The Halting Problem ATM M w M is
The Halting Problem ATM = { < M, w > | M is a TM that accepts string w } - Turing-recognizable? - Turing-decidable? [Section 4. 2]
[Section 4. 2] The Halting Problem A bit about infinite sets and their sizes ( Def 4. 12: Let A, B be sets and let is diagonalization): f: A B. We say that f - one-to-one if f(a) f(b) for every a b - onto if for every b 2 B there exists a 2 A such that f(a)=b If f is one-to-one and onto, then A, B are the and f is called correspondence. Example: N = {1, 2, 3, 4, 5, …} and {2, 4, 6, 8, …} same size
[Section 4. 2] The Halting Problem A bit about infinite sets and their sizes ( Def 4. 12: Let A, B be sets and let is diagonalization): f: A B. We say that f - one-to-one if f(a) f(b) for every a b - onto if for every b 2 B there exists a 2 A such that f(a)=b If f is one-to-one and onto, then A, B are the and f is called correspondence. Example: same size 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
The Halting Problem [Section 4. 2] Are Q (rational numbers) and R (real numbers) countable ?
The Halting Problem Cor 4. 18: There is a language that is not Turingrecognizable. [Section 4. 2]
The Halting Problem [Section 4. 2] Thm 4. 11: A TM is not decidable. Recall: A TM = { < M, w > | M is a TM that accepts string w }
The Halting Problem [Section 4. 2] Thm 4. 22: A language L is decidable iff L is Turingrecognizable and L C is Turing-recognizable (we say that L is co-Turing-recognizable). Cor 4. 23: A TMC is not Turing-recognizable.
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