A Method for Testing and Realization of Threshold Functions through Classification of Inequalities

TLDR: A method for testing and realization of threshold functions through classification of inequalities is suggested, showing that when the different inequalities involving the weights of the variables are represented in terms of their subscripts, the entire set of inequalities of any function can be classified into nine distinct types.

Abstract: A method for testing and realization of threshold functions through classification of inequalities is suggested in the present paper. The standard procedure of testing and realization of threshold functions consists in solving a system of linear inequalities in which the unknowns are the different weights to be assigned to the variables of the functions. In the paper it is first shown that when the different inequalities involving the weights of the variables are represented in terms of their subscripts only, thon, depending on the number and on the sum of the subscripts appearing on either side, the entire set of inequalities of any function can be classified into nine distinct types. The sot of inequalities is next expressed in terms of the least weight and the other different incremental weights, the knowledge of which along with that of the types of inequalities, furnishes information regarding I-realizability of the function and on the assignments of values to the different weights for integ...