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### How to prove that a language is not regular?

We learned about the class of regular languages $\mathrm{REG}$. It is characterised by any one concept among regular expressions, finite automata and left-linear grammars, so it is easy to show that a ...
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### How to prove that a language is context-free?

There are many techniques to prove that a language is not context-free, but how do I prove that a language is context-free? What techniques are there to prove this? Obviously, one way is to exhibit ...
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### Are context-free languages in $a^*b^*$ closed under complement?

The context-free languages are not closed under complement, we know that. As far as I understand, context-free languages that are a subset of $a^*b^*$ for some letters $a,b$ are closed under ...
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### Pumping lemma: if you can keep pumping, what does this tell you?

Hypothetically, let's say you are using the pumping lemma for either regular or context free languages. Now using either, you come across a case that remains true despite pumping it. In this situation,...
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### Why is the following language not context-free?

$L = \{a^n b^m | m \not= n^2 \}$ I guess I need to use Pumping Lemma for CFL in order to prove this. But I'm stuck. Assuming that $a^n b^m = uvxyz$, we know that $v$ or $y$ can not have both $a$ ...
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### Are Turing machines more powerful than pushdown automata?

I've came up with a result while reading some automata books, that Turing machines appear to be more powerful than pushdown automata. Since the tape of a Turing machine can always be made to behave ...
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### Prime number CFG and Pumping Lemma

So I have a problem that I'm looking over for an exam that is coming up in my Theory of Computation class. I've had a lot of problems with the pumping lemma, so I was wondering if I might be able to ...
### Can a two-stack PDA accept language $a^nb^mc^nd^m$ which is not context-free?
Can a two-stack PDA accept language $L=\{a^nb^mc^nd^m \mid n \geq m\}$, which has no context-free grammar? I don't believe this has a context-free grammar, but please correct me if I'm wrong.
There is a problem which I cannot solve. If you give a tip I will be very glad. Prove that following language is not context free: $L= \{ a^nb^m | \gcd(n,m) = 1 \}$. It can be proven using the ...