53 Slides--More on Decidability and Reductions

53 Slides--More on Decidability and Reductions - CS103...

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CS103 HO#53 Slides--More on Decidability and Reductions May 21, 2010 1 Theorem 4.11: A TM = { M, w | M is a Turing machine and M accepts w } is undecidable. Suppose H is a decider for A TM . Build the machine D. Run D on its own description. H: A TM D D D accepts D D D Accept Reject If D accepts D , it rejects it, and if D rejects D , it accepts it. Thus D and H cannot exist, and A TM is undecidable . D rejects or loops on D Applying the machine D to it's own description reminds us of a diagonal argument : M 1  M 2 M 3 D M 1 accept reject accept ... accept M 2 reject accept accept ... reject M 3 reject accept reject ... reject . . . D accept reject accept ... ? H D D D accepts D D D Accept Reject D rejects or loops on D 1 2 3 4 . . . M 1 1 0 0 1 M 2 0 0 0 0 M 3 1 1 0 1 M 4 0 0 1 1 . . . Another Diagonal Argument M i is the machine whose description is the i th binary string (or if that is not the description of a machine, M i is a machine that accepts no input). The i,j element is 1 if M i accepts the jth string. So the diagonal elements tell us whether a machine accepts its own description. 1 2 3 4 . . . M 1 1 0 0 1 M 2 0 0 0 0 M 3 1 1 0 1 M 4 0 0 1 1 . . . . . . Another Diagonal Argument Since there is a row and column for every string, the rows are the characteristic vectors of the languages of all Turing machines. 1 2 3 4 . . . M 1 1 0 0 1 M 2 0 0 00 M 3 1 1 0 1 M 4 0 0 1 1 . . . . . . Another Diagonal Argument Since there is a row and column for every string, the rows are the characteristic vectors of the languages of all Turing machines. 1 2 3 4 . . . M 1 0 0 0 1 M 2 0 1 0 0 M 3 1 1 1 1 M 4 0 0 1 0 . . . Now we complement the diagonal. The result cannot be the characteristic vector of the language of any Turing machine, since it differs from every row. Another Diagonal Argument Since there is a row and column for every string, the rows are the characteristic vectors of the languages of all Turing machines.
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53 Slides--More on Decidability and Reductions - CS103...

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