On a new Boolean function with applications

Consider a hypercube of 2ⁿ points described by n Boolean variables and a subcube of 2^m points, m⩽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