Recent questions tagged michael-sipser / GATE Overflow for GATE CSE

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91

Michael Sipser Edition 3 Exercise 3 Question 3 (Page No. 187)
Modify the proof of Theorem $3.16$ to obtain Corollary $3.19$, showing that a language is decidable iff some nondeterministic Turing machine decides it. (You may assume the following theorem about trees. If every node in a ... many children and every branch of the tree has finitely many nodes, the tree itself has finitely many nodes.)

Modify the proof of Theorem $3.16$ to obtain Corollary $3.19$, showing that a language is decidable iff some nondeterministic Turing machine decides it. (You may assume t...

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admin asked Oct 12, 2019

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92

Michael Sipser Edition 3 Exercise 3 Question 2 (Page No. 187)
This exercise concerns $TM\: M_{1}$, whose description and state diagram appear in Example $3.9$. In each of the parts, give the sequence of configurations that $M_{1}$ enters when started on the indicated input string. $11$ $1\#1$ $1\#\#1$ $10\#11$ $10\#10$

This exercise concerns $TM\: M_{1}$, whose description and state diagram appear in Example $3.9$. In each of the parts, give the sequence of configurations that $M_{1}$ ...

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admin asked Oct 12, 2019

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93

Michael Sipser Edition 3 Exercise 3 Question 1 (Page No. 187)
This exercise concerns $TM\: M_{2}$, whose description and state diagram appear in Example $3.7$. In each of the parts, give the sequence of configurations that $M_{2}$ enters when started on the indicated input string. $0$ $00$ $000$ $000000$

This exercise concerns $TM\: M_{2}$, whose description and state diagram appear in Example $3.7$. In each of the parts, give the sequence of configurations that $M_{2}$ e...

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admin asked Oct 12, 2019

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94

Michael Sipser Edition 3 Exercise 2 Question 59 (Page No. 160)
If we disallow $\epsilon$-rules in CFGs, we can simplify the DK-test. In the simplified test,we only need to check that each of DK’s accept states has a single rule. Prove that a CFG without $\epsilon$-rules passes the simplified DK-test iff it is a DCFG.

If we disallow $\epsilon$-rules in CFGs, we can simplify the DK-test. In the simplified test,we only need to check that each of DK’s accept states has a single rule. Pr...

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admin asked Oct 12, 2019

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95

Michael Sipser Edition 3 Exercise 2 Question 58 (Page No. 160)
Let $C = \{ww^{R} \mid w in \{0,1\}^{\ast}\}$. Prove that $C$ is not a DCFL. (Hint: Suppose that when some DPDA $P$ is started in state $q$ with symbol $x$ on the top of its stack, $P$ never pops its ... of $P's$ stack at that point cannot affect its subsequent behavior, so $P's$ subsequent behavior can depend only on $q$ and $x$.)

Let $C = \{ww^{R} \mid w in \{0,1\}^{\ast}\}$. Prove that $C$ is not a DCFL. (Hint: Suppose that when some DPDA $P$ is started in state $q$ with symbol $x$ on the top of ...

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admin asked Oct 12, 2019

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96

Michael Sipser Edition 3 Exercise 2 Question 57 (Page No. 160)
Let $B = \{a^{i}b^{j}c^{k}\mid i,j,k \geq 0\: \text{and}\: i = j\: \text{or}\: i = k \}$. Prove that $B$ is not a DCFL.

Let $B = \{a^{i}b^{j}c^{k}\mid i,j,k \geq 0\: \text{and}\: i = j\: \text{or}\: i = k \}$. Prove that $B$ is not a DCFL.

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admin asked Oct 12, 2019

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97

Michael Sipser Edition 3 Exercise 2 Question 56 (Page No. 160)
Let $A = L(G_{1})$ where $G_{1}$ is defined in Question $2.55$. Show that $A$ is not a DCFL.(Hint: Assume that $A$ is a DCFL and consider its DPDA $P$. Modify $P$ ... an accept state, it pretends that $c's$ are $b's$ in the input from that point on. What language would the modified $P$ accept?)

