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Unformatted text preview: 1 Decidability Turing Machines Coded as Binary Strings Diagonalizing over Turing Machines Problems as Languages Undecidable Problems 2 BinaryStrings from TMs We shall restrict ourselves to TMs with input alphabet {0, 1}. Assign positive integers to the three classes of elements involved in moves: 1. States: q 1 (start state), q 2 (final state), q 3 , 2. Symbols X 1 (0), X 2 (1), X 3 (blank), X 4 , 3. Directions D 1 (L) and D 2 (R). 3 Binary Strings from TMs (2) Suppose (q i , X j ) = (q k , X l , D m ). Represent this rule by string i 10 j 10 k 10 l 10 m . Key point : since integers i, j, are all > 0, there cannot be two consecutive 1s in these strings. 4 Binary Strings from TMs (2) Represent a TM by concatenating the codes for each of its moves, separated by 11 as punctuation. That is: Code 1 11Code 2 11Code 3 11 5 Enumerating TMs and Binary Strings Recall we can convert binary strings to integers by prepending a 1 and treating the resulting string as a base2 integer. Thus, it makes sense to talk about the ith binary string and about the ith Turing machine. Note : if i makes no sense as a TM, assume the ith TM accepts nothing. 6 Table of Acceptance 1 2 3 4 5 6 . . . TM i 1 2 3 4 5 6 . . . String j x x = 0 means the ith TM does not accept the jth string; 1 means it does. 7 Diagonalization Again Whenever we have a table like the one on the previous slide, we can diagonalize it....
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This note was uploaded on 03/30/2012 for the course CS 154 taught by Professor Motwani,r during the Spring '08 term at Stanford.
 Spring '08
 Motwani,R

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