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Problem 11

# In both parts, provide an analysis of the time complexity of your algorithm. a. Show that $$E Q_{\mathrm{DFA}} \in \mathrm{P}$$. b. Say that a language $$A$$ is star-closed if $$A=A^{*}$$. Give a polynomial time algorithm to test whether a DFA recognizes a star-closed language. (Note that $$E Q_{\mathrm{NFA}}$$ is not known to be in $$\mathrm{P}$$.)

Expert verified
The given problem requires us to analyze the time complexity of an algorithm in two different scenarios. a. The equivalence problem for deterministic finite automata (EQ_DFA) asks if given two DFAs, M1 and M2, the languages they recognize, L(M1) and L(M2), are the same. This problem can be solved in polynomial time using the Hopcroft-Karp algorithm, which minimizes the given DFAs and checks if the resulting DFAs are identical. The time complexity of this algorithm is $$O(n \log n)$$, where n is the number of states. Thus, EQ_DFA is in P. b. In the case of testing whether a deterministic finite automaton (DFA) recognizes a star-closed language, we simulate the DFA for each state as an initial state with input strings of length 0, 1, 2, ..., N-1 (where N is the number of states in the DFA). After reaching an accepting or a looping state, we check if the final state for each simulation is an accepting state. If this is true for all simulations, then the language recognized by the DFA is star-closed. This algorithm has a time complexity of $$O(N^3)$$, which is polynomial, providing a polynomial-time algorithm for testing whether a DFA recognizes a star-closed language.
See the step by step solution

## Step 1: Define EQ_DFA

The equivalence problem for deterministic finite automata (E_Q_(DFA)) is defined as follows: Given two deterministic finite automata M1 and M2, are the languages they recognize, L(M1) and L(M2) the same?

## Step 2: Apply Hopcroft-Karp algorithm

In order to solve E_Q_(DFA) in polynomial time, we can use the Hopcroft-Karp algorithm. This algorithm compares two DFAs by minimizing them and checking if the resulting minimal DFAs are identical. The algorithm's time complexity is $$O(n \log n)$$, where n is the number of states in the input DFAs. Because this complexity is polynomial, we can conclude that E_Q_(DFA) is in P. b. Algorithm for recognizing star-closed languages in polynomial time

## Step 1: Simulate the DFA with input strings

Given a DFA M, we will consider each state as an initial state and simulate the DFA with input strings of length 0, 1, 2,..., N-1 (where N is the number of states in M). The simulation should end when reaching either an accepting or a looping state.

## Step 2: Test for star-closed behavior

Check if the final state of the simulation from Step 1 is an accepting state for each possible initial state and input string. If it is an accepting state for all combinations, then the language recognized by the DFA is star-closed.

## Step 3: Time complexity analysis

The algorithm needs to simulate the DFA for N initial states and N-1 input strings of length at most N-1. Each simulation step takes constant time. Therefore, the time complexity is $$O(N^3)$$. Since this complexity is polynomial, we have provided a polynomial-time algorithm to test whether a DFA recognizes a star-closed language.

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