Recent questions tagged closure-property / GATE Overflow for GATE CSE

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31

Peter Linz Edition 4 Exercise 4.1 Question 20 (Page No. 110)
For a string $a_1a_2…a_n$ define the operation $shift$ as $shift(a_1a_2...a_n)=a_2a_3...a_na_1$ From this, we can define the operation on a language as $shift(L)=$ {$v:v=shift(w$ for some $w∈L$} Show that regularity is preserved under the shift operation.

For a string $a_1a_2…a_n$ define the operation $shift$ as $shift(a_1a_2...a_n)=a_2a_3...a_na_1$From this, we can define the operation on a language as...

Naveen Kumar 3

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Naveen Kumar 3 asked Apr 5, 2019

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32

Peter Linz Edition 4 Exercise 4.1 Question 19 (Page No. 110)
Define an operation $third$ on strings and languages as $third(a_1a_2a_3a_4a_5a_6...)=a_3a_6...$ with the appropriate extension of this definition to languages. Prove the closure of the family of regular languages under this operation.

Define an operation $third$ on strings and languages as $third(a_1a_2a_3a_4a_5a_6...)=a_3a_6...$with the appropriate extension of this definition...

Naveen Kumar 3

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Naveen Kumar 3 asked Apr 5, 2019

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33

Peter Linz Edition 4 Exercise 4.1 Question 18 (Page No. 110)
The $head$ of a language is the set of all prefixes of its strings, that is, $head(L)=$ {$x:xy∈L$ for some $y ∈Σ^*$} Show that the family of regular languages is closed under this operation.

The $head$ of a language is the set of all prefixes of its strings, that is, $head(L)=$ {$x:xy∈L$ for some $y ∈Σ^*$}Show that the family of regular langu...

Naveen Kumar 3

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Naveen Kumar 3 asked Apr 5, 2019

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34

Peter Linz Edition 4 Exercise 4.1 Question 17 (Page No. 110)
The $tail$ of a language is defined as the set of all suffixes of its strings, that is, $tail(L)=$ {$y:xy∈L$ for some $x∈Σ^*$} Show that if L is regular, so is $tail(L)$.

The $tail$ of a language is defined as the set of all suffixes of its strings, that is, $tail(L)=$ {$y:xy∈L$ for some $x∈Σ^*$}Show...

Naveen Kumar 3

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Naveen Kumar 3 asked Apr 5, 2019

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35

Peter Linz Edition 4 Exercise 4.1 Question 16 (Page No. 110)
Show that if the statement “If $L_1$ is regular and $L_1 ∪ L_2$ is also regular, then $L_2$ must be regular“ were true for all $L_1$ and $L_2$, then all languages would be regular.

Show that if the statement “If $L_1$ is regular and $L_1 ∪ L_2$ is also regular, then $L_2$ must be regular“were true for all $L_1$ and $L_2$, then all languages wo...

Naveen Kumar 3

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Naveen Kumar 3 asked Apr 5, 2019

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36

Peter Linz Edition 4 Exercise 4.1 Question 15 (Page No. 110)
The left quotient of a language $L_1$ with respect to $L_2$ is defined as $L_2/L_1=$ {$y:x∈L_2,xy∈L_1$} Show that the family of regular languages is closed under the left quotient with a regular language.

The left quotient of a language $L_1$ with respect to $L_2$ is defined as $L_2/L_1=$ {$y:x∈L_2,xy∈L_1$}Show that the family of regular l...

Naveen Kumar 3

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Naveen Kumar 3 asked Apr 5, 2019

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37

Peter Linz Edition 4 Exercise 4.1 Question 14 (Page No. 109)
If $L$ is a regular language, prove that the language {$uv : u ∈ L,υ ∈ L^R$} is also regular.

If $L$ is a regular language, prove that the language {$uv : u ∈ L,υ ∈ L^R$} is also regular.

