Questions tagged [pumping-lemma]

Necessary properties of formal langagues in certain classes that rely on closure against repetition of certain subwords. Make sure your question isn't covered by applying the techniques in https://cs.stackexchange.com/q/1031/755.

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Is the set of strings with equal number of 010’s and 101’s regular?

Let the language be defined over alphabet{0,1}. Can you prove this by pumping down?
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Distinction between 2 languages.One is regular the other is not

whenever it needs to be determined if langage is regular or not, I use the notion that it is impossible for a machine to "remember" an infinte states. given 2 languages:$L_1=\{(01)^{n}(10)^{n}|n \in \...
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Prove that the language $\{a^ib^i | i\geq 0\}$ is not regular? (Do we just consider $a^nb^n$, where $n$ is the pumping length?

I think to prove that $\{a^ib^i | i\geq 0\}$ is not regular, we just have to consider the string $a^nb^n$ (which is in the language) and apply the pumping lemma. But I'm not sure how to proceed using ...
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Is this proof for pumping lemma legit?

Prove that $L=\{a^{n}b^{m}c^{k}\mid n\leq(m+k)\}$ is not regular. I used the pumping lemma as follow: there exists $n\in \mathbb{N}$ $z=uvw$ $|uv|\leq n , |v|\geq1$ $uv^iw$ is a string in L, so ...
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Use the pumping lemma to show it's not regular

I just learned pumping lemma this week and got confused on this question. B={$a^{fn}$ | $f_n$ is a Fibonacci number} for $a \in Σ$. Hint: the sequence of Fibonacci numbers get increasingly further ...
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31 views

When does $p$ break the pumping lemma

I want to prove that $\{A^iB^jC^k \mid i=j \text{ or } j=k\}$ is a not a regular language using the pumping lemma. I've found that the only way to obtain a contradiction is when $x \in A^*$, $y \in B^...
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Proof that language is not regular. $L=\{w\bar{w}|w\in \{0,1\}^* and\ \bar{w}\ is\ one's\ complement\ of\ w\}$

I'm trying to proof that the following language is not regular using pumping lemma. $L=\{w\bar{w}|w\in \{0,1\}^* and\ \bar{w}\ is\ one's\ complement\ of\ w\}$ I started by stating that: $|w\bar{w}| =...
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Proving that $ L = \{ 0^{{2n}\choose{n}} : n\in\mathbb{N} \}$ is not regular

I was asked to prove that $ L = \{ 0^{{2n}\choose{n}} : n\in\mathbb{N} \}$ is not regular. I can't solve this, could anyone help me? This was an exam question from previous year. I looked your ...
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Which of these languages is regular? The Pumping Lemma seems to show none are

I've been reviewing past paper questions for an automaton course, and came across a question which effectively asks, which of these languages is regular? $$ \{\ 0^m1^{(m \times n)}0^n\ \colon\ m,n\ge ...
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Regularity of language of words of prime length [duplicate]

Is the following language regular? $$ L_{\mathit{prime}} = \{ w \in \{0,1\}^* : |w| \text{ is prime} \}. $$ I have to either provide a DFA (if the language is regular), or prove that it is not ...
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Using the pumping theorem to show that this language is not context-free

Let $\sigma = \{a,b,c\}$ and let $L = \{s | s = a^jb^jc^k\}$ where $k=i*j$ and $i,j \geq 0\}$. Using the pumping theorem, prove that $L$ is not context-free. I really don't know where to start, here. ...
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How to prove the set of powers of 2 in ternary representation to be non-regular using pumping lemma?

Given the set of natural numbers, $S = \{2^i|i\in\mathbb{N}\}$ let $L$ be the language defined as the ternary representation of all numbers in $S$. How can you prove that this is not a regular ...
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Pumping Lemma with Prime Number [closed]

$\text {Could someone please help me with this proof: }$ $L:=\left\{a^{n} d^{m} b^{k} | n, m, k \in \mathbb{N} \wedge m \text { is a prime number}\right\}$ $\text {Maybe we can say, that } w=a^{n}d^{...
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Use the pumping lemma for context free languages to prove L = {w#w | w \in {a,b}*} is not context free

I know the basics of using the pumping lemma for CFG to prove a language L is not context-free, however, the # symbol seems to be throwing me off or my understanding is not complete.
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1answer
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Are number of states in a NFA same as Pumping length?

So i was reading a post on Minimum pumping length of regular language where Yuval Filmus has proved that a pumping lemma might have lesser number of states than a minimal DFA. But What about NFA's? ...
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How can I prove the minimum pumping length of finite language?

