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Problem 10

# Let $$I N F I N I T E_{\mathrm{DFA}}=\\{\langle A\rangle \mid A$$ is a DFA and $$L(A)$$ is an infinite language $$\\} .$$ Show that INFINITE DFA is decidable.

Expert verified
The INFINITE_DFA is decidable by following these steps: (1) Simplify the DFA by removing unreachable states, (2) identify loops in the simplified DFA using a cycle detection algorithm, and (3) check if any identified loops lead to an accepting state. If there exists a loop leading to an accepting state, L(A) is infinite; otherwise, L(A) is finite.
See the step by step solution

## Step 1: Simplify the DFA

Remove any unreachable states from the DFA to simplify it, as they do not contribute to the language that the DFA accepts. This can be done using graph search algorithms, like depth-first search or breadth-first search, to discover reachable states and remove those not visited.

## Step 2: Identify loops

A language is infinite if there exist paths from the start state to a final state that can be traversed infinitely many times. Such paths will contain loops (or cycles). Perform cycle detection in the simplified DFA using Tarjan's algorithm for strongly connected components (SCC) or any other cycle-finding algorithm.

## Step 3: Determine if the loop leads to an accepting state

Now that we have identified loops in the DFA, determine if any of the identified loops lead to an accepting (final) state. If there is a loop that leads to an accepting state, the language is infinite as we can traverse the loop infinitely many times, generating infinite strings that belong to L(A). If such a loop exists, output "Yes, L(A) is infinite." If no such loop exists, output "No, L(A) is finite." In conclusion, INFINITE_DFA is decidable since we provided an algorithm that decides whether the language recognized by a given DFA is infinite or not. This algorithm involves simplifying the DFA, identifying loops, and checking if there is a loop leading to an accepting state.

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