L1 is a recursively enumerable language over Σ. An algorithm A effectively enumerates its words as w1, w2, w3, … Define another language L2 over Σ Union {#} as {wi # wj : wi, wj ∈ L1, i S1 : L1 is recursive implies L2 is recursive S2 : L2 is recursive implies L1 is recursive
Which of the following statements is true ?
(A) Both S1 and S2 are true
(B) S1 is true but S2 is not necessarily true
(C) S2 is true but S1 is not necessarily true
(D) Neither is necessarily true
Answer: (A)
Explanation: S1 is TRUE.
If L1 is recursive L2 must also be recursive. Because to check if a word w=wi#wj belongs to L2, we can give wi and wj to the decider for L1 and if both are accepted then w belongs to L1 and not otherwise.
S2 is TRUE.
With a decider for L2 we can make a decider for L1 as follows. Let w1 be the first string enumerated by algorithm A for L1. Now, to check if a word w belongs to L1, make a string w′=w1#w and give it to the decider for L2 and if accepted, then w belongs to L1 and not otherwise.
So, the answer must be (A).
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