Mind, Body, World- Foundations of Cognitive Science, 2013a

The only difference is the colouring of the points, indicating that XOR involves
making a different judgment than AND. However, this difference between graphs is
important, because now it is impossible to separate all of the black points from all of
the white points with a single straight cut. Instead, two different cuts are required,
as shown by the two dashed lines in Figure 4-2B. This means that XOR is not linearly
separable.
Linear separability defines the limits of what can be computed by a Rosenblatt
perceptron (Rosenblatt, 1958, 1962) or by a McCulloch-Pitts neuron (McCulloch
& Pitts, 1943). That is, if some pattern recognition problem is linearly separable,
then either of these architectures is capable of representing a solution to that problem.
For instance, because AND is linearly separable, it can be computed by a perceptron,
such as the one illustrated in Figure 4-3.
= 1.5
1 1
p
q
Figure 4-3. A Rosenblatt perceptron that can compute the AND operation.
This perceptron consists of two input units whose activities respectively represent
the state (i.e., either 0 or 1) of the propositions p and q. Each of these input units
sends a signal through a connection to an output unit; the figure indicates that the
weight of each connection is 1. The output unit performs two operations. First, it
computes its net input by summing the two signals that it receives (the component
of the output unit). Second, it transforms the net input into activity by applying
the Heaviside step function. The figure indicates in the second component of
the output unit that the threshold for this activation function () is 1.5. This means
that output unit activity will only be 1 if net input is greater than or equal to 1.5;
otherwise, output unit activity will be equal to 0.
If one considers the four different combinations of input unit activities that
would be presented to this device—(1,1), (1,0), (0,1), and (0,0)—then it is clear that
144 Chapter 4