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Recent questions tagged regularlanguages
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Michael Sipser Edition 3 Exercise 2 Question 44 (Page No. 158)
If $A$ and $B$ are languages, define $A \diamond B = \{xy \mid x \in A\: \text{and}\: y \in B \;\text{and} \mid x \mid = \mid y \mid \}$. Show that if $A$ and $B$ are regular languages, then $A \diamond B$ is a CFL.
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Ullman (TOC) Edition 3 Exercise 9.5 Question 2 (Page No. 418)
Show that the language $\overline{L_A}\cup \overline{L_B}$ is a regular language if and only if it is the set of all strings over its alphabet;i.e., if and only if the instance $(A,B)$ of PCP has no ... homomorphism, complementation and the pumping lemma for regular sets to show that $\overline{L_A}\cup \overline{L_B}$ is not regular.
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MadeEasy Test Series: Theory Of Computation  Regular Languages
Consider the following statements: $S_1:\{(a^n)^mn\leq m\geq0\}$ $S_2:\{a^nb^nn\geq 1\} \cup \{a^nb^mn \geq1,m \geq 1\} $ Which of the following is regular? $S_1$ only $S_2$ only Both Neither of the above
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self doubt: TOC
is union of regular language and context free language always regular?
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May 22
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Self Doubt : Ambiguity
Why is ambiguity in regular language is decidable and not decidable in CFL ? Can you give Example?
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Michael Sipser Edition 3 Exercise 2 Question 20 (Page No. 156)
Let $A/B = \{w\mid wx\in A$ $\text{for some}$ $x \in B\}.$ Show that if $A$ is context free and $B$ is regular$,$ then $A/B$ is context free$.$
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Michael Sipser Edition 3 Exercise 2 Question 18 (Page No. 156)
Let $C$ be a contextfree language and $R$ be a regular language$.$ Prove that the language $C\cap R$ is contextfree. Let $A = \{w\mid w\in \{a, b, c\}^{*}$ $\text{and}$ $w$ $\text{contains equal numbers of}$ $a’s, b’s,$ $\text{and}$ $c’s\}.$ Use $\text{part (a)}$ to show that $A$ is not a CFL$.$
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Michael Sipser Edition 3 Exercise 2 Question 17 (Page No. 156)
Use the results of $\text{Question 16}$ to give another proof that every regular language is context free$,$ by showing how to convert a regular expression directly to an equivalent contextfree grammar$.$
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Michael Sipser Edition 3 Exercise 2 Question 13 (Page No. 156)
Let $G = (V, \Sigma, R, S)$ be the following grammar. $V = \{S, T, U\}; \Sigma = \{0, \#\};$ and $R$ is the set of rules$:$ $S\rightarrow TT\mid U$ $T\rightarrow 0T\mid T0\mid \#$ $U\rightarrow 0U00\mid\#$ Describe $L(G)$ in English. Prove that $L(G)$ is not regular$.$
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Michael Sipser Edition 3 Exercise 1 Question 72 (Page No. 93)
Let $M_{1}$ and $M_{2}$ be $\text{DFA's}$ that have $k_{1}$ and $k_{2}$ states, respectively, and then let $U = L(M_{1})\cup L(M_{2}).$ Show that if $U\neq\phi$ then $U$ contains some string $s,$ where $s < max(k1, k2).$ Show that if $U\neq\sum^{*},$ then $U$ excludes some string $s,$ where $s < k1k2.$
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Apr 30
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michaelsipser
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finiteautomata
regularlanguages
proof
descriptive
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Michael Sipser Edition 3 Exercise 1 Question 71 (Page No. 93)
Let $\sum = \{0,1\}$ Let $A=\{0^{k}u0^{k}k\geq 1$ $\text{and}$ $u\in \sum^{*}\}.$ Show that $A$ is regular. Let $B=\{0^{k}1u0^{k}k\geq 1$ $\text{and}$ $u\in \sum^{*}\}.$Show that $B$ is not regular.
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Apr 30
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finiteautomata
regularlanguages
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descriptive
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Michael Sipser Edition 3 Exercise 1 Question 70 (Page No. 93)
We define the $\text{avoids}$ operation for languages $A$ and $B$ to be $\text{A avoids B = {w w ∈ A and w doesn’t contain any string in B as a substring}.}$ Prove that the class of regular languages is closed under the ${avoids}$ operation.
