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Undecidability

Undecidability - Harvard CS 121 and CSCI E-207 Lecture 15...

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Harvard CS 121 and CSCI E-207 Lecture 15: Undecidability Harry Lewis November 3, 2009 Reading: Sipser § 4.2, § 5.1.
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Harvard CS 121 & CSCI E-207 November 3, 2009 Motivation Goal : to find an explicit undecidable language By the Church–Turing 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 Turing-recognizable) 1
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Harvard CS 121 & CSCI E-207 November 3, 2009 Is every Turing-recognizable 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 = { M, w : M accepts w } . Proposition: If A TM is decidable, then every Turing-recognizable language is decidable. A TM is the hardest Turing-recognizable language.” 2
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Harvard CS 121 & CSCI E-207 November 3, 2009 A simplifying detail: every string represents some TM Let Σ be the alphabet over which TMs are represented (that is, M Σ * for any TM M ) Let w Σ * if w = M for some TM M then w represents M Otherwise w represents some fixed TM M 0 (say the simplest possible TM).
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