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Unformatted text preview: 1 Incomputable Languages Zeph Grunschlag 2 Announcements HW Due Tuesday 3 Agenda Incomputable Languages Existence Proof Explicit undecidable language: A TM Unrecognizable Language Neither recognizable nor corecognizable Example of a “real world” unsolvable problem 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? 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. 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 1to1 function f : S N from S to the natural numbers. Equivalently, an infinite set is countable if there is a bijection N S. THM: The set of bit strings {0,1}* is countable. 7 Countability of {0,1}* Proof . Intuitively we can strike up the 1to1 correspondence f : N {0,1}* by listing the strings in shortlex 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 f : Can then use induction to prove that f is 1to1 and onto. = = = = + k k k k u n f u n f n n f 1 0 ) 1 ( if , 0 1 1 ) 1 ( if , 1 if , ε ) ( 1 8 Countability of {TMLanguages} 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 1 = {recognizable languages} L 2 = { co recognizable languages} DEF: A language L over the alphabet Σ is corecognizable if Σ * L is recognizable. 9...
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This note was uploaded on 02/22/2010 for the course CS 881 taught by Professor H.f. during the Spring '10 term at Shahid Beheshti University.
 Spring '10
 H.F.

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