While the quality of the results of TEX's mathematical formula layout algorithm is convincing, its original description is hard to understand since it is presented as an imperative program with complex control flow and destructive manipulations of the data structures representing formulae. In this paper, we present a re-implementation of TEX's formula layout algorithm in the functional language SML, thereby providing a more readable description of the algorithm, extracted from the monolithical TEX system.

Arrays are probably the most widely used data structure in imperative programming languages, yet functional languages typically only support arrays in a limited manner, or prohibit them entirely. This is not too surprising, since most other mutable data structures, such as trees, have elegant immutable analogues in the functional world, whereas arrays do not. Previous attempts at addressing the problem have suffered from one of three weaknesses, either that they don't support arrays as a persistent data structure (unlike the functional analogues of other imperative data structures), or that the range of operations is too restrictive to support some common array algorithms efficiently, or that they have performance problems. Our technique provides arrays as a true functional analogue of imperative arrays with the properties that functional programmers have come to expect from their data structures. To efficiently support array algorithms from the imperative world, we provide O(1) operations for single-threaded array use. Fully persistent array use can also be provided at O(1) amortized cost, provided that the algorithm satisfies a simple requirement as to uniformity of access. For those algorithms which do not access the array uniformly or single-threadedly, array reads or updates take at most O(log n) amortized time, where n is the size of the array. Experimental results indicate that the overheads of our technique are acceptable in practice for many applications.

Functional-language compilers often perform optimizations based on beta and delta reduction. To avoid speculative optimizations that can blow up the code size, we might wish to use only shrinking reduction rules guaranteed to make the program smaller: these include dead-variable elimination, constant folding, and a restricted beta rule that inlines only functions that are called just once. The restricted beta rule leads to a shrinking rewrite system that has not previously been studied. We show some efficient normalization algorithms that are immediately useful in optimizing compilers; and we give a confluence proof for our system, showing that the choice of normalization algorithm does not affect final code quality.

Nicholas Pippenger has recently given a problem that, under two simple restrictions, can be solved in linear time by an impure Lisp program, but requires Ω(n log n) steps to be solved by any eager pure Lisp program. By showing how to solve the problem in linear time with a lazy functional program, we demonstrate that – for some problems at least – lazy evaluators are strictly more powerful than eager ones.

Almost every programmer has faced the problem of representing a tree together with a subtree that is the focus of attention, where that focus may move left, right, up or down the tree. The Zipper is Huet's nifty name for a nifty data structure which fulfills this need. I wish I had known of it when I faced this task, because the solution I came up with was not quite so efficient or elegant as the Zipper.