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Suppose i have some NPDA and there is some transition functions defined as:

$\delta(q_{1},a,A) = (q_{2}, A)$

$\delta(q_{1},a,A) = (q_{3}, Z)$

Is it allowed? I understand, that since the NPDA is nondetermenistic, it's possible to have transitions to different states with the same input character, but what about same element at the top of stack? Thanks.

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This is very well possible, the automaton nondeterministically chooses which transition is taken. So it may choose between $\lambda$-transitions and transitions reading input, all with the same symbol on top of the stack.

Sometimes this is written as $\delta(q_1,a,A)=\{ (q_2,A), (q_3,Z)\}$, where $\delta$ is considered as a function from $Q\times(\Sigma\cup\{\lambda\})\times\Gamma$ into finite subsets of $Q\times\Gamma^*$.

Alternatively $\delta$ can be seen as a finite relation in $Q\times(\Sigma\cup\{\lambda\})\times\Gamma\times Q\times\Gamma^*$, and we write $(q_1,a,A,q_2,A), (q_1,a,A,q_3,Z)\in\delta$.

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