048_Computability_theory-part_1

048_Computability_theory-part_1 - 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 1 Amir Semmo Extracted from former homeworks in the course "Theory of computation II", Summer term 2008, University of Potsdam October 6, 2008 Exercise Sheet 1 Exercise 1 Construct a Turing Machine, which decides the following language: L = { w | w ∈ { a,b,c } * and w contains the same number of occurrences for each of the symbols a , b and c } . Explain the functioning of the Turing Machine informal, but precisely enough in order to derive a formal de nition. The following description makes use of a Turing machine which has a double-sided in nite tape (in case the machine is ended to the left, we could easily write a special character to the left of the input in order to nd the beginning of the input for some of the steps depicted below). The basic idea for a Turing Machine which accepts the language is to move the head from left to right and replacing each of the symbols a , b and c with a special symbol * . After each pass the head moves back to beginning of the input. Depending on the actual state of the Turing Machine, that means which symbol was the previous one replaced, the search continues with the next symbol. At the beginning an a has been replaced, so the searching for a b is initiated. If a b has been replaced, the searching continues for a c and so on. In detail the steps are:and so on....
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048_Computability_theory-part_1 - Selected exercises with...

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