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048_Computability_theory-part_2

# 048_Computability_theory-part_2 - 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 2 Amir Semmo Extracted from former homeworks in the course "Theory of computation II", Summer term 2008, University of Potsdam October 6, 2008 Exercise Sheet 2 Exercise 1 Show that the Turing-recognizable languages are not closed under the operation of complement on condition that there exist Turing-recognizable languages, which are not decidable. Let L 1 be a Turing-recognizable, non-decidable language. Now we show, that the complement L 1 = L 2 = Σ * \ L 1 cannot be Turing-recognizable via proof by contra- diction. We assume L 2 is also Turing-recognizable. If this is the case, we can construct a new Turing Machine M 3 which simulates M 1 (recognizing L 1 ) and M 2 (recognizing L 2 ). What M 3 does is simulating these machines alternately and step by step. Since we can be sure that the input w of M 3 must be accepted by M 1 i w ∈ L 1 and is accepted by M 2 i w ∈ L 2 and L 1 ∪ L 2 = Σ * , M 3 will always halt when we de ne it in a way to accept the input when M 1 accepts and to reject when M 2 rejects. Because M 3 only accepts when M 1 accepts it is also clear that L ( M 3 ) = L ( M 1 ) = L 1 ....
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048_Computability_theory-part_2 - Selected exercises with...

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