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- W2495817859 abstract "2.1 IntroductionOne of the areas discussed in this book is the representation of Boolean functions by neural networks. Specifically, we investigate questions about the sort of Boolean functions computable by certain types of neural network, and the types of network required to compute certain classes of Boolean functions. In this chapter, therefore, we set up some notations and terminology concerning Boolean functions. Some of these will be familiar to most readers, but it is useful to recall the key ideas we shall most heavily draw upon in what follows.2.2 Boolean functions and the hypercube2.2.1 The Boolean hypercubeA Boolean function is simply a function mapping from {0, 1}n to {0, 1}. Clearly, there are 22n Boolean functions, since for each of the 2n members of {0, 1}n, the function can take one of two values. Sometimes, it is more convenient to think of a Boolean function as mapping from the set {−1, 1}n to {−1, 1}. This approach is occasionally more useful, but we shall normally use the standard convention.The domain {0, 1}n is often called the Boolean hypercube, and there are a number of useful ways of thinking about it.2.2.2 Geometry of the Boolean hypercubeThe geometry of the Boolean hypercube is sometimes important when we begin to seek answers to questions about Boolean functions computable by neural networks. A set of points in ℝn is said to be in general position if no n + 1 of them lie in an (n − 1)-dimensional hyperplane (or “flat”) of ℝn. The rigid geometrical arrangement of the points in {0, 1}n means that they are not in general position. (Indeed, no subset of 2n + 1 of the points is in general position, since at least n + 1 of them belong to the (n − 1)-dimensional flat with equation x1 = 0 or to the flat with equation x1 = 1.) This lack of general position is, as we shall see, often a complication." @default.
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- W2495817859 date "2001-01-01" @default.
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- W2495817859 title "2. Boolean Functions" @default.
- W2495817859 doi "https://doi.org/10.1137/1.9780898718539.ch2" @default.
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