Functional Pearl Research Articles

On Barron and Strachey's Cartesian Product Function

Over forty years ago, David Barron and Christopher Strachey published a startlingly elegant program for the Cartesian product of a list of lists, expressing it with a three nested occurrences of the function we now call <em>foldr</em>. This program is remarkable for its time because of its masterful display of higher-order functions and lexical scope, and we put it forward as possibly the first ever functional pearl. We first characterize it as the result of a sequence of program transformations, and then apply similar transformations to a program for the classical power set example. We also show that using a higher-order representation of lists allows a definition of the Cartesian product function where <em>foldr</em> is nested only twice.

FUNCTIONAL PEARL: A program to solve Sudoku

There's no maths involved. You solve the puzzle with reasoning and logic. Advice on how to play Sudoku, The Independent Newspaper

FUNCTIONAL PEARL: Enumerating the rationals

Every lazy functional programmer knows about the following approach to enumerating the positive rationals: generate a two-dimensional matrix (an infinite list of infinite lists), then traverse its finite diagonals (an infinite list of finite lists).

An Applicative Control-Flow Graph Based on Huet's Zipper

We are using ML to build a compiler that does low-level optimization. To support optimizations in classic imperative style, we built a control-flow graph using mutable pointers and other mutable state in the nodes. This decision proved unfortunate: the mutable flow graph was big and complex, and it led to many bugs. We have replaced it by a smaller, simpler, applicative flow graph based on Huet's [Huet, Gérard, 1997. The Zipper. Journal of Functional Programming, 7(5):549–554. Functional Pearl] zipper. The new flow graph is a success; this paper presents its design and shows how it leads to a gratifyingly simple implementation of the dataflow framework developed by [Lerner, Sorin, David Grove, and Craig Chambers. 2002. Composing dataflow analyses and transformations. Conference Record of the 29th Annual ACM Symposium on Principles of Programming Languages, in SIGPLAN Notices, 31(1):270–282].

The status of host fish populations and fish species richness in European freshwater pearl mussel (Margaritifera margaritifera) streams

Abstract The status of host fish populations and fish species richness was investigated at 36 sites of 20 extant freshwater pearl mussel populations, including the drainages of the Elbe, Danube, Rhine, Weser, Aulne, Kemijoki and Tuuloma in Germany, the Czech Republic, France and Finland, by carrying out comparative electrofishings. Brown trout (Salmo trutta f. fario) were found to be the available host fish for pearl mussels in all except one of the streams investigated with mean densities of 2861 ha−1 (range 0–8710 ha−1) and a mean biomass of 119 kg ha−1 (range 0–478 kg ha−1). Streams that had been frequently stocked with brown trout had higher trout biomass and densities of host fish than natural populations, but trout stocking had no positive effect in two of the streams investigated. Fish species richness ranged from 2 to 16 species per stream and showed a negative correlation with host fish biomass and host fish densities. Undisturbed oligotrophic pearl mussel headwater streams usually only yielded a low number of fish species. Habitat degradation can reduce competitiveness of specialized trout and result in an increased abundance of ubiquitous or atypical species. A link between the lack of juvenile pearl mussels and a lack of suitable host fish was only rarely observed. Functional pearl mussel populations with relatively high numbers of juveniles had significantly lower densities and biomass of host fish than pearl mussel populations without recent recruitment. This study suggests that 0+ host fish are not necessarily required to sustain functional pearl mussel populations. Low densities of host fish can be compensated by the higher glochidia carrying capacity of older host fish with limited previous contact with pearl mussel glochidia, by the long reproductive period of mussels, and by low mortality rates of juvenile mussels during their post‐parasitic phase. Copyright © 2005 John Wiley & Sons, Ltd.

FUNCTIONAL PEARL Derivation of a logarithmic time carry lookahead addition circuit

Using Haskell as a digital circuit description language, we transform a ripple carry adder that requires $O(n)$ time to add two $n$-bit words into a parallel carry lookahead adder that requires $O(\log n)$ time. The ripple carry adder uses a scan function to calculate carry bits, but this scan cannot be parallelized directly since it is applied to a non-associative function. Several techniques are applied in order to introduce parallelism, including partial evaluation and symbolic function representation. The derivation given here constitutes a semi-formal correctness proof, and it also brings out explicitly each of the ideas underlying the algorithm.

Generic unification via two-level types and parameterized modules

As a functional pearl, we describe an efficient, modularized implementation of unification using the state of mutable reference cells to encode substitutions. We abstract our algorithms along two dimensions, first abstracting away from the structure of the terms to be unified, and second over the monad in which the mutable state is encapsulated. We choose this example to illustrate two important techniques that we believe many functional programmers would find useful. The first of these is the definition of recursive data types using two levels: a structure defining level, and a recursive knot-tying level. The second is the use of rank-2 polymorphism inside Haskell's record types to implement a form of type parameterized modules.

FUNCTIONAL PEARL On building trees with minimum height

A common solution to the problem of handling list indexing efficiently in a functional program is to build a binary tree. The tree has the given list as frontier and is of minimum height. Each internal node of the tree stores size information (actually, the size of its left subtree) to direct the search for an element at a given position in the frontier. One application was considered in my previous pearl (Bird, 1997). There are two complementary methods for building such a tree, both of which can be implemented in linear time. One method is ‘recursive’, or top down, and works by splitting the list into two equal halves, recursively building a tree for each half, and then combining the two results. The other method is ‘iterative’, or bottom up, and works by first creating a list of singleton trees, and then repeatedly combining the trees in pairs until just one tree remains. The two methods lead to different trees, but in each case the result is a tree with smallest possible height.

Functional PearlDeduction for functional programmers

<i>Functional Pearl</i>Deduction for functional programmers