An Exercise in Polytypic Program Derivation: repmin

A program derivation is said to be polytypic if some of its parameters are data types. The repmin problem is to replace all elements of a tree of numbers by the minimum element, making only a single pass over the original tree. Here we present a polytypic derivation for that problem. The derivation has an unusual feature: when interpreted in the category of relations, the resulting program is the well-known cyclic logic program, and when interpreted in the category of functions, it is the well-known higher-order functional solution. 1 Motivation Suppose I were to show you a derivation of a shortest path algorithm, and my whole presentation was in terms of numbers, addition and minimum. Undoubtedly some of you would get up and point out that by abstracting over the operations and recording their algebraic properties, I could have derived a whole class of algorithms instead of one particular program. Indeed, such abstraction over operations is now commonly accepted as one of the hallmarks of good program derivation: generality is increased, and the presentation is not cluttered by irrelevant particularities. The purpose of this paper is to persuade you that similar beneets can be reaped by abstracting over type constructors. For concreteness, consider the well-known repmin problem 3]: it is required to change a tree t into a tree of the same shape as t, but with all elements replaced by the minimum element. There is a simple program to do that: rst nd the minimum, and then do the replacement. That simple program does however consist of two passes over the original tree t. The challenge is, therefore, to give a program that achieves the same eeect in a single pass.