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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proof with using pumping lemma

$L=\{0^n\#0^{2n}\mid n\ge 0\}$ Show that this language is iiregular. And now: Let $p$ will be length of pumping lemma. Given $w=0^p\#0^{2p}=xyz\in L$ such that. Becaues of the fact that $|xy|\le p$ ...
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A Regular Expression for all strings that…

I got a problem I have to solve, the problem says that given an alphabet $\Sigma = \{a, b, c\}$ I have to build a regular expression that describes the string with: An even number of a's. A 4k + 1 ...
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Is it possible to write an Unambiguous Grammar for Two Hard Language ?!?

I came across a very hard interview exam. It was asked wrote an unambiguous grammar for two following language, Who can hint it to solve it? 1) $L = \{a^n b^{2n} c: n\geq 0\} \cup \{a^{2n} b^n d: ...
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Regular expression. Proof.

Let $A = \{a,b,c\} $ be an alphabet. Let $\alpha $ be a regular expression. And: $$ 1) \epsilon \in \alpha \\ 2) a\alpha \subset \alpha \\ 3) b\alpha \subset \alpha $$ Prove, that: $$(a+b)^* \subset ...
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How to prove the language of all binary numbers that are prime is nonregular using pumping lemma?

How to prove the language of all binary numbers that are prime is not regular using pumping lemma? I have seen Can an infinite set of primes be a regular language or CFG? We have not studied the ...
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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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Show that language $L'$ is regular given $L$ is regular

I show you some solution and I ask you for looking at it. $L'=\{y|\exists_{z,x} xyz\in L\wedge |x|=|y|=|z|\}$ Automaton for language $L$: $M=(Q,\Sigma, \delta, q_0, F)$ For language $L':$ $M'=(Q', ...
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Understanding pumping lemma - length $p$ - in connection with other thread

I create this thread in connection with Is the following language $L$ regular? I would like to show you why I dont understand where I am wrong. I would like to ask question: $p$ - length of pumping ...
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Homomorphism. Monoids. Language. [closed]

Let $h: \{ a,b\}^* \rightarrow \{0,1\}^* $ be a homomorphism. $$h(a)= 01, h(b) =0$$ a) Determine $h(M), M = (a+b)^*$ $h((a+b)^*) = [h(a+b)]^*= [h(a) \cup h(b)]^* = (01 + 0)^*$ OK? b) Determine ...
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Trouble with induction on the length of a word

In the accepted solution of the question If L is regular, prove that $\sqrt{L}=\{w:ww\in L\}$ is regular the answerer made the claim that "What's left is to show that $δ ′ (q_{0}' ,w)=h$ , which can ...
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Language with error.

Let $L \subset \{0, 1\}^∗$ be regular language. $$L_e = \{ w | w = uxv, x \in \{0, 1\}, u\overline{x}v \in L\}$$, where $\overline{x} = 1 − x$ Prove $L_e$ is regular language. For example: If $10 ...
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1answer
21 views

construction for proof of regularity of language

$L$ is regular. $L'=\{vw:v\in L, w\notin L\}$ Show that $L'$ is regular. I ask you for controlling my construction: $M=(Q,\Sigma,\delta, q_0,F)$ for $L$ $M'=(Q,\Sigma,\delta', (q_0, F') $ ...
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23 views

$L$ is regular. Show that $Root(L) $ is also regular

Let $Root(L)=\{w \mid \exists {n\in \mathbb{N}} \text{ such that } w^n\in L\}$. How to deal with it ? I tried think about modifications connected with automat for $L$, but it failed. Help me, please.
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64 views

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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37 views

Regular language. Proof

Let $L$ be regular language over $\{0,1\}$. Prove that $L'$ is also regular: $$L' = \{ w | w \in L \mbox{ and among words of length |w|, w is the least in lexicographic order.} \}$$ To explain, for ...
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Proving if languages are regular [closed]

I want to prove that for any regular language R over the alphabet {0, 1}, the language P(R) over {0, 1,(,)} is also regular and P(R) = {(s) | s ∈ R}. I am a little new to proofs and I was wondering ...
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Is it true that if $Cycle(L)=\{uv:vu\in L\}$ is regular then $L$ is regular [closed]

Is it true that if $Cycle(L)=\{uv:vu\in L\}$ is regular then $L$ is regular ? Any clues ?
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1answer
21 views

Difference between regular and not regular languages

I don't understand the difference, in classes I've seen the pumping lemma for regular languages and I know how to apply it to demonstrate whether a language is regular or not, but I feel that I don't ...
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construct automat for language $Cycle(L)$

