Chapter4Section2_F11

Chapter4Section2_F11 - 91.304 Foundations of (Theoretical)...

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91.304 Foundations of (Theoretical) Computer Science Chapter 4 Lecture Notes (Section 4.2: The “Halting” Problem) David Martin dm@cs.uml.edu With modifications by Prof. Karen Daniels, Fall 2011 This work is licensed under the Creative Commons Attribution-ShareAlike License. To view a copy of this license, visit http://creativecommons.org/licenses/by- 1 sa/2.0/ or send a letter to Creative Commons, 559 Nathan Abbott Way, Stanford, California 94305, USA.
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Back to Σ 1 ± So the fact that Σ 1 is not closed under complement means that there exists ome language L that is not some language L that is not recognizable by any TM y Church uring thesis this means ± By Church-Turing thesis this means that no imaginable finite computer, ven with infinite memory could even with infinite memory, could recognize this language L! 2
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Non-recognizable languages ± We proceed to prove that non-Turing recognizable languages exist, in two ays: ways: ² A nonconstructive proof using Georg antor’s famous 1873 diagonalization Cantor s famous 1873 diagonalization technique, and then ² An explicit construction of such a language. 3
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A nonconstructive proof et L 0 1} * e defined by: ± Let L {0,1} be defined by: = otherwise * 1 2013 1, February on president is Obama if * 0 L Is L decidable? ± Yes ; there exists a machine M that recognizes the appropriate language; we just don’t know what machine it is right now. 4
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Learning how to count efinition et A and B be sets Then we write ± Definition Let A and B be sets. A B and say that A is equinumerous to B if there exists a one-to-one, onto function (a “ orrespondence”) correspondence ) f :A B ± Note that this is a purely mathematical efinition: the function oes not have to be definition: the function f does not have to be expressible by a Turing machine or anything like that. ± Example: { 1, 3, 2 } { six, seven, BBCCD } ± Example: N Q (textbook example 4.15) ² See next slide… 5
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Learning how to count (continued) xample: extbook example 4 15) ± Example: N Q (textbook example 4.15) 6 Source: Sipser textbook
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Countability efinition set S is ountable ± Definition A set S is countable if S is finite or S N . aying that S is countable means that ² Saying that S is countable means that you can line up all of its elements, one after another, and cover them all ² Note that R is not countable (Theorem 4.17), basically because choosing a single real number requires making infinitely many choices of what each digit in it is (see next slide). 7
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Countability (continued) heorem 4 17 ot ountable ± Theorem 4.17 : R is not countable. ± Proof Sketch: By way of contradiction, suppose R N using correspondence f . Construct R such that x is not paired with anything in N , providing a contradiction. x ) 1 , 0 ( x x is not f ( n ) for any n because it differs from f ( n ) in n th fractional digit.
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This note was uploaded on 02/13/2012 for the course CS 91.304 taught by Professor Staff during the Fall '11 term at UMass Lowell.

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Chapter4Section2_F11 - 91.304 Foundations of (Theoretical)...

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