Let $A = L(G_{1})$ where $G_{1}$ is defined in Question $2.55$. Show that $A$ is not a DCFL.(Hint: Assume that $A$ is a DCFL and consider its DPDA $P$. Modify $P$ so that...

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admin asked Oct 12, 2019

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98

Michael Sipser Edition 3 Exercise 2 Question 55 (Page No. 159)
Let $G_{1}$ be the following grammar that we introduced in Example $2.45$. Use the DK-test to show that $G_{1}$ is not a DCFG. $R \rightarrow S \mid T$ $S \rightarrow aSb \mid ab$ $T \rightarrow aTbb \mid abb$

Let $G_{1}$ be the following grammar that we introduced in Example $2.45$. Use the DK-test to show that $G_{1}$ is not a DCFG.$R \rightarrow S \mid T$$S \rightarrow aSb \...

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admin asked Oct 12, 2019

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99

Michael Sipser Edition 3 Exercise 2 Question 54 (Page No. 159)
Let G be the following grammar: $S \rightarrow T\dashv $ $T \rightarrow T aT b \mid T bT a | \epsilon$ Show that $L(G) = \{w\dashv \: \mid w\: \text{contains equal numbers of a’s and b’s} \}$. Use a proof by induction on the length of $w$. Use the DK-test to show that G is a DCFG. Describe a DPDA that recognizes L(G).

Let G be the following grammar:$S \rightarrow T\dashv $$T \rightarrow T aT b \mid T bT a | \epsilon$Show that $L(G) = \{w\dashv \: \mid w\: \text{contains equal numbers o...

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admin asked Oct 12, 2019

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100

Michael Sipser Edition 3 Exercise 2 Question 53 (Page No. 159)
Show that the class of DCFLs is not closed under the following operations: Union Intersection Concatenation Star Reversal

Show that the class of DCFLs is not closed under the following operations:UnionIntersectionConcatenationStarReversal

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admin asked Oct 12, 2019

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101

Michael Sipser Edition 3 Exercise 2 Question 52 (Page No. 159)
Show that every DCFG generates a prefix-free language.

Show that every DCFG generates a prefix-free language.

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admin asked Oct 12, 2019

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102

Michael Sipser Edition 3 Exercise 2 Question 51 (Page No. 159)
Show that every DCFG is an unambiguous CFG.

Show that every DCFG is an unambiguous CFG.

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admin asked Oct 12, 2019

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103

Michael Sipser Edition 3 Exercise 2 Question 50 (Page No. 159)
We defined the $CUT$ of language $A$ to be $CUT(A) = \{yxz| xyz \in A\}$. Show that the class of $CFLs$ is not closed under $CUT$.

We defined the $CUT$ of language $A$ to be $CUT(A) = \{yxz| xyz \in A\}$. Show that the class of $CFLs$ is not closed under $CUT$.

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admin asked Oct 12, 2019

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104

Michael Sipser Edition 3 Exercise 2 Question 49 (Page No. 159)
We defined the rotational closure of language $A$ to be $RC(A) = \{yx \mid xy \in A\}$.Show that the class of CFLs is closed under rotational closure.

We defined the rotational closure of language $A$ to be $RC(A) = \{yx \mid xy \in A\}$.Show that the class of CFLs is closed under rotational closure.

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admin asked Oct 12, 2019

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105

Michael Sipser Edition 3 Exercise 2 Question 48 (Page No. 159)
Let $\Sigma = \{0,1\}$. Let $C_{1}$ be the language of all strings that contain a $1$ in their middle third. Let $C_{2}$ be the language of all strings that contain two $1s$ ... $C_{1}$ is a CFL. Show that $C_{2}$ is not a CFL.

Let $\Sigma = \{0,1\}$. Let $C_{1}$ be the language of all strings that contain a $1$ in their middle third. Let $C_{2}$ be the language of all strings that contain two $...

