flat it gate 2

8/13/2019 Flat It Gate 2

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IT DEPARTMENT GATE MATERIAL

FORMAL LANGUAGES AND AUTOMATA THEORY

1.Consider a DFA over ={a,b}accepting all strings which have number of asdivisible by 6

and number ofbsdivisible by 8. What is the minimum number of states that the DFA will

have?

(1) 8

(2) 14

(3) 15

(4) 48

Solution: 3 [Gate-2001]

2. Consider the following languages:

Li ={wwwE {a,b}*}

L2 = {wwR w {a, b}*, wR is the reverse of w}

L3 = {021 i is an integer)

L4= {o2 i is an integer)

Which of the languages are regular?

(1) Only Li and L2

(2) Only L2, L3 and L4

(3) Only L3 and L4

(4) Only L3

Solution:1 [Gate-2001]

3. Consider the following problem X.

Given a Turing machine M over the input alphabet , any state q of M

And a word w E*, does the computation of M on w visit the state q?

Which of the following statements about X is correct?

(1) X is decidable

(2) X is undecidable but partially decidable


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(3) X is undecidable and not even partially decidable

(4) X is not a decision problem


4. The regular expression 0* (10*)* denotes the same set as

1) (1*0)*1*

2) 0 + (0 + 10)*

3) (0 + 1)* 10(0 + 1)*

4) none of theseSolution:4 [Gate-2003]

5. If the strings of a language L can be effectively enumerated in lexicographic

(i.e., alphabetic) order, which of the following statements is true ?

1) L is necessarily finite

2) L is regular but not necessarily finite

3) L is context freebut not necessarily regular

4) L is recursive but not necessarily context free


6.Which of the following suffices to convert an arbitrary CFG to an LL(1) grammar ?

1) Removing left recursion alone

2) Factoring the grammar alone

3) Removing left recursion and factoring the grammar

4) None of these


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7.Consider the following deterministic finite state automaton M .

Let S denote the set of seven bit binary strings in which the first, the fourth,

and the last bits are 1. The number of strings in S that are accepted by M is

1) 1 2) 5 3) 7 4) 8


8.Define languages L0 and L1 as follows : L0 = {< M, w, 0 > | M halts on w}

L1 = {< M, w, 1 > | M does not halts on w}

Here < M, w, i > is a triplet, whose first component. M is an encoding of a Turing

Machine, second component, w, is a string, and third component, i, is a bit.

Let L = L0 L1. Which of the followingis true ?

1) L is recursively enumerable, but is not

2) is recursively enumerable, but L is not

3) Both L and are recursive


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4) Neither L nor is recrusively enumerable


9.Consider the NFA M shown below.

Let the language accepted by M be L. Let L1 be the language accepted by the

NFA M1,obtained by changing the accepting state of M to a non-accepting state

and by changing the non-accepting state of M to accepting states. Which of the

following statements is true?

1) L1 = {0, 1}* - L

2) L1 = {0, 1}*

3) L1 L

4) L1 = L



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10.The following finite state machine accepts all those binary strings in which the

number of l's and 0's are respectively

1) divisible by 3 and 2

2) odd and even

3) even and odd

4) divisible by 2 and 3


11.The language {am bn Cm + n | m, n 1} is

1) regular

2) context-free but not regular

3) context sensitive but not context free

4) type-0 but not context sensitive


12.Consider the following grammar

G: S bS | aA | b

AbA | aB

BbB | aS | a


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Let Na (w) and Nb (w) denote the number of a's and b's in a string w

respectively. The language L(G) {a, b}+ generated by G is

1) { w | Na(w) > 3Nb(w)}

2) { w | Nb(w) > 3Nb(w)}

3) { w | Na(w) = 3k, k {0, 1, 2, ...}}

4) { w | Nb(w) = 3k, k {0, 1, 2, ...}}


13.L1 is a recursively enumerable language over. An algorithm A effectively

enumerates its words as w1, w2, w3, ... Define another language L2 over

Union {#} as {wi # wj : wi, wj L1, i


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The language recognized by M is :

1) {w {a, b}* / every a in w is followed by exactly two b's}

2) {w {a, b}* every a in w is followed by at least two b}

3) {w {a, b}* w contains the substring 'abb'}

4) {w {a, b}* w does not contain 'aa' as a substring}


15. Let Nf and Np denote the classes of languages accepted by non-

deterministic finite automata and non-deterministic push-down automata,

respectively. Let Df and Dp denote the classes of languages accepted by

deterministic finite automata and deterministic push- down automata,

respectively. Which one of the following is TRUE?

