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.

  • $\begingroup$ See here for the power of two-stack PDA, and here for techniques to show that the language is not context-free. $\endgroup$ – Raphael Apr 2 '13 at 7:15

A two-stack PDA is equivalent in computing power to a Turing machine. Since a Turing machine can accept that language (stated without proof), a two-stack PDA can as well. The actual definition of such a machine is left as an exercise :)

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  • $\begingroup$ Thanks! P.S. What did you mean about stated without proof ? Are there languages which a turing machines can't accept? $\endgroup$ – Iancovici Mar 29 '13 at 13:25
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    $\begingroup$ @echadromani a valid program that never finishes under some input (look up the halting problem for more input) $\endgroup$ – ratchet freak Mar 29 '13 at 13:55

You don't even need to know about the equivalence to a Turing Machine to decide this language and if it was the intention of this exercise to come up with the equivalence, the language is (IMHO) to easy to motivate this.

A simple two-stack PDA for this language works like this:

  1. Put $n$ on both counters, while counting $a$s.
  2. Use counter one to check the $b$s.
  3. Count up on counter one and down on counter two to check the $c$s.
  4. Check the $d$s.

I left out some details, but filling them should be easy.

By the way: You should not believe that this language is not context free, but prove it (it should be obvious which word to choose as a counter example using the pumping lemma).

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  • $\begingroup$ I don't think it's context free, I think it's easy to prove that it's not context free, therefore I ask you not to believe, but to prove. $\endgroup$ – frafl Mar 29 '13 at 19:27
  • $\begingroup$ No need for that, problem requires me to draw 2-Stack PDA for that $\endgroup$ – Iancovici Mar 29 '13 at 19:35
  • $\begingroup$ Well, you should not hand in the proof, just make sure you understand that this can't be context free. $\endgroup$ – frafl Mar 29 '13 at 19:42

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