• 1 Computational Complexity.- 1.1 Neural Networks.- 1.2 Automata: A General Introduction.- 1.3 Finite Automata.- 1.4 The Turing Machine.- 1.5 Probabilistic Turing Machines.- 1.6 Nondeterministic Turing Machines.- 1.7 Oracle Turing Machines.- 1.8 Advice Turing Machines.- 1.9 Notes.- 2 The Model.- 2.1 Variants of the Network.- 2.2 The Network’s Computation.- 2.3 Integer Weights.- 3 Networks with Rational Weights.- 3.1 The Turing Equivalence Theorem.- 3.2 Highlights of the Proof.- 3.3 The Simulation.- 3.4 Network with Four Layers.- 3.5 Real-Time Simulation.- 3.6 Inputs and Outputs.- 3.7 Universal Network.- 3.8 Nondeterministic Computation.- 4 Networks with Real Weights.- 4.1 Simulating Circuit Families.- 4.2 Networks Simulation by Circuits.- 4.3 Networks versus Threshold Circuits.- 4.4 Corollaries.- 5 Kolmogorov Weights: Between P and P/poly.- 5.1 Kolmogorov Complexity and Reals.- 5.2 Tally Oracles and Neural Networks.- 5.3 Kolmogorov Weights and Advice Classes.- 5.4 The Hierarchy Theorem.- 6 Space and Precision.- 6.1 Equivalence of Space and Precision.- 6.2 Fixed Precision Variable Sized Nets.- 7 Universality of Sigmoidal Networks.- 7.1 Alarm Clock Machines.- 7.2 Restless Counters.- 7.3 Sigmoidal Networks are Universal.- 7.4 Conclusions.- 8 Different-limits Networks.- 8.1 At Least Finite Automata.- 8.2 Proof of the Interpolation Lemma.- 9 Stochastic Dynamics.- 9.1 Stochastic Networks.- 9.2 The Main Results.- 9.3 Integer Stochastic Networks.- 9.4 Rational Stochastic Networks.- 9.5 Real Stochastic Networks.- 9.6 Unreliable Networks.- 9.7 Nondeterministic Stochastic Networks.- 10 Generalized Processor Networks.- 10.1 Generalized Networks: Definition.- 10.2 Bounded Precision.- 10.3 Equivalence with Neural Networks.- 10.4 Robustness.- 11 Analog Computation.- 11.1 DiscreteTime Models.- 11.2 Continuous Time Models.- 11.3 Hybrid Models.- 11.4 Dissipative Models.- 12 Computation Beyond the Turing Limit.- 12.1 The Analog Shift Map.- 12.2 Analog Shift and Computation.- 12.3 Physical Relevance.- 12.4 Conclusions.

"All of the three primary questions are considered: What computational models can the net simulate (within polynomial bounds)? What are the computational complexity classes that are relevant to the net? How does the net (which, after all, is an analog device) relate to Church's thesis? Moreover the power of the basic model is also analyzed when the domain of reals is replaced by the rationals and the integers." -Mathematical Reviews "Siegelmann's book focuses on the computational complexities of neural networks and making this research accessible...the book accomplishes the said task nicely." ---SIAM Review, Vol. 42, No 3.