SUFFICIENT CONDITIONS FOR SWITCHING FUNCTIONS TO BE THRESHOLD ONES AND THEIR APPLICATIONS

Abstract

In this paper we shall present two types of sufficient conditions under which a switching function can be represented by a threshold gate, i. e., a threshold function, with n variables for an arbitrary positive integer n. In deriving these sufficient conditions, a new concept called an orientating vector is introduced in this paper and it will play an important role in our discussion since it gives an insight into the structure of a threshold gate. We shall begin with notations and preliminaries in Section 2. In Section 3, we shall introduce the notion of orientating vectors in terms of which we give the sufficient conditions for threshold functions. Indeed, an orientating vector can be used to classify the set of all the input vectors Ix"), x(2), ••• , x(2n)} into two subsets where one is a set of true (i. e., on) vectors and the other a set of false (i. e., off) vectors. If we arrange all input vectors in an inverse lexical order (see Kitagawa [3]), then, for any p, 1 p_27', a classification of all the input vectors into the two sets Ix"), x(2) x(1 and fx(p+1) x(p+2), x(271)1 represents a threshold function as stated in Proposition 3.2. In Section 4, it is described that the combination of two sufficient conditions turn out to be necessary so far as p in the above classification is not greater than 4. This is the reason why the combination of these two sufficient conditions amounts to be necessary and sufficient so far as n is not greater than 3 as given in Section 5. Our results can be compared with the notion of 2-asummability due to Elgot [2] and Chow [1] which gives the necessary and sufficient condition that a switching function is a threshold function when n in not greater than 8. It is noted that the results of this paper can be used to get all the possible digraphs associated with dynamical behaviors of the neuronic equation