Let L be the set of all strings over {0, 1} whose lengths are at most three. Since L is regular, the pumping lemma holds for L, and thus there is a pumping length p associated with L. What is the ...
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If L is a regular language, how to show that cut(L) is not necessarily regular? [duplicate]

Given that L is a regular language. How do I show that cut(L) = {ac | abc ∈ L and |a| = |b| = |c|} is not necessarily regular? My thought process is that this problem can be re-written as: cut(L) = {...
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Is L2 = {a^n| n is a product of one or more primes} regular?

I am having a hard time proving the following with pumping lemma: Is $L_2 = \{a^n \mid \text{$n$ is a product of one or more primes}\}$ regular? Here's what I have so far: Suppose $L$ is regular, ...
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Did I prove the language is not regular?

I am trying to prove the following language that is not regular. I used Pumping Lemma proof and my proof goes as follows: Assume that L is regular and let p be the constant of Pumping-Lemma. This ...
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Is {a^n: n is a product of exactly two primes} regular?

I am struggling to prove the following question. $L_1 = \{a^n: n \text{ is a product of exactly two primes}\}$ I feel like the language is not regular but I am having trouble proving it. I tried ...
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Proving a language is Not Regular without using Pumping Lemma? [duplicate]

I was wondering how one would go about proving a language is Not Regular without using the traditional pumping lemma contradiction. $$L = \{ 1^k 0^n 1^n 0^k \mid k \geq 0, n \geq 0\}$$ I've seen a ...
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How to prove that L(G) is not regular by contradicting the pumping lemma?

I am trying to prove that this language is not regular by contradicting the pumping lemma. I have been reading and looking at examples but all the examples I have seen is in the for of a REGEX. I am ...
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Is Palindrome subset of a regular language regular?

Suppose we have $L$ being a regular language with alphabet $\Sigma$, if we define $M=\{ x \in \Sigma^{*} \mid xx^{R} \in L \}$, then we know that $M$ contains all half copies of palindrome strings ...
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Whether following language is linear or not?

I have a language $L= \{a^nb^nc^m : n, m \ge 0\}$. Now, I wanted to determine whether this language is linear or not. So, I came up with this grammar: $S \rightarrow A\thinspace|\thinspace Sc$ $A \...
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Whether the given language is a CFL or not?

Let $L$ be a language defined over $\Sigma = \left \{ a, b \right \}$ such that $L = \left \{ x\#y \mid x,y \in \Sigma^*, \# \text { is a constant and } x \neq y \right \}$ State whether the language ...
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63 views

How to prove this language is not regular?

I am currently learning Pumping Lemma and found this question. But I am not able to prove it not regular. L = { $0^n$ | n is power of 2}. So, here I considered w = $0^{2^n}$ where n is constant of ...
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Is there a language that pumps, but is not regular? [duplicate]

I'm looking for a concrete language that can be pumped but is not regular. I understand that closure properties can be used to further test if a language is regular/nonregular.
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Pumping lemma for L = {a^i b^j c^k: i < j < k}

I had a question regarding a specific proof I found online that I had some concerns with, I have quoted it below. Show that the language L = {a^i b^j c^k: i < j < k} is not a context-free ...
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1answer
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Pumping lemma regarding {a^2k w | w ∈ {a, b}*, |w| = k}

I had a question regarding the Pumping lemma for regular languages, I have been studying for an exam and came across the question {a^2k w | w ∈ {a, b}*, |w| = k}. In the website it lists the answer ...
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1answer
235 views

Pumping Lemma for Regular Languages with 3 variables (a^nb^mc^m)

I've been trying to understand the pumping lemma, and how to apply it to a language such as a^nb^mc^m where n >= 0 and m >= 0. The pumping lemma states that: For any regular language L, there exists ...
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294 views

Converse of pumping lemma for regular expressions

I want to come up with a language that satisfies the pumping lemma while not being a regular expression. I thought of $\{0^i1^j: i > j > 0\} $. The pumping seems to work just fine, and this is ...
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1answer
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Getting from one language to the other using closure properties(automata) [duplicate]

I am trying to deduct how i can, using closure properties, deduct that since the following language is not context free $$L=\left\{abc^{i_1}bc^{i_2}...bc^{i_{2m}}def^{j_1}ef^{j_2}..ef^{j_{2n}}ghq^{k_1}...
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Using pumping lemma to show a language is not context free(Complicated)

How can i show that the following long language is not context free using the pumping lemma? $L=\left\{abc^{i_1}bc^{i_2}...bc^{i_{2m}}def^{j_1}ef^{j_2}..ef^{j_{2n}}ghq^{k_1}hq^{k_2}...hq^{k_o}\right\}...
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1answer
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$L = \{ a^{j!} \mid j \geq1\}$ is not context free by pumping lemma

How I use the pumping lemma to prove that the language $L = \{ a^{j!} \mid j \geq1\}$ is not context-free?
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Using pumping lemma to prove irregularity of regular language - what is my error? [duplicate]

