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Unformatted text preview: Harvard CS 121 and CSCI E207 Lecture 15: Undecidability Harry Lewis November 3, 2009 Reading: Sipser 4.2, 5.1. Harvard CS 121 & CSCI E207 November 3, 2009 Motivation Goal : to find an explicit undecidable language By the ChurchTuring thesis, such a language has a membership problem that cannot be solved by any kind of algorithm We know such languages exist, by a counting argument. Every decidable language is decided by a TM There are only countably many TMs There are uncountably many languages Most languages are not decidable (or even Turingrecognizable) 1 Harvard CS 121 & CSCI E207 November 3, 2009 Is every Turingrecognizable set decidable? This would be true if there were an algorithm to solve The Acceptance Problem: Given a TM M and an input w , does M accept input w ? Formally, A TM = {h M,w i : M accepts w } . Proposition: If A TM is decidable, then every Turingrecognizable language is decidable. A TM is the hardest Turingrecognizable language. 2 Harvard CS 121 & CSCI E207 November 3, 2009 A simplifying detail: every string represents some TM Let be the alphabet over which TMs are represented (that is, h M i * for any TM M ) Let w * if w = h M i for some TM M then w represents M Otherwise w represents some fixed TM M (say the simplest possible TM)....
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This document was uploaded on 12/24/2011.
 Fall '09

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