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.