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Recent questions tagged regular-language
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151
Michael Sipser Edition 3 Exercise 1 Question 40 (Page No. 89)
Recall that string $x$ is a $\text{prefix}$ of string $y$ if a string $z$ exists where $xz = y,$ and that $x$ is a $\text{proper prefix}$ of $y$ if in addition $x\neq y.$ In each of the following parts, we define an operation on a ... $\text{NOEXTEND(A) = {w ∈ A| w is not the proper prefix of any string in A}.}$
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regular-language
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152
Michael Sipser Edition 3 Exercise 1 Question 38 (Page No. 89)
An $\text{all-NFA}$ $M$ is a $\text{5-tuple}$ $(Q, Σ, δ, q_{0}, F)$ that accepts $x\in\sum^{*}$ if every possible state that $M$ could be in after reading input $x$ is a state from $F.$ Note ... string if some state among these possible states is an accept state$.$ Prove that $\text{all-NFAs}$ recognize the class of regular languages$.$
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regular-language
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Michael Sipser Edition 3 Exercise 1 Question 37 (Page No. 89)
Let $C_{n} = \{x| x \text{ is a binary number that is a multiple of $n$}\}.$ Show that for each $n\geq1,$ the language $C_{n}$ is regular$.$
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Apr 28, 2019
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regular-language
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Michael Sipser Edition 3 Exercise 1 Question 36 (Page No. 89)
Let $B_{n} = \{a^{k}| \text{$k$ is a multiple of $n$}\}.$ Show that for each $n\geq 1,$ the language $B_{n}$ is regular$.$
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Apr 28, 2019
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admin
388
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michael-sipser
theory-of-computation
finite-automata
regular-language
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Michael Sipser Edition 3 Exercise 1 Question 35 (Page No. 89)
Let $\sum_{2}=\begin{Bmatrix}\begin{bmatrix} 0\\0 \end{bmatrix},\begin{bmatrix} 0\\1 \end{bmatrix},\begin{bmatrix} 1 \\0 \end{bmatrix},\begin{bmatrix} 1 \\1 \end{bmatrix}\end{Bmatrix}.$ Here $\sum_{2}$ contains all columns of $0's$ ... $ is the reverse of the top row of $w$}\}.$ Show that $E$ is not regular$.$
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Apr 28, 2019
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michael-sipser
theory-of-computation
finite-automata
regular-language
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Michael Sipser Edition 3 Exercise 1 Question 34 (Page No. 89)
Let $\sum_{2}=\begin{Bmatrix}\begin{bmatrix} 0\\0 \end{bmatrix},\begin{bmatrix} 0\\1 \end{bmatrix},\begin{bmatrix} 1 \\0 \end{bmatrix},\begin{bmatrix} 1 \\1 \end{bmatrix}\end{Bmatrix}.$ Here $\sum_{2}$ contains all columns of ... $\begin{bmatrix} 0\\0 \end{bmatrix}\notin D$ Show that $D$ is regular$.$
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regular-language
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157
Michael Sipser Edition 3 Exercise 1 Question 33 (Page No. 89)
Let $\sum_{2}=\begin{Bmatrix}\begin{bmatrix} 0\\0 \end{bmatrix},\begin{bmatrix} 0\\1 \end{bmatrix},\begin{bmatrix} 1 \\0 \end{bmatrix},\begin{bmatrix} 1 \\1 \end{bmatrix}\end{Bmatrix}.$ Here $\sum_{2}$ contains all columns ... $C$ is regular. $\text{(You may assume the result claimed in question $31$})$
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theory-of-computation
finite-automata
regular-language
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Michael Sipser Edition 3 Exercise 1 Question 32 (Page No. 88)
Let $\sum_{3}=\begin{Bmatrix}\begin{bmatrix} 0\\0 \\0 \end{bmatrix},\begin{bmatrix} 0\\0 \\1 \end{bmatrix},\begin{bmatrix} 0\\1 \\0 \end{bmatrix}, .,\begin{bmatrix} 1\\1 \\1 \end{bmatrix}\end{Bmatrix}.$ $\sum_{3}$ ... $B$ is regular. $\text{(Hint$:$Working with $B^{R}$ is easier. You may assume the result claimed in question $31$})$
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regular-language
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Michael Sipser Edition 3 Exercise 1 Question 31 (Page No. 88)
For any string $w = w_{1}w_{2} · · · w_{n},$ the reverse of $w,$ written $w^{R},$ is the string $w$ in reverse order$, w_{n} · · · w_{2}w_{1}.$ For any language $A,$ let $A^{R} = \{w^{R}| w\in A\}.$ Show that if $A$ is regular$,$ so is $A^{R}.$
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theory-of-computation
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regular-language
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Michael Sipser Edition 3 Exercise 1 Question 29 (Page No. 88)
Use the pumping lemma to show that the following languages are not regular. $A_{1}=\{0^{n}1^{n}2^{n}|n\geq 0\}$ $A_{2}=\{www|w\in\{a,b\}^{*}\}$ $A_{3}=\{a^{{2}^{n}}|n\geq 0\}$ $\text{(Here,}$\text{$a^{{2}^{n}}$}$ $\text{means a strings of $2^{n}$ a's.)}$
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Apr 21, 2019
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theory-of-computation
finite-automata
regular-language
pumping-lemma
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161
Michael Sipser Edition 3 Exercise 1 Question 23 (Page No. 87)
Let $B$ be any language over the alphabet $Σ.$ Prove that $B = B^{+}$ iff $BB ⊆ B.$
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Apr 21, 2019
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theory-of-computation
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proof
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Michael Sipser Edition 3 Exercise 1 Question 20 (Page No. 86)
For each of the following languages, give two strings that are members and two strings that are not members-a total of four strings for each part. Assume the alphabet $Σ = \{a,b\}$ in all parts. $a^{*}b^{*}$ $a(ba)^{*}b$ ... $aba\cup bab$ $(\epsilon\cup a)b$ $(a\cup ba\cup bb)\Sigma^{*}$
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Apr 21, 2019
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theory-of-computation
regular-language
regular-expression
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163
Michael Sipser Edition 3 Exercise 1 Question 15 (Page No. 85)
Give a counterexample to show that the following construction fails to prove $\text{Theorem 1.49,}$ the closure of the class of regular languages under the star operation$.$ Let $N_{1} = (Q_1, Σ, δ_1, q_1, F_1)$ ... , $N1,$ for which the constructed automaton $N$ does not recognize the star of $N_{1}^{'s}$ language.
