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Turing-Church Thesis and Incomputability

This text delves into the Turing-Church Thesis and the concept of computability, highlighting the fascinating realm of integer roots of polynomials. Despite some integers like x=5, y=3, z=0 providing solutions, not all integer roots exist. The essay discusses pivotal models, including Turing Machines and lambda calculus, illustrating their equivalence in defining computation. It also introduces Turing completeness, showcasing that many systems, even simple cellular automata and programming languages like Postscript, exhibit Turing-complete behavior, expanding our understanding of what machines can compute.

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Turing-Church Thesis and Incomputability

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  1. Turing-Church Thesisand Incomputability In which a few things really do not compute

  2. Integer Roots of Polynomials Turns out x=5, y=3, z=0 works. But integer roots do not always exist.

  3. Turing-Church Thesis Is About What Defines a Computation • Several famous math questions were proposed, but there was really no way to answer these questions in the negative • Several models were proposed, two of note were the Turing Machine and the lambda calculus (which is bizarre and awesome BTW) • But, as we’ve seen, a little computation goes a long way and both of these definitions of computation proved to be equivalent

  4. Turing Completeness • Another way to say “equivalent to Turing Machines”. You would say “Turns out 1-dimension cellular automata are Turing-complete” • Way more things are Turing Complete than you’d think. For example, the game of life is Turing complete. There is even a 1-D 2 color cellular automata that is Turing Complete • Often also a feature of excessively featured configuration languages like Postscript (a printer language)

  5. Surely there is nothing our magical machines can’t do?

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