Question

Let M be a pushdown automaton and let L ⊆ A∗ be a language such that L = L(M). Prove that if the length of the stack never exceeds a fixed number k ∈ N, then L is a regular language.

Answer 1

If the length of stack never exceeds value k and there are total C possible stack symbols, than there are maximum k^C possible permutations of stack content which is a finite number.

Hence we can simulate a finite automata from given push-down automata such that if there is transition in push-down automata, then this can be simulated by the transition function in finite-automata.

Thus finite automata will have maximum number of states where |Q| is number of states in push-down automata and k^C is total possible permutations of stacks in worst case.

Hence when stack size is finite, then every push-down automata can always be simulated by finite automata.