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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A proof question involving a regular set and a context free language

Claim: Let $L \subseteq \Sigma^*\{\#\}\Sigma^*$ be a context-free language, where $\# \notin \Sigma$. Suppose that for each $x \in \Sigma^*$, $\{y|x\#y \in L\}$ is finite. Then $\{y|\text{ for some } ...
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power + operator for binary

What is the specific definition for power $+$ operator in automata theory? For example, when $x$ is a binary what does it mean that $x = 0^+$. Does it mean that x is a string with at least one $0$?
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Proving following regular expressions equal to one another?

How would I go about proving the following two regular expressions are equal to one another: $$ ( a + b )^* a ( a + b )^* b( a + b )^* = (a + b)^* ab(a + b)^* $$ I can "see" why they are equal to ...
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What is the language recognized by the following deterministic finite-state automaton?

Is the answer: {w : w ∈ {0*,1*} and w contains at least 3 zero} correct?
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First and Follow for the Context Free Grammar

I am trying to understand how to calculate first and follow for given rules Let's say here are two grammars. They are quite unusual so I am not sure if I made any mistakes. ...
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Regular expression,

Question 23: The string zyyzy belong to the language, right?
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Finite Automata for regular expression

I am trying to construct finite automata for this regular expression: Every block consisting of 5 characters need to contain at least two zeros. The regular expression would look sth like this: ...
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Binary Strings: How to determine if decomposition is ambiguous

Let's say I have the following decomposition: $$\{100,10011,00110\}^*$$ How would I determine if the decomposition is ambiguous or unambiguous?
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Pumping Lemma clarification, and regularity of language of equal number of zeros and ones.

All explanations and proofs I find about the Pumping Lemma are ambiguous. So if I understand this correctly, if we can find some $p>0$, then for any string $|w| \ge p$, we should be able to split ...
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How many DFA's exist with two states over the input alphabet $\{0,1\}$?

How many DFA's exist with two states over the input alphabet $\{0,1\}$? My attempt : Input set is given. So, we have 3 parts of DFA which we can change: Start state Transition Function Final ...
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Best resources for learning about regular and context free languages

I would like to train myself when it comes to finding out if a language is regular or context free. I would be grateful for pointing what are the best places/books for training.
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Need a hint for a proof using the pumping lemma that a language is not regular

Currently I am stuck at a proof of : $A_2=\{w001;|w|_0<|w|_1 \wedge w\in \sum^*\}$ unsing only the pumping lemma. Can you give me a hint for a good start? thanks in advance.
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How to proof using only pumping lemma that something is not regular?

In formal languages I need to proof using the pumping lemma that the following is not regular: $A_1=\{1^m0^n10^n|n,m\in \mathbb{N}\}$ How to achieve that? Any help is upvotet
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Proving L is regular or not using pumping lemma

So I'm trying to prove that the language L = {$1^n$ | n is composite} is either regular or non-regular using the pumping lemma. I wanted to ask if I'm on the right track. So I assume that L is ...
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Is the complement of a given language context-free?

I have a problem with finding out if the complement of language L is context free. $L = \{ ww : w \in \{a,b\}^{*} \wedge \text{ }w \text{ number of }a\text{'s in }w \equiv \text{number of }b\text{'s ...
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Prove that if $L$ is regular, then $f(L)$ is regular too

Prove that if $L$ is regular, then $f(L)$ is regular too. $\Sigma_1$ and $\Sigma_2$ are two arbitrary alphabets, $f$ is a function that maps every symbol of $\Sigma_1$ to an element in $\Sigma_2$, ...
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A binary regular expression without 000 string [closed]

How to write a binary regular expression without 000 string My attempt to solve it was this 1.(001+ 1)* + 1.(010+ 1)* + (100+ 1)*
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For a language $L$ is $Double(L)=\{w: ww\in L\}$ regular? [duplicate]

Given a language $L$, let $Double(L)=\{w: ww\in L\}$ and $NotDouble(L)=\{w: ww\notin L\}$ If L is regular, are $Double(L)$ and $NotDouble(L)$ regular? I tried using the closure properties of ...
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formal languages - why is this regular?

