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Recent questions tagged closure-property

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GO Classes Test Series 2024 | Mock GATE | Test 14 | Question: 31
DeMorgan's Laws ensure thatClosure under intersection and complementation imply closure under union.Closure under intersection and union imply closure under ... union, intersection, and complementation implies closure under all three.
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TOC - Self Doubt
Can anyone explain $\overline{ww}$ is $CFL$ or $CSL$ And if $CFL$ can you write the equivalent $CFG$ for this ?
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#Self Doubt
Suppose L1 = CFL and L2 = Regular, We are to find out whether L1 - L2 = CFL or non CFL.I have 2 approaches to this question and I am confused ... Language is also CFL so CFL intersection CFL = non CFL.Can somebody please clarify my doubt?
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Made easy Theory of Computation
Which of them are not regular-(a) L={a^m b^n | n>=2023, m<=2023}(b) L={a^n b^m c^l | n=2023, m>2023, l>m}according made ... lang are closed under concatenation) therefore L is regular.this makes option (b) regular is it right approach ?
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ACE 2023 Test series: TOC: Basic properties
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Igate Test Series
please explain all options with example
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Michael Sipser Edition 3 Exercise 2 Question 53 (Page No. 159)
Show that the class of DCFLs is not closed under the following operations:UnionIntersectionConcatenationStarReversal
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Michael Sipser Edition 3 Exercise 2 Question 50 (Page No. 159)
We defined the $CUT$ of language $A$ to be $CUT(A) = \{yxz| xyz \in A\}$. Show that the class of $CFLs$ is not closed under $CUT$.
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Michael Sipser Edition 3 Exercise 2 Question 49 (Page No. 159)
We defined the rotational closure of language $A$ to be $RC(A) = \{yx \mid xy \in A\}$.Show that the class of CFLs is closed under rotational closure.
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CMI2019-A-1
Let $L_{1}:=\{a^{n}b^{m}\mid m,n\geq 0\: \text{and}\: m\geq n\}$ ... is:regular, but not context-freecontext-free, but not regularboth regular and context-freeneither regular nor context-free
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Peter Linz Edition 4 Exercise 4.3 Question 24 (Page No. 124)
Suppose that we know that $L_1 ∪ L_2$ and $L_1$ are regular. Can we conclude from this that $L_2$ is regular?
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Peter Linz Edition 4 Exercise 4.3 Question 23 (Page No. 124)
Is the family of regular languages closed under infinite intersection?
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Peter Linz Edition 4 Exercise 4.3 Question 22 (Page No. 124)
Consider the argument that the language associated with any generalized transition graph is regular. The language associated with such a graph is ... case, because of the special nature of $P$, the infinite union is regular.
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Peter Linz Edition 4 Exercise 4.3 Question 21 (Page No. 124)
Let $P$ be an infinite but countable set, and associate with each $p ∈ P$ a language $L_p$. The smallest set containing every $L_p$ is the union ... . Show by example that the family of regular languages is not closed under infinite union.
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Peter Linz Edition 4 Exercise 4.3 Question 16 (Page No. 123)
Is the following language regular? $L=$ {$w_1cw_2:w_1,w_2∈$ {$a,b$}$^*,w_1\neq w_2$}.
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Peter Linz Edition 4 Exercise 4.3 Question 15 (Page No. 123)
Consider the languages below. For each, make a conjecture whether or not it is regular. Thenprove your conjecture.(a) $L=$ {$a^nb^la^k:n+k+l \gt 5$}(b) $L=$ ... {$a^nb^l:n\geq 100,l\leq 100$}(g) $L=$ {$a^nb^l:|n-l|=2$}
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Peter Linz Edition 4 Exercise 4.3 Question 13 (Page No. 123)
Show that the following language is not regular.$L=$ {$a^nb^k:n>k$} $\cup$ {$a^nb^k:n\neq k-1$}.
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Peter Linz Edition 4 Exercise 4.3 Question 10 (Page No. 123)
Consider the language $L=$ {$a^n:n$ is not a perfect square}.(a) Show that this language is not regular by applying the pumping lemma directly.(b) Then show the same thing by using the closure properties of regular languages.
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Peter Linz Edition 4 Exercise 4.1 Question 26 (Page No. 110)
Let $G_1$ and $G_2$ be two regular grammars. Show how one can derive regular grammars for the languages(a) $L (G_1) ∪ L (G_2)$.(b) $L (G_1) L (G_2)$.(b) $L (G_1)^*$.
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Peter Linz Edition 4 Exercise 4.1 Question 25 (Page No. 111)
The $min$ of a language $L$ is defined as $min(L)=$ {$w∈L:$ there is no $u∈L,v∈Σ^+,$ such that $w=uv$}Show that the family of regular languages is closed under the $ min$ operation.
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Peter Linz Edition 4 Exercise 4.1 Question 24 (Page No. 111)
Define the operation $leftside$ on $L$ by $leftside(L)=$ {$w:ww^R∈L$}Is the family of regular languages closed under this operation?
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Peter Linz Edition 4 Exercise 4.1 Question 23 (Page No. 111)
Define an operation $minus5$ on a language $L$ as the set of all strings of $L$ with the fifth symbol from the left removed (strings of length less ... ). Show that the family of regular languages is closed under the $minus5$ operation.
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Peter Linz Edition 4 Exercise 4.1 Question 22 (Page No. 110)
The $\textit{shuffle}$ of two languages $L_1$ and $L_2$ ... $shuffle$ operation.
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Peter Linz Edition 4 Exercise 4.1 Question 21 (Page No. 110)
Define$exchange(a_1a_2a_3...a_{n-1}a_n)=a_na_2a_3...a_{n-1}a_1$,and$exchange(L)=$ {$v:v=exchange(w)$ for some $w∈L$}Show that the family of regular languages is closed under exchange.
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