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Generic programming: An introduction
 3rd International Summer School on Advanced Functional Programming
, 1999
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A CSP Approach To Action Systems
, 1992
"... The communicating sequential processes (CSP) formalism, introduced by Hoare [Hoa85], is an eventbased approach to distributed computing. The actionsystem formalism, introduced by Back & KurkiSuonio [BKS83], is a statebased approach to distributed computing. Using weakestprecondition formulae, M ..."
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The communicating sequential processes (CSP) formalism, introduced by Hoare [Hoa85], is an eventbased approach to distributed computing. The actionsystem formalism, introduced by Back & KurkiSuonio [BKS83], is a statebased approach to distributed computing. Using weakestprecondition formulae, Morgan [Mor90a] has defined a correspondence between action systems and the failuresdivergences model for CSP. Simulation is a proof technique for showing refinement of action systems. Using the correspondence of [Mor90a], Woodcock & Morgan [WM90] have shown that simulation is sound and complete in the CSP failuresdivergences model. In this thesis, Morgan's correspondence is extended to the CSP infinitetraces model [Ros88] in order to deal more properly with unbounded nondeterminism. It is shown that simulation is sound in the infinitetraces model, though completeness is lost in certain cases. The new correspondence is then extended to include a notion of internal action. This allows the ...
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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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.
Constructive Lattice Theory
, 1993
"... A notion of simulation of one datatype by another is defined as a constructive preorder. A calculus of datatype simulation is then developed by formulating constructive versions of leastfixedpoint theorems in lattice theory. The calculus is applied to the construction of several isomorphisms betwe ..."
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A notion of simulation of one datatype by another is defined as a constructive preorder. A calculus of datatype simulation is then developed by formulating constructive versions of leastfixedpoint theorems in lattice theory. The calculus is applied to the construction of several isomorphisms between classes of datatypes. In particular constructive adaptations of theorems in lattice theory about closure operators are shown to yield simulations and isomorphisms between monad structures, and constructive adaptations of theorems in regular algebra are shown to yield isomorphisms between list structures. A question to which any respectable theory of datatypes should provide immediate answers is when two datatypes are isomorphic, i.e. entirely equivalent modulo implementation details. A subsidiary question is when one datatype simulates another. This second question is of interest in its own right but is also important to answering the first question since isomorphism is frequently reduce...
A General Scheme for BreadthFirst Graph Traversal
 Mathematics of Program Construction. SpringerVerlag LNCS 947
, 1995
"... . We survey an algebra of formal languages suitable to deal with graph algorithms. As an example of its use we derive a general scheme for breadthfirst graph traversal. This general scheme is then applied to a reachability and a shortest path problem. 1 Introduction In books about algorithmic gra ..."
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. We survey an algebra of formal languages suitable to deal with graph algorithms. As an example of its use we derive a general scheme for breadthfirst graph traversal. This general scheme is then applied to a reachability and a shortest path problem. 1 Introduction In books about algorithmic graph theory algorithms are usually presented without formal specification and formal development. Some approaches, in contrast, provide a more precise treatment of graph algorithms, resulting in algorithms which are not only correct "in principle" but also in all details (see e.g. [Pair 70], [Backhouse, van Gasteren 92], [van den Eijnde 92], [Moller, Russling 92], [Gibbons 94] and [King, Launchbury 95]). In the present paper a uniform treatment of a class of graph algorithms is given using the algebra of formal languages and relations presented in [Moller, Russling 92], a straightened version of the framework introduced in [Moller 91]. The emphasis is not laid on the invention of new algorithm...
Category Theory as Coherently Constructive Lattice Theory
, 1998
"... ion Theorem : : : : : : : : : : : : : : : : : : : : : : : : : : : 82 6.3 The Beautiful Theorem : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 86 6.4 The Rolling Rule : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 89 6.5 The Square Theorem : : : : : : : : : : : : : : : : : ..."
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ion Theorem : : : : : : : : : : : : : : : : : : : : : : : : : : : 82 6.3 The Beautiful Theorem : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 86 6.4 The Rolling Rule : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 89 6.5 The Square Theorem : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 91 6.6 The Exchange Rule : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 94 6.7 The Diagonal Rule : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 100 6.7.1 One Half : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 100 6.7.2 The Other Half : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 103 6.8 Mutual Recursion : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 106 7 Monads 111 7.1 Introduction : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 111 7.2 Monads and Adjunctions : : : : : : : : : : : : : : : : : : : : : : : : : : : : 112 7.3 Basic Adjunction : : : : : : : : : : : : ...
Calculating Graph Algorithms for Dominance and Shortest Path ⋆
"... Abstract. We calculate two iterative, polynomialtime graph algorithms from the literature: a dominance algorithm and an algorithm for the singlesource shortest path problem. Both algorithms are calculated directly from the definition of the properties by fixedpoint fusion of (1) a least fixed poi ..."
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Abstract. We calculate two iterative, polynomialtime graph algorithms from the literature: a dominance algorithm and an algorithm for the singlesource shortest path problem. Both algorithms are calculated directly from the definition of the properties by fixedpoint fusion of (1) a least fixed point expressing all finite paths through a directed graph and (2) Galois connections that capture dominance and path length. The approach illustrates that reasoning in the style of fixedpoint calculus extends gracefully to the domain of graph algorithms. We thereby bridge common practice from the school of program calculation with common practice from the school of static program analysis, and build a novel view on iterative graph algorithms as instances of abstract interpretation.
Category Theory as Coherently Constructive Lattice Theory: An Illustration
, 1995
"... Dijkstra and Scholten have formulated a theorem stating that all disjunctivity properties of a predicate transformer are preserved by the construction of least prefix points. An alternative proof of their theorem is presented based on two fundamental fixed point theorems, the abstraction theorem ..."
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Dijkstra and Scholten have formulated a theorem stating that all disjunctivity properties of a predicate transformer are preserved by the construction of least prefix points. An alternative proof of their theorem is presented based on two fundamental fixed point theorems, the abstraction theorem and the fusion theorem, and the fact that suprema in a lattice are defined by a Galois connection. The abstraction theorem seems to be new; the fusion theorem is known but its importance does not seem to be fully recognised. The abstraction theorem, the fusion theorem, and Dijkstra and Scholten's theorem are then generalised to the context of category theory and shown to be valid. None of the theorems in this context seems to be known, although specific instances of Dijkstra and Scholten's theorem are known. The main point of the paper is to discuss the process of drawing inspiration from lattice theory to formulate theorems in category theory (first advocated by Lambek in 1968). W...
Calculating with Relations for Graph Algorithmics
, 1997
"... Much emphasis has been placed in recent years on deriving or calculating programs rather than proving them correct. Adequate calculational frameworks are needed to support such an approach. The present work explores the use of a calculus of relations to express and reason about graph properties ..."
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Much emphasis has been placed in recent years on deriving or calculating programs rather than proving them correct. Adequate calculational frameworks are needed to support such an approach. The present work explores the use of a calculus of relations to express and reason about graph properties in an algorithmic context. We take a generic program that computes a maximal set, over some universe, satisfying some predicate P and calculate two instances of it: the computation of maximal independent sets of vertices in a graph, and the computation of maximal sets of edges without cycles (i.e. maximal forests).