Threshold logic and two-person, zero-sum games

Abstract

It is shown that the problem of determining whether or not a given switching function is realizable with a single threshold gate can be reduced directly to that of determining the value of a two-person, zero-sum game. If this value is $\gt$ 1/2, the function is realizable. If the value is $\geq$ 1/2, the function cannot be realized. In the case of realizability the solution of the game gives the set of weights which accomplishes the realization. The method works equally well with functions having "don't care" conditions. When a function is not realizable a threshold gate is, nevertheless, defined together with a "zone of uncertainty." This gate is such that for a given input, gamma, if the weighted sum is above this "zone of uncertainty" then F(gamma) = 1, within the zone F(gamma;) is uncertain, and below the zone F(gamma) = 0. The larger the value of the game the smaller this "zone of uncertainty." A method for solving two-person zero-sum games is described and several examples are given.