Bart Kosko:
NEURAL NETWORKS AND FUZZY SYSTEMS (Prentice Hall, 1992)

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 A textbook on adaptive fuzzy systems that presents a unified view of neural networks and fuzzy systems. Kosko presents neural networks as stochastic gradient systems and fuzzy sets as points in unit hypercubes. All the main learning algorithms for neural networks are reviewed and formalized. It is shawn that neural computations is similar to statistics in that its goal is to approximate the function that relates a set of inputs to a set of outputs. In Kosko's formalization, a fuzzy set is a point in the unitary hypercube equivalent to Zadeh's universe of discourse, and a non-fuzzy set is one of the vertexes of such a cube. The paradoxes of classical logic occur in the middle points of the hypercube. A fuzzy set's entropy (which could be thought of as its "ambiguity") is defined by the number of violations of the law of non-contradiction compared with the number of violations of the excluded middle. Entropy is zero when both laws hold, is maximum in the center of the hypercube. Alternatively, a fuzzy set's entropy can be defined as a measure of how a set is a subset of itself. A fuzzy system is a relationship between hypercubes, a relationship of fuzzy sets into families of fuzzy sets. Fuzzy associative memories are balls of fuzzy sets into balls of fuzzy sets. Fuzzy logic, which can account for all results of the theory of probability, better represents the real world, without any need to assume the existence of randomness. For example, relative frequency is a measure of how a set is a subset of another set. Many of Physics' laws are not reversible because if they were casuality would be violated (after a transition of state probability turns into certainty and cannot be rebuilt working backwards). If they were expressed as "ambiguity", rather than probability, they would be reversible, as the ambiguity of an event remains the same before and after the event occurred. The space of neural states (the set of all possible outputs of a neural net) is identical to the power fuzzy set (the set of all fuzzy subsets of the set of neurons). A set of "n" neurons (whose signals vary continously between zero and one) defines a family of n-dimensional fuzzy sets. That space is the unitary hypercube, the set of all vectors of length "n" and coordinates in the unitary continous interval (zero to one). Hopfield's nets tend to push the state of the system towards one of the 2 to the "n" vertexes of the hypercube. This way they dynamically disambiguate fuzzy descriptions by minimizing their fuzzy entropy.