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Apr 21, 2019
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theory-of-computation
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regular-language
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164
Peter Linz Edition 4 Exercise 5.1 Question 5 (Page No. 133)
Is the language $L(G)=$ {$ab(bbaa)^nbba(ba)^n:n\geq0$} regular?
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Apr 13, 2019
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peter-linz
peter-linz-edition4
theory-of-computation
regular-language
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165
Peter Linz Edition 4 Exercise 4.3 Question 26 (Page No. 124)
Let $L=$ {$a^nb^m:n\geq100,m\leq50$}. (a) Can you use the pumping lemma to show that L is regular? (b) Can you use the pumping lemma to show that L is not regular? Explain your answers.
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Apr 12, 2019
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peter-linz
peter-linz-edition4
theory-of-computation
regular-language
pumping-lemma
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Peter Linz Edition 4 Exercise 4.3 Question 25 (Page No. 124)
In the chain code language in Exercise 24, Section 3.1, let $L$ be the set of all $w ∈$ {$u,r,l,d$}$^*$ that describe rectangles. Show that $L$ is not a regular language.
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Apr 12, 2019
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peter-linz
peter-linz-edition4
theory-of-computation
regular-language
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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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Apr 12, 2019
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peter-linz
peter-linz-edition4
theory-of-computation
regular-language
closure-property
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168
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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Apr 12, 2019
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peter-linz
peter-linz-edition4
theory-of-computation
regular-language
closure-property
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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.
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peter-linz
peter-linz-edition4
theory-of-computation
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closure-property
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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.
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theory-of-computation
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Peter Linz Edition 4 Exercise 4.3 Question 20 (Page No. 124)
Is the following language regular? $L=$ {$ww^Rv:v,w∈$ {$a,b$}$^+$}.
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Apr 12, 2019
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peter-linz-edition4
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Peter Linz Edition 4 Exercise 4.3 Question 19 (Page No. 124)
Are the following languages regular? (a) $L=$ {$uww^Rv:u,v,w∈ ${$a,b$}$^+$} (b) $L=$ {$uww^Rv:u,v,w∈ ${$a,b$}$^+$, $|u|\geq |v|$}
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156
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Peter Linz Edition 4 Exercise 4.3 Question 18 (Page No. 124)
Apply the pigeonhole argument directly to the language in $L=$ {$ww^R:w∈Σ^+$}.
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peter-linz
peter-linz-edition4
theory-of-computation
regular-language
pigeonhole-principle
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174
Peter Linz Edition 4 Exercise 4.3 Question 17 (Page No. 124)
Let $L_1$ and $L_2$ be regular languages. Is the language $L =$ {$w : w ∈ L_1, w^R ∈ L_2$ necessarily regular?
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peter-linz
peter-linz-edition4
theory-of-computation
regular-language
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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
peter-linz-edition4
theory-of-computation
pumping-lemma
regular-language
closure-property
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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) $L=$ {$a^nb^l:|n-l|=2$}
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301
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peter-linz-edition4
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regular-language
pumping-lemma
closure-property
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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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Apr 11, 2019
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282
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peter-linz
peter-linz-edition4
theory-of-computation
pumping-lemma
regular-language
closure-property
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Peter Linz Edition 4 Exercise 4.3 Question 14 (Page No. 123)
Prove or disprove the following statement: If $L_1$ and $L_2$ are non regular languages, then $L_1 ∪ L_2$ is also non regular.
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376
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peter-linz
peter-linz-edition4
theory-of-computation
regular-language
pumping-lemma
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Peter Linz Edition 4 Exercise 4.3 Question 12 (Page No. 123)
Apply the pumping lemma to show that $L=$ {$a^nb^kc^{n+k}:n\geq0,k\geq0$} is not regular.
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Apr 11, 2019
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250
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peter-linz
peter-linz-edition4
theory-of-computation
pumping-lemma
regular-language
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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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