Showing papers in "Journal of Functional Programming in 2004"

The view from the left

Abstract: Pattern matching has proved an extremely powerful and durable notion in functional programming. This paper contributes a new programming notation for type theory which elaborates the notion in various ways. First, as is by now quite well-known in the type theory community, definition by pattern matching becomes a more discriminating tool in the presence of dependent types, since it refines the explanation of types as well as values. This becomes all the more true in the presence of the rich class of datatypes known as inductive families (Dybjer, 1991). Secondly, as proposed by Peyton Jones (1997) for Haskell, and independently rediscovered by us, subsidiary case analyses on the results of intermediate computations, which commonly take place on the right-hand side of definitions by pattern matching, should rather be handled on the left. In simply-typed languages, this subsumes the trivial case of Boolean guards; in our setting it becomes yet more powerful. Thirdly, elementary pattern matching decompositions have a well-defined interface given by a dependent type; they correspond to the statement of an induction principle for the datatype. More general, user-definable decompositions may be defined which also have types of the same general form. Elementary pattern matching may therefore be recast in abstract form, with a semantics given by translation. Such abstract decompositions of data generalize Wadler's (1987) notion of ‘view’. The programmer wishing to introduce a new view of a type $\mathit{T}$, and exploit it directly in pattern matching, may do so via a standard programming idiom. The type theorist, looking through the Curry–Howard lens, may see this as proving a theorem, one which establishes the validity of a new induction principle for $\mathit{T}$. We develop enough syntax and semantics to account for this high-level style of programming in dependent type theory. We close with the development of a typechecker for the simply-typed lambda calculus, which furnishes a view of raw terms as either being well-typed, or containing an error. The implementation of this view is ipso facto a proof that typechecking is decidable.

Grammatical Framework

Abstract: Grammatical Framework (GF) is a special-purpose functional language for defining grammars. It uses a Logical Framework (LF) for a description of abstract syntax, and adds to this a notation for defining concrete syntax. GF grammars themselves are purely declarative, but can be used both for linearizing syntax trees and parsing strings. GF can describe both formal and natural languages. The key notion of this description is a grammatical object, which is not just a string, but a record that contains all information on inflection and inherent grammatical features such as number and gender in natural languages, or precedence in formal languages. Grammatical objects have a type system, which helps to eliminate run-time errors in language processing. In the same way as a LF, GF uses dependent types in abstract syntax to express semantic conditions, such as well-typedness and proof obligations. Multilingual grammars, where one abstract syntax has many parallel concrete syntaxes, can be used for reliable and meaning-preserving translation. They can also be used in authoring systems, where syntax trees are constructed in an interactive editor similar to proof editors based on LF. While being edited, the trees can simultaneously be viewed in different languages. This paper starts with a gradual introduction to GF, going through a sequence of simpler formalisms till the full power is reached. The introduction is followed by a systematic presentation of the GF formalism and outlines of the main algorithms: partial evaluation and parser generation. The paper concludes by brief discussions of the Haskell implementation of GF, existing applications, and related work.

Transformation techniques for context-sensitive rewrite systems

Abstract: Context-sensitive rewriting is a computational restriction of term rewriting used to model non-strict (lazy) evaluation in functional programming. The goal of this paper is the study and development of techniques to analyze the termination behavior of context-sensitive rewrite systems. For that purpose, several methods have been proposed in the literature which transform context-sensitive rewrite systems into ordinary rewrite systems such that termination of the transformed ordinary system implies termination of the original context-sensitive system. In this way, the huge variety of existing techniques for termination analysis of ordinary rewriting can be used for context-sensitive rewriting, too. We analyze the existing transformation techniques for proving termination of context-sensitive rewriting and we suggest two new transformations. Our first method is simple, sound, and more powerful than the previously proposed transformations. However, it is not complete, i.e., there are terminating context-sensitive rewrite systems that are transformed into non-terminating term rewrite systems. The second method that we present in this paper is both sound and complete. All these observations also hold for rewriting modulo associativity and commutativity.

