CONVERGENCE OF AN ITERATIVE ALGORITHM FOR COMPUTING PARAMETERS OF MULTI-VALUED THRESHOLD FUNCTIONS

A k-valued threshold function is defined as f (x(1), ... ,x(n)) = i is an element of {0,1, ... ,k - 1} double left right arrow double left right arrow b(i) <= L (x(1), ... , x(n)) < b(i+1) where L (x(1), ... , x(n)) = a(1)x(1) + a(2)x(2) + ... + a(n)x(n), is a linear form in variables x(1), ... , x(n), with the values in {0, 1, ... , k - 1} and coefficients a(1), ... , a(n) in and b(0), ... ,b(k) are some thresholds for L in R, b(0) < b(1) < ... < b(k). A. V. Burdelev and V. G. Nikonov have created and published in J. Computational Nanotechnology (2017, no.1, pp. 7-14) an iterative algorithm for computing co-efficients a(1), ... , a(n) and thresholds b(0), ... , b(k) for any k-valued threshold function f(x(1), ... , x(N)) given by its values f(c(1), ... , c(n)) for all (c(1) ... c(n)) in {0, ... , k - 1}(n). In computer experiment they showed the convergence of this algorithm on many different examples. Here, we present a theoretical proof of this algorithm convergence on each k-valued threshold function for a finite number of steps (iterations). The proof is very much similar to the geometrical proof of perceptron convergence theorem by M. Minsky and S. Papert.