A language L satisfies the pumping lemma for regular languages and also the pumping lemma for context free languages.Which of the following statements about L is true ?
A. L is necessarily a regular language.
B. L is necessarily a CFL but not Regular.
C. L is necessarily a non-regular.
D. None
I'll clear where I'm having doubt. If L satisfies pumping lemma for regular languages then it is not necessarily regular. Same with context free. So it can be Regular or non-regular. CFL or non-CFL. Answer given is B but in my opinion it should be D. Can anyone point out what I'm missing.
Answer B is definitely not right. Try the language Σ*, which is absolutely regular and definitely context-free. It also passes the conditions of both pumping lemmas. Therefore, it's definitely not the case that the language is context-free but not regular.
Both pumping lemmas give necessary conditions for a language to be regular or context-free, rather than sufficient conditions for those languages to be regular or context-free. Therefore, if a language passes both of the pumping lemmas, it might be regular and it might be context-free, but there's no guarantee that this must necessarily be the case.
I'm pretty sure D is the correct choice here.
Hope this helps!