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I need to create a PDA that accepts by empty stack and accepts the language formed by strings over the alphabet $\{a, b\}$ of the form: $uw$, where $w$ is the string $u$ reversed and doubled. So, for example, if $u=ab$ then $w=bbaa$ (the $ab$ is reversed then its characters are doubled).

Examples of strings in the language: $\epsilon$, $aaa$, $baaabb$, $abbbbbbaa$, etc.

I'm really struggling with setting up a PDA that satisfies the criteria. Any help is appreciated.

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    $\begingroup$ Please tell if there are any conditions on string u and w... it would be better if you define the language than give an example of it for a comprehensive answer $\endgroup$ – Haseeb Hassan Asif Nov 12 '20 at 4:46
  • $\begingroup$ $u$ can be any string made up of $a$'s and $b$'s $\endgroup$ – Fraser Nov 12 '20 at 14:39
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The PDA operates as follows :

  1. Push the symbols you read to the stack

  2. Non-deterministically guess when u ends and w starts

  3. Now for the rest of the string , we consider 2 symbols by 2 symbols , if the 2 symbols are not the same reject , if the 2 symbols are the same , compare them to top of stack , if they are not the same symbol reject , if they are the same pop the stack

  4. After reading the entire input , if the stack is empty accept , else reject

Ex : s = ab bbaa , we push a,b to stack , then we guess u ends and w starts , now we read b , then b , both are same symbol , we look at top of stack we have b we pop the b , now the stack has a , and we will read aa , repeat the above again and we accept

If you get how it works , you may want to try to sketch the PDA

Here is a sketch of the PDA to check your answer :

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  • $\begingroup$ Thank you! Your explanation is very helpful. I'm still struggling to put it all together but I'm getting there. Your use of the symbol #, I'm assuming that's equivalent to the use of $Z_{0}$ used in my textbook to denote the empty stack? $\endgroup$ – Fraser Nov 12 '20 at 14:59
  • $\begingroup$ Yes , some books use Z0 and some use $$ , but you should remember that any special symbol can be used to do this check , by special we mean that it is in Γ but not in Σ , and is only used to do the check (these conditions guarantee that no problem occurs) , however Z0 is preferable since some languages contain # or $ , so in this PDA feel free to replace # by Z0 , if you have any other questions please feel free to ask $\endgroup$ – Anazz Nov 12 '20 at 15:47

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