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The Set Theory of Arithmetic Decomposition
August 1990 (vol. 39 no. 8)
pp. 993-1005

The set theory of arithmetic decomposition is a method of designing complex addition/subtraction circuits at any radix using strictly positional, sign-local number systems. The specification of an addition circuit is simply an equation that describes the inputs and the outputs as weighted digit sets. Design is done by applying a set of rewrite rules known as decomposition operators to the equation. The order in which and weight at which each operator is applied maps directly to a physical implementation, including both multiple-level logic and connectivity. The method is readily automated, and has been used to design some higher radix arithmetic circuits. It is possible to compute the cost of a given adder before the detailed design is complete.

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Index Terms:
subtraction circuits; set theory; arithmetic decomposition; strictly positional; sign-local number systems; addition circuit; equation; inputs; outputs; weighted digit sets; rewrite rules; decomposition operators; multiple-level logic; connectivity; radix arithmetic circuits; digital arithmetic; logic circuits; logic design; many-valued logics; set theory.
Citation:
T.M. Carter, J.E. Robertson, "The Set Theory of Arithmetic Decomposition," IEEE Transactions on Computers, vol. 39, no. 8, pp. 993-1005, Aug. 1990, doi:10.1109/12.57037
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