lect17-tm3

lect17-tm3 - 1 Decidability Turing Machines Coded as Binary...

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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 Binary-Strings from TM’s ◆ We shall restrict ourselves to TM’s 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 TM’s – (2) ◆ Suppose δ (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 1’s in these strings. 4 Binary Strings from TM’s – (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 TM’s and Binary Strings ◆ Recall we can convert binary strings to integers by prepending a 1 and treating the resulting string as a base-2 integer. ◆ Thus, it makes sense to talk about “the i-th binary string” and about “the i-th Turing machine.” ◆ Note : if i makes no sense as a TM, assume the i-th 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 i-th TM does not accept the j-th 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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lect17-tm3 - 1 Decidability Turing Machines Coded as Binary...

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