The complexity of central slice functions

Abstract

Let f be a monotone Boolean function over X = { x 1,…, x n }. The k-slice of f is the function f k = ( f^ T k n ) vT k+1 n , where T k n is the kth threshold function. Berkowitz has shown that sufficiently large superlinear lower bounds on the monotone network complexity of k k 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 1 2 n slice functions and show that, for three basic monotone Boolean NP-complete functions, this slice is also NP-complete. In addition the 1 2 n- slice of some Boolean matrix functions F is studied, and is proved to be no easier to compute than F.