Let L1 and L2 be any deterministic context-free language and R be any regular language. Then, which of the following is correct ?
I. L1 ∪ L2 is deterministic context-free.
II. L1′ is context-free.
III. L1-R is context-free.
IV. L1 ∩ L2 deterministic context-free.
(A) I, II and IV only
(B) I and III only
(C) II and IV only
(D) II and III only
Explanation: The set of deterministic context-free languages is closed under the following operations:
- inverse homomorphism
- right quotient with a regular language
- pre: pre(L) is the subset of all strings having a proper prefix that also belongs to L.
- min: min(L) is the subset of all strings do not have a proper prefix in L.
- max: max(L) is the subset of all strings that are not the prefix of a longer string in L.
The set of deterministic context-free language is not closed under the following operations:
- Kleene star
- ε-free morphism
- Mirror image
statements (II) and (III) only are correct.
Option (D) is true.
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