I'm studying for a test on formal languages and automata. I came upon the following question (translating, so i apologize for the non-formal english): $L_1$ is the language composed of all words ...
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String recursive definition corner case

I need your assistance with a corner case of this problem: Find a recursive definition for the strings of odd length that start with "a" and end with "b" over the alphabet $\Sigma$={a,b}. I've ...
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What is the nature of given language?

$$L=\{a^n b^n :n\geq0, n\neq100 \}$$ I just wanted to know that through pda. How will we make sure that $n\neq100$ or say I put a restriction that $n\geq100$. How to design a PDA using these ...
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Question about deterministic finite automaton and accepting states

For $n \in \mathbb N$, an "$n-$DFA" is an automaton with exactly $n$ accepting states. Let $\Sigma=\{0,1\}$. Prove that the set of the languages that can be accepted by "$1-$DFA" is a subset of the ...
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Question about deterministic finite automaton (DFA) [closed]

For $n \in \mathbb N$, an "$n-$DFA" is an automaton with exactly $n$ accepting states. Let $\Sigma=\{0,1\}$. Prove that the language $\mathcal L=\{0,00,0000\}$ cannot be accepted by any $2-$DFA.
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Representation of regular languages by monoids [closed]

I'm interested in representation of regular languages by monoids, and in particular of how to use this kind of representation to get a recognizer. I have found some references on the web, but does ...
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Language with middle third removed

Originating from Sipser's book: Let $A$ be any language, define $A_{{1\over3}-{1\over3}}$ be the subset of strings of $A$ whose middle third is removed. The solution I came across makes the ...
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which of these languages are regular sets?

$$ L_1 = \{a^p b^q\ |\ p+q \ge 10^6\} \\ L_2 = \{a^m b^n\ |\ m-n \ge 10^6\} $$ According to me both of these languages require comparison between number of $a$'s and $b$'s so both of them should ...
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Determining whether a given language is regular, and finding a regular expression

I know there are a lot of questions similar to this one, Proving a language is regular is just one example. However, I have not managed to find an answer that really answers my question. I'm currently ...
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Is there a subtle difference between NOEXTEND(A) vs NOPREFIX(A)?

My question originates from Sipser's book. Let A be a language with the DFA $(Q, \Sigma, \delta, q_{0}, F)$ and define: NOPREFIX(A) = {w $\in$ A| no proper prefix of w is a member of A} NOEXTEND(A) ...
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Construction of DFA using an odd bit of language

I am working through a lecture and it constructs a DFA using the language: $$\{w\mid w\textsf{ is any string not in }(ab^+)^\ast\}$$ What does the $(ab^+)$ mean?
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If $A$ is regular, is the language $\{x \;\mid\; \exists y : |y| = |x|^2, xy \in A\}$ regular?

Here is the question: Let $A$ be any regular set over some alphabet $\Sigma$. Is the language $$ L = \{x \;\mid\; \exists y : |y| = |x|^2, xy \in A\} $$ necessarily regular? I am unable to ...
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Converting Automata To Regular Expression Using State Removal Method

From the following automaton this solution is given: $$(a\mid b)^*aa(ba)^*a(a\mid b)^*$$ But when I try to convert this automaton into a regular expression I always end up with the wrong ...
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How to prove non-regularity of a language from the non-regularity of another language?

How can I prove that $L_1=\{a^nb^m\mid n\ne m\}$ is not regular based on the fact that the language $L_2=\{a^nb^n\mid n\in\Bbb N\}$ is not regular? Thank you
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Is $L_1 = \{w ∈ {0,1}∗ | \text{w has at least as many occurrences of (110)’s as (011)’s}\}$ regular?

