Specifying a positive threshold function via extremal points

Abstract

An extremal point of a positive threshold Boolean function f is either a maximal zero or a minimal one. It is known that if f depends on all its variables, then the set of its extremal points completely specifies f within the universe of threshold functions. However, in some cases, f can be specified by a smaller set. The minimum number of points in such a set is the specification number of f . It was shown in [S.-T. Hu. Threshold Logic, 1965] that the specification number of a threshold function of n variables is at least n + 1. In [M. Anthony, G. Brightwell, and J. Shawe-Taylor. On specifying Boolean functions by labelled examples. Discrete Applied Mathematics, 1995] it was proved that this bound is attained for nested functions and conjectured that for all other threshold functions the specification number is strictly greater than n+1. In the present paper, we resolve this conjecture negatively by exhibiting threshold Boolean functions of n variables, which are non-nested and for which the specification number is n + 1. On the other hand, we show that the set of extremal points satisfies the statement of the conjecture, i.e. a positive threshold Boolean function depending on all its n variables has n + 1 extremal points if and only if it is nested. To prove this, we reveal an underlying structure of the set of extremal points.