Hypercomputation: Computing Beyond the Church-Turing Barrier (Google eBook)

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Springer Science & Business Media, Dec 10, 2008 - Computers - 270 pages
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Hypercomputation is a relatively new theory of computation that is about computing methods and devices that transcend the so-called Church-Turing thesis. This book will provide a thorough description of the field of hypercomputation covering all attempts at devising conceptual hypermachines and all new promising computational paradigms that may eventually lead to the construction of a hypermachine. Readers of this book will get a deeper understanding of what computability is and why the Church-Turing thesis poses an arbitrary limit to what can be actually computed. Hypercomputing is in and of itself quite a novel idea and as such the book will be interesting in its own right. The most important features of the book, however, will be the thorough description of the various attempts of hypercomputation: from trial-and-error machines to the exploration of the human mind, if we treat it as a computing device.
  

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Contents

I Introduction
1
12 From Computation to Hypercomputation
6
13 Why Bother with Hypercomputation?
9
II On the ChurchTuring Thesis
11
22 General Recursive Functions
15
23 Recursive Relations and Predicates
17
24 The ChurchTuring Thesis
20
III Early Hypercomputers
25
VII Computing Real Numbers
113
711 Type2 Machines
114
712 Computable Topologies
117
713 Type2 Computability of Real Numbers
119
714 The Arithmetic Hierarchy of Real Numbers
120
715 Computable Real Functions
121
72 Indeterministic Multihead Type2 Machines
123
73 BSSMachines
125

312 A Model of the Human Mind
27
32 TAEComputability
30
33 Inductive Turing Machines
33
34 Extensions to the Standard Model of Computation
37
35 Exotic Machines
40
36 On Pseudorecursiveness
42
IV InfiniteTime Turing Machines
45
42 InfiniteTime Turing Machines
48
421 How the Machines Operate
49
422 On the Power of InfiniteTime Machines
52
423 Clockable Ordinals
55
424 On InfiniteTime Halting Problems
56
425 Machines with Only One Tape
57
427 Posts Problem for Supertasks
59
43 InfiniteTime Automata
60
44 Building Infinite Machines
61
45 Metaphysical Foundations for Computation
63
V Interactive Computing
69
52 Interaction Machines
72
53 Persistent Turing Machines
75
54 Site and Internet Machines
77
55 Other Approaches
81
VI Hyperminds
85
61 Mathematics and the Mind
86
612 The Argument from Infinitary Logic
96
613 The Modal Argument
97
62 Philosophy and the Mind
100
622 The Chinese Room Argument Revisited
102
63 Neurobiology and the Mind
104
64 Cognition and the Mind
109
731 FiniteDimensional Machines
126
732 Machines over a Commutative Ring
129
733 Parallel Machines
130
74 RealNumber RandomAccess Machines
131
75 Recursion Theory on the Real Numbers
133
VIII Relativistic and Quantum Hypercomputation
137
82 SAD Machines
140
83 Supertasks near Black Holes
144
84 Quantum Supertasks
148
IX Natural Computation and Hypercomputation
165
92 Models of Analog Computation
169
93 On Undecidable Problems of Analysis
174
94 Noncomputability in Computable Analysis
178
95 The Halting Function Revisited
180
96 Neural Networks and Hypercomputation
183
97 An Optical Model of Computation
184
98 Fuzzy Membrane Computing
189
99 Analog XMachines
193
A The P NP Hypothesis
199
B Intractability and Hypercomputation
203
C Socioeconomic Implications
205
D A Summary of Topology and Differential Geometry
209
D2 Vector Spaces and Lie Algebras
210
Definitions
212
D4 Banach and Hilbert Spaces
215
D5 Manifolds and Spacetime
217
References
221
Name Index
235
Subject Index
239
Copyright

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About the author (2008)

Apostolos Syropoulos is a computer scientist. He has been instrumental in the spread of TeX and other related document preparation tools inside and outside Greece and is currently working as a computer educator.

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