Necessary conditions that a switching function be, in particular, a threshold function have been introduced by Paull and McCluskey [11]; named and studied at some length by Winder [13], [14], [15] and Muroga, et al. [8]; and generalized by Elgot [3], Winder [14], [15] and Gabelman [4]. The basic conditions are the k-monotonicities and have been used not only to recognize threshold functions but also to determine realizing weights and threshold.
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Hamming et al. (1965) studied this question.