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P' contains all the moves of P besides the following.

1. The first move of P' is to go to the initial configuration of P by pushing the start symbol Z₀' of P'on top of the stock. From this point onward P' simulates the behavior of P (using P's moves) on any input string. Even if P empties its stack without accepting the input, P' will not empty its stock because of the new start symbol that was pushed on to the top of the stock initially.
2. If P enters an accepting state, P' simply enters the state P'.
3. On state P' , the DPDA P' erases all the stock symbols without bothering the input eventually emptying its stock. So, P' accepts a string X where P accepts it and vice versa.

Now leema 1,2,3 together gives us the following theorem.

Theorem: A language L is accepted by a DPDA by empty stock if and only if it has the prefix property and is accepted by some DPDA by final state.

 

 

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