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    Hello everybody,

    I would like to announce that my book on hypercomputation
has been just published. The table of contents follows:

Contents
Preface V
Chapter 1 Introduction 1
1.1 On Computing and Its Limits . . . . . . . . . . . . . . . . . . . 1
1.2 From Computation to Hypercomputation . . . . . . . . . . . . 6
1.3 Why Bother with Hypercomputation? . . . . . . . . . . . . . . 9
Chapter 2 On the Church–Turing Thesis 11
2.1 TuringMachines . . . . . . . . . . . . . . . . . .. . . . . . . . . 11
2.2 General Recursive Functions . . . . . . . . . . .. . . . . . . . . 15
2.3 Recursive Relations and Predicates . . . . . . . . . . . . . . . . 17
2.4 The Church–Turing Thesis . . . . . . . .. . . . . . . . . . . . . 20
Chapter 3 Early Hypercomputers 25
3.1 Trial-and-ErrorMachines . . . . . . . . . . . . . . . . . . . . . 25
3.1.1 Extending Recursion Theory . . . . . . . . . . . . . . . 25
3.1.2 A Model of the Human Mind . . . . . . . . . . . . . . . 27
3.2 TAE-Computability . . . . . . . . . . . . . . . . . . . . . . . . . 30
3.3 Inductive Turing Machines . . . . . . . . . . . . . . . . . . . . 33
3.4 Extensions to the StandardModel of Computation . . . . . . 37
3.5 Exotic Machines . . . . .. . . . . . . . . . . . . . . . . . . . . . . 40
3.6 On Pseudorecursiveness . . . . . . . . . . . . . . . . . . . . . . 42
Chapter 4 Infinite-Time TuringMachines 45
4.1 On Cardinal and Ordinal Numbers . . . . . . . . . .. . . . . . . 45
4.2 Infinite-Time Turing Machines . . . . . . . . . . . .. . . . . . . 48
4.2.1 How the Machines Operate . . . . . . . . . . . . . . . . 49
4.2.2 On the Power of Infinite-Time Machines . . . . . . . . 52
4.2.3 Clockable Ordinals . . . . . . . . . . . . . . . . . . . . . 55
4.2.4 On Infinite-Time Halting Problems . . . . . . . . . . . 56
4.2.5 Machines with Only One Tape . . . . . . . . . . . . . . 57
4.2.6 Infinite-Time Machines with Oracles . . . . . . . . . . . 57
4.2.7 Post’s Problem for Supertasks . . . . . . . . . . . . . . . 59
4.3 Infinite-Time Automata . . . . . . . . . . . . . . . . . . . . . . 60
XI
XII Contents
4.4 Building Infinite Machines . . . . . . . . . . . . . . . . . . . . 61
4.5 Metaphysical Foundations for Computation . . . . . . . . . . 63
Chapter 5 Interactive Computing 69
5.1 Interactive Computing and Turing Machines . . . . . . . . . . 69
5.2 InteractionMachines . . . . . . . . . . . . . . . . . . . . . . . . 72
5.3 Persistent Turing Machines . . . . . .. . . . . . . . . . . . . . . 75
5.4 Site and Internet Machines . . . . . . . . . . . . . . . . . . . . 77
5.5 Other Approaches . . . . . . . . . .. . . . . . . . . . . . . . . . . 81
Chapter 6 Hyperminds 85
6.1 Mathematics and the Mind . . . . . . . . . . . . . . . . . . . . 86
6.1.1 The Pure Gödelian Argument . . . . . . . . . . . . . . . 86
6.1.2 The Argument from Infinitary Logic . . . . . . . . . . . 96
6.1.3 The Modal Argument . . . . . . . . . . . . . . . . . . . . 97
6.2 Philosophy and the Mind . . . . . . . . . . . . . . . . . . . . . . 100
6.2.1 Arguments Against Computationalism . . . . . . . . . . 100
6.2.2 The Chinese Room Argument Revisited . . . . . . . . . 102
6.3 Neurobiology and the Mind . . . . . . . . . . . . . . . . . . . . 104
6.4 Cognition and the Mind . . . . . . . . . . . . . . . . . . . . . . 109
Chapter 7 Computing Real Numbers 113
7.1 Type-2 Theory of Effectivity . . . . . . . . . . . . . . . . . . . . 113
7.1.1 Type-2 Machines . . . . . . . . . . . . . . . . . . . . . . . 114
7.1.2 Computable Topologies . . . . . . . . . . . . . . . . . . . 117
7.1.3 Type-2 Computability of Real Numbers . . . . . . . . . 119
7.1.4 The Arithmetic Hierarchy of Real Numbers . . . . . . 120
7.1.5 Computable Real Functions . . . . . . . . . . . . . . . . 121
7.2 IndeterministicMultihead Type-2 Machines . . . . . . . . . . 123
7.3 BSS-Machines . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
7.3.1 Finite-DimensionalMachines . . . . . . . . . . . . . . . 126
7.3.2 Machines over a Commutative Ring . . . . . . . . . . . 129
7.3.3 Parallel Machines . . . . . .. . . . . . . . . . . . . . . . . 130
7.4 Real-Number Random-AccessMachines . . . . . . . . . . . . 131
7.5 Recursion Theory on the Real Numbers . . . . . . . . . . . . . 133
Chapter 8 Relativistic and Quantum Hypercomputation 137
8.1 Supertasks in Relativistic Spacetimes . . . . . . . . . . . . . . 137
8.2 SAD Machines . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
8.3 Supertasks near Black Holes . . . . . . . . . . . . .. . . . . . . 144
8.4 Quantum Supertasks . . . . . . . . . . . . . . . . .. . . . . . . . 148
8.5 Ultimate Computing Machines . . . . . . . . . . . . . . . . . . 152
8.6 Quantum Adiabatic Computation . . . . . . . . . . .. . . . . . 154
8.7 Infinite Concurrent Turing Machines . . . . . . . . . . . . . . 162
Contents XIII
Chapter 9 Natural Computation and Hypercomputation 165
9.1 Principles of Natural Computation . . . . . . . . . . . .. . . . . 165
9.2 Models of Analog Computation . . . . . . . . . . . . . .. . . . . 169
9.3 On Undecidable Problems of Analysis . . . . . . . . . . . . . . 174
9.4 Noncomputability in Computable Analysis . . . . . . . . .. . . 178
9.5 The Halting Function Revisited . . . . . . . . . . . . . . .. . . . 180
9.6 Neural Networks and Hypercomputation . . . . . . . . . . .. . 183
9.7 An Optical Model of Computation . . . . . . . . . . . . . . . . 184
9.8 FuzzyMembrane Computing . . . . . . . . . . . . . . . . . . . 189
9.9 Analog X-Machines . . . . . . . . . . . . . . . . . . . . . . . . . 193
Appendix A The P = NP Hypothesis 199
Appendix B Intractability and Hypercomputation 203
Appendix C Socioeconomic Implications 205
Appendix D A Summary of Topology and Differential Geometry 209
D.1 Frames . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 
209
D.2 Vector Spaces and Lie Algebras . . . . . . . . . . . . . . . . . . 210
D.3 Topological Spaces: Definitions . . . . . . . . . . . . . . . . . . 212
D.4 Banach and Hilbert Spaces . . . . . . . . . . . . . . . . . . . . 215
D.5 Manifolds and Spacetime . . . . . . . . . . . . . . . . . . . . . 217
References 220
Name Index 235
Subject Index 239


Sincerely,
A.S.

----------------------
Apostolos Syropoulos
Xanthi, Greece
http://obelix.ee.duth.gr/~apostolo
http://asyropoulos.wordpress.com


      

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