# Techniques to prove a language is not DCFL

I know that DCFL is closed under complementation and intersection with regular languages. By using these we can prove that a language is not DCFL.

Are there any other techniques that will help me to prove a language L is not DCFL?

• Well, using other closure properties of DCFL would work similarly, but I'm guessing you are after something else? Are you aware of direct proofs? (I guess most will have the form "assume $L$ was DCFL", anyway.) – Raphael Feb 11 '15 at 11:37
• The accepted answer to cs.stackexchange.com/questions/10974/… is also relevant here. – Klaus Draeger Feb 11 '15 at 15:36
• Thanks for your reply @Raphael.But I don't know the direct proofs.Can you please tell me or send me some link where I can find some examples of direct proof. – anand nayak Feb 12 '15 at 6:34

As an example, $\{ a^n b^m \mid n\ge1, m=n \mbox{ or } m=2n \}$ is not DCFL; which is seen as follows.
Assume it is. Consider a computation of its DPDA $\cal A$ on input $a^nb^{2n}$. As $a^nb^n$ belongs to the language, the computation must enter a final state halfway the $b$'s. Now rewire $\cal A$ such that from the first final state entered we move to a copy of $\cal A$ where all letters $b$ are changed into $c$. Only keep final states in this copy that are reached after reading at least one $c$. Then the new automaton will accept $\{ a^nb^nc^n\mid n\ge 1\}$, which (we know) is not context-free. Contradiction, $\cal A$ cannot exist.
• Changing the automaton in such proofs is dangerous. In particular, it's not completely clear why the new automaton accepts this language. The usual way would be to create a contradicting compution in the original $\mathcal{A}$. – Raphael Feb 17 '15 at 7:43