Show that the class of context-free languages is closed under the regular operation union. For simplicity, you may assume that the alphabets of G1 and G2 are the same.
[Hint: Use a constructive proof. Start with the formal definitions, G1 = (V1 ,∑, R1,S1) and G2 = (V2, ∑, R2, S2) and derive the formal definition of G∪.]
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Show that the class of context-free languages is closed under the regular operation union. For simplicity,...
4. (Closure) Show that the class of context-free languages is closed under the star operation.
Show that the class of context-free languages is not closed under difference. Use either of the following facts: a. The class of context-free languages is not closed under intersection. b. The language {ww | w ∈ {a,b}*} is not a CFL.
Automata Question
(3) Show that the family of deterministic context-free languages is not closed under union and intersection.
Show that the family of context-free languages is closed under reversal.
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.
Show using a cross-product construction that the class of regular languages is closed under set difference. You do not need an inductive proof, but you should convincingly explain why your construction works.
Show using a cross-product construction that the class of regular languages is closed under set difference. You do not need an inductive proof, but you should convincingly explain why your construction works.
2. (15) Show using a cross-product construction that the class of regular languages is closed under set difference. You do not need an inductive proof, but you should convincingly explain why your construction works.
(20 pt.) Prove that the class of regular languages is closed under reverse. That is, show that if ? is a regular language, then ?? = {?? | ? ∈ ?} is also regular. Hint: given a DFA ? = (?, Σ, ?, ?0, ?) that recognizes ?, construct a new NFA ? = (?′, Σ, ?′, ?0′, ?′)that recognizes ?? and justify why your construction is correct.