The structure and interpretation of the computer science curriculum

Abstract: Twenty years ago Abelson and Sussman's Structure and Interpretation of Computer Programs radically changed the intellectual landscape of introductory computing courses. Instead of teaching some currently fashionable programming language, it employed Scheme and functional programming to teach important ideas. Introductory courses based on the book showed up around the world and made Scheme and functional programming popular. Unfortunately, these courses quickly disappeared again due to shortcomings of the book and the whimsies of Scheme. Worse, the experiment left people with a bad impression of Scheme and functional programming in general. In this pearl, we propose an alternative role for functional programming in the first-year curriculum. Specifically, we present a framework for discussing the first-year curriculum and, based on it, the design rationale for our book and course, dubbed How to Design Programs. The approach emphasizes the systematic design of programs. Experience shows that it works extremely well as a preparation for a course on object-oriented programming.

FUNCTIONAL PEARL Pickler combinators

Abstract: The tedium of writing pickling and unpickling functions by hand is relieved using a combinator library similar in spirit to the well-known parser combinators. Picklers for primitive types are combined to support tupling, alternation, recursion, and structure sharing. Code is presented in Haskell; an alternative implementation in ML is discussed.

The risks and benefits of teaching purely functional programming in first year

Abstract: We argue that teaching purely functional programming as such in freshman courses is detrimental to both the curriculum as well as to promoting the paradigm. Instead, we need to focus on the more general aims of teaching elementary techniques of programming and essential concepts of computing. We support this viewpoint with experience gained during several semesters of teaching large first-year classes (up to 600 students) in Haskell. These classes consisted of computer science students as well as students from other disciplines. We have systematically gathered student feedback by conducting surveys after each semester. This article contributes an approach to the use of modern functional languages in first year courses and, based on this, advocates the use of functional languages in this setting.

Two-level types and parameterized modules

Abstract: In this paper, we describe two techniques for the efficient, modularized implementation of a large class of algorithms. We illustrate these techniques using several examples, including efficient generic unification algorithms that use reference cells to encode substitutions, and highly modular language implementations. We chose these examples to illustrate the following important techniques that we believe many functional programmers would find useful. First, defining recursive data types by splitting them into two levels: a structure defining level, and a recursive knot-tying level. Second, the use of rank-2 polymorphism inside Haskell's record types to implement a kind of type-parameterized modules. Finally, we explore techniques that allow us to combine already existing recursive Haskell data-types with the highly modular style of programming proposed here.

Linear lambda calculus and PTIME-completeness

Abstract: We give transparent proofs of the PTIME-completeness of two decision problems for terms in the λ-calculus. The first is a reproof of the theorem that type inference for the simply-typed λ-calculus is PTIME-complete. Our proof is interesting because it uses no more than the standard combinators Church knew of some 70 years ago, in which the terms are linear affine – each bound variable occurs at most once. We then derive a modification of Church's coding of Booleans that is linear, where each bound variable occurs exactly once. A consequence of this construction is that any interpreter for linear λ-calculus requires polynomial time. The logical interpretation of this consequence is that the problem of normalizing proofnets for multiplicative linear logic (MLL) is also PTIME-complete.

Parsing permutation phrases

Abstract: A permutation phrase is a sequence of elements (possibly of different types) in which each element occurs exactly once and the order is irrelevant. Some of the permutable elements may be optional. We show how to extend a parser combinator library with support for parsing such free-order constructs. A user of the library can easily write parsers for permutation phrases and does not need to care about checking and reordering the recognized elements. Applications include the generation of parsers for attributes of XML tags and Haskell's record syntax.

Composition of functions with accumulating parameters

Abstract: Many functional programs with accumulating parameters are contained in the class of macro tree transducers. We present a program transformation technique that can be used to solve the efficiency problems due to creation and consumption of intermediate data structures in compositions of such functions, where classical deforestation techniques fail. To do so, given two macro tree transducers under appropriate restrictions, we construct a single macro tree transducer that implements the composition of the two original ones. The imposed restrictions are more liberal than those in the literature on macro tree transducer composition, thus generalising previous results.

