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048_Computability_theory-part_5

# 048_Computability_theory-part_5 - Selected exercises with...

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Unformatted text preview: Selected exercises with solutions on Computability theory in the field of the Theory of computation Part 5 Amir Semmo Extracted from former homeworks in the course "Theory of computation II", Summer term 2008, University of Potsdam October 6, 2008 Exercise Sheet 5 Exercise 1 For a Turing Machine M let φ M : N-→ N be the function on natural numbers, which is calculated by M . Prove by diagonalization, that the set B = {h M i | M is a Turing Machine, where φ M is de ned on the whole interval and does not have the function value of at no position } is not Turing-recognizable. We will prove by contradiction, where we show with the help of the diagonalization method, that when there is a Turing Machine M with the language B , a Turing Machine exists, which is also in B but is not recognized by M . By this we assume that the Turing Machine M recognizes B ( L ( M ) = B ). Because M recognizes B , there exists an enumerator, which enumerates all h M i i of the following table: 1 2 3 4 5 ... j h M i b 00 b 01 b 02 b 03 b 04 b 05 ... b j h M 1 i b 10 b 11 b 12 b 13 b 14 b 15 ... b 1 j h M 2 i b 20 b 21 b 22 b 23 b 24 b 25 ... b 2 j h M 3 i b 30 b 31 b 32 b 33 b 34 b 35 ... b 3 j h M 4 i b 40 b 41 b 42 b 43 b 44 b 45 ... b...
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048_Computability_theory-part_5 - Selected exercises with...

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