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.

A special issue of the Journal of Functional Programming will be devoted to the use of functional programming in theorem proving. The submission deadline is 31 August 1997.

The histories of theorem provers and functional languages have been deeply intertwined since the advent of Lisp. A notable example is the ML family of languages, which are named for the meta language devised for the LCF theorem prover, and which provide both the implementation platform and interaction facilities for numerous later systems (such as Coq, HOL, Isabelle, NuPrl). Other examples include Lisp (as used for ACL2, PVS, Nqthm) and Haskell (as used for Veritas).

This special issue is devoted to the theory and practice of using functional languages to implement theorem provers and using theorem provers to reason about functional languages. Topics of interest include, but are not limited to:

– architecture of theorem prover implementations

– interface design in the functional context

– limits of the LCF methodology

– impact of host language features

– type systems

– lazy vs strict languages

– imperative (impure) features

– performance problems and solutions

– problems of scale

– special implementation techniques

– term representations (e.g. de Bruijn vs name carrying vs BDDs)

– limitations of current functional languages

– mechanised theories of functional programming

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.

Among the many flavours of balanced binary trees, Braun trees (Braun and Rem, 1983) are perhaps the most circumscribed. For any given node of a Braun tree, the left subtree is either exactly the same size as the right subtree, or one element larger. Braun trees always have minimum height, and the shape of each Braun tree is completely determined by its size. In return for this rigor, algorithms that manipulate Braun trees are often exceptionally simple and elegant, and need not maintain any explicit balance information.

Braun trees have been used to implement both flexible arrays (Braun and Rem, 1983; Hoogerwoord, 1992; Paulson, 1996) and priority queues (Paulson, 1996; Bird, 1996). Most operations involving a single element (e.g. adding, removing, inspecting or updating an element) take O(log n) time, since the trees are balanced. We consider three algorithmically interesting operations that manipulate entire trees. First, we give an O(log2n) algorithm for calculating the size of a tree. Second, we show how to create a tree containing n copies of some element x in O(log n) time. Finally, we describe an order-preserving algorithm for converting a list to a tree in O(n) time. This last operation is not nearly as straightforward as it sounds!

Given two ascending lists xs and ys of combined length greater than n, consider the computation of

formula here

The standard function merge merges two ascending sequences and (!!) denotes list indexing. With a lazy evaluator the computation takes O(n) steps; with an eager one it takes O(p+q) steps, where p=length xs and q=length ys. Now in functional programming it is more efficient to index a tree than a list, so the question arises: can we find a faster solution if xs and ys are each represented by a tree? Somewhat surprisingly the answer is yes: if xs and ys are each represented by balanced binary search trees, then the computation can be reduced to O(log p+log q) steps. This is despite the fact that there is no known method for merging two binary search trees in better than linear time. The details, presented below, depend on a subtle relationship between merging and indexing.

The popular method of enumerating the primes is the Sieve of Eratosthenes. It can be programmed very neatly in a lazy functional language, but runs rather slowly. A little-known alternative method is the Wheel Sieve, originally formulated as a fast imperative algorithm for obtaining all primes up to a given limit, assuming destructive access to a bit-array. This article describes functional variants of the wheel sieve that enumerate all primes as a lazy list.

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.

A special issue of the Journal of Functional Programming will be devoted to functional programming and computational complexity. The submission deadline is 1st November 1997.