Continuation−Passing Style‚ Defunctionalization‚ Accumulations‚ and Associativity

Abstract

Reynolds showed us how to use continuation-passing style and defunctionalization to transform a recursive interpreter for a language into an abstract machine for programs in that language. The same techniques explain other programming tricks, including zippers and accumulating parameters. Buried within all those applications there is usually a hidden appeal to the algebraic property of associativity. Our purpose in this paper is to entice associativity out of the shadows and into the limelight. We revisit some well-known applications (factorial, fast reverse, tree flattening, and a compiler for a simple expression language) to spotlight their dependence on associativity. We replay developments of these programs through a series of program transformations and data refinements, justified by equational reasoning. Understanding the crucial role played by associativity clarifies when continuation-passing style and defunctionalization can help and when they cannot, and may prompt other applications of these techniques.