Recent questions tagged finite-automata / GATE Overflow for GATE CSE

0 votes

0 answers

211

Michael Sipser Edition 3 Exercise 1 Question 65 (Page No. 93)
Prove that for each $n > 0,$ a language $B_{n}$ exists where $B_{n}$ is recognizable by an $\text{NFA}$ that has $n$ states, and if $B_{n} = A_{1}\cup...\cup A_{k},$ for regular languages $A_{i},$ then at least one of the $A_{i}$ requires a $\text{DFA}$ with exponentially many states$.$

Prove that for each $n 0,$ a language $B_{n}$ exists where$B_{n}$ is recognizable by an $\text{NFA}$ that has $n$ states, andif $B_{n} = A_{1}\cup...\cup A_{k},$ for reg...

admin

305 views

admin asked Apr 30, 2019

0 votes

0 answers

212

Michael Sipser Edition 3 Exercise 1 Question 64 (Page No. 92)
Let $N$ be an $\text{NFA}$ with $k$ states that recognizes some language $A.$ Show that if $A$ is nonempty, A contains some string of length at most k. Show, by giving an example, that $\text{part (a)}$ is not necessarily ... ;s shortest member strings are of length exponential in $k.$ Come as close to the bound in $(c)$ as you can$.$

Let $N$ be an $\text{NFA}$ with $k$ states that recognizes some language $A.$Show that if $A$ is nonempty, A contains some string of length at most k.Show, by giving an e...

admin

591 views

admin asked Apr 30, 2019

0 votes

0 answers

213

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 $B\subseteqq C\subseteqq D.$

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...

admin

346 views

admin asked Apr 30, 2019

0 votes

0 answers

214

Michael Sipser Edition 3 Exercise 1 Question 62 (Page No. 92)
Let $\sum =\{a, b\}.$ For each $k\geq 1,$ let $D_{k}$ be the language consisting of all strings that have at least one a among the last $k$ symbols$.$Thus $D_{k}=\sum^{*}a(\sum \cup \epsilon)^{k-1}$.Describe a $\text{DFA}$ with at most $k+ 1$ states that recognizes $D_{k}$ in terms of both a state diagram and a formal description.

Let $\sum =\{a, b\}.$ For each $k\geq 1,$ let $D_{k}$ be the language consisting of all strings that have at least one a among the last $k$ symbols$.$Thus $D_{k}=\sum^{*}...

admin

224 views

admin asked Apr 30, 2019

0 votes

0 answers

215

Michael Sipser Edition 3 Exercise 1 Question 61 (Page No. 92)
Let $Σ = \{a, b\}.$ For each $k\geq 1,$ let $C_{k}$ be the language consisting of all strings that contain an a exactly $k$ places from the right-hand end$.$ Thus $C_{k}=\sum^{*}a\sum^{k-1}.$ Prove that for each $k,$ $\text{no DFA}$ can recognize $C_{k}$ with fewer than $2^{k}$ states.

Let $Σ = \{a, b\}.$ For each $k\geq 1,$ let $C_{k}$ be the language consisting of all strings that contain an a exactly $k$ places from the right-hand end$.$ Thus $C_{k}...

admin

295 views

admin asked Apr 30, 2019

0 votes

1 answer

216

Michael Sipser Edition 3 Exercise 1 Question 60 (Page No. 92)
Let $Σ = \{a, b\}.$ For each $k\geq 1,$ let $C_{k}$ be the language consisting of all strings that contain an a exactly $k$ places from the right-hand end$.$ Thus $C_{k}=\sum^{*}a\sum^{k-1}.$ Describe an $\text{NFA}$ with $k + 1$ states that recognizes $C_{k}$ in terms of both a state diagram and a formal description$.$

Let $Σ = \{a, b\}.$ For each $k\geq 1,$ let $C_{k}$ be the language consisting of all strings that contain an a exactly $k$ places from the right-hand end$.$ Thus $C_{k}...

admin

2.6k views

admin asked Apr 30, 2019

0 votes

0 answers

217

Michael Sipser Edition 3 Exercise 1 Question 59 (Page No. 92)
Let $M = (Q, Σ, δ, q_{0}, F)$ be a $\text{DFA}$ and let $h$ be a state of $M$ called its $\text{ home }.$ A $\text{synchronizing sequence}$ for $M$ and $h$ is a string $s\in\sum^{*}$ ... $\text{DFA,}$ then it has a synchronizing sequence of length at most $k^{3}.$Can you improve upon this bound$?$

Let $M = (Q, Σ, δ, q_{0}, F)$ be a $\text{DFA}$ and let $h$ be a state of $M$ called its $\text{“home”}.$ A $\text{synchronizing sequence}$ for $M$ and $h$ is a str...

admin

620 views

admin asked Apr 30, 2019

0 votes

0 answers

218

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 ... a set $A$ for which $B_{2}(A)$ is regular but $B_{3}(A)$ is not regular$.$ Prove that your example works.

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,...

admin

333 views

admin asked Apr 30, 2019

0 votes

1 answer

219

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.

