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Unformatted text preview: 1 More Undecidable Problems Rices Theorem Posts Correspondence Problem Some Real Problems 2 Properties of Languages Any set of languages is a property of languages. Example : The infiniteness property is the set of infinite languages. 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 TMs M such that L(M) has property P. 4 Trivial Properties There are two ( trivial ) properties P for which L P is decidable. 1. The alwaysfalse property , which contains no RE languages. 2. The alwaystrue property , which contains every RE language. Rices Theorem : For every other property P, L P is undecidable. 5 Plan for Proof of Rices 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. 6 Closure of Recursive Languages Under Complementation If L is a language with alphabet *, then the complement of L is *  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. 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. 8 Reductions A 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. 9 TMs as Transducers We have regarded TMs as acceptors of strings. But we could just as well visualize TMs as having an output tape , where a string is written prior to the TM halting. Such a TM translates its input to its output. 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....
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 Spring '08
 Motwani,R

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