On the number of multilinear partitions and the computing capacity of multiple-valued multiple-threshold perceptrons

We introduce the concept of multilinear partition of a point set V⊂Rⁿ and the concept of multilinear separability of a function f:V→K={0,...,k-1}. Based on well-known relationships between linear partitions and minimal pairs, we derive formulae for the number of multilinear partitions of a point set in general position and of the set K². The (n,k,s)-perceptrons partition the input space V into s+1 regions with s parallel hyperplanes. We obtain results on the capacity of a single (n,k,s)-perceptron, respectively, for V⊂Rⁿ in general position and for V=K². Finally, we describe a fast polynomial-time algorithm for counting the multilinear partitions of K².