Recent questions tagged regular-languages

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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.

asked 1 day ago in Theory of Computation by Lakshman Patel RJIT Veteran (50.8k points) | 3 views

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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.

asked Jul 26 in Theory of Computation by Lakshman Patel RJIT Veteran (50.8k points) | 32 views

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MadeEasy Test Series: Theory Of Computation - Regular Languages
Consider the following statements: $S_1:\{(a^n)^m|n\leq m\geq0\}$ $S_2:\{a^nb^n|n\geq 1\} \cup \{a^nb^m|n \geq1,m \geq 1\} $ Which of the following is regular? $S_1$ only $S_2$ only Both Neither of the above

asked May 26 in Theory of Computation by Hirak Active (3.4k points) | 256 views

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self doubt: TOC
is union of regular language and context free language always regular?

asked May 22 in Theory of Computation by Hirak Active (3.4k points) | 50 views

+1 vote

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Self Doubt : Ambiguity
Why is ambiguity in regular language is decidable and not decidable in CFL ? Can you give Example?

asked May 10 in Theory of Computation by logan1x (407 points) | 117 views

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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$.$

asked May 4 in Theory of Computation by Lakshman Patel RJIT Veteran (50.8k points) | 13 views

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Michael Sipser Edition 3 Exercise 2 Question 18 (Page No. 156)
Let $C$ be a context-free language and $R$ be a regular language$.$ Prove that the language $C\cap R$ is context-free. 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$.$

asked May 4 in Theory of Computation by Lakshman Patel RJIT Veteran (50.8k points) | 12 views

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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 context-free grammar$.$

asked May 4 in Theory of Computation by Lakshman Patel RJIT Veteran (50.8k points) | 11 views

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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$.$

asked May 4 in Theory of Computation by Lakshman Patel RJIT Veteran (50.8k points) | 27 views

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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 $|s| < k1k2.$

asked Apr 30 in Theory of Computation by Lakshman Patel RJIT Veteran (50.8k points) | 48 views

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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.

asked Apr 30 in Theory of Computation by Lakshman Patel RJIT Veteran (50.8k points) | 29 views

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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.

asked Apr 30 in Theory of Computation by Lakshman Patel RJIT Veteran (50.8k points) | 21 views

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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 $\text{CUT}.$

asked Apr 30 in Theory of Computation by Lakshman Patel RJIT Veteran (50.8k points) | 35 views

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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.

asked Apr 30 in Theory of Computation by Lakshman Patel RJIT Veteran (50.8k points) | 11 views

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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 $B\subseteqq C\subseteqq D.$

asked Apr 30 in Theory of Computation by Lakshman Patel RJIT Veteran (50.8k points) | 10 views

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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{xz|for 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.

asked Apr 30 in Theory of Computation by Lakshman Patel RJIT Veteran (50.8k points) | 7 views

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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{x|for some y,|x|=|y| and xy $\in$ A\}}.$ Show that if $A$ is regular,then so is $A_{\frac{1}{2}-}.$

asked Apr 30 in Theory of Computation by Lakshman Patel RJIT Veteran (50.8k points) | 19 views

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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.

asked Apr 30 in Theory of Computation by Lakshman Patel RJIT Veteran (50.8k points) | 27 views

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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 ... $\epsilon$ $1^{*}01^{*}01^{*}$ $10(11^{*}0)^{*}0$ $1011$ $\sum^{*}$

asked Apr 30 in Theory of Computation by Lakshman Patel RJIT Veteran (50.8k points) | 33 views

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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.

asked Apr 30 in Theory of Computation by Lakshman Patel RJIT Veteran (50.8k points) | 45 views

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Michael Sipser Edition 3 Exercise 1 Question 53 (Page No. 91)
Let $\sum=\{0,1,+,=\}$ and $ADD=\{x=y+z|x,y,z$ $\text{are binary integers,and}$ $x$ $\text{is the sum of}$ $y$ $\text{and}$ $z\}.$ Show that $\text{ADD}$ is not a regular.

asked Apr 30 in Theory of Computation by Lakshman Patel RJIT Veteran (50.8k points) | 10 views

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Michael Sipser Edition 3 Exercise 1 Question 49 (Page No. 90)
Let $B=\{1^{k}y|y\in\{0,1\}^{*}$ $\text{ and y contains at least}$ $k$ $1's,$ $\text{for every}$ $k\geq 1\}.$ Show that $B$ is a regular language. Let $C=\{1^{k}y|y\in\{0,1\}^{*}$ $\text{ and y contains at most}$ $k$ $1's,$ $\text{for every}$ $k\geq 1\}.$ Show that $C$ isn’t a regular language.

asked Apr 30 in Theory of Computation by Lakshman Patel RJIT Veteran (50.8k points) | 23 views

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Michael Sipser Edition 3 Exercise 1 Question 48 (Page No. 90)
Let $\sum = \{0,1\}$ and let $D = \{w|w$ $\text{contains an equal number of occurrences of the sub strings 01 and 10}\}.$ 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.

asked Apr 30 in Theory of Computation by Lakshman Patel RJIT Veteran (50.8k points) | 8 views

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Michael Sipser Edition 3 Exercise 1 Question 47 (Page No. 90)
Let $\sum=\{1,\#\}$ and let $Y=\{w|w=x_{1}\#x_{2}\#...\#x_{k}$ $\text{for}$ $k\geq 0,$ $\text{each}$ $ x_{i}\in 1^{*},$ $\text{and}$ $x_{i}\neq x_{j}$ $\text{for}$ $i\neq j\}.$ Prove that $Y$ is not regular.

asked Apr 30 in Theory of Computation by Lakshman Patel RJIT Veteran (50.8k points) | 8 views

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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. $\{0^{n}1^{m}0^{n}|m,n\geq 0\}$ $\{0^{m}1^{n}|m\neq n\}$ $\{w|w\in\{0,1\}^{*} \text{is not a palindrome}\}$ $\{wtw|w,t\in\{0,1\}^{+}\}$

asked Apr 30 in Theory of Computation by Lakshman Patel RJIT Veteran (50.8k points) | 14 views

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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.

asked Apr 30 in Theory of Computation by Lakshman Patel RJIT Veteran (50.8k points) | 9 views

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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.

asked Apr 30 in Theory of Computation by Lakshman Patel RJIT Veteran (50.8k points) | 6 views

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Michael Sipser Edition 3 Exercise 1 Question 43 (Page No. 90)
Let $A$ be any language. Define $\text{DROP-OUT(A)}$ to be the language containing all strings that can be obtained by removing one symbol from a string in $A.$ Thus, $\text{DROP-OUT(A) = $\{xz| xyz\in A$ where $x, ... $\text{Theorem 1.47.}$

asked Apr 30 in Theory of Computation by Lakshman Patel RJIT Veteran (50.8k points) | 12 views

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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.

asked Apr 29 in Theory of Computation by Lakshman Patel RJIT Veteran (50.8k points) | 15 views

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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.

asked Apr 29 in Theory of Computation by Lakshman Patel RJIT Veteran (50.8k points) | 11 views

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