# Tagged Questions

Regular languages are formal languages which are recognized by a finite automaton. It is equivalently the languages which are expressible as a regular expression. In addition to these two, there are several other equivalent definitions.

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### If $L$ is regular, prove that $\sqrt{L}=\left\{ w : ww\in L\right\}$ is regular

Let $L$ be a regular language. Prove that $\sqrt{L}:=\left\{ w : ww\in L\right\}$ is also a regular language. I suppose I need to modify state machine for $L$ to accept $\sqrt{L}$, but I've been ...
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### Why isn't this a regular language?

I'm stuck as to figuring out why $L_1$={$n^p$ | $p$ = a prime number} is not a regular language but $L_2$={$n^p$ | $p$ = a prime number bounded by some fixed number f} is. I can see that $L_2$ is a ...
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### Regular expression 00 or 11 not both

Can anyone help me with this question: I know it before, but I have tried to solve it myself and didnt succeed. what is the regular expression for this language: L=all words that have 00 or 11 but not ...
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### Formally prove that every finite language is regular

I know how to prove this informally, but don't know what the formal proof should look like.
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### show that language $L'$ is regular (given $L$ regular)

Let $L$ be a regular language. Show that $L'=\{x \mid\exists_{y,z} xyz\in L \text{ and }|x|=|y|=|z|\}$ is also regular. Firstly I show my idea. When you accept it I will try to formalize it. Every ...
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### Is $L_1$ context free language?

Let $L = L_1 \cap L_2$, where $L_1$ and $L_2$ are languages as defined below: $L_1= \left \{ a^m b^mca^nb^m \mid m,n \geq 0 \right \}$ $L_2=\left \{ a^i b^j c^k \mid i,j,k \geq 0 \right \}$ Then L ...
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### Are regular languages necessarily deterministic context-free languages?

The original problem Suppose M is DCFL (Deterministic Context Free Language) and N is a regular language. Answer the following questions and justify your answers. a) Is M-N necessarily context-free? ...
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### A regular expression for the words that don't contain the sequence $ab$ over $\{a,b,c\}$

The following is an exercise in a book I am reading: Let $\Sigma=\{a,b,c\}$, define $L$ to be the language of all words over $\Sigma$ that do not contain $ab$ as a sub-word. Find a regular ...
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### Possible solution for Sipser 1.63

Sipser's question 1.63: Let A be an infinite regular language. Prove that A can be split into two infinite disjoint regular subsets. Is my solution correct? Since $A$ is infinite and ...
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### Prove the following language is not regular

The set of strings of 0's and 1's, beginning with a 1, such that when interpreted as an integer, that integer is prime. I'm assuming the best way to move forward is to use the pumping lemma. I'm ...
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### Pumping Lemma to show a language is not regular

Let $\Sigma = \{a, b\}$. Use the Pumping Lemma to show that $\mathcal L = \{ a^pab^q: p < q \}$ is not regular. Not sure how to apply PL here, if someone can give some direction.
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### Regular languages- homomorphism.

Let $h: \{ a,b,c,d \}^* \rightarrow \{a,b\}^*$ be a homomorphism such, that $h(a) = aa, h(b) = ab, h(c) = ba, h(d) = b$ . Determine: $h^{-1}((bab)^*ba^*b).$ I have trying do it by 4 hours. I don't ...
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### Why is $S(L)$ regular?

Let $L$ be a regular language, and $\Sigma$ be its alphabet. Then, the language $S(L) = \{y \in \Sigma^*~|~xy \in L \text{ for some string }x \in \Sigma^*\}$ is also regular. I am trying to ...
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### Pumping Lemma Excercise

I am having a really tough time proving the following language is not regular using the pumping lemma. This whole time I have been working on pumping lemma problems, the variable of the power has been ...
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### Is regularity is preserved under reversal?

When talking about languages and regular languages. Can I say that reversal preserved regularity since if the language L is regular, we can generate it by right linear grammar. Therefore, the ...
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### Prove that this language is not regular

I need to prove that the language $$L = \{a^nb^mc^k|n+k\neq3m\}$$ is not regular. Any ideas how I can do that?
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### Non regular language that satisfies pumping lemma

Let $$L = \{ ww^rx \mid w,x \in \{a,b \}^+\}$$ where $\{a,b\}^+$ means the set of words over $\{a,b \}$ that has at least length 1, and $w^r$ is the reverse of $w$. I'm trying to prove that this ...
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### Can an infinite set of primes be a regular language or CFG?

At first glance, it seems like the pumping lemmas should somehow "easily" show that an infinite set of primes (say, written in binary) cannot be a regular language or context-free grammar. But I don't ...
Let $A, B \subset \Sigma^*$ be languages. If the concatenation product $AB$ is regular, are $A$ and $B$ necessarily regular? I'm inclined to think this is true since the regular language $AB$ has a ...