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This short book considers selected topics involving the interplay between certain areas of discrete mathematics and the simplest types of artificial neural networks. Graph theory, some partially ordered set theory, computational complexity, and discrete probability theory are among the mathematical topics involved.
The aim of this book is to give those interested in discrete mathematics a taste of the large, active, and expanding field of artificial neural network theory. The book might be best regarded as a series of extended essays on topics involving neural networks, discrete mathematics, and Boolean functions. A book of this length can only touch on some of the very many interesting issues involved, and those that are considered can all be explored much more deeply. Inevitably, therefore, the book focuses—and only briefly—on topics which have been of most interest to me, and I apologize if the selection appears idiosyncratic. Pointers are given in each chapter to further reading on the given topics and on related topics, and the reader is encouraged to pursue these leads.
Artificial Neural Networks
Boolean Functions
Threshold Functions
Number of Threshold Functions
Sizes of Weights for Threshold Functions
Threshold Order
Threshold Networks and Boolean Functions
Specifying Sets
Neural Network Learning
Probabilistic Learning
VC-dimensions of Neural Networks
The Complexity of Learning
Boltzmann Machines and Combinatorial Optimization