For every 4 ‘c’ in the input push a ‘c’ in the stack (Push down automata)

Question

Given the following language: L = {$a^{2m}$ $c^{4n}$ $d^{n}$ $b^{m}$ : m,n >= 0} I’m trying to design a PDA. My aproach is:

-for every 2 ‘a’ push an ‘a’

-for every 4 ‘c’ push a ‘c’

-then pop them to solve the 2nd half of the language

This is the first part to count 2 ‘a’, but the ‘c’part is not very clear:

Any hints or solutions for the problem: ‘’for every 4 ‘c’ push a ‘c’ on the stack‘’

Thanks!

Answer 1

Use states to record how many 'c' you have seen (0 to 3), use transitions that don't change the stack to move among them when reading 'c'; on a 'c' in a state recording 3 'c', push a 'c' and go back to 0 'c' seen. Can mix/combine freely with e.g. counting the 'a' or 'b'.

Answer 2

You can recognise the language $L = \{c^{4n}|n\geq 0\}$ with a normal DFA. Use this automata as a base and just use one designated state, that pushes a C onto your stack, for which I would choose the first one. For every transition one c is consumed from the word.

$\{(q_1, c, S, CS, q_2) , (q_2, c, S, S, q_3), (q_3, c, S,S,q_4), (q_4, c, S,S,q_1), (q_1, d, C, \varepsilon, q_d)\} \subset \Delta $ with $S \in \Gamma$