On monotone simulations of nonmonotone networks

Abstract

We consider the following problem: given some n-argument monotone Boolean function, f( X n) , with formal arguments X n={x 1,…,x n} , compute f using the 2 n+1 inputs X n ∪{ f 0, f 1,…, f n }. Here f k is the k-slice of f, i.e. the n-argument monotone Boolean function f k( X n=(f ∧ T n k) ∨ T n k+1 , where T n k is the n-argument monotone Boolean function which takes the value 1 iff at least k of its arguments are 1. It is easy to see that if nonmonotone operations are permitted then O( n) gates are sufficient by using the relation f = V n k=0 (f k∧ T n k+1 ) . The properties of slice functions imply th efficient monotone solutions would allow superlinear lower bounds on the combinational complexity of f to be obtained from large enough lower bounds on the monotone complexity of f. Since negation is known to be superpolynomially powerful, some monotone functions must have superpolynomial complexity even if all the slice functions are given as extra inputs. However it is possible that efficient simulations, using slice functions, exist for restricted classes of monotone functions. In this paper we examine a broad class of monotone Boolean functions, proving that for almost all of the functions in the class, no such simulation exists, and that in a very weak sense negation is exponentially powerful. In contrast to this an example of an efficient construction is given, again for a natural class of monotone Boolean functions.