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 la...

admin

945 views

admin asked Apr 30, 2019

0 votes

0 answers

220

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.

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...

admin

288 views

admin asked Apr 30, 2019

1 votes

0 answers

221

Michael Sipser Edition 3 Exercise 1 Question 52 (Page No. 91)
$\text{Myhill-Nerode theorem.}$ Refer to $\text{Question 51}.$Let $L$ be a language and let $X$ be a set of strings. Say that $X$ is $\text{pairwise distinguishable}$ by $L$ if every two distinct strings in $X$ are ... $\text{DFA}$ recognizing it$.$

$\text{Myhill–Nerode theorem.}$ Refer to $\text{Question 51}.$Let $L$ be a language and let $X$ be a set of strings. Say that $X$ is $\text{pairwise distinguishable}$ b...

admin

392 views

admin asked Apr 30, 2019

0 votes

0 answers

222

Michael Sipser Edition 3 Exercise 1 Question 51 (Page No. 90)
Let $x$ and $y$ be strings and let $L$ be any language. We say that $x$ and $y$ are $\text{distinguishable}$ by $L$ if some string $z$ exists whereby exactly one of the strings $xz$ and $yz$ ... $≡L$ is an equivalence relation. A $\text{palindrome}$ is a string that reads the same forward and backward.

Let $x$ and $y$ be strings and let $L$ be any language. We say that $x$ and $y$ are $\text{distinguishable}$ by $L$ if some string $z$ exists whereby exactly one of the s...

admin

308 views

admin asked Apr 30, 2019

1 votes

0 answers

223

Michael Sipser Edition 3 Exercise 1 Question 50 (Page No. 90)
Read the informal definition of the finite state transducer given in Question $24.$ Prove that $\text{no FST}$ can output $w^{R}$ for every input $w$ if the input and output alphabets are $\{0,1\}.$

Read the informal definition of the finite state transducer given in Question $24.$ Prove that $\text{no FST}$ can output $w^{R}$ for every input $w$ if the input and out...

admin

195 views

admin asked Apr 30, 2019

0 votes

1 answer

224

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.

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...

admin

451 views

admin asked Apr 30, 2019

1 votes

0 answers

225

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.

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 ...

admin

255 views

admin asked Apr 30, 2019

0 votes

0 answers

226

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.

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\...

admin

322 views

admin asked Apr 30, 2019

0 votes

0 answers

227

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

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 compleme...

admin

499 views

admin asked Apr 30, 2019

0 votes

0 answers

228

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.

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.

admin

205 views

admin asked Apr 30, 2019

0 votes

0 answers

229

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.

Let $B$ and $C$ be languages over $\sum = \{0, 1\}.$ Define $B\overset{1}{\leftarrow} C = \{w\in B|$ $\text{for some}$ $y\in C$, $\text{strings}$ $w$ $\text{and}$ $y$ $\...

admin

209 views

admin asked Apr 30, 2019

0 votes

0 answers

230

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 $A.$ ... $\text{Theorem 1.47.}$

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, $\t...

admin

676 views

admin asked Apr 30, 2019

0 votes

0 answers

231

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.

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_...

admin

339 views

admin asked Apr 28, 2019

0 votes

0 answers

232

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.

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} ∈...

admin

473 views

admin asked Apr 28, 2019

0 votes

0 answers

233

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 ... $\text{NOEXTEND(A) = {w ∈ A| w is not the proper prefix of any string in A}.}$

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 $x\neq y.$...

admin

365 views

admin asked Apr 28, 2019

0 votes

0 answers

234

Michael Sipser Edition 3 Exercise 1 Question 39 (Page No. 89)
The construction in $\text{Theorem 1.54}$ shows that every $\text{GNFA}$ is equivalent to a $\text{GNFA}$ with only two states$.$ We can show that an opposite phenomenon occurs for $\text{DFAs.}$ Prove that for every $k > 1,$ a ... exists that is recognized by a $\text{DFA}$ with $k$ states but not by one with only $k − 1$ states$.$

The construction in $\text{Theorem 1.54}$ shows that every $\text{GNFA}$ is equivalent to a $\text{GNFA}$ with only two states$.$ We can show that an opposite phenomenon ...

admin

380 views

admin asked Apr 28, 2019

0 votes

1 answer

235

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$.$

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...

admin

2.0k views

admin asked Apr 28, 2019

0 votes

1 answer

236

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$.$

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$.$

admin

976 views

admin asked Apr 28, 2019

0 votes

2 answers

237

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$.$

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$.$

admin

406 views

admin asked Apr 28, 2019

0 votes

0 answers

238

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$.$

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}...

admin

178 views

admin asked Apr 28, 2019

0 votes

0 answers

239

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$.$

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}...

admin

240 views

admin asked Apr 28, 2019

0 votes

0 answers

240

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 $31$})$

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}...

admin

260 views

admin asked Apr 28, 2019

tags	tag:apple
author	user:martin
title	title:apple
content	content:apple
exclude	-tag:apple
force match	+apple
views	views:100
score	score:10
answers	answers:2
is accepted	isaccepted:true
is closed	isclosed:true