## Hypercomputation: Computing Beyond the Church-Turing BarrierHypercomputation 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

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 |

221 | |

235 | |

239 | |