Properties and Transformations of Single Threshold Element Functions

A large number of Boolean functions may be realized as the output of a single threshold element; however, not all functions are so realizable. Those F
_n
which are, are said to belong to the class R. No easily applicable criteria have been formulated which enable an unequivocal determination to be made of whether a given F
_n
ϵ R. There are, however, certain transformations allowable on those F
_n
ϵ R which exhibit a closure property, that is, the transformation results in an F
_p
ϵ R. There are also certain properties of the form of F
_n
ϵ R which can be recognized easily. A knowledge of these permissible transformations and observations of the form of a given F
_n
will, in many instances, allow for the desired. determination. This paper, for the most part, examines those permissible transformations and properties of form.