PDA for the language of words $uv$ such that $|u| \geq |v|$ and $v$ contains 1

Question

Consider the language $\{ uv : \text{$|u| \ge |v|$ and $v$ contains a 1}\}$.

I am unable to understand how to accept this language using a PDA. How to check the length condition as well as check if there's a 1 in the $v$ portion?

The question has previously been posted here but I couldn't figure out much from the answers to that. PDA recognising all strings with a $1$ in the second half

I’m voting to close this question because it has been asked and answered before. — Yuval Filmus, Jan 18 '21 at 10:59
@Yuval Filmus I still am stuck. I am not able to solve it. You can vote to close this, that's okay. But can you help me with this question? — user766787, Jan 18 '21 at 11:01
The first step is to link to the previous post, and to explain why you couldn’t understand any of the answers. Any new answer should be added there. — Yuval Filmus, Jan 18 '21 at 11:03
Why am I not able to see anything? I am getting notified of comments but I don't see them here.. what is happening? — user766787, Jan 18 '21 at 12:38
Does this answer your question? PDA recognising all strings with a $1$ in the second half — xskxzr, Jan 31 '21 at 01:18

score 1 · Accepted Answer · answered Jan 18 '21 at 12:55

1

A pushdown automaton accepting this language proceeds as follows:

Push a marker $S$ to the top of the stack.
Push $A$ to the stack for each input symbol read.
At some point, nondeterministically transition to the second phase.
Pop $A$ from the stack for each input symbol read, marking whether you ever read $1$.
Fail if you encounter $S$.
Succeed if you successfully read the entire input, and encountered $1$ during the second phase.

It is also easy to give a context-free grammar for this language, using $\Sigma$ as a shortcut for the alphabet: \begin{align} &S \to \Sigma S \mid \Sigma S \Sigma \mid \Sigma T 1 \\ &T \to \Sigma T \Sigma \mid \epsilon \end{align}

answered Jan 18 '21 at 12:55

Yuval Filmus

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Thank you so much. I have tried to draw a pda. But it's accepting some strings that it should not. I will think about it more. – user766787 Jan 18 '21 at 17:52
Well then, you have made some mistake. Unfortunately you cannot expect us to debug your PDA. – Yuval Filmus Jan 18 '21 at 17:53
Yes, I understand. Thanks. – user766787 Jan 18 '21 at 17:53
Just one more question, even if it's a nondeterministic pda, no branch should accept the string 10010000, right? – user766787 Jan 18 '21 at 17:56
The PDA should accept a string in some branch iff it is in the language. – Yuval Filmus Jan 18 '21 at 17:57
Okay. Understood. Thank you. – user766787 Jan 18 '21 at 17:58

PDA for the language of words $uv$ such that $|u| \geq |v|$ and $v$ contains 1

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