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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Proving language is not context-free with pumping lemma

I'm trying to prove that this language is not context free using pumping lemma. I am having difficulty as to where to even start on this. $$\{c^{2i} d^j b^{2j} d^k c^{3j} \mid i,j,k \ge 0\}$$
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$Prove\; that\; L=a^i b^j c^k: i\le j\le k$ i s not context free language

Proof-: Assume L is CFL. Let p is pumping constant for L. w exists in L such that |w|$\ge p$ Let w=$a^p b^p c^p$ |w|$\ge$3p so everything is fine. Now let us see all decompositions of w such that-: vy$...
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I can't visualize what happens when we pump v and y in pumping lemma for $a^n b^n c^n$

If you need some context-: https://www.andrew.cmu.edu/user/ko/pdfs/lecture-11.pdf around page 7. Case 1-: Say vxy contains ab So when I pump v and y, what will get pumped? And how the result would be. ...
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Understanding about pumping lemma for regular language-confusions of beginner-:

I want to understand how is this proof working. What I know-: Pumping lemma for regular language-: Let $L$ be regular language. Then there exists a constant $n$ which depends on $L$ such that for ...
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I need a pumping lemma for context free lenguages for this example? [duplicate]

Prove is that a context free lenguage or not $L= $ { $0^k 1^l 0^k; k\geq l; (k,l) ∈ N ∪ 0$ }
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Can I solve pumping lemma for context free language proofs using examples?

Say I need to prove that $L=${$a^n b^n c^n; n\geq 1$} is not context free language I take n=3. w=aaabbbccc Here |w|=9. we know by pumping lemma-: |vxy| $\leq$n so vxy=abb |vy| $\geq$1 so vy=ab Hence I ...
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Check Proof Using Pumping Lemma to Show Language Not Regular

Please check my proof where I use the pumping lemma to show that the language $B=\{0^n1^n | n≥0\}$ is not regular. I'll state the pumping lemma here for clarity: Pumping lemma If $A$ is a regular ...
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Dividing a String According to the Pumping Lemma

I have some questions about how a string can be divided into pieces according to the pumping lemma. I am learning from Michael Sipser’s book Introduction to the Theory of Computation, 3rd Edition. He ...
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Summary of Pumping Lemma Application

For my own understanding I would like to summarize how to use the pumping lemma to show that a language is not regular. The pumping lemma is defined as follows. Pumping lemma If $A$ is a regular ...
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What is the minimum pumping lemma length of $01^*0^*1$?

I've taken the following steps to prove that the minimum pumping length (PL) of the above language, $L= 01^*0^*1$: Set a PL. I chose $p=2$ Choose a string from $L$ where $|w|\geq p$, I chose $w=011$. ...
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Show that $\{xy : x \in \{a\}^*, y \in \{b\}^*, |x| = |y|\}$ is a not a regular language

I have been asked as an exercise how to prove that this is not a regular language. first I tried to use the pumping lemma, but I got stucked. Th erxercise hust said to prove thata this isn't a regular ...
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How to use pumping lemma on languages that do not follow a strict structure?

Let me preface this by saying, I do NOT want an example of a proof, I would merely like pointers as to how I could approach this problem. For example, I have a language: $$L = \{w \mid w \in \{0, 1\}^*...
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Implication of the Pumping lemma

I'm reading Hopcroft and Ullman's '79 edition of "Introduction to Automata theory, Languages, and Computation". In chapter 3, the authors say "The lemma[sic] does not state that every ...
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Proving that $\{ a^i b^j c^{\max(i,j)} \}$ is not context-free

Prove that $L$ is not a Context-free language, where $$L = \{ a^{i} b^{j}c^{h}\mid i,j,h\in \mathbb{N} \wedge h = \max(i,j)\}.$$ I have an idea: It can be divided into two situations: When $i < j$...
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How to prove that $L = \{w\in\{a,b\}^*\mid w = uav \text{ and } |u| = |v|\}$ is not a regular language

$L = \{w\in\{a,b\}^*\mid w = uav \text{ and } |u| = |v|\}$ I know to use the pump lemma, but I don’t know how to use it correctly.
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Check if a language is context free [duplicate]

Check whether the following language is context-free. If yes, a suitable grammar should be given; if no, the pumping lemma should be used as a tool. $$L=\{a^ib^jc^k \mid i, j, k \in N \text{ and } i &...
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Proof that $\{a^ib^jc^k\mid i,j,k\in\mathbb{N}, i<k<j\}$ is not context-free using the Pumping Lemma

$$ L=\{a^ib^jc^k \;| \;i, j, k \in \mathbb{N} \; \text{and} \; i <k<j\} $$ I need to show that this language is not context-free with the help of the Pumping Lemma. My first intuition is, that ...
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Use the CFL pumping lemma to show that this language(0^p where p is prime) is not context free

L = {0^p |p is a prime}. So was looking at the explanation at the bottom of page 5 of the following website: https://www.ics.uci.edu/~goodrich/teach/cs162/hw/HW5Sols.pdf I want to make sure that I ...
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How to choose word for pumping lemma for $a^kb^{2k}a^k$?

