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13
Datatype Laws without Signatures
, 1996
"... ing from syntax. Conventionally an equation for algebra ' is just a pair of terms built from variables, the constituent operations of ' , and some fixed standard operations. An equation is valid if the two terms are equal for all values of the variables. We are going to model a syntactic t ..."
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Cited by 22 (6 self)
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ing from syntax. Conventionally an equation for algebra ' is just a pair of terms built from variables, the constituent operations of ' , and some fixed standard operations. An equation is valid if the two terms are equal for all values of the variables. We are going to model a syntactic term as a morphism that has the values of the variables as source. For example, the two terms ` x ' and ` x join x ' (with variable x of type tree ) are modeled by morphisms id and id \Delta id ; join of type tree ! tree . So, an equation for ' is modeled by a pair of terms (T '; T 0 ') , T and T 0 being mappings of morphisms which we call `transformers'. This faces us with the following problem: what properties must we require of an arbitrary mapping T in order that it model a classical syntactic Datatype Laws without Signatures 7 term? Or, rather, what properties of classical syntactic terms are semantically essential, and how can we formalise these as properties of a transformer T ? Of course...
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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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.
Structured Parallel Programming: Theory meets Practice
 Research Directions in Computer Science
, 1995
"... We address the issue of what should be the proper relationship between theoretical computer science and practical computing. Starting from an analysis of what we perceive of as the failure of formally based research to have as much impact on practical computing as is merited we propose a diagnos ..."
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Cited by 10 (2 self)
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We address the issue of what should be the proper relationship between theoretical computer science and practical computing. Starting from an analysis of what we perceive of as the failure of formally based research to have as much impact on practical computing as is merited we propose a diagnosis based on the way formally based research is conducted and the way it is envisaged that results from these areas will be translated into practice. We suggest that it is the responsibility of practitioners of theoretical computer science to work more closely with the practical areas in order to identify ways in which their ideas can be used to augment current practice rather than seeking to replace it. As a case in point we examine functional programming and its relationship to programming parallel machines. We introduce a development, structured parallel programming, that seeks to combine the theoretical advantages of functional programming with established practice in these area...
From Transformations to Methodology in Parallel Program Development: A Case Study
 Microprocessing and Microprogramming
, 1996
"... The BirdMeertens formalism (BMF) of higherorder functions over lists is a mathematical framework supporting formal derivation of algorithms from functional specifications. This paper reports results of a case study on the systematic use of BMF in the process of parallel program development. We dev ..."
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Cited by 4 (1 self)
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The BirdMeertens formalism (BMF) of higherorder functions over lists is a mathematical framework supporting formal derivation of algorithms from functional specifications. This paper reports results of a case study on the systematic use of BMF in the process of parallel program development. We develop a parallel program for polynomial multiplication, starting with a straightforward mathematical specification and arriving at the target processor topology together with a program for each processor of it. The development process is based on formal transformations; design decisions concerning data partitioning, processor interconnections, etc. are governed by formal type analysis and performance estimation rather than made ad hoc. The parallel target implementation is parameterized for an arbitrary number of processors; for the particular number, the target program is both time and costoptimal. We compare our results with systolic solutions to polynomial multiplication.
Separation of Correctness and Complexity in Algorithm Design
, 1993
"... this paper we propose a new approach to the design of algorithms. This approach is based on the view that all algorithms are composed of a computation and a control component, and that these components can be designed separately. The computation component is responsible for the correctness of an alg ..."
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Cited by 3 (3 self)
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this paper we propose a new approach to the design of algorithms. This approach is based on the view that all algorithms are composed of a computation and a control component, and that these components can be designed separately. The computation component is responsible for the correctness of an algorithm. It embodies the computational knowledge about a problem domain that is needed to solve the corresponding problem. The control component governs complexity aspects of the solution method by directing the usage of the computational knowledge. This bisection allows the problem of how to construct an algorithm to be split into two smaller problems: "What are the elementary units of computational knowledge for the problem at hand?" and "In what order should these units be used to (efficiently) obtain a solution." This way, the concerns of correctness and complexity are separated. We assert that there are many advantages in separating correctness and complexity in algorithm design. Hence, we examine properties of programming formalisms that influence the possibility of separating the design of the computation and control components of an algorithm.
Using underspecification in the derivation of some optimal partition algorithms
, 1990
"... Indeterminacy is inherent in the specification of optimal partition (and many more) algorithms, even though the algorithms themselves may be fully determinate. Indeterminacy is a notoriously hard phenomenon to deal with in a purely functional setting. In the paper “A Calculus Of Functions for Progra ..."
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Indeterminacy is inherent in the specification of optimal partition (and many more) algorithms, even though the algorithms themselves may be fully determinate. Indeterminacy is a notoriously hard phenomenon to deal with in a purely functional setting. In the paper “A Calculus Of Functions for Program Derivation ” R. S. Bird tries to handle it by using underspecified functions. (Other authors have proposed to use ‘indeterminate ’ functions, and to use relations instead of functions.) In this paper we redo Bird’s derivation of the Leery and Greedy algorithm while being very precise about underspecification, and still staying in the functional framework. It turns out that Bird’s theorems are not exactly what one would like to have, and what one might understand from his wording of the theorems. We also give a derivation in the BirdMeertens style of a (linear time) optimal partition algorithm that was originally found by J. C. S. P. van der Woude. 1
Efficient Functional Programming Communication Functions on the AP1000
, 1994
"... One problem of parallel computing is that parallel computers vary greatly in architecture so that a program written to run efficiently on a particular architecture, when porting to a different architecture, would often need to be changed and adapted substantially in order to run with reasonable perf ..."
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One problem of parallel computing is that parallel computers vary greatly in architecture so that a program written to run efficiently on a particular architecture, when porting to a different architecture, would often need to be changed and adapted substantially in order to run with reasonable performance on the target architecture. Porting with performance is, hence, labourintensive and costly. A method of parallel programming using the BirdMeertens Formalism where programs are formulated as compositions of (mainly) higher order functions on some data type in the data parallel functional style has been proposed as a solution. The library of (mainly) higherorder functions in which all communication and parallelism in a program is embedded could (it is argued) be implemented efficiently on different parallel architectures. This gives the advantage of portability between different architectures with reasonable and predictable performance without change in program source. ...
Homo and Catamorphisms, Reductions and Maps An Overview
, 1990
"... The classical notion of homomorphism, the BirdMeertens notion of homomorphism — here called catamorphism — having reduction and map as a special case, are defined formally and investigated with respect to their calculational properties. All this is done both in terse category theoretic terminology, ..."
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The classical notion of homomorphism, the BirdMeertens notion of homomorphism — here called catamorphism — having reduction and map as a special case, are defined formally and investigated with respect to their calculational properties. All this is done both in terse category theoretic terminology, as well as spelled out in more conventional, less abstract, concepts. This note reflects my current understanding of other people’s work, in particular the ideas of Lambert Meertens. 1
Belfast BT7 1NN,
, 1995
"... functional specification 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 ..."
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functional specification 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.