Question

Let n be the number of states in a DFA. Show that A_{DFA} ∈ P.

Answer 1

We have to show that all the states (n) in the DFA given, exist in P.

How to re-term it in order to make it simple to prove? We can say we have to show that-

All the states in a DFA that can be reached from initial state 'i' and get accepted would ∈ P. But, how to show it?

This can be proved by using DFS (Depth first search) or BFS(breadth first search).

Please note that DFS would take more time to complete whereas BFS would take up more space, so it depends on you which method you want to choose if you implement it.

So, using DFS/BFS on the deterministic finite automata, starting from initial state 'i', keep on traversing to find accepting states.

If you ever find a non-accepting state which is reachable, deny it. This means that the DFA doesn't accept a certain pattern.

Once you reach a particular final state which is reachable, backtrack and search for other accepting states. Thus, all the 'n' states get covered and for all reachable states from initial state 'i', we have successfully proved that A_{DFA} ∈ P.

Using DFS/BFS, we were able to show this in polynomial time.