I have to show that the language $ \mathcal {L} = \{a ^ k b ^ {2k} a ^ k: k \geq 0 \} $ is not a regular language. So that's what I want to use the pumping motto for. What I could do is this: let $ \ ...
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Pumping Lemma for $\mathcal{L} = \{ \omega \omega^R a^{|\omega|} : \omega \in \{a,b\}^* \} $

I have to show that this language is not context free $\mathcal{L} = \{ \omega \omega^R a^{|\omega|} : \omega \in \{a,b\}^* \} $, where the $R$ corresponds to the reverse. For this I will use the ...
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Show that a language with union is not regular by using pumping lemma

Given the language $L:= { \{ c^{2k} w \ \vert \ k \ge 1, \ w \in \{a,b,c\}^* \ and \ \vert w\vert_a \ = \ \vert w\vert_b \} \ \cup \ \{ a,b \}^* }$ I'm really unsure how to even start because of the ...
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Minimum pumping length of finite language

Background Let L = {aa}. We know that the minimum pumping length of L is |aa| + 1 = 3. For this length all the three conditions of the pumping lemma vacuously hold true. Doubt Let L = {aa, aab}. Is it ...
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Prove $L =\{0^{2^n}\mid n \geqslant 0\}$ is not context free [duplicate]

Here $0^j$ means $0$ repeated $j$ times e.g. $0^2$ is $00$. So to prove this I was asked to use the pumping lemma. So let $m$ be the pumping length and assume $L$ is a CFL by contradiction. We can ...
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Can a non-regular language $L$ have a non regular $L^*$?

I have been looking around and i cant seem to find an example of such case that a non-regular $L$ has a non regular $L^*$. Is it possible? If so, can you provide an example of such case please?
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Prove $\{xy \mid |x|=|y|, x \neq y\}$ is not a linear language

Show the language $$L = \{xy \mid |x| = |y|, x\neq y\}$$ is not linear. I've seen and proved a pumping lemma for linear languages, mentioned here: If $L$ is linear then there exists a constant $p$ ...
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Proving $\{ a^n b^m \mid n \leq m^2 \}$ is not context-free using pumping lemma

I am working on a pumping lemma question and trying to prove that the following is not context-free, but I can't finish the proof. The language is $$L = \{ a^n b^m \mid n \leq m^2 \}$$ Assume Demon ...
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Prove with pumping lemma that the language { $a^n b^n b^m a^m | n ≠ m $ } is not context free

I'm having a trouble proving it to be non-context-free. For example, if I take w = $a^k b^k b^{k+1} a^{k+1}$, it would be problematic if the partition of $vxy$ with $|v| = |y|$ was in the $ b^{k+1} a^{...
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Proving that $L=\{ w \mid \lvert w \rvert$ is prime $\}$* is a regular language

I'm trying to prove that the following languague is a regular language: $L=\{ w \mid \lvert w \rvert$ is prime $\}$* What I have thought is to divide each word $w \in L$ into subwords of length 2 if ...
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Using pumping lemma to prove that $L = \{ a^ib^j \mid \lvert i - j \rvert \le 2 \} $ is irregular

Given the following language: $L = \{ a^ib^j \mid \lvert i - j \rvert \le 2 \} $ I am trying to prove that it is not regular. On the one hand my intuition tells me that the language is non-regular as ...
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Proving Irregularity of $L = \{ a^mb^nb^n \mid nm \ge 3 \} $

I'm trying to prove the irregularity of the following language: $$L = \{ a^mb^nb^n \mid nm \ge 3 \} $$ I tried to demonstrate that it doesn't verifies the Pumping Lemma but for all words I tried it ...
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Existence of a CFL $L$ such that $\sqrt{L}$ is not CFL

Does there exist a CFL L such that the language defined as $L' = \sqrt{L} = \{w | ww \in L\}$ is not CFL? I feel that there is no such $L$ but obviously, I am unable to prove it. I am sorry but I have ...
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Is the language $\{a^n b^m : 1000|nm \}$ regular?

We have a language $$ L = \{a^n b^m \mid 1000|nm \} $$ Is this language regular? I'm trying to disprove this using the Pumping Lemma, but it didn't work. assume I say $x=a^{h}$ and $y=a^{t}$ and $z =...
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Is the language $\{a^n b^m \mid 2n + 3m \le 1000 \}$ regular?

