# It is decidable whether a pushdown automaton will accept a word? [duplicate]

I'm asking myself if the problem of decide whether a push down automaton will accept a word is decidable.

I would say that you can simulate a push down automaton with a Turing Machine and, if it doesn't loop forever, accept the word when the automaton accepts, or reject when it does. In this case, the problem of decide if a push down automaton accept a word would be semi-decidable. But I'm not sure if that is correct and also, I can't find a way to prove if it is decidable or not.

Any ideas here?

Pushdown automata are nondeterministic, and there is no a priori bound on the length of computation (imagine $$\epsilon$$-transitions that add and remove the same element from the stack). Therefore it's not clear how to perform your simulation.