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14
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...
The Third Homomorphism Theorem
, 1995
"... The Third Homomorphism Theorem is a folk theorem of the constructive algorithmics community. It states that a function on lists that can be computed both from left to right and from right to left is necessarily a list homomorphismit can be computed according to any parenthesization of the list. ..."
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Cited by 26 (3 self)
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The Third Homomorphism Theorem is a folk theorem of the constructive algorithmics community. It states that a function on lists that can be computed both from left to right and from right to left is necessarily a list homomorphismit can be computed according to any parenthesization of the list. We formalize and prove the theorem, and use it to improveanO#n 2 # sorting algorithm to O#n log n#. 1Introduction List homomorphisms are those functions on #nite lists that promote through list concatenationthat is, functions h for which there exists a binary operator # such that, for all #nite lists x and y , h #x ++ y# = hx#hy where `++' denotes list concatenation. Such functions are ubiquitous in functional programming. Some examples of list homomorphisms are: # the identity function id ; # the map function map f , which applies a given function f to every elementof a list; # the function concat , which concatenates a list of lists into a single long list; # the function ...
Systematic Extraction and Implementation of DivideandConquer Parallelism
 Programming languages: Implementation, Logics and Programs, Lecture Notes in Computer Science 1140
, 1996
"... Homomorphisms are functions that match the divideandconquer paradigm and thus can be computed in parallel. Two problems are studied for homomorphisms on lists: (1) parallelism extraction: finding a homomorphic representation of a given function; (2) parallelism implementation: deriving an efficien ..."
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Cited by 18 (5 self)
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Homomorphisms are functions that match the divideandconquer paradigm and thus can be computed in parallel. Two problems are studied for homomorphisms on lists: (1) parallelism extraction: finding a homomorphic representation of a given function; (2) parallelism implementation: deriving an efficient parallel program that computes the function. A systematic approach to parallelism extraction proceeds by generalization of two sequential representations based on traditional cons lists and dual snoc lists. For some nonhomomorphic functions, e.g., the maximum segment sum problem, our method provides an embedding into a homomorphism. The implementation is addressed by introducing a subclass of distributable homomorphisms and deriving for them a parallel program schema, which is time optimal on the hypercube architecture. The derivation is based on equational reasoning in the BirdMeertens formalism, which guarantees the correctness of the parallel target program. The approach is illustrated with function...
Extracting and Implementing List Homomorphisms in Parallel Program Development
 Science of Computer Programming
, 1997
"... this paper, we study functions called list homomorphisms, which represent a particular pattern of parallelism. ..."
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Cited by 12 (0 self)
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this paper, we study functions called list homomorphisms, which represent a particular pattern of parallelism.
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.
Parallelizing Functional Programs by Generalization
 Journal of Functional Programming
, 1997
"... List homomorphisms are functions that are parallelizable using the divideandconquer paradigm. We study the problem of finding a homomorphic representation of a given function, based on the BirdMeertens theory of lists. A previous work proved that to each pair of leftward and rightward sequential ..."
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Cited by 9 (1 self)
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List homomorphisms are functions that are parallelizable using the divideandconquer paradigm. We study the problem of finding a homomorphic representation of a given function, based on the BirdMeertens theory of lists. A previous work proved that to each pair of leftward and rightward sequential representations of a function, based on cons and snoclists, respectively, there is also a representation as a homomorphism. Our contribution is a mechanizable method to extract the homomorphism representation from a pair of sequential representations. The method is decomposed to a generalization problem and an inductive claim, both solvable by term rewriting techniques. To solve the former we present a sound generalization procedure which yields the required representation, and terminates under reasonable assumptions. We illustrate the method and the procedure by the parallelization of the scanfunction (parallel prefix). The inductive claim is provable automatically.
List Homomorphic Parallel Algorithms for Bracket Matching
 Department of Computer Science, University of Edinburgh
, 1993
"... We present a family of parallel algorithms for simple language recognition problems involving bracket matching. The algorithms are expressed in the BirdMeertens Formalism, exploiting only list operations which are inherently massively parallel. Our intention is to illustrate the practical efficacy ..."
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Cited by 9 (0 self)
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We present a family of parallel algorithms for simple language recognition problems involving bracket matching. The algorithms are expressed in the BirdMeertens Formalism, exploiting only list operations which are inherently massively parallel. Our intention is to illustrate the practical efficacy with which such algorithms can be derived and expressed given the support of a well understood theoretical foundation. One of the variants produced is of particular interest in that it exploits the same theoretical result twice to produce nested parallelism. 1 Introduction In [8], we investigated an informal methodology for the generation of parallel algorithms based upon exploitation of a fundamental result from the BirdMeertens "theory of lists". Our main example was an algorithm for the maximum segment sum problem. In this report we provide further examples of the approach. For completeness, the remainder of this section and sections 2 and 3 repeat the introductory material from [8]. Re...
Construction of List Homomorphisms by Tupling and Fusion
, 1996
"... List homomorphisms are functions which can be efficiently computed in parallel since they ideally suit the divideandconquer paradigm. However, some interesting functions, e.g., the maximum segment sum problem, are not list homomorphisms. In this paper, we propose a systematic way of embedding them ..."
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Cited by 5 (4 self)
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List homomorphisms are functions which can be efficiently computed in parallel since they ideally suit the divideandconquer paradigm. However, some interesting functions, e.g., the maximum segment sum problem, are not list homomorphisms. In this paper, we propose a systematic way of embedding them into list homomorphisms so that parallel programs are derived. We show, with an example, how a simple, and "obviously" correct, but possibly inefficient solution to the problem can be successfully turned into a semantically equivalent almost homomorphism by means of two transformations: tupling and fusion.
Parallelizing Functional Programs by Term Rewriting
, 1997
"... List homomorphisms are functions that can be computed in parallel using the divideandconquer paradigm. We study the problem of finding a homomorphic representation of a given function, based on the BirdMeertens theory of lists. A previous work proved that to each pair of leftward and rightward se ..."
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Cited by 2 (2 self)
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List homomorphisms are functions that can be computed in parallel using the divideandconquer paradigm. We study the problem of finding a homomorphic representation of a given function, based on the BirdMeertens theory of lists. A previous work proved that to each pair of leftward and rightward sequential representations of a function, based on cons and snoclists, respectively, there is also a representation as a homomorphism. Our contribution is a mechanizable method to extract the homomorphism representation from a pair of sequential representations. The method is decomposed to a generalization problem and an inductive claim, both solvable by term rewriting techniques. To solve the former we present a sound generalization procedure which yields the required representation, and terminates under reasonable assumptions. We illustrate the method and the procedure by the parallelization of the scanfunction (parallel prefix). The inductive claim is provable automatically. Keywords: P...