Variants of the npda model. We have defined that an npda accepts if both the pushdown

store is empty and the state is accepting. There are obvious variants of this definition:

accept by $($reaching an$)$ accepting state.

Accept by $($reaching an$)$ empty pushdown store.

Show that all these are equivalent, that is

$(1)$ if an npda $M$ accepts $L$ by accepting state, then there is an npda $M^{\text{'}}$ which accepts $L$ by

accepting state and empty pushdown store.

$(2)$ If an npda $M$ accepts $L$ by empty pushdown store, then there is an npda $M^{\text{'}}$ which accepts

$L$ by accepting state and empty pushdown store.