Neural Networks and Analog Computation: Beyond the Turing Limit

1 Computational Complexity.- 1.1 Neural Networks.- 1.2 Automata: A General Introduction.- 1.2.1 Input Sets in Computability Theory.- 1.3 Finite Automata.- 1.3.1 Neural Networks and Finite Automata.- 1.4 The Turing Machine.- 1.4.1 Neural Networks and Turing Machines.- 1.5 Probabilistic Turing Machines.- 1.5.1 Neural Networks and Probabilistic Machines.- 1.6 Nondeterministic Turing Machines.- 1.6.1 Nondeterministic Neural Networks.- 1.7 Oracle Turing Machines.- 1.7.1 Neural Networks and Oracle Machines.- 1.8 Advice Turing Machines.- 1.8.1 Circuit Families.- 1.8.2 Neural Networks and Advice Machines.- 1.9 Notes.- 2 The Model.- 2.1 Variants of the Network.- 2.1.1 A "System Diagram" Interpretation.- 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.- 3.2.1 Cantor-like Encoding of Stacks.- 3.2.2 Stack Operations.- 3.2.3 General Construction of the Network.- 3.3 The Simulation.- 3.3.1 P-Stack Machines.- 3.4 Network with Four Layers.- 3.4.1 A Layout Of The Construction.- 3.5 Real-Time Simulation.- 3.5.1 Computing in Two Layers.- 3.5.2 Removing the Sigmoid From the Main Layer.- 3.5.3 One Layer Network Simulates TM.- 3.6 Inputs and Outputs.- 3.7 Universal Network.- 3.8 Nondeterministic Computation.- 4 Networks with Real Weights.- 4.1 Simulating Circuit Families.- 4.1.1 The Circuit Encoding.- 4.1.2 A Circuit Retrieval.- 4.1.3 Circuit Simulation By a Network.- 4.1.4 The Combined Network.- 4.2 Networks Simulation by Circuits.- 4.2.1 Linear Precision Suffices.- 4.2.2 The Network Simulation by a Circuit.- 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.- 7.1.1 Adder Machines.- 7.1.2 Alarm Clock and Adder Machines.- 7.2 Restless Counters.- 7.3 Sigmoidal Networks are Universal.- 7.3.1 Correctness of the Simulation.- 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.- 9.1.1 The Model.- 9.2 The Main Results.- 9.2.1 Integer Networks.- 9.2.2 Rational Networks.- 9.2.3 Real Networks.- 9.3 Integer Stochastic Networks.- 9.4 Rational Stochastic Networks.- 9.4.1 Rational Set of Choices.- 9.4.2 Real Set of Choices.- 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 Discrete Time 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.