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Chapter 3

Chapter 3. Turing Machines. Deeply Understanding TMs. To deeply understand how TMs really work, consider this simple question. How does a TM swap two tape symbols? Consider a tape alphabet of {a, b, _} Consider this tape (_ _ _ a _ _ b _ _...) How would you swap them?. Hilbert’s Problem.

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Chapter 3

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  1. Chapter 3 Turing Machines

  2. Deeply Understanding TMs • To deeply understand how TMs really work, consider this simple question. • How does a TM swap two tape symbols? • Consider a tape alphabet of {a, b, _} • Consider this tape (_ _ _ a _ _ b _ _...) • How would you swap them?

  3. Hilbert’s Problem

  4. Hilbert’s Problem • Does a polynomial have integral roots?

  5. Origin of true algorithms • Hilbert stated: • “a process according to which it can be determined by a finite number of operations.” • He assumed such an algorithm existed, but someone just needed to find it. • But apparently, no such algorithm exists • Hilbert’s problem is algorithmically unsolveable.

  6. Church-Turing thesis

  7. Hilbert’s problem as a language • D is not decidable • D is however recognizable

  8. Turing-recognizable

  9. Turing-recognizable • Essentially means any language that is recursively enumerable • Positive examples lead to acceptance for some Turing machine.

  10. Algorithms = Turing Machines

  11. Algorithms = Turing Machines

  12. Algorithms = Turing Machines

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