Tight bounds for the multiplicative complexity of symmetric functions

Authors:
Joan Boyar;René Peralta
Affiliations:
Department of Mathematics and Computer Science, University of Southern Denmark, Odense, Denmark;Information Technology Laboratory, National Institute of Standards and Technology, USA
Venue:
Theoretical Computer Science
Year:
2008

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Abstract

The multiplicative complexity of a Boolean function f is defined as the minimum number of binary conjunction (AND) gates required to construct a circuit representing f, when only exclusive-or, conjunction and negation gates may be used. This article explores in detail the multiplicative complexity of symmetric Boolean functions. New techniques that allow such exploration are introduced. They are powerful enough to give exact multiplicative complexities for several classes of symmetric functions. In particular, the multiplicative complexity of computing the Hamming weight of n bits is shown to be exactly n-H^N(n), where H^N(n) is the Hamming weight of the binary representation of n. We also show a close relationship between the complexities of basic symmetric functions and the fractal known as Sierpinski's gasket.