Recent questions tagged context-free-grammar / GATE Overflow for GATE CSE

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61

TIFR CSE 2020 | Part B | Question: 6
Consider the context-free grammar below ($\epsilon$ denotes the empty string, alphabet is $\{a,b\}$): $S\rightarrow \epsilon \mid aSb \mid bSa \mid SS.$ What language does it generate? $(ab)^{\ast} + (ba)^{\ast}$ $(abba) {\ast} + (baab)^{\ast}$ ... of the form $a^{n}b^{n}$ or $b^{n}a^{n},n$ any positive integer Strings with equal numbers of $a$ and $b$

Consider the context-free grammar below ($\epsilon$ denotes the empty string, alphabet is $\{a,b\}$):$$S\rightarrow \epsilon \mid aSb \mid bSa \mid SS.$$What language doe...

admin

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admin asked Feb 10, 2020

2 votes

4 answers

62

ISRO2020-39
The language which is generated by the grammar $S \rightarrow aSa \mid bSb \mid a \mid b$ over the alphabet of $\{a,b\}$ is the set of Strings that begin and end with the same symbol All odd and even length palindromes All odd length palindromes All even length palindromes

The language which is generated by the grammar $S \rightarrow aSa \mid bSb \mid a \mid b$ over the alphabet of $\{a,b\}$ is the set ofStrings that begin and end with the ...

Satbir

2.1k views

Satbir asked Jan 13, 2020

3 votes

0 answers

63

Michael Sipser Edition 3 Exercise 5 Question 36 (Page No. 242)
Say that a $CFG$ is minimal if none of its rules can be removed without changing the language generated. Let $MIN_{CFG} = \{\langle G \rangle \mid \text{G is a minimal CFG}\}$. Show that $MIN_{CFG}$ is $T-$recognizable. Show that $MIN_{CFG}$ is undecidable.

Say that a $CFG$ is minimal if none of its rules can be removed without changing the language generated. Let $MIN_{CFG} = \{\langle G \rangle \mid \text{G is a minimal CF...

admin

633 views

admin asked Oct 20, 2019

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0 answers

64

Michael Sipser Edition 3 Exercise 5 Question 32 (Page No. 241)
Prove that the following two languages are undecidable. $OVERLAP_{CFG} = \{\langle G, H\rangle \mid \text{G and H are CFGs where}\: L(G) \cap L(H) \neq \emptyset\}$. $PREFIX-FREE_{CFG} = \{\langle G \rangle \mid \text{G is a CFG where L(G) is prefix-free}\}$.

Prove that the following two languages are undecidable.$OVERLAP_{CFG} = \{\langle G, H\rangle \mid \text{G and H are CFGs where}\: L(G) \cap L(H) \neq \emptyset\}$.$PREF...

admin

455 views

admin asked Oct 20, 2019

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0 answers

65

Michael Sipser Edition 3 Exercise 5 Question 21 (Page No. 240)
Let $AMBIG_{CFG} = \{\langle G \rangle \mid \text{G is an ambiguous CFG}\}$. Show that $AMBIG_{CFG}$ is undecidable. (Hint: Use a reduction from $PCP$ ... $a_{1},\dots,a_{k}$ are new terminal symbols. Prove that this reduction works.)

Let $AMBIG_{CFG} = \{\langle G \rangle \mid \text{G is an ambiguous CFG}\}$. Show that $AMBIG_{CFG}$ is undecidable. (Hint: Use a reduction from $PCP$. Given an instance...

admin

395 views

admin asked Oct 19, 2019

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66

Michael Sipser Edition 3 Exercise 5 Question 2 (Page No. 239)
Show that $EQ_{CFG}$ is co-Turing-recognizable.

Show that $EQ_{CFG}$ is co-Turing-recognizable.

admin

175 views

admin asked Oct 17, 2019

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0 answers

67

Michael Sipser Edition 3 Exercise 5 Question 1 (Page No. 239)
Show that $EQ_{CFG}$ is undecidable.

Show that $EQ_{CFG}$ is undecidable.

admin

164 views

admin asked Oct 17, 2019

1 votes

0 answers

68

Michael Sipser Edition 3 Exercise 4 Question 31 (Page No. 212)
Say that a variable $A$ in $CFL\: G$ is usable if it appears in some derivation of some string $w \in G$. Given a $CFG\: G$ and a variable $A$, consider the problem of testing whether $A$ is usable. Formulate this problem as a language and show that it is decidable.

Say that a variable $A$ in $CFL\: G$ is usable if it appears in some derivation of some string $w \in G$. Given a $CFG\: G$ and a variable $A$, consider the problem of te...

admin

457 views

admin asked Oct 17, 2019

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0 answers

69

Michael Sipser Edition 3 Exercise 4 Question 29 (Page No. 212)
Let $C_{CFG} = \{\langle G, k \rangle \mid \text{ G is a CFG and L(G) contains exactly $k$ strings where $k \geq 0$ or $k = \infty$}\}$. Show that $C_{CFG}$ is decidable.

