the stack size in a PDA M can grow to hold at most k symbols. What kind of language is L(M)?

Question

the stack size in a PDA M can grow to hold at most k symbols. What kind of language is L(M)? Prove your claim.

I think the answer for this is this: The language L(M) is a regular language if it can hold at most k symbols. Since it can hold at most k symbols, the stack size is therefore finite. Furthermore, we can create a DFA that will accept L(M), and therefore we can conclude that this language L(M) is regular.

But I am having a debate with my friends whether this is correct or not

Answer 1

You are correct, such automata accept the regular languages.

One way to argue this is by showing how to construct an NFA for any of these limited-stack PDAs. The construction is easy, as you have suggested and has been suggested in the comments:

1. take your limited-stack PDA
2. for every state q and every possible stack configuration S (there can be no more than (|E| + 1)^k of them for sure) in the limited-stack PDA, define state qS in the NFA
3. for every transition (q, S, s, q', S') where q is the initial state, S is the initial stack configuration, s is the transition symbol, q' is the transition state and S' is the stack configuration after the transition, add a transition from qS to q'S' in the NFA on symbol s. Note: do this for empty/epsilon/lambda transitions as well
4. make the initial state of the NFA the state q0S0 the one where q0 is the initial state of the limited-stack PDA and S0 is the empty stack
5. make the accepting states of the PDA those states qaSa where qa is accepting in the limited-stack PDA (and/or if you are accepting by empty stack, Sa is the empty stack).