Models of Massive Parallelism

1. Turing Computability and Complexity.- 1.1 Models of Sequential Computation.- 1.1.1 A Simple Model: the Finite-State Machine.- 1.1.2 Turing Machines.- 1.2 Complexity.- 1.2.1 Nondeterministic Computations.- 1.2.2 Randomized Algorithms.- 1.2.3 Parallel Computation.- 1.3 Cellular Machines.- 1.4 Prerequisites.- References.- 2. Cellular Automata.- 2.1 Finite-State Automata.- 2.2 Regular Graphs.- 2.3 Local Rules and Global Maps.- 2.3.1 Cellular Spaces.- 2.3.2 Local Rules.- 2.3.3 Global Maps and Dynamical Systems.- 2.4 Fundamental Questions.- 2.5 Notation.- 2.6 Problems.- 2.7 Notes.- References.- 3. Linear Cellular Automata.- 3.1 Linear Rules.- 3.2 Basic Properties.- 3.2.1 Global Injectivity and Surjectivity Modulo m.- 3.2.2 Self-reproduction with Linear Automata.- 3.2.3 Linear Automata on Rings and Semigroups.- 3.3 Global Dynamics via Fractals.- 3.4 The Role of Linear Rules.- 3.5 Problems.- 3.6 Notes.- References.- 4. Semi-totalistic Automata.- 4.1 Semi-totalistic Rules.- 4.1.1 An Example: Conway's Game of LIFE.- 4.1.2 Nomenclature for Totalistic Rules.- 4.2 Construction and Computation Universality.- 4.2.1 Computation Universality of LIFE.- 4.2.2 Constructibility and Self-reproduction.- 4.2.3 Provable Computation Universality.- 4.3 Restricted Totalistic Rules.- 4.4 Threshold Automata.- 4.5 Problems.- 4.6 Notes.- References.- 5. Decision Problems.- 5.1 Algorithmic and Dynetic Problems.- 5.2 ID Euclidean Automata.- 5.3 2D Euclidean Automata.- 5.3.1 Reversibility is Unsolvable.- 5.3.2 Surjectivity is Unsolvable.- 5.4 Noneuclidean Automata.- 5.5 Complexity Questions.- 5.6 Problems.- 5.7 Notes.- References.- 6. Neural and Random Boolean Networks.- 6.1 Types of Generalizations.- 6.2 Other Parallel Models.- 6.3 Summary of Results.- 6.4 Proofs.- 6.4.1 A Hierarchy.- 6.4.2 A Universal Neural Network.- 6.4.3 Equivalence of Cellular Automata and Neural Networks.- 6.4.4 Equivalence of Neural Networks and Random Networks.- 6.4.5 The Stability Problem is Neurally Unsolvable.- 6.5 Problems.- 6.6 Notes.- References.- 7. General Properties.- 7.1 Metric Preliminaries.- 7.1.1 Metrics and Topologies.- 7.1.2 Convergence and Continuity.- 7.2 Basic Results.- 7.2.1 The Moore-Myhill Theorem.- 7.2.2 Nondeterministic Cellular Automata.- 7.3 Injeetivity, Surjectivity and Local Reversibility.- 7.4 Some Generalizations.- 7.4.1 Neural and Random Networks.- 7.4.2 Combinatorial Generalizations on Euclidean Spaces.- 7.5 Problems.- 7.6 Notes.- References.- 8. Classification.- 8.1 Finite Networks.- 8.1.1 The Difficulties.- 8.1.2 Complexity of Classifying Finite Networks.- 8.2 Wolfram Classification.- 8.3 Classification via Limit Sets.- 8.3.1 Culik-Yu's Classes and Ishii's Classes.- 8.3.2 About Entropy.- 8.4 Mean Field Theory.- 8.5 Local Structure Theory.- 8.5.1 Zeroth-order and First-order Local Structure Theories.- 8.5.2 Higher-order Local Structure Theories.- 8.6 Other Classifications.- 8.7 Problems.- 8.8 Notes.- References.- 9. Asymptotic Behavior.- 9.1 Linear Rules.- 9.1.1 Linear Automata on Tori.- 9.1.2 Linear Automata on the Line.- 9.2 Exact Solution.- 9.3 Simulation in Continuous Systems.- 9.3.1 Discrete Computation by Continuous Systems.- 9.3.2 Nonlocal Properties.- 9.3.2.1 The o-transform.- 9.3.2.2 Sarkovskii's Theorem.- 9.4 Observability.- 9.4.1 Observability of the Identity.- 9.4.2 Toggle Rules and the Extension Property.- 9.4.3 Observability of Linear Cellular Automata.- 9.4.4 Observability in Neural Networks.- 9.5 Problems.- 9.6 Notes.- References.- 10. Some Inverse Problems.- 10.1 Signals and Synchronization.- 10.1.1 Synchronization of a Line.- 10.1.2 Synchronization of a Network.- 10.1.3 Signals in Dimension 1.- 10.1.4 Clocks.- 10.2 Formal Language Recognition.- 10.2.1 Models.- 10.2.2 A Speedup Theorem.- 10.3 Picture Languages.- 10.3.1 The Issue of Representation.- 10.3.2 ?-Languages in Spaces of Linear Growth.- 10.3.3 2D-Euclidean Languages.- 10.3.4 Recognition over Spaces of Exponential Growth.- 10.3.5 Recognition over Spaces of Subexponential Growth.- 10.4 Problems.- 10.5 Notes.- References.- 11. Real Computation.- 11.1 Representation and Primitives.- 11.2 Exact Computation.- 11.2.1 Constant-time Computation.- 11.2.2 Variable-time Computation.- 11.3 Approximate Computation by Neural Nets.- 11.3.1 Relative Shadowing.- 11.3.2 Shadowing Bases.- 11.4 Problems.- 11.5 Notes.- References.- 12. A Bibliography of Applications.- 12.1 Physics.- 12.2 Chemistry.- 12.3 Biology.- 12.4 Computer Science.- 12.5 Artificial Intelligence and Cognitive Science.- 12.6 Miscellaneous.- References.- Author Index.- Symbol Index.