Let $C_{CFG} = \{\langle G, k \rangle \mid \text{ G is a CFG and L(G) contains exactly $k$ strings where $k \geq 0$ or $k = \infty$}\}$. Show that $C_{CFG}$ is decidable...

admin

262 views

admin asked Oct 17, 2019

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0 answers

70

Michael Sipser Edition 3 Exercise 4 Question 28 (Page No. 212)
Let $C = \{ \langle G, x \rangle \mid \text{G is a CFG $x$ is a substring of some $y \in L(G)$}\}$. Show that $C$ is decidable. (Hint: An elegant solution to this problem uses the decider for $E_{CFG}$.)

Let $C = \{ \langle G, x \rangle \mid \text{G is a CFG $x$ is a substring of some $y \in L(G)$}\}$. Show that $C$ is decidable. (Hint: An elegant solution to this problem...

admin

182 views

admin asked Oct 17, 2019

0 votes

0 answers

71

Michael Sipser Edition 3 Exercise 4 Question 15 (Page No. 212)
Show that the problem of determining whether a CFG generates all strings in $1^{\ast}$ is decidable. In other words, show that $\{\langle { G \rangle} \mid \text{G is a CFG over {0,1} and } 1^{\ast} \subseteq L(G) \}$ is a decidable language.

Show that the problem of determining whether a CFG generates all strings in $1^{\ast}$ is decidable. In other words, show that $\{\langle { G \rangle} \mid \text{G is a C...

admin

561 views

admin asked Oct 17, 2019

0 votes

0 answers

72

Michael Sipser Edition 3 Exercise 4 Question 14 (Page No. 211)
Let $\Sigma = \{0,1\}$. Show that the problem of determining whether a $CFG$ generates some string in $1^{\ast}$ is decidable. In other words, show that $\{\langle {G \rangle}\mid \text{G is a CFG over {0,1} and } 1^{\ast} \cap L(G) \neq \phi \}$ is a decidable language.

Let $\Sigma = \{0,1\}$. Show that the problem of determining whether a $CFG$ generates some string in $1^{\ast}$ is decidable. In other words, show that $\{\langle {G \ra...

admin

192 views

admin asked Oct 17, 2019

0 votes

0 answers

73

Michael Sipser Edition 3 Exercise 4 Question 4 (Page No. 211)
Let $A\varepsilon_{CFG} = \{ \langle{ G }\rangle \mid G\: \text{is a CFG that generates}\: \epsilon \}.$Show that $A\varepsilon_{CFG}$ is decidable.

Let $A\varepsilon_{CFG} = \{ \langle{ G }\rangle \mid G\: \text{is a CFG that generates}\: \epsilon \}.$Show that $A\varepsilon_{CFG}$ is decidable.

admin

193 views

admin asked Oct 15, 2019

0 votes

0 answers

74

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...

admin

298 views

admin asked Oct 12, 2019

1 votes

0 answers

75

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 \...

admin

242 views

admin asked Oct 12, 2019

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0 answers

76

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...

admin

247 views

admin asked Oct 12, 2019

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0 answers

77

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.

admin

243 views

admin asked Oct 12, 2019

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0 answers

78

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.

admin

227 views

admin asked Oct 12, 2019

1 votes

0 answers

79

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...

admin

780 views

admin asked Oct 12, 2019

0 votes

0 answers

80

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 ...

admin

422 views

admin asked Oct 12, 2019

0 votes

0 answers

81

Ullman (Compiler Design) Edition 2 Exercise 4.4 Question 10 (Page No. 233)
Show how, having filled in the table as in Question $4.4.9$, we can in $O(n)$ time recover a parse tree for $a_{1}a_{2}\cdot\cdot\cdot a_{n}$. Hint: modify the table so it records, for each nonterminal $A$ in each table entry $T_{ij}$, some pair of nonterminals in other table entries that justified putting $A$ in $T_{ij}$.

Show how, having filled in the table as in Question $4.4.9$, we can in $O(n)$ time recover a parse tree for $a_{1}a_{2}\cdot\cdot\cdot a_{n}$. Hint: modify the table so i...

admin

203 views

admin asked Aug 20, 2019

0 votes

0 answers

82

Ullman (Compiler Design) Edition 2 Exercise 4.4 Question 9 (Page No. 232)
Every language that has a context-free grammar can be recognized in at most $O(n^{3})$ time for strings of length $n$. A simple way to do so,called the Cocke- Younger-Kasami (or CYK) algorithm is based on dynamic programming. ... in the table, how do you determine whether $a_{l}a_{2}\cdot\cdot\cdot a_{n}$ is in the language?

Every language that has a context-free grammar can be recognized in at most $O(n^{3})$ time for strings of length $n$. A simple way to do so,called the Cocke- Younger-Ka...

admin

282 views

admin asked Aug 20, 2019

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0 answers

83

Ullman (Compiler Design) Edition 2 Exercise 4.2 Question 3 (Page No. 207)
Design grammars for the following languages: The set of all strings of $0's$ and $1's$ such that every $0$ is immediately followed by at least one $1$. The set of all strings of $0's$ and $1's$ that are palindromes; ... $1's$ of the form $xy$, where $x\neq y$ and $x$ and $y$ are of the same length.

