4. (Closure) Show that the class of context-free languages is closed under the star operation.
Solution: Star operation basically involves a unary operator * that operates on a set of symbols or strings, ∑, and produces an infinite set of all possible strings of all possible lengths as output. As far as Context-Free Languages are concerned, they are generated by the Context-Free Grammars of form C -> ρ (where C ∈ N and ρ ∈ (T ∪ N)* and N is a non-terminal and T is a terminal).
Proof: Let us say there is a language L1 that is context-free, therefore its star closure L1* would always be context-free as shown below.
L1 = {
anbn | n >= 0 }
L1* = { anbn | n >= 0 }* is also context
free.
L1* is context-free because it is nothing but a superset of all the strings that can be created out of the elements present in context-free language L1 and it is already known that L1 is a collection of the strings which follow the context-free grammar. Therefore L1* is a context-free language as well.
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4. (Closure) Show that the class of context-free languages is closed under the star operation.
Show that the family of context-free languages is closed under reversal.
Automata Question (3) Show that the family of deterministic context-free languages is not closed under union and intersection.
Explain the answer QUESTION 8 The classes of languages P and NP are closed under certain operations, and not closed under others, just like classes such as the regular languages or context-free languages have closure properties. Decide whether P and NP are closed under each of the following operations. 1. Union. 2. Intersection. 3. Intersection with a regular language. 4. Concatenation 5. Kleene closure (star). 6. Homomorphism. 7. Inverse homomorphism. Then, select from the list below the true statement. OP...
1. Show that the following languages are context-free. You can do this by writing a context free grammar or a PDA, or you can use the closure theorems for context-free languages. For example, you could show that L is the union of two simpler context-free languages. (b) L {0, 1}* - {0"1" :n z 0}
1. Show that the following languages are context-free. You can do this by writing a context free grammar or a PDA, or you can use the closure theorems for context-free languages. For example, you could show that L is the union of two simpler context-free languages. (d) L = {0, 1}* - L1, where L1 is the language {1010010001…10n-110n1 : n n ≥ 1}.
Q.5 Are the context-free languages closed under reversal? Answer yes or no, and explain. (Reversal means to form the language containing the reverse of every string in the original.)
Prove that the class of regular languages is closed under intersection. That is, show that if ? and ? are regular languages, then ? ∩ ? = {? | ? ∈ ? ??? ? ∈ ?} is also regular. Hint:givenaDFA? =(?,Σ,?,?,?)thatrecognizes?andaDFA? =(?,Σ,?,?,?)that11111 22222 recognizes ?, construct a new DFA ? = (?, Σ, ?, ?0, ?) that recognizes ? ∩ ? and justify why your construction is correct.
Closure properties of P and NP. (a) Is P closed under union, intersection, concatenation, complement and star? Just answer ”yes” or ”no” for each operation. (b) Is NP closed under union, intersection, concatenation, complement and star? Just answer ”yes” or "no" for each operation.
Automata Prove that regular languages are closed under difference, using an indirect proof (leveraging the closure of other set operators).