Neural and Automata Networks: Dynamical Behavior and Applications

ISBN-13: 9789401067249 / Angielski / Miękka / 2011 / 264 str.

E. Goles; Servet Martinez

Opis

"Et moi, ..., si j'avait Sll comment en revenir. One sennce mathematics has rendered the human race. It has put common sense back je n'y serais point alle.' Jules Verne whe", it belongs, on the topmost shelf next to the dusty canister labelled 'discarded non- The series is divergent; therefore we may be smse'. able to do something with it. Eric T. Bell O. Heaviside Mathematics is a tool for thought. A highly necessary tool in a world where both feedback and non- linearities abound. Similarly, all kinds of parts of mathematics serve as tools for other parts and for other sciences. Applying a simple rewriting rule to the quote on the right above one finds such statements as: 'One service topology has rendered mathematical physics ...'; 'One service logic has rendered com- puter science ...'; 'One service category theory has rendered mathematics ...'. All arguably true. And all statements obtainable this way form part of the raison d'!ltre of this series.

Spis treści

1. Automata Networks.- 1.1. Introduction.- 1.2. Definitions Regarding Automata Networks.- 1.3. Cellular Automata.- 1.4. Complexity Results for Automata Networks.- 1.5. Neural Networks.- 1.6. Examples of Automata Networks.- 1.6.1. XOR Networks.- 1.6.2. Next Majority Rule.- 1.6.3. Multithreshold Automaton.- 1.6.4. The Ising Automaton.- 1.6.5. Bounded Neural Network (BNN).- 1.6.6. Bounded Majority Network.- 2. Algebraic Invariants on Neural Networks.- 2.1. Introduction.- 2.2. K-Chains in 0-1 Periodic Sequences.- 2.3. Covariance in Time.- 2.4. Algebraic Invariants of Synchronous Iteration on Neural Networks.- 2.5. Algebraic Invariants of Sequential Iteration on Neural Networks.- 2.6. Block Sequential Iteration on Neural Networks.- 2.7. Iteration with Memory.- 2.8. Synchronous Iteration on Majority Networks.- 3. Lyapunov Functionals Associated to Neural Networks.- 3.1. Introduction.- 3.2. Synchronous Iteration.- 3.3. Sequential Iteration.- 3.4. Tie Rules for Neural Networks.- 3.5. Antisymmetrical Neural Networks.- 3.6. A Class of Symmetric Networks with Exponential Transient Length for Synchronous Iteration.- 3.7. Exponential Transient Classes for Sequential Iteration.- 4. Uniform One and Two Dimensional Neural Networks.- 4.1. Introduction.- 4.2. One-Dimensional Majority Automata.- 4.3. Two-Dimensional Majority Cellular Automata.- 4.3.1. 3-Threshold Case.- 4.3.2. 2-Threshold Case.- 4.4. Non-Symmetric One-Dimensional Bounded Neural Networks.- 4.5. Two-Dimensional Bounded Neural Networks.- 5. Continuous and Cyclically Monotone Networks.- 5.1. Introduction.- 5.2. Positive Networks.- 5.3. Multithreshold Networks.- 5.4. Approximation of Continuous Networks by Multithreshold Networks.- 5.5. Cyclically Monotone Networks.- 5.6. Positive Definite Interactions. The Maximization Problem.- 5.7. Sequential Iteration for Decreasing Real Functions and Optimization Problems.- 5.8. A Generalized Dynamics.- 5.9. Chain-Symmetric Matrices.- 6. Applications on Thermodynamic Limits on the Bethe Lattice.- 6.1. Introduction.- 6.2. The Bethe Lattice.- 6.3. The Hamiltonian.- 6.4. Thermodynamic Limits of Gibbs Ensembles.- 6.5. Evolution Equations.- 6.6. The One-Site Distribution of the Thermodynamic Limits.- 6.7. Distribution of the Thermodynamic Limits.- 6.8.Period ? 2 Limit Orbits of Some Non Linear Dynamics on $$ \mathbb_{ + }^ $$.- 7. Potts Automata.- 7.1. The Potts Model.- 7.2. Generalized Potts Hamiltonians and Compatible Rules.- 7.2.1. Majority Networks.- 7.2.2. Next Majority Rule.- 7.2.3. Median Rule.- 7.2.4. Threshold Functions.- 7.3. The Complexity of Synchronous Iteration on Compatible Rules.- 7.3.1. Logic Calculator.- 7.3.2. Potts Universal Automaton.- 7.4. Solvable Classes for the Synchronous Update.- 7.4.1. Maximal Rules.- 7.4.1.1. Majority Networks.- 7.4.1.2. Local Coloring Rules.- 7.4.2. Smoothing Rules.- 7.4.3. The Phase Unwrapping Algorithm.- References.- Author and Subject Index.