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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 that Closure under intersection and complementation imply closure under union. Closure under intersection and union imply closure under complementation. Closure under union and complementation imply closure ... Closure under any two of union, intersection, and complementation implies closure under all three.

GO Classes asked in Theory of Computation Feb 5

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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 ?

Jiten008 asked in Theory of Computation Oct 24, 2023

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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 which is wrong: L1 - L2 = L1 intersection L2' L2 being Regular L2' is also Regular so CFL ... being Regular L2' is also Regular and every Regular Language is also CFL so CFL intersection CFL = non CFL. Can somebody please clarify my doubt?

Sunnidhya Roy asked in Theory of Computation Dec 11, 2022

by Sunnidhya Roy

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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 easy (b) is the answer but can we do like this- Let L1= {a^n |n=2023} ... ) and so L2 is regular L=L1.L2 (regular lang are closed under concatenation) therefore L is regular.this makes option (b) regular is it right approach ?

Shreya2002 asked in Theory of Computation Dec 1, 2022

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ACE 2023 Test series: TOC: Basic properties

abhinowKatore asked in Theory of Computation Oct 18, 2022

by abhinowKatore

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1 answer

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Igate Test Series
please explain all options with example

SKMAKM asked in Theory of Computation Jul 20, 2022

by SKMAKM

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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: Union Intersection Concatenation Star Reversal

admin asked in Theory of Computation Oct 12, 2019

by admin

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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 $CUT$.

admin asked in Theory of Computation Oct 12, 2019

by admin

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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.

admin asked in Theory of Computation Oct 12, 2019

by admin

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CMI2019-A-1
Let $L_{1}:=\{a^{n}b^{m}\mid m,n\geq 0\: \text{and}\: m\geq n\}$ and $L_{2}:=\{a^{n}b^{m}\mid m,n\geq 0\: \text{and}\: m < n\}.$ The language $L_{1}\cup L_{2}$ is: regular, but not context-free context-free, but not regular both regular and context-free neither regular nor context-free

gatecse asked in Theory of Computation Sep 13, 2019

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11

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?

Naveen Kumar 3 asked in Theory of Computation Apr 12, 2019

by Naveen Kumar 3

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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?

Naveen Kumar 3 asked in Theory of Computation Apr 12, 2019

by Naveen Kumar 3

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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 $L=\bigcup_{p∈P} L(r_p)$, where $P$ ... is regular. Show that in this case, because of the special nature of $P$, the infinite union is regular.

Naveen Kumar 3 asked in Theory of Computation Apr 12, 2019

by Naveen Kumar 3

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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 over the infinite set $P$; it will be denoted by $U_{p∈p}L_p$. Show by example that the family of regular languages is not closed under infinite union.

Naveen Kumar 3 asked in Theory of Computation Apr 12, 2019

by Naveen Kumar 3

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1 vote

1 answer

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Peter Linz Edition 4 Exercise 4.3 Question 16 (Page No. 123)
Is the following language regular? $L=$ {$w_1cw_2:w_1,w_2∈$ {$a,b$}$^*,w_1\neq w_2$}.

Naveen Kumar 3 asked in Theory of Computation Apr 12, 2019

by Naveen Kumar 3

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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. Then prove your conjecture. (a) $L=$ {$a^nb^la^k:n+k+l \gt 5$} (b) $L=$ {$a^nb^la^k:n \gt 5,l> 3,k\leq l$} (c) $L=$ {$a^nb^l:n/l$ is an integer} (d) $L=$ ... $L=$ {$a^nb^l:n\geq 100,l\leq 100$} (g) $L=$ {$a^nb^l:|n-l|=2$}

Naveen Kumar 3 asked in Theory of Computation Apr 12, 2019

by Naveen Kumar 3

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Peter Linz Edition 4 Exercise 4.3 Question 13 (Page No. 123)
Show that the following language is not regular. $L=$ {$a^nb^k:n>k$} $\cup$ {$a^nb^k:n\neq k-1$}.

Naveen Kumar 3 asked in Theory of Computation Apr 11, 2019

by Naveen Kumar 3

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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.

Naveen Kumar 3 asked in Theory of Computation Apr 11, 2019

by Naveen Kumar 3

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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) $L (G_1)^*$.

Naveen Kumar 3 asked in Theory of Computation Apr 5, 2019

by Naveen Kumar 3

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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.

Naveen Kumar 3 asked in Theory of Computation Apr 5, 2019

by Naveen Kumar 3

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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?

Naveen Kumar 3 asked in Theory of Computation Apr 5, 2019

by Naveen Kumar 3

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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 than five are left unchanged). Show that the family of regular languages is closed under the $minus5$ operation.

Naveen Kumar 3 asked in Theory of Computation Apr 5, 2019

by Naveen Kumar 3

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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 $L_2$ is defined as $\textit{shuffle}(L_1,L_2)= \{w_1v_1w_2v_2w_3v_3...w_mv_m:w_1w_2w_3….w_m∈L_1, v_1v_2...v_m∈L_2,\text{ for all }w_i,v_i∈Σ^*\}.$ Show that the family of regular languages is closed under the $shuffle$ operation.

Naveen Kumar 3 asked in Theory of Computation Apr 5, 2019

by Naveen Kumar 3

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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.

Naveen Kumar 3 asked in Theory of Computation Apr 5, 2019

by Naveen Kumar 3

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