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Multilinear Algebra and Chess Endgames
 of No Chance: Combinatorial Games at MRSI
, 1996
"... Abstract. This article has three chief aims: (1) To show the wide utility of multilinear algebraic formalism for highperformance computing. (2) To describe an application of this formalism in the analysis of chess endgames, and results obtained thereby that would have been impossible to compute usi ..."
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Abstract. This article has three chief aims: (1) To show the wide utility of multilinear algebraic formalism for highperformance computing. (2) To describe an application of this formalism in the analysis of chess endgames, and results obtained thereby that would have been impossible to compute using earlier techniques, including a win requiring a record 243 moves. (3) To contribute to the study of the history of chess endgames, by focusing on the work of Friedrich Amelung (in particular his apparently lost analysis of certain sixpiece endgames) and that of Theodor Molien, one of the founders of modern group representation theory and the first person to have systematically numerically analyzed a pawnless endgame. 1.
The Derivation of a Hierarchy of Algorithms for Pattern Matching on Arrays
 Proceedings ATABLE92, Second international workshop on array structures
, 1992
"... This paper derives a hierarchy of algorithms for pattern matching on arrays in the BirdMeertens calculus for program transformation. In this calculus, both specifications and algorithms are functions, and a few highlevel theorems are used as transformation rules. An algorithm is derived from its s ..."
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This paper derives a hierarchy of algorithms for pattern matching on arrays in the BirdMeertens calculus for program transformation. In this calculus, both specifications and algorithms are functions, and a few highlevel theorems are used as transformation rules. An algorithm is derived from its specification by means of a calculation which typically consists of a sequence of equalities, each an instantiation of a highlevel theorem or a definition. Aspects of the BirdMeertens calculus can be found in [4], [5], [9], [14], [15], and [13]. The laws we use in the derivation are derived from the definition of the data type
An exercise in Transformational Programming: Backtracking and BranchandBound
 Science of Computer Programming
, 2004
"... We present a formal derivation of program schemes that are usually called Backtracking programs and BranchandBound programs. The derivation consists of a series of transformation steps, specifically algebraic manipulations, on the initial specification until the desired programs are obtained. T ..."
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We present a formal derivation of program schemes that are usually called Backtracking programs and BranchandBound programs. The derivation consists of a series of transformation steps, specifically algebraic manipulations, on the initial specification until the desired programs are obtained. The wellknown notions of linear recursion and tail recursion are extended, for structures, to elementwise linear recursion and elementwise tail recursion; and a transformation between them is derived too.
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
A Framework for Refining Functional Specifications into Parallel Reconfigurable Hardware Implementations
, 2005
"... Reconfigurable logic devices such as the FPGA have brought about a revolution in the field of hardware design. The reduction in development costs has had a huge impact on broadening the scope of applications for which a hardware implementation is a realistic possibility. Current FPGA devices run to ..."
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Reconfigurable logic devices such as the FPGA have brought about a revolution in the field of hardware design. The reduction in development costs has had a huge impact on broadening the scope of applications for which a hardware implementation is a realistic possibility. Current FPGA devices run to many millions of gates, giving a huge potential for efficiency gains, benefiting from the inherently parallel nature of hardware circuits. These devices continue to grow in size, to the end that we can now seriously consider implementing even large scale systems purely in reconfigurable logic. Despite these advances, we find ourselves somewhat lacking in the tools and methodologies required to fully exploit this potential. Issues of hardware implementation and parallelism introduce significant complexity into the design process. We argue that without the correct approach, not only will this potential be under used, but the inherent complexity will undermine people’s