Say that a PDA is $\varepsilon$-free if it contains no $\varepsilon$ transitions (that is to say, $\varepsilon$ is not in the recognized string symbols even if it still is a stack symbol), but it may instead push any finite number of symbols to the stack during a single transition. Does every PDA have an equivalent $\varepsilon$-free PDA?
I tried thinking about how to eliminate $\varepsilon$ transitions from a given PDA. If the $\varepsilon$ transition goes from a state to itself, that means that you are either pushing or popping an arbitrary number of elements onto or off of the stack, before processing a character in the string. That doesn't seem to have an equivalent operation if you must process a character at each step, so it seems like they should not be equivalent.
But I struggle to think of an example language which is decided by a PDA but for which there is no corresponding $\varepsilon$-free PDA. Say that the PDA initially loads the stack with any number of $X$ which the $\varepsilon$-free PDA could not do. Then you try to do something with this arbitrary number of $X$, like maybe check that the total number of 0s and 1s is the number of them in the stack? Eh, that just accepts every string, and there's an equivalent for that.
Well, I've been thinking along similar lines for a while and not coming up with much. And I've been thinking about how to systematically eliminate $\varepsilon$ transitions and also not coming up with something that seems like it will always produce an equivalent.
The wiki page for the Greibach normal form also mentions that any CFL can be accepted by a real-time PDA.
– Knogger Mar 02 '24 at 21:11