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lect18-tm4 - RicesTheorem SomeRealProblems 1 x languages x...

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1 More Undecidable Problems Rice’s Theorem Post’s Correspondence Problem Some Real Problems
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2 Properties of Languages Any set of languages  is a  property   of  languages. Example : The infiniteness property is the  set of infinite languages.
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3 Properties of Langauges – (2) As always, languages must be defined by  some descriptive device. The most general device we know is the  TM. Thus, we shall think of a property as a  problem  about Turing machines. Let L P  be the set of binary TM codes for  TM’s M such that L(M) has property P. 
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4 Trivial Properties There are two ( trivial  ) properties P for  which L P  is decidable. 1. The  always-false property , which  contains no RE languages. 2. The  always-true property , which contains  every RE language. Rice’s Theorem : For every other  property P, L P  is undecidable.
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5 Plan for Proof of Rice’s Theorem 1. Lemma needed : recursive languages  are closed under complementation. 2. We need the technique known as  reduction , where an algorithm  converts instances of one problem to  instances of another. 3. Then, we can prove the theorem.
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6 Closure of Recursive Languages  Under Complementation If L is a language with alphabet  Σ *, then  the  complement   of L is  Σ * - L. Denote the complement of L by L c . Lemma : If L is recursive, so is L c . Proof : Let L = L(M) for a TM M. Construct M’ for L c . M’ has one final state, the new state f.
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7 Proof  – Concluded M’ simulates M. But if M enters an accepting state, M’  halts without accepting. If M halts without accepting, M’ instead  has a move taking it to state f. In state f, M’ halts.
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8 Reductions reduction   from language L to  language L’ is an algorithm (TM that  always halts) that takes a string w and  converts it to a string x, with the  property that: x is in L’ if and only if w is in L.
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9 TM’s as  Transducers We have regarded TM’s as acceptors of  strings. But we could just as well visualize TM’s  as having an  output tape , where a string  is written prior to the TM halting. Such a TM translates its input to its  output.
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10 Reductions – (2) If we reduce L to L’, and L’ is decidable,  then the algorithm for L’ + the algorithm  of the reduction shows that L is also  decidable. Used in the contrapositive : If we know L  is not decidable, then L’ cannot be  decidable.
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11 Reductions –  Aside This form of reduction is not the most  general.
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