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L18 - Incomputable Languages Zeph Grunschlag 1...

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1 Incomputable Languages Zeph Grunschlag
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2 Announcements  HW Due Tuesday
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3 Agenda Incomputable Languages Existence Proof Explicit undecidable language:  A TM   Unrecognizable Language Neither recognizable nor  corecognizable Example of a “real world” unsolvable  problem
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4 Incomputable Languages Are there questions of a computational nature  which cannot be answered methodically?  We  saw last time that the computability viewpoint  is to encode algorithmic problems as formal  languages and then ask:  Are there undecidable languages?  Are there languages which are not even  recognizable?  Can we construct a specific language that’s  undecidable?  Are there “useful” undecidable languages?
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5 Incomputable Languages Existential Proof We start first with an indirect proof that an  undecidable language exists.  In fact, we can  show that a language exists that is neither  recognizable nor  corecognizable :  I.e., there  is no TM which halts on all positive instances,  and  there is no TM which halts on all negative  instances so we have no way of necessarily  telling when an arbitrary string is or isn’t in the  language. The proof is “ existential”  in that a counting  argument is used to show that the language  must exist, but we don’t know a priori what  the language is.
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6 Incomputable Languages Existence Proof The idea is simple:  We show that there are many  more languages than Turing machines. Every TM can be encoded in binary by some  string.  Consequently the cardinality 1  of the set  of TM’s is no greater than that of {0,1}*.  Recall  the notion of  countable  :   A set  S  is countable if  there is a 1-to-1 function    from  S   to the  natural numbers.  Equivalently, an infinite set is  countable if there is a bijection    S. THM: The set of bit strings {0,1}* is countable.
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7 Countability of {0,1}* Proof .  Intuitively we can strike up the 1-to-1  correspondence  N    {0,1}* by listing the  strings in short-lex order: ε  0 1 00 01 10 11 000 001 010 011 100… 1 2 3 4  5   6   7   8    9     10   11  12… Can even give recursive definition for  : Can then use induction to prove that  is 1-to-1  and onto. = - = - = = + k k k k u n f u n f n n f 10 ) 1 ( if , 01 1 ) 1 ( if , 0 1 if , ε ) ( 1
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8 Countability of  {TM-Languages} Consequently, as every TM is described by a  bit string, there can only be as many TM’s  as bit strings.  In particular, the following  sets are countable: L = {recognizable languages} L = { co recognizable languages} DEF: A language  over the alphabet  Σ  is  corecognizable  if   Σ *- L  is recognizable.
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9 Countability of 
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