# PDA accepting of a specific symmetric language

Assume we have PDA that accepts a specific symmetric language on $$\{a,b\}^*$$.

if we have $$a$$ This side of the string, on the other side of the string we have $$aa$$. and if we have $$b$$ This side of the string, on the other side of the string we have $$bb$$.

Examples of words in this language (middle point emphasized):

$$ab\, bbaa$$

$$aaa\, aaaaaa$$

$$bab\, bbaabb$$

What is a PDA accepting this language?

I can find a grammar for this language but I can't find a PDA.

The grammar: $$S \to aSaa \mid bSbb \mid aaa\mid bbb$$

In this case, start with the PDA for $$\{a^n b^n \colon n > 0\}$$, instead of storing one $$A$$ on the stack for each input $$a$$, store two, and the same for $$b$$. When reaching the middle, start discounting and accept by empty stack.