048_Computability_theory-part_3

048_Computability_theory-part_3 - 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 3 Amir Semmo Extracted from former homeworks in the course "Theory of computation II", Summer term 2008, University of Potsdam October 7, 2008 Exercise Sheet 3 Exercise 1 Prove or disprove the following statement: "For each lambda expression s , which has a normal form, we can say: If s-→ t 1-→ t 2-→ ... is a sequence of β-reductions, then at most a nite number of expressions in { t 1 ,t 2 ,... } are not α-convertible, that means the set of equivalence classes { t 1 ,t 2 ,... } / ≡ is nite." We will disprove the statement above. First we take a look at the following lambda expression: ( λ a b · a ) ( λ x · x ) T We know that this expression can be reduced in two steps to the following expres- sion: ( λ a b · a ) ( λ x · x ) T-→ ( λ b · ( λ x · x )) T-→ λ x · x and λ x · x is clearly in normal form. Now we set T as the expression: T = (( λ t · ttt ) ( λ t · ttt )) As we can see, T is a lambda expression which does not have a normal form, because if we reduce T , we always get a longer expression compared to the initial expression., we always get a longer expression compared to the initial expression....
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048_Computability_theory-part_3 - Selected exercises with...

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