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On a New Boolean Function with Applications
March 1999 (vol. 48 no. 3)
pp. 296-310

Abstract—Consider a hypercube of $2^n$ points described by $n$ Boolean variables and a subcube of $2^m$ points, $m \leq n$. As is well-known, the Boolean function with value 1 in the points of the subcube can be expressed as the product (AND) of $n-m$ variables. The standard synthesis of arbitrary functions exploits this property. We extend the concept of subcube to the more powerful pseudocube. The basic set is still composed of $2^m$ points, but has a more general form. The function with value 1 in a pseudocube, called pseudoproduct, is expressed as the AND of $n-m$ EXOR-factors, each containing at most $m+1$ 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 $n$-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.

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Index Terms:
Pseudocube, pseudoproduct, EXOR-factor, Boolean function, algebraic expression, logical design.
Fabrizio Luccio, Linda Pagli, "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
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