On the Co-absence of Input Terms in Higher Order Neuron Representation of Boolean Functions

Abstract

Boolean functions (BFs) can be represented by using polynomial functions when −1 and +1 are used represent True and False respectively. The coefficients of the representing polynomial can be obtained by exact interpolation given the truth table of the BF. A more parsimonious representation can be obtained with so called polynomial sign representation, where the exact interpolation is relaxed to allow the sign of the polynomial function to represent the BF value of True or False. This corresponds exactly to the higher order neuron or sigma-pi unit model of biological neurons. It is of interest to know what is the minimal set of monomials or input lines that is sufficient to represent a BF. In this study, we approach the problem by investigating the (small) subsets of monomials that cannot be absent as a whole from the representation of a given BF. With numerical investigations, we study low dimensional BFs and introduce a graph representation to visually describe the behavior of the two-element monomial subsets as to whether they cannot be absent from any sign representation. Finally, we prove that for any n-variable BF, any three-element monomial set cannot be absent as a whole if and only if all the pairs from that set has the same property. The results and direction taken in the study may lead to more efficient algorithms for finding higher order neuron representations with close-to-minimal input terms for Boolean functions.