Question

This question adheres to the notation used in the proof of Theorem 2.2.1 in the textbook. a) (10 points) Consider the NFA M={K,Σ,Δ,s,F} below. For each state q∈K calculate E(q). b) (30 points) The following is a set of instructions to convert an NFA M into a DFA M . It is known that M does not have any empty transitions from its starting state. Identify in which steps the instructions go astray. State your reasons and correct the steps. Let M={K,Σ,Δ,s,F} and M ={K Σ δs F 1. Define K as the set consisting of all subsets of K. 2. Define the alphabet Σ as precisely the set Σ. 3. Define the set of starting states, s as the set whose only element is s. 4. Define the set of final states, F as those elements of K which consists of only the states q∈F. 5. Define the transition function δ as taking two inputs: an element Q of K and an element of a of Σ

. The function returns the set whose elements are precisely those states p in K for which there exists a q∈Q and (q,a,p)∈Δ.

Answer 1

To calculate the epsilon closure E(q) for each state q in an NFA, start by initializing E(q) with the state q itself. Then, check all transitions in Δ to add any states that can be reached from q by following only epsilon transitions. In the instructions to convert an NFA M into a DFA M, the definition of the set of final states, F, is incorrect. Instead of selecting the elements of K that consist of only the states q∈F, you should select the elements of K that intersect with F.

Step-by-step explanation:

1. To calculate the epsilon closure E(q) for each state q in K, you need to find all states that can be reached from q by following only epsilon transitions. Start by initializing E(q) with the state q itself. Then, check all transitions (q, epsilon, p) in Δ. If a transition is found, add state p to E(q) and repeat the process for p. Keep track of each state that is visited to avoid duplicates.
2. In step 4 of the instructions to convert an NFA M into a DFA M, the definition of the set of final states, F, is incorrect. Instead of selecting the elements of K that consist of only the states q∈F, you should select the elements of K that intersect with F. In other words, F should be defined as F = {Q ∈ K : Q ∩ F ≠ ∅}.