Parallel Parsing Processes

Abstract: We derive a combinator library for non-deterministic parsers with a monadic interface, by means of successive refinements starting from a specification The choice operator of the parser implements a breadth-first search rather than the more common depth-first search, and can be seen as a parallel composition between two parsing processes The resulting library is simple and efficient for “almost deterministic” grammars, which are typical for programming languages and other computing science applications

Type-FUNCTIONAL PEARL safe cast

Abstract: Comparing two types for equality is an essential ingredient for an implementation of dynamic types. Once equality has been established, it is safe to cast a value from one type to another. In a language with run-time type analysis, implementing such a procedure is fairly straightforward. Unfortunately, this naive implementation destructs and rebuilds the argument while iterating over its type structure. However, by using higher-order polymorphism, a casting function can treat its argument parametrically. We demonstrate this solution in two frameworks for ad-hoc polymorphism: intensional type analysis and Haskell type classes.

The recursive record semantics of objects revisited

Abstract: In a call-by-value language, representing objects as recursive records requires using an unsafe fixpoint. We design, for a core language including extensible records, a type system which rules out unsafe recursion and still supports the construction of a principal type for each typable term. We illustrate the expressive power of this language with respect to object-oriented programming by introducing a sub-language for “mixin-based” programming.

Building reliable, high-performance networks with the Nuprl proof development system

Abstract: Proof systems for expressive type theories provide a foundation for the verification and synthesis of programs. But despite their successful application to numerous programming problems there remains an issue with scalability. Are proof environments capable of reasoning about large software systems? Can the support they offer be useful in practice? In this article we answer this question by showing how the NUPRL proof development system and its rich type theory have contributed to the design of reliable, high-performance networks by synthesizing optimized code for application configurations of the ENSEMBLE group communication toolkit. We present a type-theoretical semantics of OCAML, the implementation language of ENSEMBLE, and tools for automatically importing system code into the NUPRL system. We describe reasoning strategies for generating verifiably correct fast-path optimizations of application configurations that substantially reduce end-to-end latency in ENSEMBLE. We also discuss briefly how to use NUPRL for checking configurations against specifications and for the design of reliable adaptive network protocols.

Functional programming with the FC++ library

Abstract: We describe the FC++ library, a rich library supporting functional programming in C++. Prior approaches to encoding higher order functions in C++ have suffered with respect to polymorphic functions from either lack of expressiveness or high complexity. In contrast, FC++ offers full and concise support for higher-order polymorphic functions through a novel use of C++ type inference. The FC++ library has a number of useful features, including a generalized mechanism to implement currying in C++, a “lazy list” class which enables the creation of “infinite data structures”, a subtype polymorphism facility, and an extensive library of useful functions, including a large part of the Haskell Standard Prelude. The FC++ library has an efficient implementation. We show the results of a number of experiments which demonstrate the value of optimizations we have implemented. These optimizations have improved the run-time performance by about an order of magnitude for some benchmark programs that make heavy use of FC++ lazy lists. We also make an informal performance comparison with similar programs written in Haskell.

Dependent types ensure partial correctness of theorem provers

Abstract: Static type systems in programming languages allow many errors to be detected at compile time that wouldn't be detected until runtime otherwise. Dependent types are more expressive than the type systems in most programming languages, so languages that have them should allow programmers to detect more errors earlier. In this paper, using the Twelf system, we show that dependent types in the logic programming setting can be used to ensure partial correctness of programs which implement theorem provers, and thus avoid runtime errors in proof search and proof construction. We present two examples: a tactic-style interactive theorem prover and a union-find decision procedure.

Derivation of a logarithmic time carry lookahead addition circuit

Abstract: 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.

Linearization of the lambda-calculus and its relation with intersection type systems

Abstract: In this paper we present a notion of expansion of a term in the lambda-calculus which transforms terms into linear terms. This transformation replaces each occurrence of a variable in the original term by a fresh variable taking into account non-trivial implications in the structure of the term caused by these simple replacements. We prove that the class of terms which can be expanded is the same of terms typable in an Intersection Type System, i.e. the strongly normalizable terms. We then show that expansion is preserved by weak-head reduction, the reduction considered by functional programming languages.

Proof, Language and Interaction, Essays in Honour of Robin Milner eds Gordon Plotkin, Colin Sterling, Mads Tofte, Foundations of Computing Series, MIT Press 2000

Abstract: This Festschrift in honour of the scientific life and achievements of Robin Milner consists of 24 papers by various authors plus a biography of Robin Milner by the editors. The volume is well suited to the aim of the series in Foundations of Computing which is to provide a forum in which important research topics can be presented and placed in perspective for researchers, students and practitioners alike.