Design grammars for the following languages:The set of all strings of $0's$ and $1's$ such that every $0$ is immediately followed by at least one $1$.The set of all strin...

admin

438 views

admin asked Aug 17, 2019

1 votes

0 answers

84

Ullman (Compiler Design) Edition 2 Exercise 4.2 Question 2 (Page No. 206 - 207)
Repeat Question $4.2.1$ for each of the following grammars and strings: $S\rightarrow 0S1\mid 01$ with string $000111$. $S\rightarrow +SS\mid \ast SS\mid a$ with string $+\ast aaa$ ... $bfactor\:\rightarrow\:not\:bfactor\mid (bexpr)\mid true\mid false$

Repeat Question $4.2.1$ for each of the following grammars and strings: $S\rightarrow 0S1\mid 01$ with string $000111$.$S\rightarrow +SS\mid \ast SS\mid a$ with string $+...

admin

749 views

admin asked Aug 17, 2019

4 votes

1 answer

85

Ullman (Compiler Design) Edition 2 Exercise 4.2 Question 1 (Page No. 206)
Consider the context-free grammar:$S\rightarrow SS + \mid SS {\ast} \mid a$and the string $aa + a{\ast}$. Give a leftmost derivation for the string. Give a rightmost derivation for the string. ... for the string. Is the grammar ambiguous or unambiguous? Justify your answer. Describe the language generated by this grammar.

Consider the context-free grammar:$$S\rightarrow SS + \mid SS {\ast} \mid a$$and the string $aa + a{\ast}$.Give a leftmost derivation for the string.Give a rightmost deri...

admin

10.9k views

admin asked Aug 7, 2019

1 votes

0 answers

86

Ullman (Compiler Design) Edition 2 Exercise 2.2 Question 6 (Page No. 52)
Construct a context-free grammar for roman numerals.

Construct a context-free grammar for roman numerals.

admin

284 views

admin asked Jul 26, 2019

0 votes

0 answers

87

Ullman (Compiler Design) Edition 2 Exercise 2.2 Question 4 (Page No. 51 - 52)
Construct unambiguous context-free grammars for each of the following languages. In each case show that your grammar is correct. Arithmetic expressions in postfix notation. Left-associative lists of identifiers separated by commas. Right- ... $(d)$.

Construct unambiguous context-free grammars for each of the following languages. In each case show that your grammar is correct. Arithmetic expressions in postfix notatio...

admin

659 views

admin asked Jul 26, 2019

1 votes

2 answers

88

Ullman (Compiler Design) Edition 2 Exercise 2.2 Question 3 (Page No. 51)
Which of the grammars are ambiguous? $S\rightarrow 0S1 \mid 01$ $S\rightarrow +SS \mid -SS \mid a$ $S\rightarrow S(S)S \mid \epsilon$ $S\rightarrow aSbS \mid bSaS \mid \epsilon$ $S\rightarrow a \mid S+S \mid SS \mid S^{\ast} \mid (S)$

Which of the grammars are ambiguous? $S\rightarrow 0S1 \mid 01$$S\rightarrow +SS \mid -SS \mid a$$S\rightarrow S(S)S \mid \epsilon$$S\rightarrow aSbS \mid bSaS \mid \epsi...

admin

1.2k views

admin asked Jul 26, 2019

0 votes

0 answers

89

Ullman (Compiler Design) Edition 2 Exercise 2.2 Question 2 (Page No. 51)
What language is generated by the following grammars? In each case justify your answer. $S\rightarrow 0S1 \mid 01$ $S\rightarrow +SS \mid -SS \mid a$ $S\rightarrow S(S)S \mid \epsilon$ $S\rightarrow aSbS \mid bSaS \mid \epsilon$ $S\rightarrow a \mid S+S \mid SS \mid S^{\ast} \mid (S)$

What language is generated by the following grammars? In each case justify your answer. $S\rightarrow 0S1 \mid 01$$S\rightarrow +SS \mid -SS \mid a$$S\rightarrow S(S)S \m...

admin

333 views

admin asked Jul 26, 2019

0 votes

0 answers

90

Ullman (Compiler Design) Edition 2 Exercise 2.2 Question 1 (Page No. 51)
Consider the context-free grammar $S\rightarrow SS+\mid SS^{\ast}\mid a$ Show how the string $aa+a^{\ast}$ can be generated by this grammar. Construct a parse tree for this string. What language does this grammar generate? Justify your answer.

Consider the context-free grammar$S\rightarrow SS+\mid SS^{\ast}\mid a$Show how the string $aa+a^{\ast}$ can be generated by this grammar.Construct a parse tree for this ...

admin

176 views

admin asked Jul 26, 2019

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