By Nils J Nilsson

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USING VERSION SPACES FOR LEARNING Chapter 4 Neural Networks In chapter two we de ned several important subsets of Boolean functions. Suppose we decide to use one of these subsets as a hypothesis set for supervised function learning. We next have the question of how best to implement the function as a device that gives the outputs prescribed by the function for arbitrary inputs. In this chapter we describe how networks of non-linear elements can be used to implement various input-output functions and how they can be trained using supervised learning methods.

The total squared error (over all patterns in a training set, , containing m patterns) is then: "= m X nX +1 i=1 j =1 (di ; xij wj )2 We want to choose the weights wj to minimize this squared error. One way to nd such a set of weights is to start with an arbitrary weight vector and move it along the negative gradient of " as a function of the weights. Since " is quadratic in the wj , we know that it has a global minimum, and thus this steepest descent procedure is guaranteed to nd the minimum. Each component of the gradient is the partial derivative of " with respect to one of the weights.

There is no closed-form expression for the number of linearly separable functions of n dimensions, but the following table gives the numbers for n up to 6. n 1 2 3 4 5 6 Boolean Linearly Separable Functions Functions 4 4 16 14 256 104 65,536 1,882 4:3 109 94,572 1:8 1019 15,028,134 Muroga, 1971] has shown that (for n > 1) there are no more than 2n2 linearly separable functions of n dimensions. 3 Summary The diagram in Fig. 6 shows some of the set inclusions of the classes of Boolean functions that we have considered.