Let $L_1 = \{w ∈ \{0,1\}^∗ | \text{w has at least as many occurrences of (110)’s as (011)’s}\}$. Let $L_2=\{w ∈ \{0,1\}^∗ | \text{ w has at least as many ...
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Prove about $NFA$ and pumping lemma

The question: Let it be $L$ a regular language. few definitions: $p(L)$-the minimum natural number so that $L$ fulfills the pumping lemma. $n(L)$- minimal NFA that accepts $L$. $m(L)$- $Rank(L)$, the ...
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Concatenation of unknown language

$$If \space L_1L_2\space is \space regular, then \space L_2L_1 \space is\space regular$$ Is this statement correct? I can't seem to find any counter example. Besides, what is a good way of tackling ...
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Regularity of a language

$$ L = \{w \mid w \text{ does not contain } 000\} $$ $$ L_2 = \{w \mid xwy \in L \text{ for some } x,y \in (0+1)^*\} $$ Is $L_2$ regular? I am thinking regular language is closed under concatenation, ...
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Regular Pumping Lemma

$$\begin{align*} L&=\left\{b^5w:w\in\{a,b\}^*,\big(2n_a(w)+5n_b(w)\big)\bmod 3=0\right\}\\ L&=\left\{(ab)^na^k:n>k,k\ge 0\right\} \end{align*}$$ Determine if each language is regular ...
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Regular Expressions with Repetition

I'm learning about regular expressions and how they represent regular languages of an alphabet. Conceptually, I'm having trouble imagining what a regular expression would look like, representing a ...
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Prove that relates to pumping lemma that I am not sure about

So, I will define like in my last post (for a regular language $L$): We will define $p(L)$ to be the minimal natural number so that a language $L$ fulfill the pumping lemma. We will also define ...
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How to prove that a simple NFA is minimal, without any algorithm?

First, I will present the question I was doing: We will define $p(L)$ to be the minimal natural number so that a language L fulfill the pumping lemma. We will also define $n(L)$ to be the minimal NFA ...
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Do the initial segments of the strings of a regular language form a regular language?

Let's say you have a set of strings $R$. A string $s$ is part of my language $S$ iff there is a string $r \in R$ such that $s$ is an initial segment of $r$ (you can get $s$ by removing characters from ...
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Is the resulting language regular?

If $L$ is a regular language then is $L'=\{w \mid wx \in L \text{ for some string }x\}$ regular? First step is understand $L'$. So it is a subset of $L$ that contains strings with a certain prefix?
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Cocke-Younger-Kasami (CYK) Proving a word is in a language

Using CYK algorithm I need to figure out whether the word abbabb is a word of the language of the following grammar. I think I have completed the problem correctly but I'm not sure, I'm hoping ...
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What does arbitrary number mean?

A FSM (Finite State Machine) can be designed to add two integers of any arbitrary length (arbitrary number of digits). Is it true ? My attempt : Arbitrary length means variable length, and there ...
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Identify the class of language?

Given a set $$S=\{x∣ \text{there is an x-block of 5's in the decimal expansion of π}\}$$ (Note: x-block is a maximal block of x successive 5's). Identify class of language? Somewhere it ...
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Proving that the language $\mathscr L$ is non regular using the pumping lemma

I need to prove that the language $\mathscr L=\{\text{all the binary words such that the number of ones divide the number of zeros}\}$ is non regular using the pumping lemma For example: ...
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Is $r(^∗)=r^∗$ valid regular expression?

Which of the following regular expression identities is/are TRUE? $r(^∗)=r^∗$ $(r^∗s^∗)=(r+s)^∗$ $(r+s)^∗=r^∗+s^∗$ $ r^∗s^∗=r^∗+s^∗$ My attempt : I can't say anything, but it should be ...
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Proving that $\mathscr L=\{0^n \big|\text{n is the square of a natural number }\}$ is non regular using the pumping lemma

I need to prove that the language $\mathscr L=\{0^n \big|\text{n is the square of a natural number}\}$ is non regular using the pumping lemma My try: $\mathscr ...
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Proving that the language $\{w\in \{a,b\}^* \big|\#_a(w)< \#_b (w)\}$ is non regular using the pumping lemma

I need to prove that the language $\mathscr L=\{w\in \{a,b\}^* \big|\#_a(w)< \#_b (w)\}$ is non regular using the pumping lemma My try: $\{a,b\}^*=\{\epsilon,a,b,aa,ab,ba,bb,aaa,aab,\dots\}$ ...
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Read-only Turing machine recognizes only regular languages?

Show that the Turing machines, which have a read only input tape and constant size work tape, recognize precisely the class of regular languages. According to wiki : A read-only Turing machine or ...