Continuation-passing style, defunctionalization, accumulations, and associativity

Context: 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. Inquiry: Buried within all those applications there is usually a hidden appeal to the algebraic property of associativity. Approach: Our purpose in this paper is to entice associativity out of the shadows and into the limelight. Knowledge: 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. Grounding: We replay developments of these programs through a series of program transformations and data refinements, justified by equational reasoning. Importance: 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.