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Issue No.03 - March (1999 vol.48)

pp: 296-310

DOI Bookmark: http://doi.ieeecomputersociety.org/10.1109/12.754996

ABSTRACT

<p><b>Abstract</b>—Consider a hypercube of <tmath>$2^n$</tmath> points described by <tmath>$n$</tmath> Boolean variables and a subcube of <tmath>$2^m$</tmath> points, <tmath>$m \leq n$</tmath>. As is well-known, the Boolean function with value 1 in the points of the subcube can be expressed as the product (AND) of <tmath>$n-m$</tmath> variables. The standard synthesis of arbitrary functions exploits this property. We extend the concept of subcube to the more powerful <it>pseudocube</it>. The basic set is still composed of <tmath>$2^m$</tmath> points, but has a more general form. The function with value 1 in a pseudocube, called <it>pseudoproduct</it>, is expressed as the AND of <tmath>$n-m$</tmath> EXOR-factors, each containing at most <tmath>$m+1$</tmath> variables. Subcubes are special cases of pseudocubes and their corresponding pseudoproducts reduce to standard products. An arbitrary Boolean function can be expressed as a sum of pseudoproducts (SPP). This expression is in general much shorter than the standard sum of products, as demonstrated on some known benchmarks. The logical network of an <tmath>$n$</tmath>-bit adder is designed in SPP, as a relevant example of application of this new technique. A class of symmetric functions is also defined, particularly suitable for SPP representation.</p>

INDEX TERMS

Pseudocube, pseudoproduct, EXOR-factor, Boolean function, algebraic expression, logical design.

CITATION

Fabrizio Luccio, "On a New Boolean Function with Applications",

*IEEE Transactions on Computers*, vol.48, no. 3, pp. 296-310, March 1999, doi:10.1109/12.754996