Unformatted text preview: L 1 · L 2 is Turingrecognizable. 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 Turingrecognizable 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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 Fall '07
 CharikarandChazelle
 Logic, Alan Turing

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