By Nils J Nilsson

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Usually, the sigmoid function used is f (s) = 1+e1 −s , where s is the input and f is the output. 18: A Sigmoid Function 1 [Russell & Norvig 1995, page 595] attributes the use of this idea to [Bryson & Ho 1969]. 56 CHAPTER 4. NEURAL NETWORKS We show the network containing sigmoid units in place of TLUs in Fig. 19. (j) The output of the i-th sigmoid unit in the j-th layer is denoted by fi . ) −s 1+e i First Layer X(0) ... j-th Layer X(1) ... (k-1)-th Layer X(j) ... Wi(1) fi(1) Wi(j) fi(j) (1) bi bi(j) wli(j) si(1) si(j) ...

F threshold weight wn+1 n+1 f = thresh( ! 1: A Threshold Logic Unit (TLU) The n-dimensional feature or input vector is denoted by X = (x1 , . . , xn ). When we want to distinguish among different feature vectors, we will attach subscripts, such as Xi . The components of X can be any real-valued numbers, but we often specialize to the binary numbers 0 and 1. The weights of a TLU are represented by an n-dimensional weight vector, W = (w1 , . . , wn ). Its components are real-valued numbers (but we sometimes specialize to integers).

2. 4: The Version Graph Upon Seeing (1, 0, 1) and (1, 0, 0) Boundary sets are important because they provide an alternative to representing the entire version space explicitly, which would be impractical. Given only the boundary sets, it is possible to determine whether or not any hypothesis (in the prescribed class of Boolean functions we are using) is a member or not of the version space. This determination is possible because of the fact that any member of the version space (that is not a member of one of the boundary sets) is more specific than some member of the general boundary set and is more general than some member of the specific boundary set.