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admin asked Oct 12, 2019

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106

Michael Sipser Edition 3 Exercise 2 Question 47 (Page No. 159)
Let $\Sigma = \{0,1\}$ and let $B$ be the collection of strings that contain at least one $1$ in their second half. In other words, $B = \{uv \mid u \in \Sigma^{\ast}, v \in \Sigma^{\ast}1\Sigma^{\ast}\: \text{and} \mid u \mid \geq \mid v \mid \}$. Give a PDA that recognizes $B$. Give a CFG that generates $B$.

Let $\Sigma = \{0,1\}$ and let $B$ be the collection of strings that contain at least one $1$ in their second half. In other words, $B = \{uv \mid u \in \Sigma^{\ast}, v...

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admin asked Oct 12, 2019

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107

Michael Sipser Edition 3 Exercise 2 Question 46 (Page No. 158)
Consider the following CFG $G:$ $S \rightarrow SS \mid T$ $T \rightarrow aT b \mid ab$ Describe $L(G)$ and show that $G$ is ambiguous. Give an unambiguous grammar $H$ where $L(H) = L(G)$ and sketch a proof that $H$ is unambiguous.

Consider the following CFG $G:$$S \rightarrow SS \mid T$$T \rightarrow aT b \mid ab$Describe $L(G)$ and show that $G$ is ambiguous. Give an unambiguous grammar $H$ where ...

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admin asked Oct 12, 2019

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108

Michael Sipser Edition 3 Exercise 2 Question 45 (Page No. 158)
Let $A = \{wtw^{R}\mid w,t\in\{0,1\}^{\ast}\:\text{and} \mid w \mid = \mid t \mid \}$. Prove that $A$ is not a CFL.

Let $A = \{wtw^{R}\mid w,t\in\{0,1\}^{\ast}\:\text{and} \mid w \mid = \mid t \mid \}$. Prove that $A$ is not a CFL.

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admin asked Oct 12, 2019

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109

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.

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 re...

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admin asked Oct 12, 2019

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110

Michael Sipser Edition 3 Exercise 2 Question 43 (Page No. 158)
For strings $w4 and $t,4 write $w \circeq t$ if the symbols of $w$ are a permutation of the symbols of $t.$ In other words, $w \circeq t$ if $t$ ... a regular language is context free. What happens in part $(a)$ if $\Sigma$ contains three or more symbols? Prove your answer.

For strings $w4 and $t,4 write $w \circeq t$ if the symbols of $w$ are a permutation of the symbols of $t.$ In other words, $w \circeq t$ if $t$ and $w4 have the same sy...

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111

Michael Sipser Edition 3 Exercise 2 Question 42 (Page No. 158)
Let $Y = \{w\mid w = t_{1}\#t_{2}\#\dots t_{k} \:\text{for}\: k\geq 0,\text{each}\: t_{i}\in 1^{\ast}, \text{and}\: t_{i}\neq t_{j} \text{whenever}\: i\neq j\}$.Here $\Sigma = \{1,\#\}$. Prove that $Y$ is not context free.

Let $Y = \{w\mid w = t_{1}\#t_{2}\#\dots t_{k} \:\text{for}\: k\geq 0,\text{each}\: t_{i}\in 1^{\ast}, \text{and}\: t_{i}\neq t_{j} \text{whenever}\: i\neq j\}$.Here $\Si...

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112

Michael Sipser Edition 3 Exercise 2 Question 41 (Page No. 158)
Read the definitions of NOPREFIX(A) and NOEXTEND(A) in Question $1.40$. Show that the class of CFLs is not closed under NOPREFIX. Show that the class of CFLs is not closed under NOEXTEND.

Read the definitions of NOPREFIX(A) and NOEXTEND(A) in Question $1.40$.Show that the class of CFLs is not closed under NOPREFIX.Show that the class of CFLs is not closed ...