Constructed product result analysis for Haskell

Abstract: Compilers for ML and Haskell typically go to a good deal of trouble to arrange that multiple arguments can be passed efficiently to a procedure. For some reason, less effort seems to be invested in ensuring that multiple results can also be returned efficiently. In the context of the lazy functional language Haskell, we describe an analysis, Constructed Product Result (CPR) analysis, that determines when a function can profitably return multiple results in registers. The analysis is based only on a function's definition, and not on its uses (so separate compilation is easily supported) and the results of the analysis can be expressed by a transformation of the function definition alone. We discuss a variety of design issues that were addressed in our implementation, and give measurements of the effectiveness of our approach across a substantial benchmark set. Overall, the pricesperformance ratio is good: the benefits are modest in general (though occasionally dramatic), but the costs in both complexity and compile time, are low.

Enumerating the strings of regular languages

Abstract: Haskell code is developed for two ways to list the strings of the language defined by a regular expression: directly by set operations and indirectly by converting to and simulating an equivalent automaton. The exercise illustrates techniques for dealing with infinite ordered domains and leads to an effective standard form for nondeterministic finite automata.

Inverting the Burrows—Wheeler transform

Abstract: The Burrows–Wheeler Transform is a string-to-string transform which, when used as a preprocessing phase in compression, significantly enhances the compression rate However, it often puzzles people how the inverse transform is carried out In this pearl we to exploit simple equational reasoning to derive the inverse of the Burrows–Wheeler transform from its specification We also outline how to derive the inverse of two more general versions of the transform, one proposed by Schindler and the other by Chapin and Tate

Functional chart parsing of context-free grammars

Abstract: This paper implements a simple and elegant version of bottom-up Kilbury chart parsing (Kilbury, 1985; Wiren, 1992). This is one of the many chart parsing variants, which are all based on the data structure of charts. The chart parsing process uses inference rules to add new edges to the chart, and parsing is complete when no further edges can be added. One novel aspect of this implementation is that it doesn't have to rely on a global state for the implementation of the chart. This makes the code clean, elegant and declarative, while still having the same space and time complexity as the standard imperative implementations.

Concurrent distinct choices

Abstract: An injective finite mapping is an abstraction common to many programs. We describe the design of an injective finite mapping and its implementation in Curry, a functional logic language. Curry supports the concurrent asynchronous execution of distinct portions of a program. This condition prevents passing from one portion to another a structure containing a partially constructed mapping to ensure that a new choice does not violate the injectivity condition. We present some motivating problems and we show fragments of programs that solve these problems using our design and implementation.

Composing fractals

Abstract: This paper describes a simple but flexible family of Haskell programs for drawing pictures of fractals such as Mandelbrot and Julia sets. Its main goal is to showcase the elegance of a compositional approach to program construction, and the benefits of a clean separation between different aspects of program behavior. Aimed at readers with relatively little experience of functional programming, the paper can be used as a tutorial on functional programming, as an overview of the Mandelbrot set, or as a motivating example for studies in computability.

Global variables in Haskell

Abstract: Haskell today provides good support not only for a functional programming style, but also for an imperative one. Elements of imperative programming are needed in applications such as web servers, or to provide efficient implementations of well-known algorithms, such as many graph algorithms. However, one element of imperative programming, the global variable, is surprisingly hard to emulate in Haskell. We discuss several existing methods, none of which is really satisfactory, and finally propose a new approach based on implicit parameters. This approach is simple, safe, and efficient, although it does reveal weaknesses in Haskell's present type system.

Escape from Zurg: an exercise in logic programming

Abstract: In this Pearl we illustrate with an example that modern functional programming languages like Haskell can be used effectively for programming search problems, in contrast to the widespread belief that Prolog is much better suited for tasks like these.

The Functional “C” experience

Abstract: A functional programming language can be taught successfully as a first language, but if there is no follow up the students do not appreciate the functional approach. Following discussions concerning this issue at the 1995 FPLE conference (Hartel & Plasmeijer, 1995), we decided to develop such a follow up by writing a book that teaches C to students who can write simple functional programs. This paper summarises the essence of our approach, which is based on program transformation, and presents our experience teaching functional C at the Universities of Southampton and Bristol.