The threshold order of a Boolean function

Abstract

The notion of a threshold function as a Boolean function for which there is a hyperplane in R n which separates the true vectors from the false vectors of the function is generalized to the case in which more general surfaces may be needed for the separation. A Boolean function is said to be a threshold function of order m if the surface required to separate the true from the false vectors is a polynomial of degree m . We have used such functions in the problem of extrapolating partially defined Boolean functions. For each dimension n , there are exactly two threshold functions of order 0 (the constant functions), and it is shown that there are exactly two threshold functions of order n (the parity functions ). It is also shown that a Boolean function is a threshold function of order at least m if some sequence of “contractions” and “projections” of it to a (generally) lower dimensional space is a threshold function of order m . This leads to a generalization of a result of threshold function theory that complete monotonicity is a necessary condition for a Boolean function to be a threshold function (of order 0 or 1). A more recent result of threshold function theory, namely that the threshold recognition problem for a general Boolean function is NP-complete is shown to hold true for the order- m recognition problem as well. Also, the characterization of threshold functions in terms of summability is generalized to a characterization of threshold functions of any order. Finally, we present the result of a calculation in which the numbers of threshold functions of all orders for dimension n ≤ 4 are determined exactly, and statistical estimates are given for n = 5, 6, and 7. This leads to the supposition that the vast majority of threshold functions have order near n /2 for n even and ( n ± 1)/2 for n odd, as well as to some more precise conjectures.