Let f be a monotone Boolean function over X = {x1,…,xn}. The k-slice of f is the function fk = (f^Tkn)vTk+1n, where Tkn is the kth threshold function.
Berkowitz has shown that sufficiently large superlinear lower bounds on the monotone network complexity of kk imply lower bounds of the same order on the combinational complexity of f, and that if f has large combinational complexity, then some slice of f must have large monotone complexity. However, this latter result does not specify any particular slice and it is known that some (nontrivial) slice functions of NP-complete predicates have linear complexity. In this paper we consider the slice functions and show that, for three basic monotone Boolean NP-complete functions, this slice is also NP-complete.
In addition the of some Boolean matrix functions F is studied, and is proved to be no easier to compute than F.