Deterministic pushdown automaton, stack start symbol, end-of-string delimiter

1. Find a DPDA for the complement of the language aⁿ b²ⁿ. (Hint: Start with Figure 5.3 and add a stack start symbol (S) and use an end-of-string delimiter ($). Add states and edges as needed to make this explicitly deterministic per Definition 5.2. Make sure you empty the stack even for strings not in the language. You dont have to include processing for impossible cases; e.g., you may assume that no symbols appear after the ending $. Then form the complement of the DPDA. Only states where the stack will be emptied need to be accepting.)

Trace the strings aabbb and aabbbbb to show that they are accepted by your complement. Note: There will be more states than we've seen in any previous examples.