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The Nature of Computation$
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Cristopher Moore and Stephan Mertens

Print publication date: 2011

Print ISBN-13: 9780199233212

Published to Oxford Scholarship Online: December 2013

DOI: 10.1093/acprof:oso/9780199233212.001.0001

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The Grand Unified Theory of Computation

The Grand Unified Theory of Computation

Chapter:
(p.223) Chapter 7 The Grand Unified Theory of Computation
Source:
The Nature of Computation
Author(s):

Moore Cristopher

Publisher:
Oxford University Press
DOI:10.1093/acprof:oso/9780199233212.003.0007

This chapter explores a universal notion of computation, first by describing Charles Babbage's vision of a mechanical device that can perform any calculation as well as David Hilbert's dream of a mechanical procedure capable of proving or refuting any mathematical claim. It then considers the universality of modern computers and the undecidability of certain problems, explores diagonalization and the Halting Problem, and discusses Kurt Gödel's Incompleteness Theorem. It also looks at the three models of computation proposed in the early twentieth century — partial recursive functions, the lambda-calculus, and Turing machines — and show that they are all equivalent to each other and can carry out any conceivable computation. The chapter concludes by considering universal computation and undecidability in tilings of the plane, products of fractions, and the motions of a chaotic system.

Keywords:   computation, Charles Babbage, David Hilbert, universality, undecidability, Halting Problem, Kurt Gödel, Incompleteness Theorem, partial recursive functions, lambda-calculus

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