This looks quite complicated, but can be simplified. A variable that does not
occur on the left side of any production must be useless, so we can immediately
eliminate (q 0 Aq 0 ) and (q 0 Aq 2 ). Also, by looking at the transition graph of the
modified npda, we see that there is no path from q 1 to q 0 , from q 1 to q 1 , from q 1
to q 3 , and from q 2 to q 2 , which makes the associated variables also useless.
When we eliminate all these useless productions, we are left with the much
shorter grammar
with start variable (q 0 zq 2 ).