In this paper, we show how lazy functional programs can be compiled for the Java Virtual Machine using a mapping between a version of the 〈v, G〉-machine and the Java Virtual Machine. This mapping is elegant – the description is entirely straightforward – and efficient – using it, both code size and execution speed are of the same order of magnitude as those obtained with a traditional functional language bytecode interpreter. In future, our work could serve as the basis of an interface between Haskell and Java.

The future annotation of MultiLisp provides a simple method for taming the implicit parallelism of functional programs. Prior research on future has concentrated on implementation and design issues, and has largely ignored the development of a semantic characterization of future. This paper considers an idealized functional language with futures and presents a series of operational semantics with increasing degrees of intensionality. The first semantics defines future to be a semantically transparent annotation. The second semantics interprets a future expression as a potentially parallel task. The third semantics explicates the coordination of parallel tasks by introducing placeholder objects and touch operations.

We use the last semantics to derive a program analysis algorithm and an optimization algorithm that removes provably redundant touch operations. Experiments with the Gambit compiler indicate that this optimization significantly reduces the overhead imposed by touch operations.

We present a general method to transform a compositional specification of a specializer for a functional programming language into a set of combinators that can be used to perform the same specialization more efficiently. The main transformation steps are the transition to higher-order abstract syntax and untagging. All transformation steps are proved correct. The resulting combinators can be implemented in any functional language, typed or untyped, pure or impure. They may also be considered as forming a domain-specific language for meta-programming. We demonstrate the generality of the method by applying it to several specializers of increasing strength. We demonstrate its efficiency by comparing it with a traditional specialization system based on self-application.

The LCF system was the first mechanical theorem prover to be user-programmable via a metalanguage, ML, from which the functional programming language Standard ML has been developed. Paulson has demonstrated how a modular rewriting engine can be implemented in LCF. This provides both clarity and flexibility. This paper shows that the same modular approach (using higher-order functions) allows transparent optimisation of the rewriting engine; performance can be improved while few, if any, changes are required to code written using these functions. The techniques described have been implemented in the HOL system, a descendant of LCF, and some are now in daily use. Comparative results are given. Some of the techniques described, in particular ones to avoid processing parts of a data structure that do not need to be changed, may be of more general use in functional programming and beyond.

This paper was guest-edited by Harry Mairson and Bruce Kapron, for our intended Special Issue on Functional Programming and Computational Complexity. Other papers submitted for the special issue were either out-of-scope or otherwise unsuitable for JFP. Even though only one paper met their high standards, this did not make Harry and Bruce's job any easier, and we thank them for their efforts.

In previous work the author has introduced a lambda calculus SLR with modal and linear types which serves as an extension of Bellantoni–Cook's function algebra BC to higher types. It is a step towards a functional programming language in which all programs run in polynomial time. While this previous work was concerned with the syntactic metatheory of SLR in this paper we develop a semantics of SLR in terms of Chu spaces over a certain category of sheaves from which it follows that all expressible functions are indeed in PTIME. We notice a similarity between the Chu space interpretation and CPS translation which as we hope will have further applications in functional programming.

In functional programming, fold is a standard operator that encapsulates a simple pattern of recursion for processing lists. This article is a tutorial on two key aspects of the fold operator for lists. First of all, we emphasize the use of the universal property of fold both as a proof principle that avoids the need for inductive proofs, and as a definition principle that guides the transformation of recursive functions into definitions using fold. Secondly, we show that even though the pattern of recursion encapsulated by fold is simple, in a language with tuples and functions as first-class values the fold operator has greater expressive power than might first be expected.

Recursive block decomposition algorithms (also known as quadtree algorithms when the blocks are all square) have been proposed to solve well-known problems such as matrix addition, multiplication, inversion, determinant computation, block LDU decomposition and Cholesky and QR factorization. Until now, such algorithms have been seen as impractical, since they require leading submatrices of the input matrix to be invertible (which is rarely guaranteed). We show how to randomize an input matrix to guarantee that submatrices meet these requirements, and to make recursive block decomposition methods practical on well-conditioned input matrices. The resulting algorithms are elegant, and we show the recursive programs can perform well for both dense and sparse matrices, although with randomization dense computations seem most practical. By ‘homogenizing’ the input, randomization provides a way to avoid degeneracy in numerical problems that permits simple recursive quadtree algorithms to solve these problems.

Functional logic languages with a sound and complete operational semantics are mainly based on an inference rule called narrowing. Narrowing extends functional evaluation by goal solving capabilities, as in logic programming. Due to the huge search space of simple narrowing, steadily improved narrowing strategies have been developed in the past. Needed narrowing is currently the best narrowing strategy for first-order functional logic programs due to its optimality properties wrt the length of derivations and the number of computed solutions. In this paper, we extend the needed narrowing strategy to higher-order functions and λ-terms as data structures. By the use of definitional trees, our strategy computes only independent solutions. Thus, it is the first calculus for higher-order functional logic programming which provides for such an optimality result. Since we allow higher-order logical variables denoting λ-terms, applications go beyond current functional and logic programming languages. We show soundness and completeness of our strategy with respect to LNT reductions, a particular form of higher-order reductions defined via definitional trees. A general completeness result is only provided for terminating rewrite systems due to the lack of an overall theory of higher-order reduction which is outside the scope of this paper.

