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It is easy to write a circuit specification in a pure functional language so that execution of the specification simulates the circuit. It's harder to make an executable specification generate the circuit's netlist without using impure language features. The difficulty is that a circuit specification evaluates to a graph isomorphic to the circuit, so the specification of a circuit with feedback will evaluate to a circular (or infinite) graph. That prevents a naive graph traversal algorithm written in a pure functional language from terminating. This paper solves the problem by requiring the circuit specification to name components explicitly. With suitable higher order functions, the naming can be achieved without placing an undue burden on the circuit designer. This approach clarifies the distinction between transformations that preserve both the behaviour and structure of a circuit and transformations that preserve the behaviour while possibly changing the structure. It a...

This paper considers the expression and derivation of efficient data parallel programs for SIMD and MIMD machines. It is shown that efficient parallel programs must utilise both sequential and parallel computation; these are termed hybrid programs. The Bird--Meertens formalism, a calculus of higher order functions, is used to derive and express programs. Our goal is to derive efficient parallel programs for a variety of machines by: starting with an abstract specification, deriving an abstract algorithm and successively refining this to more efficient and machine dependent algorithms incorporating greater implementation detail. Nested data structures are used to express hybrid algorithms. Using this technique efficient accumulate (scan/parallel prefix) algorithms are derived for SIMD and MIMD machines. 1 Introduction The main reason for parallel programming is to achieve high performance. Unfortunately designing and writing efficient parallel programs, especially for MIMD machines, i...

The main contribution of this thesis is a study of the dynamic programming and greedy strategies for solving combinatorial optimization problems. The study is carried out in the context of a calculus of relations, and generalises previous work by using a loop operator in the imperative programming style for generating feasible solutions, rather than the fold and unfold operators of the functional programming style. The relationship between fold operators and loop operators is explored, and it is shown how to convert from the former to the latter. This fresh approach provides additional insights into the relationship between dynamic programming and greedy algorithms, and helps to unify previously distinct approaches to solving combinatorial optimization problems. Some of the solutions discovered are new and solve problems which had previously proved difficult. The material is illustrated with a selection of problems and solutions that is a mixture of old and new. Another contribution is the invention of a new calculus, called the graph calculus, which is a useful tool for reasoning in the relational calculus and other non-relational calculi. The graph

This paper presents a formal semantics for vectors and matrices, suitable for static type-checking. This is not available in apl, which produces run-time type errors, or in the usual functional languages, where matrices are typically implemented by lists of lists. Here, a matrix is a vector of vectors. Vectors are distinguished from lists by requiring that vector computations determine the length of the result from that of the argument, without reference to values. This leads to a two-level semantics, with values above and shapes below. Each operation must then specify its action on shapes as well as its action on values. Vectors and matrices inherit much of their structure from lists. In particular, the monadic structure given by singleton lists and the flattening of lists of lists extends in this way. Some new constructions, such as transposition of matrices, have no list counterpart. The power of this calculus for vector and matrix algebra is sufficient to represent the discrete Fou...

this paper we show that the latter axiomatization is equal (in a categorical sense) to an enrichment of the former. The approach by Freyd (see definitions 2.1 and 2.2) is a very smooth axiomatization of the notion of category of relations. But, as Carboni observed [Car93], the theory is rather rigid. In particular, the modular law is not satisfactory from a theoretical point of view: it can not even be stated unless one has an involution which is the "identity on objects", which in nature never occurs unless one is already in a category of relations. Hence Carboni and Walters proposed another axiomatization (definition 2.4 below) which is "more categorical". So both approaches have their strong points, and the result of this paper allows to exploit them both without any penalty at all. This seems to be particular relevant since categories of relations have been used recently in theoretical computer science to model nondeterministic programs [HJ86, She90, Man85]. 2 Two categories of relations