(e) Give an example if a recursively enumerable language whose complement is recursively enumerable. Give a precise definition of this language and explain your answer. If such a language does not exist, explain why.
Added by Vicenta G.
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e., it will halt and accept) but may either reject or run forever on strings not in the language. The complement of a language L, denoted as L', consists of all strings not in L. Show more…
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(a) Consider the set of languages for which some Turing machine exists which recognizes the language. Is this set countable or uncountable? Explain your answer. (b) Are there languages which can be recognized by Turing machines, but not by finite state machines? Give reasons for your answer, and at least one example if there are any. (c) Are there languages which cannot be recognized by Turing machines? Give reasons for your answer, and at least one example if there are any.
Adi S.
Akash M.
(a) For every language L over Σ at least one of L and L^C is Turing-recognizable. (b) If languages L_1 and L_2 over Σ are both regular with L_1 ⊆ L_2, then a language L that is sandwiched between them, that is L_1 ⊆ L ⊆ L_2 is Turing-recognizable. (c) For every language L over Σ that is Turing-recognizable, at least one of L and L^C is Turing-decidable. Prove the following statement: (d) Let T_1 and T_2 be Turing Machines such that L(T_2)^C ⊆ L(T_1). If, on every input word w, T_1 takes longer to compute than T_2, then L(T_2) is Turing-decidable.
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