Realizing Boolean functions on disjoint sets of variables

Abstract

For switching functions f let C(f) be the combinational complexity of f. We prove that for every ε>0 there are arbitrarily complex functions f:{0,1} n →{0,1} n such that C( f× f)⩽ (1+ ε) C( f) and arbitrarily complex functions f:{0,1} n →{0,1} such that C( v∘( fxf)⩽ (1+ ε) C( f). These results and the techniques developed to obtain them are used to show that Ashenhurst decomposition of switching functions does not always yield optimal circuits, and to prove a new result concerning the gap between circuit size and monotone circuit size.