
Computation and its Limits
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Content
- Cover
- Contents
- 1 Introduction
- 1.1 Overview
- 1.2 Summary
- 1.3 Acknowledgements
- 2 What is computation?
- 2.1 The apparent mystery of maths
- 2.2 Counting sheep
- 2.3 Counting materialised in our own bodily movements
- 2.4 From 'aides-memoire' to the first digital calculating devices
- 3 Mechanical computers and their limits
- 3.1 Antikythera
- 3.2 Late mechanical computers
- 3.3 Analogue mechanical multiply/accumulate
- 3.4 Mechanizing the abacus
- 4 Logical limits to computing
- 4.1 Introduction
- 4.2 Propositional logic
- 4.3 Set theory
- 4.4 Predicate logic
- 4.5 Recursion
- 4.6 Peano arithmetic
- 4.7 Paradoxes
- 4.8 Arithmetizing mathematics and incompleteness
- 4.9 Infinities
- 4.10 Real numbers and Cantor diagonalization
- 4.11 Turing machines
- 4.12 Universal TM and undecidability
- 4.13 Computational procedures
- 4.14 The Church-Turing thesis
- 4.15 Machines, programs, and expressions
- 5 Heat, information, and geometry
- 5.1 The triumph of digital computation
- 5.2 Analogue computing with real numbers
- 5.3 What memories are made of
- 5.4 Power consumption as a limit
- 5.5 Entropy
- 5.6 Shannon's information theory
- 5.7 Landauer's limit
- 5.8 Non-entropic computation
- 5.9 Interconnection
- 6 Quantum computers
- 6.1 Foundations of quantum theory
- 6.2 The quantum rules
- 6.3 Qubits
- 6.4 Entanglement and quantum registers
- 6.5 Quantum computers
- 6.6 Quantum algorithms
- 6.7 Building a quantum computer
- 6.8 Physical limits to real number representations
- 6.9 Error rates in classical and quantum gates
- 7 Beyond the logical limits of computing?
- 7.1 Introduction
- 7.2 Oracles, complexity, and tractability
- 7.3 Beyond the Turing Machine?
- 7.4 Numberology
- 7.5 What is real about the reals?
- 7.6 Real measurement
- 7.7 Back to Turing
- 7.8 Reservations about Cantor
- 8 Hypercomputing proposals
- 8.1 Infinite Turing Machines
- 8.2 Infinitely precise analogue computers
- 8.3 Wegner and Eberbach's super-Turing computers
- 8.4 Interaction Machines
- 8.5 p-Calculus
- 8.6 $-Calculus
- 8.7 Conclusions
- Bibliography
- Index
- A
- B
- C
- D
- E
- F
- G
- H
- I
- J
- K
- L
- M
- N
- O
- P
- Q
- R
- S
- T
- U
- V
- W
- Y
- Z
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