We have automat $A$ for language $L$. Construct automat for $Cycle(L)$ where $Cycle(L)=\{uv:vu\in L\}$. I have a problem with this exercise. Help me, please. Edit $A = (Q_A, \Sigma, \delta, q_0, ...
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1answer
70 views

Clarification of the statement of the Pumping Lemma

In class we were told that Pumping Lemma states: "Let A be a regular language over $\Sigma$. Then there exists k such that for any words $x,y,z\in\Sigma^{*}$, such that $w=xyz\in A$ and $\lvert ...
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operations which are closest under regularity - other seeing

We have: $L_1, L_2, $ regular and $L_3$ irreguar. Now: $L_1\cap L_2$ is regular. $L_1\cap L_4 = L_3$ Can I say that $L_4$ is irregular ? The same question about $\cdot, \cup$
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Is the following language $L$ regular?

$L=\{ww^Rv\mid v,w\in \{a,b\}^+\}$ Is $L$ regular ? Edit Is it ok? Let $p$ will be length of pumping lemma. Then, $a^pb(a^pb)^Rv\in L$. When $a^p(a^pb)^Rv = xyz$ $p\ge |y|\ge 1 $ So ...
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How to show that a language is regular?

Let $L$ be a regular language over $\Sigma$. Show that: $$\left\{ x_1x_2 \dotsm x_k \mid x_1,x_2,\ldots , x_k \in \Sigma, \exists y_1, y_2, \ldots, y_k \in \Sigma: x_1y_1\dotsm x_ky_k \in L ...
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37 views

Proving an operation is closed under regular languages

Following operation is defined over languages where $n \in \mathbb{N} :$ $L \ominus n = \lbrace s \in \sum^* | \exists s^{'} \in \sum^* (length(s^{'})=n,ss^{'} \in L) \rbrace$ Meaning that $L ...
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How to prove that Pumping lemma can't be used to prove regular languages.

I need a prove that pumping lemma can't be used to prove regular language. Pumping lemma is only used for proving non-regular language, but I need to show that how it can't be used to prove regular ...
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11 views

regularity of language $D(L)$, $L$ is regular

Show that $D(L)$ is regular, $L$ is regular. $D(L) = \{w|w\in \Sigma^* \wedge ww^R\in L\}$ Let assume that M has only one state accepting. Therefore, $M'$ has only one state start. ($M'$ recognizes ...
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prove that language is regular (other language is regular)

Let $L$ is regular. Prove that $L'$ is regular. $ L'=\{uv: u\cdot rev(v)\in L\}$ $rev(v)=v^{-1}$ Idea: $L^{-1}$ is regular and recognized by $R$, and $L$ by $M$. Let's assume that $M$ is NFA such ...
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How to Solve Complex Non-Regular Language? [closed]

$L_5={\{ c^n a^m b^p,n+m=p,p≥6}\}$ where $∑=(a,b,c)$ I need little help, I was practicing Pumping lemma, and I encountered this language, I saw these conditions and I got totally confused, what to do ...
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Isomporphism two languages.

Let $A$ be an alphabet. Let $X,L \subset A^*$ $L$ is regular. Let $$X^{-1}L := \{ w \in A^* \mid \exists x \in X\ \ xw \in L \} $$ $$LX^{-1} := \{ w \in A^* \mid \exists x \in X\ \ wx \in L \} $$ ...
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find automata that accepts following language

Find automata, that accepts following language: $L\subseteq\{0,1\}^* $ $$L=\{w|w\ \text{dosen't contatain four 1's in 7 consecutive symbols}\} $$ My only proposition is: It is only fragment, but it ...
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1answer
39 views

Prove that if $L$ is regular then $f(L)$ is regular

Prove that if $L$ is regular then $f(L)$ is regular. $$f(L)=\{w: \text{every prefix of $w$ of odd length $\in$ L } \}$$ So my attemption is: Let $M= (Q, \Sigma, \delta, q_0,F) $ will be DFA ...
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Prove that language is regular given other regular language

Let $L\subseteq A^*$ be regular. Prove that $L'$ is regular where $L' = \{vw \mid \exists u\in A^*\ vuw \in L\wedge |u|=|w|+|v|\}$ Help me please. It is very hard for me, I don't know how to start. ...
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Convert the regular expression to a NFA

I have to convert the following regular expressions to a NFA: $$(0 \cup 1)^{\star} 000 (0 \cup 1)^{\star}$$ $$(((00)^{\star} (11)) \cup 01)^{\star}$$ $$\emptyset^{\star}$$ $$a(abb)^{\star} \cup ...
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1answer
32 views

Write the regular expression of the language that the DFA accepts.

