A representation changer is a function that converts a concrete representation of an abstract value into a different concrete representation of that value. Many useful functions can be recognised as representation changers; examples include compilers and arithmetic functions such as addition and multiplication. Functions that can be specified as the right inverse of other functions are special cases of representation changers. In recent years, a number of authors have used a relational calculus to derive representation changers from their specifications. In this paper, we show that the generality of relations is not essential, and representation changers can be derived within the more basic setting of functional programming. We illustrate our point by deriving a carry-save adder and a base-converter, two functions which have previously been derived relationally.

The tree-drawing problem is to produce a ‘tidy’ mapping from elements of a tree to points in the plane. In this paper, we derive an efficient algorithm for producing tidy drawings of trees. The specification, the starting point for the derivations, consists of a collection of intuitively appealing criteria satisfied by tidy drawings. The derivation shows constructively that these criteria completely determine the drawing. Indeed, the criteria completely determine a simple but inefficient algorithm for drawing a tree, which can be transformed into an efficient algorithm using just standard techniques and a small number of inventive steps.

The algorithm consists of an upwards accumulation followed by a downwards accumulation on the tree, and is further evidence of the utility of these two higher-order tree operations.