We have a language $$ L = \{a^n b^m \mid 2n + 3m \le 1000 \} $$ Is this language regular? I'm trying to disprove this using the Pumping Lemma, but it didn't work. assume I say x = $x=a^{h}$ and $y=a^{...
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Proving that the language $\{ w^n\mid w \in \{0,1\}^∗, \, n \ge 2 \}$ is not regular

I'm trying to prove that the following language is not regular: $$\{ w^n\mid w \in \{0,1\}^∗, \, n \ge 2 \}$$ I'm trying to prove this with the pumping lemma, but I'm kind of confused because $w$ is ...
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Irregularity of $\{ w_1 aa w_2 \mid |w_1| \neq |w_2| \}$

I'm currently struggling to come up with a proof that the following language is irregular: $$L_2 := \{w_1aaw_2 \in \Sigma^* \mid w_1, w_2\in\Sigma^* \land |w_1| \ne |w_2|\}$$ where $\Sigma = \{a, b\}$....
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Pumping lemma for an involved non context free language

Hi I'm trying to show $C=\{wzzw^R|w,z\in\{0,1\}^+\}$ is not a context-free language.( I have this believe because $C=\{ww|w\in\{0,1\}^+\}$ is not a context free language.) I'm really struggling to ...
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Proving a language with $(ab)^n$ is not regular with pumping lemma?

I have been working to understand the pumping lemma better, but I am quite stuck at proving these two languages is not regular: \begin{align} L_1 &= \{(ab)^n c^m \mid n\ge 1, m\ge 2n \} \\ L_2 &...
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Is $\{a^mb^nc^n \mid m,n \geq 0\} \cup \{b,c\}^*$ regular or not?

Show if $L = \{a^mb^nc^n \mid m,n \geq 0\} \cup \{b,c\}^*$ is regular or not. My attempt: I think the Pumping lemma won't work in that constellation, so I'm working with "The intersection of ...
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How do I apply the Pumping Lemma to prove that this language is not regular?

I am trying to teach myself Automata theory. I have hard time with the Pumping Lemma, so I am trying to work through examples. I stumbled upon this example, but it doesn't have steps how to solve it. ...
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Pumping lemma for regular languages. Proof

Please help me understand the following $L = \{ a | a ∈ \{0, 1\}^∗, |a| = k ≥ 4, a = a_1a_2...a_{k−1}a_k, ∃i ∈ N, 1 ≤ i < k : a_i = a_{i+1} \}$ To prove: The language $L$ has regular pumping ...
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Pumping Lemma Proof (Type of wcw language)

I have the language $L = \{ dkd\space \mid d \in \{a,b\}^*, k \in \{a,b\} \}$ and i have to show that it's non-regular using the pumping lemma. The structure of the language i think can be explained ...
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Cardinality of sets and strings -> confused

I have a question regarding the cardinality of sets and strings. If $ \Sigma^* $ is empty, the cardinality is 1, because the empty word $ \varepsilon $ is counted. If $ \Sigma^+ $ is empty, the ...
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What is wrong with this proof that proves that 0*1* is not a regular language?

I know why cases 1 and 2 are wrong because our language can have different numbers of 0's and 1's. But I'm not sure how case 3 can be proved wrong for our language. Exercise 1.30: Describe the error ...
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Proving that $ \{u\#v\#w \mid u,v,w \in {a,b,c}*, |u|_a = |v|_b = |w|_c\}$ isn't context-free

I have a question about the pumping lemma for context-free languages. I understand the conditions of the pumping lemma. Assume $L$ is context-free. Let $n>0$ be the pumping length given by the ...
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Why does the Pumping Lemma Constraint |xy| ≤ p mean that y can't be 1 in the string 0p1p

I am trying to get my head around the Pumping Lemma to prove a language is non-regular. I am reading the Sipser text book and he gives the following example. Let B be the language $\{0^n 1^n | n \ge 0\...
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How to prove a language isn't necessarily regular? [duplicate]

Assuming we have a regular language $L$, how can we prove that $L'= \{ xz \mid \exists y : xyz \in L \text{ and } |x|=|y|=|z|\}$ isn't necessarily regular. So far I can't come up with much for how to ...
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Subexponential size of string to prove $\{xy : x,y \in \{0,1\}^\star, |x| = |y|, x \ne y\}$ is not regular?

In the standard proof of this language not being regular using the Pumping Lemma for Regular languages, one picks $0^p 1^p 0^{p+p!} 1^p$ where $p$ is the pumping constant and using that can derive the ...
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Is $L:=\{a^k \mid k \text{ is prime}\}$ regular?

For this exercise the pumping lemma should be used. My instructor gave me a tip it should start with $w:= a^{prime(n)}$ where prime is a while program returning the nth prime number. This does make ...
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Pumping lemma: why x in ∣xy∣ ≤ p?

Looking at the pumping lemma, I've noticed that in the string $xy^pz$, there seems to be no rule explicitly stated for $x$ and $z$. If I understand correctly, $x$ and $z$ are basically anything on the ...
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Is there a human-friendly version of the Pumping-Lemma?

I found this on Wikipedia and I'm confused by the parenthesis in the notation not that it doesn't make sense to me but is there a more natural human version? And im generally confused about all the ...

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