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Example of perceptrons:
Here as an example function in which the AND boolean function outputs a 1 only but if both inputs are 1 and where the OR function only outputs a 1 then if either inputs are 1. Here obviously these relate to the connectives we studied in first order logic. But in the following two perceptrons can represent the AND and OR boolean functions respectively:
However one of the major impacts of Minsky and Papert's book was to highlight the fact that perceptions cannot learn a particular boolean function that called XOR. In fact this function outputs a 1 if the two inputs are not the same. It means to see why XOR cannot be learned and try or write down a perception to do the job. So here the following diagram highlights the notion of linear reparability in Boolean functions that explains why they can't be learned by perceptions:
Practically in each case we've plotted the values taken use of the Boolean function where the inputs are particular values as: (-1,-1);(1,-1);(-1,1) and (1,1). But just because the AND function there is only one place whereas a 1 is plotted, namely where both inputs are 1. Moreover we could draw the dotted line to separate the output -1s from the 1s. So than we were able to draw a similar line in the OR case. Its means there, that we can draw these lines, we say that these functions are linearly separable. Remember there that it is not possible to draw any line for the XOR plot: but where you try then you never get a clean split into 1s and -1s.
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