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A cost calculus for parallel functional programming
 Journal of Parallel and Distributed Computing
, 1995
"... Abstract Building a cost calculus for a parallel program development environment is difficult because of the many degrees of freedom available in parallel implementations, and because of difficulties with compositionality. We present a strategy for building cost calculi for skeletonbased programmin ..."
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Abstract Building a cost calculus for a parallel program development environment is difficult because of the many degrees of freedom available in parallel implementations, and because of difficulties with compositionality. We present a strategy for building cost calculi for skeletonbased programming languages which can be used for derivational software development and which deals in a pragmatic way with the difficulties of composition. The approach is illustrated for the BirdMeertens theory of lists, a parallel functional language with an associated equational transformation system. Keywords: functional programming, parallel programming, program transformation, cost calculus, equational theories, architecture independence, BirdMeertens formalism.
The BirdMeertens Formalism as a Parallel Model
 Software for Parallel Computation, volume 106 of NATO ASI Series F
, 1993
"... The expense of developing and maintaining software is the major obstacle to the routine use of parallel computation. Architecture independent programming offers a way of avoiding the problem, but the requirements for a model of parallel computation that will permit it are demanding. The BirdMeertens ..."
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Cited by 41 (0 self)
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The expense of developing and maintaining software is the major obstacle to the routine use of parallel computation. Architecture independent programming offers a way of avoiding the problem, but the requirements for a model of parallel computation that will permit it are demanding. The BirdMeertens formalism is an approach to developing and executing dataparallel programs; it encourages software development by equational transformation; it can be implemented efficiently across a wide range of architecture families; and it can be equipped with a realistic cost calculus, so that tradeoffs in software design can be explored before implementation. It makes an ideal model of parallel computation. Keywords: General purpose parallel computing, models of parallel computation, architecture independent programming, categorical data type, program transformation, code generation. 1 Properties of Models of Parallel Computation Parallel computation is still the domain of researchers and those ...
Elements of a Relational Theory of Datatypes
 Formal Program Development, volume 755 of Lecture Notes in Computer Science
, 1993
"... The "Boom hierarchy" is a hierarchy of types that begins at the level of trees and includes lists, bags and sets. This hierarchy forms the basis for the calculus of total functions developed by Bird and Meertens, and which has become known as the "BirdMeertens formalism". This paper describes a hie ..."
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Cited by 35 (0 self)
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The "Boom hierarchy" is a hierarchy of types that begins at the level of trees and includes lists, bags and sets. This hierarchy forms the basis for the calculus of total functions developed by Bird and Meertens, and which has become known as the "BirdMeertens formalism". This paper describes a hierarchy of types that logically precedes the Boom hierarchy. We show how the basic operators of the BirdMeertens formalism (map, reduce and filter) can be introduced in a logical sequence by beginning with a very simple structure and successively refining that structure. The context of this work is a relational theory of datatypes, rather than a calculus of total functions. Elements of the theory necessary to the later discussion are summarised at the beginning of the paper. 1 Introduction This paper reports on an experiment into the design of a programming algebra. The algebra is an algebra of datatypes oriented towards the calculation of polymorphic functions and relations. Its design d...
An Exploration of the BirdMeertens Formalism
 In STOP Summer School on Constructive Algorithmics, Abeland
, 1989
"... Two formalisms that have been used extensively in the last few years for the calculation of programs are the Eindhoven quantifier notation and the formalism developed by Bird and Meertens. Although the former has always been applied with ultimate goal the derivation of imperative programs and th ..."
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Cited by 32 (3 self)
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Two formalisms that have been used extensively in the last few years for the calculation of programs are the Eindhoven quantifier notation and the formalism developed by Bird and Meertens. Although the former has always been applied with ultimate goal the derivation of imperative programs and the latter with ultimate goal the derivation of functional programs there is a remarkable similarity in the formal games that are played. This paper explores the BirdMeertens formalism by expressing and deriving within it the basic rules applicable in the Eindhoven quantifier notation. 1 Calculation was an endless delight to Moorish scholars. They loved problems, they enjoyed finding ingenious methods to solve them, and sometimes they turned their methods into mechanical devices. (J. Bronowski, The Ascent of Man. Book Club Associates: London (1977).) 1 Introduction Our ability to calculate  whether it be sums, products, differentials, integrals, or whatever  would be woefull...
Generic Downwards Accumulations
 Science of Computer Programming
, 2000
"... . A downwards accumulation is a higherorder operation that distributes information downwards through a data structure, from the root towards the leaves. The concept was originally introduced in an ad hoc way for just a couple of kinds of tree. We generalize the concept to an arbitrary regular d ..."
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Cited by 19 (3 self)
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. A downwards accumulation is a higherorder operation that distributes information downwards through a data structure, from the root towards the leaves. The concept was originally introduced in an ad hoc way for just a couple of kinds of tree. We generalize the concept to an arbitrary regular datatype; the resulting denition is coinductive. 1 Introduction The notion of scans or accumulations on lists is well known, and has proved very fruitful for expressing and calculating with programs involving lists [4]. Gibbons [7, 8] generalizes the notion of accumulation to various kinds of tree; that generalization too has proved fruitful, underlying the derivations of a number of tree algorithms, such as the parallel prex algorithm for prex sums [15, 8], Reingold and Tilford's algorithm for drawing trees tidily [21, 9], and algorithms for query evaluation in structured text [16, 23]. There are two varieties of accumulation on lists: leftwards and rightwards. Leftwards accumulation ...