We present a hierarchical classification of specializations of the divide-and-conquer paradigm. The aim is to identify a subclass of divide-and-conquer algorithms with an efficient parallel implementation which can be viewed as a static space-time mapping. The specializations impose a balanced call tree, a fixed degree of the problem division, and elementwise operations. The correctness of our compile-time transformations is proved by equational reasoning in Haskell; recursion and iteration are handled by induction. We demonstrate the practicality of the skeleton by some examples, one of which is Strassen's matrix multiplication.

In this paper we demonstrate that a broad class of higher-order functional programs can be transformed into semantically equivalent multidimensional intensional programs that contain only nullary variable definitions. The proposed algorithm systematically eliminates user-defined functions from the source program, by appropriately introducing context manipulation (i.e. intensional) operators. The transformation takes place in M steps, where M is the order of the initial functional program. During each step the order of the program is reduced by one, and the final outcome of the algorithm is an M-dimensional intensional program of order zero. As the resulting intensional code can be executed in a purely tagged-dataflow way, the proposed approach offers a promising new technique for the implementation of higher-order functional languages.

We adopt the untyped imperative object calculus of Abadi and Cardelli as a minimal setting in which to study problems of compilation and program equivalence that arise when compiling object-oriented languages. We present both a big-step and a small-step substitution-based operational semantics for the calculus. Our first two results are theorems asserting the equivalence of our substitution-based semantics with a closure-based semantics like that given by Abadi and Cardelli. Our third result is a direct proof of the correctness of compilation to a stack-based abstract machine via a small-step decompilation algorithm. Our fourth result is that contextual equivalence of objects coincides with a form of Mason and Talcott's CIU equivalence; the latter provides a tractable means of establishing operational equivalences. Finally, we prove correct an algorithm, used in our prototype compiler, for statically resolving method offsets. This is the first study of correctness of an object-oriented abstract machine, and of operational equivalence for the imperative object calculus.

This paper discusses the principles of implementing an LCF-style proof assistant using a purely functional metalanguage. Two approaches are described; in both, signatures are treated as ordinary values, rather than as mutable components within an abstract datatype. The first approach treats the object logic as a partial algebra and represents it as a partial datatype, that is, a datatype in which the domains of the constructors are restricted by predicate functions. This results in a compact, executable specification of the logic. The second approach uses an abstract type to allow an efficient representation of the logic, whilst keeping the same interface. A case study describes how these principles were put into practice in implementing a fairly complex dependently-sorted logic.

To some readers functional pearls may seem like London buses: you can wait ages for one and then two or three come along at once. In the following pages you will find no less than three pearls for your enjoyment and instruction. There are a number of others a little way down the road, so I trust that your pleasure will continue for some time yet.

If my calculations are correct, twenty-seven pearls have been published in the eight and a half years since JFP was inaugurated. There was something of a desert in 1994 and 1995, but since January 1997 they have been appearing in virtually every issue. When I was asked to contribute a regular column to JFP in 1990, the general idea was for something in style that was a mixture of Jon Bentley's Programming Pearls column for the Communications of the ACM, and Martin Rem's Small Programming Exercises section in the Science of Computer Programming. Articles could be on any subject, but they had to be polished, preferably short, and to show something of the unique opportunities a functional perspective could offer to the study of computing. In my own case that usually meant the use of functional notation to derive programs from their specifications, to optimise them, and to reason about their properties in an equational style. Over the years it has emerged that a functional approach is also extremely attractive in the presentation of ideas about data structures, and some very intriguing pearls have been about the use of elegant data structures to achieve efficiency. But there have also been pearls about particular applications, like parsing, pretty-printing, theorem proving, and unification. In fact, practically every area of computing science can benefit from the clarity and brevity a functional approach can offer. That is why I believe the importance of the subject goes beyond the activity of simply writing code in a functional language.

I knew from the outset that I wasn't going to write all of the pearls, and believed that other contributors would soon emerge. Fortunately, I was right: I have written about a third, but some twenty contributors have composed the rest. Authors like writing pearls perhaps because they like responding to the opportunities and challenges provided by the format. All pearls are refereed (including my own), and there is often a fair amount of polishing as a result of the reviewing stage. I would like to record my gratitude to the authors of the pearls so far published (Stephen Adams, Lennart Augustsson, Koen Claessen, Martin Erwig, Jeremy Gibbons, Ralf Hinze, Rob Hoogerwoord, Gerard Huet, Graham Hutton, Patrik Jansson, Johan Jeuring, Andrew Kennedy, Jamie Leifer, Doug Mcllroy, Eric Meijer, Chris Okasaki, Mikael Rittri, Colin Runciman, Bernard Sufrin, Dan Synek) and to ask them, and others, to continue in their good work.