I am given a DFA and I have tried to write the regular expression of the language that it accepts. This is the DFA that I am given: I have found some words that the DFA accepts: ...
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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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83 views

Prove that language $L=\{a^ib^j:\gcd(i,j)=1\}$ is irregular

Prove that language $L=\{a^ib^j:\gcd(i,j)=1\}$ is irregular. I have tried for a long time, but I haven't managed to solved it. Someone help me ?
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Are there two non-regular languages whose concatenation is regular?

Here is my proof: Consider non-regular languages $$L_1 = \{a^i b^j | i \neq j\} $$ and $$L_2 = \{b^n a^m | n \neq m\}$$ Then the concatenation of these two languages would be $$L_1 L_2 = \{a^i ...
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Prove that the language $\{bin(p) \mid p\ \text{is prime}\}$ is not regular (prime numbers)

Prove that the language $\{bin(p) \mid p\ \text{is prime}\}$ is not regular, where $bin(p)$ denotes the binary representation of $p$. I should use the pumping lemma. But I have a problem. Could you ...
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Is regular following language?

I try to prove that the language $L$ is not regular: $$ L = \{w\in(a+b)^*:\#_a(u)>2009\#_b(u)\ \text{for every nonempty prefix u of word w} \} $$ Note: $\#_a(u)$ means the number of symbols $a$ ...
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prove that language is not regular (prime numbers)

$$\sum_{p\,\in\,\text{Prime}}(cb^*)^p + (b+c)^*cc(b+c)^*$$ Show that language is not regular. We see that there are two possibilities: $p$ (prime) blocks of $b's$ separated by $c$ or any string of ...
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Proving Reversal of a Language in Recursive Way

We define the $reverse$ of a string as follows: $(x_1x_2...x_n)^R=x_nx_{n-1}...x_1$ where $x_1,x_2,...,x_n \in \Sigma$. We can also define the reverse of a language by $L^R= \lbrace s' | \exists s ...
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19 views

Clarification on Lemma problem

I have trouble understanding this question. I have no idea where to start. Let $A$ be the set of palindromes over $\{a, b\}$. Suppose you are trying to prove that $A$ is not regular using the ...
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show that language is regular

Let $B_n = \{a^k\ |\text{ where } k\text{ is a multiple of } n\}$. Show that for each $n\ge 1$ the $B_n$ language is regular. My proposition of solution: What about it ?
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Show that a language is regular

Show that language $B$ is regular: $$B = \left\{1^ky\mid y\in \{0,1\}^*\right\} $$ $y$ contains $\ge k$ symbols $1$ So I try in following way - I'll draw DFA: What about my solution? Is it good?
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Show that the language is regular - Closure

For languages $A$ and $B$, let the perfect shuffle of $A$ and $B$ be the language $$L=\{w \ \mid \ w=a_1 b_1 \dots a_k b_k, \text{ where } a_1 \cdots a_k \in A \text{ and } b_1 \cdots b_k \in B, ...
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31 views

Show that the language is regular

Let $$B_n=\{a^k \ \mid \ k \text{ is a multiple of } n\}$$ Show that for each $n \geq 1$, the language $B_n$ is regular. $$$$ Could you give me some hints how we coukd show this?? Do we have ...
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24 views

Construct the DFA of the language

I have to construct a DFA for the language $$\{w \mid w \text{ has exactly two } a's \text{ and at leat two } b's\}$$ To construct it we have to construct first the DFA's for the languages $$\{ w ...
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2answers
46 views

Checking Understanding of DFA Regular Operations - Intersection and Star

I'm currently taking a Logics course, and trying to understand the regular operations, intersection and star. I have a question regarding the work I have done so far. Given the following ...
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1answer
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$L_{a}= \{w_{1}cw_{2} : w_{1},w_{2} \in \{a,b\}^{\ast}, w_{1} = w_{2}\}$ [closed]

Why the following is not context free? Anyone could describe it for me. $L_{a}= \{w_{1}cw_{2} : w_{1},w_{2} \in \{a,b\}^{\ast}, w_{1} = w_{2}\}$
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4answers
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Regular Expression for Simple Language

I'm having trouble writing a regular expression given the following $\{a, b, c\}$ which produces the set of strings of length 3. I don't really understand how to restrict the length of the string. ...