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048_Computability_theory-part_8

# 048_Computability_theory-part_8 - 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 8 Amir Semmo Extracted from former homeworks in the course "Theory of computation II", Summer term 2008, University of Potsdam October 6, 2008 Exercise Sheet 8 Exercise 1 For a partial computable function f : N-→ N , let f be de ned as: f ( x ) =    f ( x ) if f is de ned at x else. Obviously the function f is de ned on the whole intervall of N . Show that the function f cannot be partial computable for every function f . We take a look at the partial computable function f , based on a Turing Machine for the language HALT TM : f ( n ) =    1 if n = h M,w i and M halts on input w ⊥ else. What f does, is extracting the Gödel number for the Turing Machine M and con- catenating the word w ( w ∈ Σ * ), which is also coded as a natural number ( note: every word can be encoded as a unique natural number )....
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048_Computability_theory-part_8 - Selected exercises with...

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