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The Grand Unified Theory of Computation
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AbstractThis 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.
Title: The Grand Unified Theory of Computation
Description:
AbstractThis 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.
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