1. Chapter 3
Turing Machines

14. 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?

15. Hilbert’s Problem

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

17. 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.

18. Church-Turing thesis

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

20. Turing-recognizable

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

22. Algorithms = Turing Machines

23. Algorithms = Turing Machines

24. Algorithms = Turing Machines