Induction details for proof of closure of regular languages under unions

I was reading M. Sipser, Introduction to the theory of computation 3ed. where he presents a proof by construction that the class of regular languages is closed under unions (Theorem 1.25). However, he omits the induction details for a formal proof of correctness. I tried filling in the details myself but it was a mess; how is this usually done?

Prove that regular languages are closed under kleen

Given $ L$ is a regular language, how can i prove that $ L^*$ is a regular language too? I’ve constructed a NFA which contains a new start state that has an $ \epsilon$ -transition to the original starting state, and also connected all the final states to the original starting state with an $ \epsilon$ -transition.

How can i formally prove that this is correct?

Regular Expression Kleene Star application – using 0s and 1s

Have a question from an online course I am trying to take for a certificate. Came across a few questions like this and need help working through the problem and the rationale to get the solution. Any help appreciated.

Consider the regular expressions R1 = (01)*(0*1*)(10) and R2 = (101)(01). Which of following is true?

10 in R1 and 101 R2

10 in R1 and 1000 R2

010 in R1 and 100 R2

010 not in R1 and 100 not in R2

McNaughton-Yamada Algotihm (1960) Regular Expression to DFA without passing through NFA

I have a homework problem here. It asks me to use the McNaughton-Yamada algorithm to first convert a regular expression to a DFA, and then minimize this using a partition argument. I can do the latter. My problem is that I cannot access any actual reference on the algorithm. Their original paper is behind a paywall at IEEE that my university does not have access to.

The algorithm went something like this: 1. For each symbol in the expression, given them a subscript from left to right increasing by one for each instance of that symbol. For example, the expression, aa* would receive a_1 a_2^*.

  1. We proceed to construct a diagram based on the possible lengths of words.

If done appropriately, this produces a DFA. I think the labeling in (1) is to help label the states.

Feel free to come up with your own example if you decide to give an answer. I won’t provide any problem here because there is no guarantee that it isn’t actually my homework exercise.

Arden’s Rule, DFA & NFA to regular expressions

I have been trying to figure out the Arden’s Rule and the equational method to transform DFA’s & NFA’s to RE. I know what the rule state:

if x = s + xr
then x = sr*, with $ s,r\in$ Regular Expressions

With that said, when I’m trying to transform one DFA in a RE this questions pop:

For example regarding this DFA

  1. The $ \epsilon$ is added in the entry stage A or in the final stage D and A ?

  2. The equations should be written regarding the transitions in or out of a given state

    2.1 For example A = $ \epsilon$ + 0B + 1C or A = $ \epsilon$ + 0C

  3. Can the equational method and Arden’s Rule be applied to a NFA with multiple initial states ?

Final thoughts, I have been trying out and it seems that when we count the transitions out of a state the $ \epsilon$ should be added to the final state. When we count the transitions into a state the $ \epsilon$ should be added to the initial state.

Keep in mind that I SERIOUSLY doubt my conclusions and I really need some help.

Prove the following language is regular?

Assume $ L_1$ is a regular language, and define:

$ $ L = \{wcv ∈ \{a, b, c\}^* \mid |w|_a + 2|v|_b ≡ 3 \bmod 5, w, v ∈ L_1\}.$ $

Show that $ L$ is regular.

I first tried to prove by showing that the pumping lemma holds true, then learned that it was not a double implication and can only be used to prove languages are not regular.

Then I tried to draw an NFA, but didn’t make any progress.

What’s a good way to prove that a language like this is regular?

Proving a language is not regular using Myhill Nerode Theorem

Let $ L = \{\alpha\in\{a,b,c\}^{*} \mid \alpha \text{ is palindrome}\}$ , show that $ L$ is not regular using Myhill-Nerode relation.

I don’t know how to show that $ L$ has infinite equivalence classes because $ \alpha$ is a palindrome. I tried to use something like this, but I don’t know if its correct:

$ \alpha \equiv_{L} \beta \iff \alpha (aba)^k \in{L} \iff \beta (aba)^k \in{L}$ $ \forall k \in \mathbb N$ which implies that for every k there exists an equivalence class because the repetition of aba k times.

Converting DFA to Regular Expression Using State Removal

I’m trying to convert the following NFA to a regular expression.

I’ve attached my work below and end up with the expression $ aa^*bb^*$ . As far as I can tell, this doesn’t seem correct but I’ve been working at it for quite a while. Can anyone tell me where I went wrong? And if it happens to be correct, can you tell me why?

Thanks a lot in advance.


Upon further work, I came up with the regular expression: $ aa^*b(b\cup aaa^*b)^*$ . This seems like the correct response.