On the complexity of slice functions

Abstract

By a result of Berkowitz the monotone circuit complexity of slice functions cannot be much larger than the circuit (combinational) complexity of these functions for arbitrary complete bases. This result strengthens the importance of the theory of monotone circuits. We show in this paper that monotone circuits for slice functions can be understood as special circuits called set circuits. Here disjunction and conjunction are replaced by set union and set intersection. All known methods for proving lower bounds on the monotone complexity of Boolean functions do not work in their present form for slice functions. Furthermore we show that the canonical slice functions of the Boolean convolution, the Nechiporuk Boolean sums and the clique function can be computed with linear many gates.