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admin asked Oct 12, 2019

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113

Michael Sipser Edition 3 Exercise 2 Question 40 (Page No. 158)
Say that a language is prefix-closed if all prefixes of every string in the language are also in the language. Let $C$ be an infinite, prefix-closed, context-free language. Show that $C$ contains an infinite regular subset.

Say that a language is prefix-closed if all prefixes of every string in the language are also in the language. Let $C$ be an infinite, prefix-closed, context-free languag...

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admin asked Oct 10, 2019

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114

Michael Sipser Edition 3 Exercise 2 Question 39 (Page No. 158)
For the definition of the shuffle operation. 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 context-free languages is not closed under shuffle.

For the definition of the shuffle operation. For languages $A$ and $B,$ let the $\text{shuffle}$ of $A$ and $B$ be the language $\{w| w = a_{1}b_{1} \ldots a_{k}b_{k},$ ...

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admin asked Oct 10, 2019

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115

Michael Sipser Edition 3 Exercise 2 Question 38 (Page No. 158)
For the definition of the perfect shuffle operation. 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},$ ... k} ∈ B,$ each $a_{i}, b_{i} ∈ Σ$}}.$ Show that the class of context-free languages is not closed under perfect shuffle.

For the definition of the perfect shuffle operation. For languages $A$ and $B,$ let the $\text{perfect shuffle}$ of $A$ and $B$ be the language$\text{{$w| w = a_{1}b_{1} ...

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admin asked Oct 10, 2019

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116

Michael Sipser Edition 3 Exercise 2 Question 37 (Page No. 158)
Prove the following stronger form of the pumping lemma, where in both pieces $v$ and $y$ must be nonempty when the string $s$ is broken up$.$If $A$ is a context-free language, then there is a number $k$ where, if $s$ is any string in $A$ of ... $i\geq 0,uv^{i}xy^{i}z\in A,$ $v\neq\epsilon$ and $y\neq\epsilon,$and $\mid vxy\mid\leq k.$

Prove the following stronger form of the pumping lemma, where in both pieces $v$ and $y$ must be nonempty when the string $s$ is broken up$.$If $A$ is a context-free lang...

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admin asked May 4, 2019

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117

Michael Sipser Edition 3 Exercise 2 Question 36 (Page No. 158)
Give an example of a language that is not context free but that acts like a $CFL$ in the pumping lemma$.$ Prove that your example works$.$ $\text{(See the analogous example for regular languages in Question 54.)}$

Give an example of a language that is not context free but that acts like a $CFL$ in the pumping lemma$.$ Prove that your example works$.$ $\text{(See the analogous examp...

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admin asked May 4, 2019

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118

Michael Sipser Edition 3 Exercise 2 Question 35 (Page No. 157)
Let $G$ be a $CFG$ in Chomsky normal form that contains $b$ variables$.$ Show that if $G$ generates some string with a derivation having at least $2^{b}$ steps$, L(G)$ is infinite$.$

Let $G$ be a $CFG$ in Chomsky normal form that contains $b$ variables$.$ Show that if $G$ generates some string with a derivation having at least $2^{b}$ steps$, L(G)$ is...

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admin asked May 4, 2019

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119

Michael Sipser Edition 3 Exercise 2 Question 34 (Page No. 157)
Let $G = (V, \Sigma, R, S)$ be the following grammar. $V = \{S, T, U\}; \Sigma = \{0, \#\};$ and $R$ is the set of rules$:$ $S\rightarrow TT\mid U$ $T\rightarrow 0T\mid T0\mid \#$ ... existence of a pumping length $p$ for $B.$ What is the minimum value of $p$ that works in the pumping lemma$?$ Justify your answer$.$

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 T...

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admin asked May 4, 2019

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120

Michael Sipser Edition 3 Exercise 2 Question 33 (Page No. 157)
Show that $F = \{a^{i}b^{j}\mid i = kj$ $\text{for some positive integer $k$\}}$ is not context free$.$

Show that $F = \{a^{i}b^{j}\mid i = kj$ $\text{for some positive integer $k$\}}$ is not context free$.$

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admin asked May 4, 2019

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