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The application of a sequence notation to the design of systolic computations

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Abstract

The sequence notation suggested in a previous paper provides a tool for the clear and precise specification of systolic computations. Namely, it separates the static and dynamic levels of the specification. At the static level, the topology of the network and the function of each cell are described by a system of causal equations on sequences, and at the dynamic level, the data flow is described by the elements of the individual sequences.

In this paper, we describe a technique for the transformation of a given algorithm into a system of causal sequence equations/input-output description which specifies a systolic computation. The basic idea of the technique is to pack arrays of variables along one or more dimensions into sequences. Doing this, however, may result in a system of equations that is not causal, and hence, a transformation of indices in the original algorithm may be essential in order to guarantee causality (the positive increment of time).

The derivation of index transformations from the data dependence vectors of an algorithm was discussed in the literature. However, data dependence vectors do not carry any information about absolute values of the indices, and hence, allow only the derivation of linear transformations. In order to overcome this problem, we suggest a method for the derivation of the index transformation from 〈defined, used〉 pairs. These pairs retain information about the absolute values of the indices, and thus allow for nonlinear transformations.

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Melhem, R., Guerra, C. The application of a sequence notation to the design of systolic computations. BIT 29, 409–427 (1989). https://doi.org/10.1007/BF02219228

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