Using the Perceptron Algorithm to Find Consistent Hypotheses

The perceptron learning algorithm yields quite naturally an algorithm for finding a linearly separable boolean function consistent with a sample of such a function. Using the idea of a specifying sample, we give a simple proof that this algorithm is not efficient, in general. A boolean function t defined on {0, 1} is linearly separable if there are α ∈ R and θ ∈ R such that t(x) = { 1 if 〈α, x〉 ≥ θ 0 if 〈α, x〉 < θ, where 〈α, x〉 is the standard inner product of α and x. Given such α and θ, we say that t is represented by [α, θ] and we write t← [α, θ]. The vector α is known as the weight-vector, and θ is known as the threshold. This class of functions is the set of functions computable by the simple boolean perceptron (see [8, 9, 6]), and we shall denote it by BPn.