8. Undecidability | Summary and Q&A

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October 6, 2021
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8. Undecidability

TL;DR

The existence of unrecognizable and undecidable languages is proven, highlighting the limitations of automata theory.

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Key Insights

  • 👍 The diagonalization method is a powerful technique used to prove problems as undecidable or unrecognizable.
  • ❓ The complement of a recognizable language is not necessarily recognizable.

Transcript

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Questions & Answers

Q: How does Cantor's diagonalization method apply to automata theory?

Cantor's diagonalization method is used in automata theory to prove certain problems as undecidable or unrecognizable. It is a powerful tool that shows there are limits to what can be achieved in automata theory.

Q: Why is A,TM unrecognizable?

A,TM is undecidable, but it is recognizable because there is a Turing machine that can accept an input and halt if the input is a valid Turing machine description. However, its complement, A,TM-complement, is unrecognizable because if both A,TM and A,TM-complement were recognizable, A,TM would be decidable, leading to a contradiction.

Q: How can the complement of a recognizable language be unrecognizable?

If both a language and its complement are recognizable, then the language is decidable. Therefore, if we know that a language is recognizable and undecidable, its complement must be unrecognizable. These results are based on the properties and limitations of automata theory.

Summary & Key Takeaways

  • The diagonalization method, proposed by Georg Cantor, is introduced as a way to compare the sizes of infinite sets.

  • The acceptance problem for Turing Machines is proven to be undecidable using the diagonalization method.

  • The complement of the acceptance problem for Turing Machines (A,TM) is shown to be unrecognizable, further highlighting the limitations of automata theory.

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