Linear read-once and related Boolean functions

TL;DR

This paper characterizes linear read-once Boolean functions as those with exactly n+1 extremal points, explores their relation to read-once and Chow functions, and disproves a conjecture about their minimal specification number.

Contribution

It provides a characterization of linear read-once functions via extremal points and minimal forbidden subfunctions, and challenges a previous conjecture on their specification number.

Findings

Linear read-once functions have exactly n+1 extremal points.

The class of linear read-once functions is the intersection of read-once and Chow functions.

A threshold non-linear read-once function with minimal specification number n+1 is constructed.

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 func- tions 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. conjec- tured 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.