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An Undecidable Language

Explore concepts of undecidable languages, one-to-one and onto functions, infinite sets, and countability in the context of Turing machines. Learn about the undecidability of ATM and the distinction between Turing-recognizable and Turing-decidable languages.

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An Undecidable Language

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  1. [Section 4.2] An Undecidable Language • ATM = { <M,w> | M is a TM that accepts string w } • Turing-recognizable? • Turing-decidable?

  2. [Section 4.2] An Undecidable Language • A bit about infinite sets and their sizes (diagonalization): • Def 4.12: Let A,B be sets and let f:AB. We say that f is • one-to-one if f(a)  f(b) for every a  b • onto if for every b2B there exists a2A such that f(a)=b • If f is one-to-one and onto, then A,B are the same size and f is called correspondence. • Example:N= {1,2,3,4,5,…} and {2,4,6,8,…}

  3. [Section 4.2] An Undecidable Language • A bit about infinite sets and their sizes (diagonalization): • Def 4.12: Let A,B be sets and let f:AB. We say that f is • one-to-one if f(a)  f(b) for every a  b • onto if for every b2B there exists a2A such that f(a)=b • If f is one-to-one and onto, then A,B are the same size and f is called correspondence. • Example:N= {1,2,3,4,5,…} and {2,4,6,8,…} • Def 4.14: A set is countable if it is finite or has the same size as N.

  4. [Section 4.2] An Undecidable Language Are Q (rational numbers) and R (real numbers) countable ?

  5. [Section 4.2] An Undecidable Language Cor 4.18: There is a language that is not Turing-recognizable.

  6. [Section 4.2] An Undecidable Language Thm 4.11: ATM is not decidable. Recall: ATM = { <M,w> | M is a TM that accepts string w }

  7. [Section 4.2] An Undecidable Language Thm 4.22: A language L is decidable iff L is Turing-recognizable and LC is Turing-recognizable (we say that L is co-Turing-recognizable). Cor 4.23: ATMC is not Turing-recognizable.

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