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finiteautomata
regularlanguages
proof
descriptive
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Michael Sipser Edition 3 Exercise 1 Question 68 (Page No. 93)
In the traditional method for cutting a deck of playing cards, the deck is arbitrarily split two parts, which are exchanged beforereassembling the deck. In a more complex cut, called $\text{Scarne's cut,}$ the deck is broken into three parts ... $ CUT(CUT(B)).}$ Show that the class of regular languages is closed under $\text{CUT}.$
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regularlanguages
scarnescut
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Michael Sipser Edition 3 Exercise 1 Question 67 (Page No. 93)
Let the rotational closure of language $A$ be $RC(A) = \{yx xy ∈ A\}.$ Show that for any language $A,$ we have $RC(A) = RC(RC(A)).$ Show that the class of regular languages is closed under rotational closure.
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Michael Sipser Edition 3 Exercise 1 Question 63 (Page No. 92)
Let $A$ be an infinite regular language. Prove that $A$ can be split into two infinite disjoint regular subsets. Let $B$ and $D$ be two languages. Write $B\subseteqq D$ if $B\subseteq D$ and $D$ contains infinitely many ... regular languages where $B\subseteqq D,$ then we can find a regular language $C$ where $B\subseteqq C\subseteqq D.$
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Michael Sipser Edition 3 Exercise 1 Question 58 (Page No. 92)
If $A$ is any language,let $A_{\frac{1}{2}\frac{1}{3}}$ be the set of all strings in $A$ with their ,middle thirds removed so that $A_{\frac{1}{2}\frac{1}{3}}=\{\text{xzfor some y,x=y=z and xyz $\in$ A\}}.$ Show that if $A$ is regular,then $A_{\frac{1}{2}\frac{1}{3}}$ is not necessarily regular.
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regularlanguages
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descriptive
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Michael Sipser Edition 3 Exercise 1 Question 57 (Page No. 92)
If $A$ is any language,let $A_{\frac{1}{2}}$ be the set of all first halves of strings in $A$ so that $A_{\frac{1}{2}}=\{\text{xfor some y,x=y and xy $\in$ A\}}.$ Show that if $A$ is regular,then so is $A_{\frac{1}{2}}.$
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michaelsipser
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regularlanguages
proof
descriptive
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Michael Sipser Edition 3 Exercise 1 Question 56 (Page No. 91)
If $A$ is a set of natural numbers and $k$ is a natural number greater than $1,$ let $B_{k}(A)=\{\text{w w is the representation in base k of some number in A\}}.$ Here, we do not allow leading $0's$ in the representation of a ... a set $A$ for which $B_{2}(A)$ is regular but $B_{3}(A)$ is not regular$.$ Prove that your example works.
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finiteautomata
regularlanguages
proof
descriptive
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Michael Sipser Edition 3 Exercise 1 Question 55 (Page No. 91)
The pumping lemma says that every regular language has a pumping length $p,$ such that every string in the language can be pumped if it has length $p$ or more. If $p$ is a pumping length for language $A,$ so is any length $p^{'}\geq p.$ The minimum pumping ... $\epsilon$ $1^{*}01^{*}01^{*}$ $10(11^{*}0)^{*}0$ $1011$ $\sum^{*}$
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michaelsipser
theoryofcomputation
regularlanguages
pumpinglemma
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descriptive
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Michael Sipser Edition 3 Exercise 1 Question 54 (Page No. 91)
Consider the language $F=\{a^{i}b^{j}c^{k}i,j,k\geq 0$ $\text{and if}$ $ i = 1$ $\text{then} $ $ j=k\}.$ Show that $F$ is not regular. Show that $F$ acts like a regular language in the pumping lemma. ... three conditions of the pumping lemma for this value of $p.$ Explain why parts $(a)$ and $(b)$ do not contradict the pumping lemma.
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finiteautomata
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pumpinglemma
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descriptive
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Michael Sipser Edition 3 Exercise 1 Question 53 (Page No. 91)
Let $\sum=\{0,1,+,=\}$ and $ADD=\{x=y+zx,y,z$ $\text{are binary integers,and}$ $x$ $\text{is the sum of}$ $y$ $\text{and}$ $z\}.$ Show that $\text{ADD}$ is not a regular.
