381hw11solns - CS381 Prove that the halting problem for...

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CS381 Homework 11: Problem 1 Prove that the halting problem for Turing machine is undecidable. We can think of the Turing machine for the following pseudocode. The function halt takes two para- meters as input. The first parameter is a program and the second is the argument to that program. We send breakHalt in as the argument to itself. The program therefore says that if breakHalt doesn’t halt then we are done, otherwise we loop forever. This creates a contradiction, therefore the halting problem is undecidable. function breakHalt(string s) if halt(s, s) = false return true else loop forever 1
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Rice’s Theorem: Every nontrivial property of the RE languages is undecidable. Proof: Assume Ø, the empty language is not in the nontrivial property P. Let L be a nonempty language in P and M L be a Turing Machine accepting L. Explanation: The reduction from L u to L P will be sufficient in proving L P is undecidable since we’ve already established that L u is. We need to construct a Turing Machine M’ that takes as input (M,w) which is just another Turing Machine and a string.
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