Naveen Kumar 3

226 views

Naveen Kumar 3 asked Apr 4, 2019

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38

Peter Linz Edition 4 Exercise 4.1 Question 13 (Page No. 109)
If $L$ is a regular language, prove that $L_1 =$ {$uv : u ∈ L, |υ| = 2$} is also regular.

If $L$ is a regular language, prove that $L_1 =$ {$uv : u ∈ L, |υ| = 2$} is also regular.

Naveen Kumar 3

193 views

Naveen Kumar 3 asked Apr 4, 2019

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39

Peter Linz Edition 4 Exercise 4.1 Question 12 (Page No. 109)
Suppose we know that $L_1 ∪ L_2$ is regular and that $L_1$ is finite. Can we conclude from this that $L_2$ is regular?

Suppose we know that $L_1 ∪ L_2$ is regular and that $L_1$ is finite. Can we conclude from this that $L_2$ is regular?

Naveen Kumar 3

188 views

Naveen Kumar 3 asked Apr 4, 2019

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40

Peter Linz Edition 4 Exercise 4.1 Question 11 (Page No. 109)
Show that $L_1 = L_1L_2/L_2$ is not true for all languages $L_1$ and $L_2$.

Show that $L_1 = L_1L_2/L_2$ is not true for all languages $L_1$ and $L_2$.

Naveen Kumar 3

209 views

Naveen Kumar 3 asked Apr 4, 2019

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41

Peter Linz Edition 4 Exercise 4.1 Question 10 (Page No. 109)
Let $L_1 = L (a^*baa^*)$ and $L_2 = L (aba^*)$. Find $L_1/L_2$.

Let $L_1 = L (a^*baa^*)$ and $L_2 = L (aba^*)$. Find $L_1/L_2$.

Naveen Kumar 3

172 views

Naveen Kumar 3 asked Apr 4, 2019

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42

Peter Linz Edition 4 Exercise 4.1 Question 9 (Page No. 109)
Which of the following are true for all regular languages and all homomorphisms? (a) $h (L_1 ∪ L_2)= h (L_1) ∩ h (L_2)$. (b) $h (L_1 ∩ L_2)= h (L_1) ∩ h (L_2)$. (c) $h (L_1L_2) = h (L_1) h (L_2)$.

Which of the following are true for all regular languages and all homomorphisms?(a) $h (L_1 ∪ L_2)= h (L_1) ∩ h (L_2)$.(b) $h (L_1 ∩ L_2)= h (L_1) ∩ h (L_2)$.(c) ...

Naveen Kumar 3

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Naveen Kumar 3 asked Apr 4, 2019

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43

Peter Linz Edition 4 Exercise 4.1 Question 8 (Page No. 109)
Define the $complementary$ or $(cor)$ of two languages by $cor(L_1,L_2)=$ {$w:w∈\overline{L_1}$ or $w∈\overline{L_2}$} Show that the family of regular languages is closed under the $cor$ operation.

Define the $complementary$ or $(cor)$ of two languages by $cor(L_1,L_2)=$ {$w:w∈\overline{L_1}$ or $w∈\overline{L_2}$}Show that the family of regul...

Naveen Kumar 3

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Naveen Kumar 3 asked Apr 4, 2019

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44

Peter Linz Edition 4 Exercise 4.1 Question 7 (Page No. 109)
The $nor$ of two languages is $nor(L_1,L_2)=$ {$w:w∉L_1$ and $w∉L_2$} Show that the family of regular languages is closed under the $nor$ operation.

The $nor$ of two languages is $nor(L_1,L_2)=$ {$w:w∉L_1$ and $w∉L_2$}Show that the family of regular languages is closed under the $nor$ operati...

