048_Computability_theory-part_7

048_Computability_theory-part_7 - 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 7 Amir Semmo Extracted from former homeworks in the course "Theory of computation II", Summer term 2008, University of Potsdam October 6, 2008 Exercise Sheet 7 Exercise 1 Proof the following two propositions: 1. A language A is Turing-recognizable, if and only if A ≤ f A TM . 2. A language A is decidable, if and only if A ≤ f L (0 * 1 * ) . First we prove the implication (a) to (b) of the rst proposition: Let A be Turing- recognizable ( A ⊆ Σ * ). Since A is Turing-recognizable, there exist a Turing Machine M recognizing A ( L ( M ) = A ). Now we de ne our reduction f from A to A TM as follows: f ( w ) = h M,w i , where w ∈ Σ * and M is the Turing Machine for the language A . Let F be the Turing Machine, calculating f : TM F = "On input w : 1. Return h M,w i and halt." Obviously F halts on every input w and returns h M,w i . By this f is a computable function. Additionally we can say: If the input w is in A , h M,w i is in A TM , because M accepts w . On the other hand, if h M,w i is in A TM , M accepts w and w is in A , because L ( M ) = A . From this it follows that A ≤ f A TM ....
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This note was uploaded on 02/22/2010 for the course CS 881 taught by Professor H.f. during the Spring '10 term at Shahid Beheshti University.

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048_Computability_theory-part_7 - Selected exercises with...

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