GATE | GATE IT 2006 | Question 81

Let L be a regular language. Consider the constructions on L below:
repeat (L) = {ww | w ∈ L}
prefix (L) = {u | ∃v : uv ∈ L}
suffix (L) = {v | ∃u uv ∈ L}
half (L) = {u | ∃v : | v | = | u | and uv ∈ L}
Which of the constructions could lead to a non-regular language?
(A) Both I and IV
(B) Only I
(C) Only IV
(D) Both II and III


Answer: (B)

Explanation:  

  • repeat(L) should not be confused with concatenation as there is a specific order to it. Only same strings are concatenated with each other and not all. Moreover, double word language is not even a CFG [non-regular].
  • prefix(L) is a regular language – all the states in L’s DFA could be made final, giving result to a DFA which would accept prefix(L) [regular]
  • suffix(L) is a regular language as a NFA could be constructed which would accept suffix(L). Every state in L’s DFA can get an incident # – edge from start state – this NFA would accept suffix(L) [regular].
  • half(L) is a regular language. It is just a language containing even length strings.

So, option (B) is correct.

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