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Apr 30
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finiteautomata
regularlanguages
proof
descriptive
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Michael Sipser Edition 3 Exercise 1 Question 49 (Page No. 90)
Let $B=\{1^{k}yy\in\{0,1\}^{*}$ $\text{ and y contains at least}$ $k$ $1's,$ $\text{for every}$ $k\geq 1\}.$ Show that $B$ is a regular language. Let $C=\{1^{k}yy\in\{0,1\}^{*}$ $\text{ and y contains at most}$ $k$ $1's,$ $\text{for every}$ $k\geq 1\}.$ Show that $C$ isn’t a regular language.
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finiteautomata
regularlanguages
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descriptive
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Michael Sipser Edition 3 Exercise 1 Question 48 (Page No. 90)
Let $\sum = \{0,1\}$ and let $D = \{ww$ $\text{contains an equal number of occurrences of the sub strings 01 and 10}\}.$ Thus $101\in D$ because $101$ contains a single $01$ and a single $10,$ but $1010\notin D$ because $1010$ contains two $10's$ and one $01.$ Show that $D$ is a regular language.
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theoryofcomputation
finiteautomata
regularlanguages
proof
descriptive
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Michael Sipser Edition 3 Exercise 1 Question 47 (Page No. 90)
Let $\sum=\{1,\#\}$ and let $Y=\{ww=x_{1}\#x_{2}\#...\#x_{k}$ $\text{for}$ $k\geq 0,$ $\text{each}$ $ x_{i}\in 1^{*},$ $\text{and}$ $x_{i}\neq x_{j}$ $\text{for}$ $i\neq j\}.$ Prove that $Y$ is not regular.
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theoryofcomputation
finiteautomata
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proof
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25
Michael Sipser Edition 3 Exercise 1 Question 46 (Page No. 90)
Prove that the following languages are not regular. You may use the pumping lemma and the closure of the class of regular languages under union, intersection,and complement. $\{0^{n}1^{m}0^{n}m,n\geq 0\}$ $\{0^{m}1^{n}m\neq n\}$ $\{ww\in\{0,1\}^{*} \text{is not a palindrome}\}$ $\{wtww,t\in\{0,1\}^{+}\}$
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Apr 30
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finiteautomata
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proof
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Michael Sipser Edition 3 Exercise 1 Question 45 (Page No. 90)
Let $\text{A/B = {w wx ∈ A for some x ∈ B}}.$ Show that if $A$ is regular and $B$ is any language, then $\text{A/B}$ is regular.
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finiteautomata
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descriptive
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Michael Sipser Edition 3 Exercise 1 Question 44 (Page No. 90)
Let $B$ and $C$ be languages over $\sum = \{0, 1\}.$ Define $B\overset{1}{\leftarrow} C = \{w\in B$ $\text{for some}$ $y\in C$, $\text{strings}$ $w$ $\text{and}$ $y$ $\text{contain equal numbers of}$ $1’s\}.$ Show that the class of regular languages is closed under the $\overset{1}{\leftarrow}$operation.
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finiteautomata
regularlanguages
descriptive
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Michael Sipser Edition 3 Exercise 1 Question 43 (Page No. 90)
Let $A$ be any language. Define $\text{DROPOUT(A)}$ to be the language containing all strings that can be obtained by removing one symbol from a string in $A.$ Thus, $\text{DROPOUT(A) = $\{xz xyz\in A$ where $x, ... $\text{Theorem 1.47.}$
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theoryofcomputation
finiteautomata
regularlanguages
proof
descriptive
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Michael Sipser Edition 3 Exercise 1 Question 42 (Page No. 89)
For languages $A$ and $B,$ let the $\text{shuffle}$ of $A$ and $B$ be the language $\{w w = a_{1}b_{1} \ldots a_{k}b_{k},$ where $ a_{1} · · · a_{k} ∈ A $ and $b_{1} · · · b_{k} ∈ B,$ each $a_{i}, b_{i} ∈ Σ^{*}\}.$ Show that the class of regular languages is closed under shuffle.
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Apr 29
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finiteautomata
regularlanguages
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Michael Sipser Edition 3 Exercise 1 Question 41 (Page No. 89)
For languages $A$ and $B,$ let the $\text{perfect shuffle}$ of $A$ and $B$ be the language $\text{{$w w = a_{1}b_{1} · · · a_{k}b_{k},$ where $a_{1} · · · a_{k} ∈ A$ and $b_{1} · · · b_{k} ∈ B,$ each $a_{i}, b_{i} ∈ Σ$}}.$ Show that the class of regular languages is closed under perfect shuffle.
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