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

space, an entire row of a truth table is represented as a point on a graph. The coordinates
of a point in a pattern space are determined by the truth values of the input
propositions. The colour of the point represents the truth value of the operation
computed over the inputs.
Figure 4-2A illustrates the pattern space for the AND operation of Table 4-1.
Note that it has four graphed points, one for each row of the truth table. The coordinates
of each graphed point—(1,1), (1,0), (0,1), and (0,0)—indicate the truth values
of the propositions p and q. The AND operation is only true when both of these
propositions are true. This is represented by colouring the point at coordinate (1,1)
black. The other three points are coloured white, indicating that the logical operator
returns a “false” value for each of them.
1
1
q
q
A
0
B
0
0 p 1
0 p
1
Figure 4-2. (A) Pattern space for AND; (B) Pattern space for XOR.
Pattern spaces are used for digital pattern recognition by carving them into decision
regions. If a point that represents a pattern falls in one decision region, then it is
classified in one way. If that point falls in a different decision region, then it is classified
in a different way. Learning how to classify a set of patterns involves learning
how to correctly carve the pattern space up into the desired decision regions.
The AND problem is an example of a linearly separable problem. This is
because a single straight cut through the pattern space divides it into two decision
regions that generate the correct pattern classifications. The dashed line in Figure
4-2A indicates the location of this straight cut for the AND problem. Note that the
one “true” pattern falls on one side of this cut, and that the three “false” patterns fall
on the other side of this cut.
Not all problems are linearly separable. A linearly nonseparable problem is one
in which a single straight cut is not sufficient to separate all of the patterns of one
type from all of the patterns of another type. An example of a linearly nonseparable
problem is the XOR problem, whose pattern space is illustrated in Figure 4-2B.
Note that the positions of the four patterns in Figure 4-2B are identical to the positions
in Figure 4-2A, because both pattern spaces involve the same propositions.
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