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

Hilbert’s Problem Does a polynomial have integral roots?

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.

Church-Turing thesis

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

Turing-recognizable

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

Algorithms = Turing Machines

Algorithms = Turing Machines

Algorithms = Turing Machines