Chow Parameters in Threshold Logic

This paper is a broad treatment of Chow parameters-a set of n+l integers which can be abstracted from any given n-argument switching function. Basic properties and alternative definitions of these numbers are established and correlated with several earlier works in the subject. The main results are as follows: The class of "unique" functions-those with unique Chow parameter N-tuples-lies properly between the classes of threshold functions and the class of completely monotonic functions. The class of "extremal" functions-with locally minimal or maximal single parameters-lies properly between the class of unique functions and the class of unate functions (and these inclusions cannot be tightened in terms of other k-monotonicities). A closely related question recently raised is settled. Quadratic bounds and an infinite family of linear bounds, all tight, are obtained. A smooth well-behaved surface exists which encloses only the Chow parameters of nonthreshold functions and whose tangent hyperplanes define realizations of the function whose parameters lie outside the point of tangency.