This theoretical pearl is about the closed term model of pure untyped lambda-terms modulo β-convertibility. A consequence of one of the results is that for arbitrary distinct combinators (closed lambda terms) M, M′, N, N′ there is a combinator H such that

formula here

The general result, which comes from Statman (1998), is that uniformly r.e. partitions of the combinators, such that each ‘block’ is closed under β-conversion, are of the form {H−1{M}}M∈ΛΦ. This is proved by making use of the idea behind the so-called Plotkin-terms, originally devised to exhibit some global but non-uniform applicative behaviour. For expository reasons we present the proof below. The following consequences are derived: a characterization of morphisms and a counter-example to the perpendicular lines lemma for β-conversion.

The design of theorem provers, especially in the LCF-prover family, has strongly profited from functional programming. This paper attempts to develop a metaphor suited to visualize the LCF-style prover design, and a methodology for the implementation of graphical user interfaces for these provers and encapsulations of formal methods. In this problem domain, particular attention has to be paid to the need to construct a variety of objects, keep track of their interdependencies and provide support for their reconstruction as a consequence of changes. We present a prototypical implementation of a generic and open interface system architecture, and show how it can be instantiated to an interface for Isabelle, called IsaWin, as well as to a tailored tool for transformational program development, called TAS.

We derive a confluent λ-calculus with a catch/throw mechanism (called λct-calculus) from Parigot's λμ-calculus. We also present several translations from one calculus into the other which are morphisms for the reduction. We use them to show that the λct-calculus is a retract of λμ-calculus (these calculi are isomorphic if we consider only convertibility). As a by-product, we obtain the subject reduction property for the λct-calculus, as well as the strong normalization for λct-terms typable in the second order classical natural deduction.

This article proposes a new language mechanism for data-parallel processing of dynamically allocated recursively defined data. Different from the conventional array-based data- parallelism, it allows parallel processing of general recursively defined data such as lists or trees in a functional way. This is achieved by representing a recursively defined datum as a system of equations, and defining new language constructs for parallel transformation of a system of equations. By integrating them with a higher-order functional language, we obtain a functional programming language suitable for describing data-parallel algorithms on recursively defined data in a declarative way. The language has an ML style polymorphic type system and a type sound operational semantics that uniformly integrates the parallel evaluation mechanism with the semantics of a typed functional language. We also show the intended parallel execution model behind the formal semantics, assuming an idealized distributed memory multicomputer.

“I have no data yet. It is a capital mistake to theorise before one has data.” Sir Arthur Conan Doyle The Adventures of Sherlock Holmes

de Bruijn notation is a coding of lambda terms in which each occurrence of a bound variable x is replaced by a natural number, indicating the ‘distance’ from the occurrence to the abstraction that introduced x. One might suppose that in any datatype for representing de Bruijn terms, the distance restriction on numbers would have to be maintained as an explicit datatype invariant. However, by using a nested (or non-regular) datatype, we can define a representation in which all terms are well-formed, so that the invariant is enforced automatically by the type system. Programming with nested types is only a little more difficult than programming with regular types, provided we stick to well-established structuring techniques. These involve expressing inductively defined functions in terms of an appropriate fold function for the type, and using fusion laws to establish their properties. In particular, the definition of lambda abstraction and beta reduction is particularly simple, and the proof of their associated properties is entirely mechanical.

The close relationship between writing programs and proving theorems has frequently been cited as an advantage of functional programming languages. We illustrate the interplay between programming and proving in the development of a program for regular expression matching. The presentation is inspired by Lakatos's method of proofs and refutations in which the attempt to prove a plausible conjecture leads to a revision not only of the proof, but of the theorem itself. We give a plausible implementation of a regular expression matcher that contains a flaw that is uncovered in an attempt to prove its correctness. The failure of the proof suggests a revision of the specification, rather than a change to the code. We then show that a program meeting the revised specification is nevertheless sufficient to solve the original problem.

Functional programming languages are an excellent tool for teaching algorithms and data structures. This paper explains binomial heaps, a beautiful data structure for priority queues, using the functional programming language Haskell (Peterson and Hammond, 1997). We largely follow a deductive approach: using the metaphor of a tennis tournament we show that binomial heaps arise naturally through a number of logical steps. Haskell supports the deductive style of presentation very well: new types are introduced at ease, algorithms can be expressed clearly and succinctly, and Haskell's type classes allow to capture common algorithmic patterns. The paper aims at the level of an undergraduate student who has experience in reading and writing Haskell programs, and who is familiar with the concept of a priority queue.