I have a vital misunderstanding of the pumping lemma. In the following example I show an example of using it on a regular language to come to incorrect conclusions. What am I doing wrong? L={ab}, ...
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Confused about 3rd rule of CFG pumping lemma

Let $L = \{\space ww \space | \space w \in \{0,1\}^*$} (need to prove that $L$ is not CFL) Assuming $L$ is CFL we can use the PL and split $s=uvxyz$ and we choose $s = 0^p1^p0^p1^p$ where $p$ is the ...
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Show that the Language L is not regular (pumping lemma) [closed]

$L = \{cda^nb^n\mid n\in \Bbb N\} \cup \{a,b,d\}^*$ Assuming $L$ is regular then there exist a pumping length $n$ for $L$. Lets use w = $cda^nb^n$. Thus $w \in L$ and $|w| = 2n+2$ $\implies$ $|w| \...
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What does pump down means in this solution?

Problem text (from Sipser's "Introduction to the Theory of Computation"): 2.42 Let $E = \{1,\#\}$ and $Y = \{ w \mid w = t_1\#t_2\# ...... \#t_k \, \text{for $k \geq 0$, each $t_i \in 1^*$, and $...
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1answer
156 views

Closure properties of a non-regular language under complement? [duplicate]

Assume I have L1 which is a regular language, so we know since regular language is closed under complement, the complement of L1 is also a regular language. But let's say if the complement of L1 is a ...
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1answer
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Does |xy| ≤ p in the pumping lemma count for all i?

While learning about the pumping lemma, I came across the following question: Given the language L is $ a^n(0|1)^* $ with $ c_0 \cdot c_1 = n $, where $ c_0 $ indicates the amount of zeros present, ...
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Prove $\{abc : a+b=c\}$ is not context-free using pumping lemma

I have the following alphabet: $Σ = {0, 1, . . . , 9}$ and the Language $L$ defined as: $L = \{ abc | a + b = c\} $ where substrings $a$, $b$ and $c$ are interpreted as ordinary integers. My answer ...
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Pumping lemma — Is this enough to prove that the language L is not regular?

I have a language $L$: $$L = \{w : a^ib^j; i > j \}$$ I need to prove this language is not regular using Pumping Lemma. I'm wondering if I'm doing it correctly: $L$ Is assumed to be regular. A ...
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2answers
61 views

Pumping lemma occurrence of c > d

I'm trying to prove a language is not regular through using pumping lemma, but can't seem to come up with any way of doing it. The alphabet is: $$ \Sigma = \{c, d\} $$ The language is: $$ A = \{z ...
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180 views

Proving that L is not regular using closure properties

I need to show that the following language is not regular. $$L = \{\ ab^jc^j\ |\ j \geq 0\ \}\ \cup\ \{\ a^ib^jc^k\ |\ i, j, k \geq 0 \ and\ i \neq 1\ \}$$ There is also a hint that it cannot be ...
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1answer
75 views

Prove that $L = \{ xy \in \{a , b \}\textbf{*} \mid |x|_a = 2|y|_b \}$ is not regular

Prove that $L = \{ xy \in \{a,b\}^* \mid |x|_a = 2|y|_b \}$ is not regular. I have already tried to prove it by using the pumping lemma and reduction to absurdity, but have been unsuccesful on both. ...
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1answer
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What is the importance of the condition “| xy | < p” in pumping lemma? [duplicate]

Let L be a language. w $\in$ L , and w could be broken in xyz. Then if L is regular, there exists a pumping length p such that: |y| $\gt$ 0 |xy| $\le$ p $\forall$ i $\ge$ 0, xy$^i$z $\in$ L I ...
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1answer
420 views

Irregularity of language of prefixes of decimal expansion of pi

Let $L_{\pi}$ be the language consisting of prefixes of the decimal expansion of $\pi$: $$L_\pi = \{3, 31, 314, 3141, 31415, 314159, \ldots\}.$$ Prove that Lπ is not DFA-recognizable. You may use the ...
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2answers
92 views

Regularity of a language contains more 1's than 0's

The language of all bitstrings with more 1s than 0s, i.e., $ A = \{x: 2\Sigma_{i}^{|x|} x_{i} > |x|\}$ is regular. I know I should apply Pumping Lemma and the proof as well, what I cannot ...
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1answer
83 views

Confused about pumping lemma, What i'm missing? [duplicate]

When I apply pumping lemma on this language: ${L=\{010^n:n\ge0\}}$ over the alphabet ${\Sigma =\{0,1\}}$ I get that it is non-regular despite the fact that it is regular. let ${n=4}$, then $w=010000$...
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4answers
172 views

Proving non-regularity of $\{a^p \mid p \in \text{Prime} \}$ without pumping lemma

I was wondering whether it is possible to prove $\{a^p \mid p \in \text{Prime} \}$ is a non-regular language without using the pumping lemma. I'm having trouble choosing an alphabet that completes the ...

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