PDA to accept language with more a's than b's and c's
Note the language where the number of $a$'s is larger than both the number of $b$'s and the number of $c$'s is not context-free, and cannot be accepted by a PDA.

Formally the language can be written as $L = \{ x\in\{a,b,c\}^*\mid \#_a(x) &gt; \#_b(x) \text{ and }\#_a(x) &gt; \#_c(x) \}$.

Intuitively we cannot recognize $L$ using a single pushdown as the count of $a$'s has to be used twice, but in comparing with the number of $b$'s and the number of $c$'s. That is no formal argument, there might be clever tricks avoiding the direct comparison of numbers. Yet there is a possible water tight proof, it uses the pumping lemma for context-free languages.