Naveen Kumar 3

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Naveen Kumar 3 asked Apr 4, 2019

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45

Peter Linz Edition 4 Exercise 4.1 Question 6 (Page No. 109)
The symmetric difference of two sets $S_1$ and $S_2$ is defined as $S_1 θ S_2 =$ {$x: x ∈ S_1$ or $x ∈ S_2,$ but $x$ is not in both $S_1$ and $S_2$}. Show that the family of regular languages is closed under symmetric difference.

The symmetric difference of two sets $S_1$ and $S_2$ is defined as $S_1 θ S_2 =$ {$x: x ∈ S_1$ or $x ∈ S_2,$ but $x$ is not in both $S_1$ and $S_2$}.Show that the fa...

Naveen Kumar 3

150 views

Naveen Kumar 3 asked Apr 4, 2019

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46

Peter Linz Edition 4 Exercise 4.1 Question 5 (Page No. 109)
Show that the family of regular languages is closed under finite union and intersection, that is, if $L_1,L_2,…, L_n$ are regular, then $L_U=\bigcup_{i=1,2,..,n}$L_i$ and $L_I=\bigcap_{i=1,2,3,...,n}L_i$ are also regular.

Show that the family of regular languages is closed under finite union and intersection, that is, if $L_1,L_2,…, L_n$ are regular, then $L_U=\bigcup_{i=1,2,.....

Naveen Kumar 3

158 views

Naveen Kumar 3 asked Apr 4, 2019

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47

Peter Linz Edition 4 Exercise 4.1 Question 3 (Page No. 108)
“The family of regular languages is closed under difference.” Provide constructive proof for this argument.

“The family of regular languages is closed under difference.”Provide constructive proof for this argument.

Naveen Kumar 3

150 views

Naveen Kumar 3 asked Apr 4, 2019

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1 answer

48

Peter Linz Edition 4 Exercise 4.1 Question 2 (Page No. 108)
Find nfa's that accept (a) $L ((a + b) a^*) ∩ L (baa^*)$. (b) $L (ab^*a^*) ∩ L (a^*b^*a)$.

Find nfa's that accept(a) $L ((a + b) a^*) ∩ L (baa^*)$.(b) $L (ab^*a^*) ∩ L (a^*b^*a)$.

Naveen Kumar 3

466 views

Naveen Kumar 3 asked Apr 4, 2019

1 votes

1 answer

49

MadeEasy Subject Test 2019: Theory of Computation - Closure Property
Consider the following Statements : There Exist a non-deterministic CFL whose reversal is DCFL. There exist a non regular CSL whose Kleene Closure is regular. Which of following are True ? Explain with reasons.

Consider the following Statements :There Exist a non-deterministic CFL whose reversal is DCFL.There exist a non regular CSL whose Kleene Closure is regular.Which of follo...

Na462

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Na462 asked Jan 13, 2019

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50

MadeEasy Test Series: Theory Of Computation - Closure Property
L1 is regular, L2 and L3 are CFL L1 is regular, L2 is CFL and L3 is CSL L1 is CFL but not regular,L2 is CSL but not CFL,L3 is CFL L1, L2 and L3 are CFL

L1 is regular, L2 and L3 are CFLL1 is regular, L2 is CFL and L3 is CSLL1 is CFL but not regular,L2 is CSL but not CFL,L3 is CFLL1, L2 and L3 are CFL

Sambhrant Maurya

525 views

Sambhrant Maurya asked Jan 9, 2019

4 votes

1 answer

51

GATE Overflow | Mock GATE | Test 1 | Question: 56
Consider the following language $L$: $L=\{<M,x,k> \mid M \text{ is a Turing Machine and M does not halt on x within k steps} \}$ The language family to which $L$ belongs is not closed under? Intersection Homomorphism Set Difference Complementation

Consider the following language $L$:$L=\{<M,x,k \mid M \text{ is a Turing Machine and M does not halt on x within k steps} \}$The language family to which $L$ belongs is ...

