hw10 - L 1 L 2 is Turing-recognizable Problem 3 Let S be...

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Computer Science 340 Reasoning about Computation Homework 10 Due at the beginning of class on Wednesday, December 5, 2007 Problem 1 Give an implementation level description of a Turing machine that decides the following language over alphabet { 0 , 1 } : { w | w does not contain twice as many 0 s as 1 s } . Read the Section on “Terminology For Describing Turing Machines” in the notes (pages 144-147) in order to clarify what an implementation level description is. Problem 2 The concatenation operation on languages L 1 and L 2 is defined as follows: L 1 · L 2 = { x 1 x 2 | x 1 L 1 , x 2 L 2 } In other words L 1 · L 2 consists of all strings obtained by attaching a string from L 1 in front of a string from L 2 . Suppose that languages L 1 and L 2 are Turing-recognizable. Show that
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Unformatted text preview: L 1 · L 2 is Turing-recognizable. Problem 3 Let S be the set of all finite subsets of integers. Show that S is countable. Problem 4 Let C be a language. Prove that C is Turing-recognizable iff a decidable language D exists such that C = { x | ∃ y ( h x, y i ∈ D ) } . Hint: For the “only if” part, suppose M is a Turing machine that recognizes C , i.e. M accepts input x iff x ∈ C . Consider modifying M by giving it additional information (the string y ) as input so that the modified machine halts always. For every x ∈ C , there should be some value of y that causes the modified machine to halt and accept....
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