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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 p ..."
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Cited by 36 (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...
Parallelization in Calculational Forms
 In 25th ACM Symposium on Principles of Programming Languages
, 1998
"... The problems involved in developing efficient parallel programs have proved harder than those in developing efficient sequential ones, both for programmers and for compilers. Although program calculation has been found to be a promising way to solve these problems in the sequential world, we believe ..."
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Cited by 33 (25 self)
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The problems involved in developing efficient parallel programs have proved harder than those in developing efficient sequential ones, both for programmers and for compilers. Although program calculation has been found to be a promising way to solve these problems in the sequential world, we believe that it needs much more effort to study its effective use in the parallel world. In this paper, we propose a calculational framework for the derivation of efficient parallel programs with two main innovations:  We propose a novel inductive synthesis lemma based on which an elementary but powerful parallelization theorem is developed.  We make the first attempt to construct a calculational algorithm for parallelization, deriving associative operators from data type definition and making full use of existing fusion and tupling calculations. Being more constructive, our method is not only helpful in the design of efficient parallel programs in general but also promising in the construc...
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.
Calculating Accumulations
, 1999
"... this paper, we shall formulate accumulations as higher order catamorphisms , and propose several general transformation rules for calculating accumulations (i.e., finding and manipulating accumulations) by calculationbased (rather than a searchbased) program transformation methods. Some examples ..."
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Cited by 16 (6 self)
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this paper, we shall formulate accumulations as higher order catamorphisms , and propose several general transformation rules for calculating accumulations (i.e., finding and manipulating accumulations) by calculationbased (rather than a searchbased) program transformation methods. Some examples are given for illustration.
A library of constructive skeletons for sequential style of parallel programming
 In InfoScale ’06: Proceedings of the 1st international conference on Scalable information systems, volume 152 of ACM International Conference Proceeding Series
, 2006
"... With the increasing popularity of parallel programming environments such as PC clusters, more and more sequential programmers, with little knowledge about parallel architectures and parallel programming, are hoping to write parallel programs. Numerous attempts have been made to develop highlevel pa ..."
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Cited by 14 (8 self)
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With the increasing popularity of parallel programming environments such as PC clusters, more and more sequential programmers, with little knowledge about parallel architectures and parallel programming, are hoping to write parallel programs. Numerous attempts have been made to develop highlevel parallel programming libraries that use abstraction to hide lowlevel concerns and reduce difficulties in parallel programming. Among them, libraries of parallel skeletons have emerged as a promising way towards this direction. Unfortunately, these libraries are not well accepted by sequential programmers, because of incomplete elimination of lowerlevel details, adhoc selection of library functions, unsatisfactory performance, or lack of convincing application examples. This paper addresses principle of designing skeleton libraries of parallel programming and reports implementation details and practical applications of a skeleton library SkeTo. The SkeTo library is unique in its feature that it has a solid theoretical foundation based on the theory of Constructive Algorithmics, and is practical to be used to describe various parallel computations in a sequential manner. 1.
The Automated Transformation of Abstract Specifications of Numerical Algorithms into Efficient Array Processor Implementations
 Science of Computer Programming
, 1997
"... We present a set of program transformations which are applied automatically to convert abstract functional specifications of numerical algorithms into efficient implementations tailored to the AMT DAP array processor. The transformations are based upon a formal algebra of a functional array form, wh ..."
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Cited by 12 (5 self)
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We present a set of program transformations which are applied automatically to convert abstract functional specifications of numerical algorithms into efficient implementations tailored to the AMT DAP array processor. The transformations are based upon a formal algebra of a functional array form, which provides a functional model of the array operations supported by the DAP programming language. The transformations are shown to be complete. We present specifications and derivations of two example algorithms: an algorithm for computing eigensystems and an algorithm for solving systems of linear equations. For the former, we compare the execution performance of the implementation derived by transformation with the performance of an independent, manually constructed implementation; the efficiency of the derived implementation matches that of the manually constructed implementation.
Computing Downwards Accumulations on Trees Quickly
, 1995
"... Downwards passes on binary trees are essentially functions which pass information down a tree, from the root towards the leaves. Under certain conditions, a downwards pass is both `efficient' (computable in a functional style in parallel time proportional to the depth of the tree) and `manipula ..."
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Cited by 9 (3 self)
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Downwards passes on binary trees are essentially functions which pass information down a tree, from the root towards the leaves. Under certain conditions, a downwards pass is both `efficient' (computable in a functional style in parallel time proportional to the depth of the tree) and `manipulable' (enjoying a number of distributivity properties useful in program construction); we call a downwards pass satisfying these conditions a downwards accumulation. In this paper, we show that these conditions do in fact yield a stronger conclusion: the accumulation can be computed in parallel time proportional to the logarithm of the depth of the tree, on a Crew Pram machine.
(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 "BirdMee ..."
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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...
Promotional Transformation on Monadic Programs
, 1995
"... this paper, we propose a new theory on monadic catamorphism bymoving Fokkinga's assumption on the monad to the condition of a map between monadic algebras so that our theory is valid for arbitrary monads including, for example, the state monad that is not allowed in Fokkinga's theory. Our ..."
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Cited by 9 (0 self)
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this paper, we propose a new theory on monadic catamorphism bymoving Fokkinga's assumption on the monad to the condition of a map between monadic algebras so that our theory is valid for arbitrary monads including, for example, the state monad that is not allowed in Fokkinga's theory. Our theory covers Fokkinga's as a special case. Moreover, Meijer and Jeuring's informal transformation rules of monadic programs in their case study is actually an instance of our general promotion theorem.
Making Formality Work For Us
 EATCS Bulletin
, 1989
"... Formal reasoning is notoriously long and arduous; in order to use it to reason effectively in the construction of programs it is, therefore, paramount that we design our notations to be both clear and economical. Taking examples from AI, from imperative programming, from the use of the BirdMeer ..."
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Cited by 7 (2 self)
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Formal reasoning is notoriously long and arduous; in order to use it to reason effectively in the construction of programs it is, therefore, paramount that we design our notations to be both clear and economical. Taking examples from AI, from imperative programming, from the use of the BirdMeertens formalism and from category theory we demonstrate how the right choice of what to denote and how it is denoted can make significant improvements to formal calculations. Brief mention is also made of the connection between economical notation and properties of type. 1 2 Foreword Earlier this year I was an invited speaker at the 5th British Computer Society Theoretical Computer Science Colloquium held at Royal Holloway and Bedford New College, London. Before you is the text of my lecture, almost but not quite as given at the conference. (Perhaps the best way to describe the present paper is as the lecture that I should have given.) The publication of the text of the lecture will, ...