Example : The language has the prefix properly; since if , then no proper prefix of can be in L. This is because the symbol c identifies the mid-point of the string . In many proper prefix of , the symbol c will not be the mid-point of that prefix.

Again, consider the language It is quite obvious that there are infinitely many pairs of stings in one of which is a prefix of the other e.g. both are in and is a proper prefix of . This is a regular language and still not accepted by any DPDA by empty stack.

It is to be noted that prefix property is not a severe restriction. Because we can always introduce a special end marker, say , not in at the end of every string of a language L to convert it to a language with prefix property. That is is a language with prefix property.

Assume for contradiction that the language L accepted by the DPDA by empty stack does not have the prefix property. Hence, there must exist two strings x and xy (with ) such that P accepts both. Then we have