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Yes, a dpda can have dead state

Basically a dpda is nothing but a dfa with 1 stack as a memory

Just like dfa is moves from 1 state to another and is state dependent with an addition of being dependent on symbol on the top of stack

The only way to accept a given string is either by landing at final state or by accepting empty stack symbol. There is no was to reject a string other than to be in a nonfinal state.

So to reject a string which will never be true u have to enter into a dead state

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