Acceptance by final state and empty stack

We have already proved in case of NPDA that the two methods of acceptance (by empty stack and final state) are equivalent. That is, a language L has an NPDA that accepts by final state if and only if some NPDA accepts it by empty stack. But this is not true for DPDAs. The language recognizing capability of DPDA s that accept by empty stack is much less than that of the other. This is proved in the following lemma.

Lemma1 : If a language L is accepted by a DPDA by empty stack, then L has the “prefix properly”.

Before giving the proof of the above lemma we first define the “prefix properly” of a language.

Lemma1 : If a language L is accepted by a DPDA by empty stack, then Lhas the “prefix properly”.

Before giving the proof of the above lemma we first define the “prefix properly” of a language.

Definition : A language L is said to have the prefix properly if whenever , no proper prefix of x is in L