Question

1. Show that ifis a DFA that recognizes language$\mathit{B}$, swapping the accept and non accept states inyields a new DFA recognizing the complement of$\mathit{B}$. Conclude that the class of regular languages is closed under complement.
2. Show by giving an example that ifM is an NFA that recognizes language C swapping the accept and non accept states in Mdoesn’t necessarily yield a new NFA that recognizes the complement of C. Is the class of languages recognized by NFAs closed under complement? Explain your answer.

Short answer

In solution (a),whole complement class of regular languages is complete.

In solution (b), NFAs identify a class of languages that is closed under complement.

Step-by-step solution

Explain DFA.

DFA , is the Deterministic Finite Automata, that has the finite state that can be reached on transition. The deterministic finite automata , has the knowledge about the state that will be reached.

Step 2:Show that if M is a DFA that recognizes language B .

(a)

is a DFA that recognizes' B language.Let M be just the updated DFA, with accept as well as non-accept states exchanged in M .Assume M takes an integer x as input. Run $M^{\text{'}}$upon x, $M^{\text{'}}$will almost certainly attain its acceptable state.Both accept but also non-accept statuses of both the computers M and $M^{\text{'}}$have already been switched.

If runs on x, M rejects, reject. If $M^{\text{'}}$acceptsx ,M rejectsx . If M accepts X, $M^{\text{'}}$doesn't at all acceptx .If, $x.B=B$then $x\in B$equals complements of B as well as vice versa.

As a result,$M^{\text{'}}$ would allow strings something M does not accept.As a result,$M^{\text{'}}$ identifies languages that are complements to .Because M recognises an appropriate language B, $M^{\text{'}}$recognisesB 's complementing, which would be likewise normal; hence, whole complement class of regular languages is complete.

Therefore, The class of regular languages is closed under complement.

Is the class of languages recognized by NFAs closed under complement? Explain your answer.

(b)

Assumeis an NFA which understands the languageC.

M’sstate diagramis as follows,

To get accompanying $M^{\text{'}}$if simply switch the approve with non allow states of M.

Consider that class of languages recognised by NFAs is exactly the same as the class recognised by DFAs.

Therefore, yes class of languages recognized by NFAs closed under complement.