Program Calculation Properties of Continuous Algebras
, 1991
"... Defining data types as initial algebras, or dually as final coalgebras, is beneficial, if not indispensible, for an algebraic calculus for program construction, in view of the nice equational properties that then become available. It is not hard to render finite lists as an initial algebra and, ..."
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Cited by 19 (0 self)
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Defining data types as initial algebras, or dually as final coalgebras, is beneficial, if not indispensible, for an algebraic calculus for program construction, in view of the nice equational properties that then become available. It is not hard to render finite lists as an initial algebra and, dually, infinite lists as a final coalgebra. However, this would mean that there are two distinct data types for lists, and then a program that is applicable to both finite and infinite lists is not possible, and arbitrary recursive definitions are not allowed. We prove the existence of algebras that are both initial in one category of algebras and final in the closely related category of coalgebras, and for which arbitrary (continuous) fixed point definitions ("recursion") do have a solution. Thus there is a single data type that comprises both the finite and the infinite lists. The price to be paid, however, is that partiality (of functions and values) is unavoidable.
Exploiting symmetry on parallel architectures
, 1995
"... This thesis describes techniques for the design of parallel programs that solvewellstructured problems with inherent symmetry. Part I demonstrates the reduction of such problems to generalized matrix multiplication by a groupequivariant matrix. Fast techniques for this multiplication are described ..."
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Cited by 10 (1 self)
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This thesis describes techniques for the design of parallel programs that solvewellstructured problems with inherent symmetry. Part I demonstrates the reduction of such problems to generalized matrix multiplication by a groupequivariant matrix. Fast techniques for this multiplication are described, including factorization, orbit decomposition, and Fourier transforms over nite groups. Our algorithms entail interaction between two symmetry groups: one arising at the software level from the problem's symmetry and the other arising at the hardware level from the processors' communication network. Part II illustrates the applicability of our symmetryexploitation techniques by presenting a series of case studies of the design and implementation of parallel programs. First, a parallel program that solves chess endgames by factorization of an associated dihedral groupequivariant matrix is described. This code runs faster than previous serial programs, and discovered a number of results. Second, parallel algorithms for Fourier transforms for nite groups are developed, and preliminary parallel implementations for group transforms of dihedral and of symmetric groups are described. Applications in learning, vision, pattern recognition, and statistics are proposed. Third, parallel implementations solving several computational science problems are described, including the direct nbody problem, convolutions arising from molecular biology, and some communication primitives such as broadcast and reduce. Some of our implementations ran orders of magnitude faster than previous techniques, and were used in the investigation of various physical phenomena.
(Relational) Programming Laws in the Boom Hierarchy of Types
 Mathematics of Program Construction
, 1992
"... . In this paper we demonstrate that the basic rules and calculational techniques used in two extensively documented program derivation methods can be expressed, and, indeed, can be generalised within a relational theory of datatypes. The two methods to which we refer are the socalled "BirdMeertens ..."
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Cited by 9 (1 self)
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. In this paper we demonstrate that the basic rules and calculational techniques used in two extensively documented program derivation methods can be expressed, and, indeed, can be generalised within a relational theory of datatypes. The two methods to which we refer are the socalled "BirdMeertens formalism" (see [22]) and the "DijkstraFeijen calculus" (see [15]). The current paper forms an abridged, though representative, version of a complete account of the algebraic properties of the Boom hierarchy of types [19, 18]. Missing is an account of extensionality and the socalled crossproduct. 1 Introduction The "BirdMeertens formalism" (to be more precise, our own conception of it) is a calculus of total functions based on a small number of primitives and a hierarchy of types including trees and lists. The theory was set out in an inspiring paper by Meertens [22] and has been further refined and applied in a number of papers by Bird and Meertens [8, 9, 11, 12, 13]. Its beauty deriv...
Categorical Data Types
 IN SECOND WORKSHOP ON ABSTRACT MODELS FOR PARALLEL COMPUTATION
, 1993
"... An ideal abstract model for parallel computation must carefully balance requirements for effective software engineering with requirements for efficient implementation. Models based on sets of fixed communication/computation patterns satisfy these requirements but, in general, the sets of patterns ar ..."
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Cited by 7 (1 self)
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An ideal abstract model for parallel computation must carefully balance requirements for effective software engineering with requirements for efficient implementation. Models based on sets of fixed communication/computation patterns satisfy these requirements but, in general, the sets of patterns are chosen arbitrarily. Categorical data types are a way of building such models while automatically generating operations, equations, and a guarantee of completeness. We illustrate this construction, and its usefulness for practical problems, by building the type of chemical molecules and showing how molecular properties can be computed in parallel.
The derivation of online algorithms, with an application to finding palindromes
 Algorithmica
, 1994
"... Abstract. A theory for the derivation of online algorithms is presented. The algorithms are derived in the BirdMeertens calculus for program transformations. This calculus provides a concise functional notation for algorithms, and a few powerful theorems for proving equalities of functions. The th ..."
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Cited by 7 (4 self)
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Abstract. A theory for the derivation of online algorithms is presented. The algorithms are derived in the BirdMeertens calculus for program transformations. This calculus provides a concise functional notation for algorithms, and a few powerful theorems for proving equalities of functions. The theory for the derivation of online algorithms is illustrated with the derivation of an algorithm for finding palindromes. An online lineartime random access machine (RAM) algorithm for finding the longest palindromic substring in a string is derived, For the purpose of finding the longest palindromic substring, all maximal palindromic substrings are computed. The list of maximal palindromes obtained in the computation of the longest palindrome can be used for other purposes such as finding the largest palindromic rectangle in a matrix and finding the shortest partition of a string into palindromes. Key Words. Derivation of online algorithms, Transformational programming, BirdMeertens calcu