Ruturaj Mohanty

1.7k views

Ruturaj Mohanty asked Dec 27, 2018

4 votes

1 answer

52

Closure Properties
What is difference between Σ* and L* ? Which is true ? S1 : Σ* – {ϵ} = Σ+ S2 : L* – {ϵ} = L+ .

What is difference between Σ* and L* ? Which is true ?S1 : Σ* – {ϵ} = Σ+S2 : L* – {ϵ} = L+ .

anurag sharma

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anurag sharma asked Dec 24, 2018

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53

proving a language is regular based on two regular language over same $\Sigma$ (complicated)
Hello, I've encountered with a difficult question i don't know how to solve. the question is about proving that a language is regular based on two language over the same $\Sigma$. the questions goes ... know it's complicated and would appreciate help with it. thank you very much your much appreciated help!

Hello,I’ve encountered with a difficult question i don’t know how to solve. the question is about proving that a language is regular based on two language over the sa...

csenoob

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csenoob asked Dec 8, 2018

1 votes

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54

Closure Property
If L1 is CSL and L2 is CFL, then which of the following is correct ? A.L1' - L2 is CSL always B. L1 - L2' is CSL always C. L1 intersection Regular is Regular always D. L1.L2 is CSL but not CFL

If L1 is CSL and L2 is CFL, then which of the following is correct ?A.L1' - L2 is CSL alwaysB. L1 - L2' is CSL alwaysC. L1 intersection Regular is Regular alwaysD. L1.L2...

Na462

1.9k views

Na462 asked Sep 9, 2018

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55

Relationship b/w closure and decidability?
Is there any relationship b/w closure and decidability? Thanks

Is there any relationship b/w closure and decidability?Thanks

surbhijain93

309 views

surbhijain93 asked Sep 9, 2018

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56

self doubt
CAN SOMEONE EXPLAIN ME WITH AN EXAMPLE THAT WHY DCFL CLOSED UNDER COMPLEMENATION AND CFL NOT CLOSED UNDER COMPLEMENATION?

CAN SOMEONE EXPLAIN ME WITH AN EXAMPLE THAT WHY DCFL CLOSED UNDER COMPLEMENATION AND CFL NOT CLOSED UNDER COMPLEMENATION?

sushmita

176 views

sushmita asked Sep 9, 2018

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2 answers

57

Theory of computation - Closure Property
Cfl is closed under intersection with regular language. Then resultant languages will be regular or cfl ? Let X is cfl Y is regular language L=X intersection Y Then L is what?

Cfl is closed under intersection with regular language.Then resultant languages will be regular or cfl ?Let X is cflY is regular language L=X intersection YThen L is what...

Dhoomketu

603 views

Dhoomketu asked May 10, 2018

4 votes

2 answers

58

ISRO2018-25
$CFG$ (Context Free Grammar) is not closed under: Union Complementation Kleene star Product

$CFG$ (Context Free Grammar) is not closed under: UnionComplementationKleene starProduct

Arjun

1.6k views

Arjun asked Apr 22, 2018

27 votes

6 answers

59

GATE CSE 2018 | Question: 7
The set of all recursively enumerable languages is: closed under complementation closed under intersection a subset of the set of all recursive languages an uncountable set

The set of all recursively enumerable languages is:closed under complementationclosed under intersectiona subset of the set of all recursive languagesan uncountable set

gatecse

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gatecse asked Feb 14, 2018

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60

Kenneth Rosen Edition 6th Exercise 7.4 (Page No. 488)
Let R be a relation on a set A. R may or may not have some property P, such as reflexivity, symmetry, or transitivity. If there is a relation S with property P containing R such that S is a subset of every relation with ... R, then S is called the closure Relations of R with respect to P. can someone explain this definition in simple words?

Let R be a relation on a set A. R may or may not have some property P, such as reflexivity, symmetry, or transitivity. If there is a relation S with property P containing...

Mk Utkarsh

424 views

Mk Utkarsh asked Feb 14, 2018

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