Linear read-once and related Boolean functions

Abstract

It is known that a positive Boolean function f depending on n variables has at least n+1 extremal points, i.e. minimal ones and maximal zeros. We show that f has exactly n+1 extremal points if and only if it is linear read-once.The class of linear read-once functions is known to be the intersection of the classes of read-once and threshold functions. Generalizing this result we show that the class of linear read-once functions is the intersection of read-once and Chow functions. We also find the set of minimal read-once functions which are not linear read-once and the set of minimal threshold functions which are not linear read-once. In other words, we characterize the class of linear read-once functions by means of minimal forbidden subfunctions within the universe of read-once and the universe of threshold functions.Within the universe of threshold functions the importance of linear read-once functions is due to the fact that they attain the minimum value of the specification number, which is n+1 for functions depending on n variables. In 1995 Anthony et al. conjectured that for all other threshold functions the specification number is strictly greater than n+1. We disprove this conjecture by exhibiting a threshold non-linear read-once function depending on n variables whose specification number is n+1.