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There is a tool called JFLAP, which, among other things, can analyze push-down automata, and find non-determinism.

In this example it is detecting non-determinism in state q0: Non-deterministic push-down automaton

The first symbol in the transition represents the symbol read as input; the second symbol represents the symbol extracted from the stack; and the third symbol is the symbol pushed to the stack. λ represents the empty symbol, so this is an empty transition without checking the stack or pushing anything to it.

I am surprised, as that state seems to fulfill the conditions for determinism for push-down automatons (if only because it only contains a single transition!). I would expect the next state to be q1 under any circumstance.

In comparison, JFLAP doesn't find any non-determinism here: Deterministic push-down automaton

Mind you, the transition is the same, it only changes that this one adds something to the stack. Am I missing something or is JFLAP wrong in the first instance?

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All deterministic push-down automatons are also nondeterministic push-down automatons, so it is correct to say that your automaton is nondeterministic.

That said you are correct in saying that your first push-down automaton is also deterministic, according to the definition on Wikipedia.

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