. However, ANF is difficult to implement effectively and, as we formalize, difficult to extend with new lexically scoped constructs such as scoped region-based allocation. The problem, as has often been observed, is that normalization of commuting conversions is hard.

This traditional view of ANF that normalizing commuting conversions is hard, found in formal models and informed by high-level calculi, is wrong. By studying the low-level intensional aspects of ANF, we can derive a normal form in which normalizing commuting conversion is easy, does not require join points, or code duplication, or renormalization after inlining, and is easily extended with new lexically scoped effects. We formalize the connection between ANF and monadic form and their intensional properties, derive an imperative ANF, and design a compiler pipeline from an untyped 𝜆-calculus with scoped regions, to monadic form, to a low-level imperative monadic form in which A-normalization is trivial and safe for regions. We prove that any such compiler preserves, or optimizes, stack and memory behaviour compared to ANF. Our formalization reconstructs and systematizes pragmatic choices found in practice, including current productionready compilers.

The main take-away from this work is that, in general, monadic form should be preferred over ANF, and A-normalization should only be done in a low-level imperative intermediate form. This maximizes the advantages of each form, and avoids all the standard problems with ANF.