1) Df Nf and Dp Np

2) Df Nf and Dp = Np

3) Df = Nf and Dp = Np

4) Df = Nf and Dp Np


16.Consider the languages


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L1 = {anbncm | n, m > 0} and L2 = {a

nbmcm | n, m > 0} Which one of the following

statements is FALSE?

1) L1 L2 is a context-free language

2) L1 L2 is a context-free language

3) L1 and L2 are context-free language

4) L1 L2 is a context sensitive language


17. Consider the languages

L1 = {wwR |w {0, 1}*}

L2 = {w # wR | w {0, 1}*}, where # is a special symbol

L3 = {ww | w (0, 1}*)

Which one of the following is TRUE?

1) L1 is a deterministic CFL


3) L3 is a CFL, but not a deterministic CFL



18. The following diagram represents a finite state machine which takes as input

a binary number from the least significant bit.


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Which one of the following is TRUE?

1) It computes 1's complement of the input number

2) It computes 2's complement of the input number

3) It increments the input number

4) It decrements the input number


19. If s is a string over (0 + 1)*, then let n0 (s) denote the number of 0's in s and n1(s)

the number of 1's in s. Which one of the following languages is not regular ?

1) L = {s (0 + 1)* | n0 (s) is a 3-digit prime}

2) L = {s (0 + 1)* | for every prefix s' of s, | n0 (s') - n1(s') | 2}

3) L = {s (0 + 1)* | n0 (s) - n1(s) 4}

4) L = {s (0 + 1)* | n0 (s) mod 7 = n1(s) mod 5 = 0}


20. For s (0 + 1) * let d(s) denote the decimal value of s (e.g. d (101) = 5).

Let L = {s (0 + 1)*

| d (s) mod 5 = 2 and d (s) mod 7 4) Which one of the followingstatements is true ?

1) L is recursively enumerable, but not recursive


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2) L is recursive, but not context-free

3) L is context-free, but not regular

4) L is regular


21. Consider the following statements about the context-

free grammar, G = (S SS, S ab, S ba, S )

I. G is ambiguous

II. G produces all strings with equal number of a's and b's

III. G can be accepted by a deterministic PDA

Which combination below expresses all the true statements about G?

1) I only

2) I and III only

3) II and III only

4) I, II, and III


22. Let L1 be regular language, L2 be a deterministic context-free

language and L3 a recursively enumerable, but not recursive, language.

Which one of the following statements is false ?

1) L1 L2 is a deterministic CFL

2) L3 L1 is recursive

3) L1 L2 is context free

4) L1 L2 L3 is recursively enumerable



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23. Consider the regular language L = (111 + 11111)*. The minimum number of states

in any

DFA accepting this languages is

1) 3 2) 5 3) 8 4) 9


24.Which of the following problems is undecidable?

1) Membership problem for CFGs.

2) Ambiguity problem for CFGs.

3) Finiteness problem for FSAs.

4) Equivalence problem for FSAs.


25.Which of the following is TRUE?

1) Every subset of a regular set is regular

2) Every finite subset of a non-regular set is regular

3) The union of two non-regular sets is not regular

4) Infinite union of finite sets is regular


26.A minimum state deterministic finite automaton accepting the languageL = {w | w {0, 1}*, number of 0s and 1s in w are divisible by 3 and 5, respectively}

has


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1) 15 states

2) 11 states

3) 10 states

4) 9 states


27.The language L = {0i 21i | i 0) over the alphabet {0, 1, 2} is

1) not recursive

2) is recursive and is a deterministic CFL

3) is a regular language

4) is not a deterministic CFL but a CFL


28.Which of the following languages is regular ?

1) {wwR | w {0, 1}+}

2) {wwRx | x, w {0, 1}+}

3) {wxwR | x, w {0, 1}+}

4) {xwwR | x, w {0, 1}+}


29.Consider the grammar with non-terminals N = {S, C, S1), terminals T = {a, b, i, t, e},

with S as the start symbol, and the following set of rules


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S iCtSS1|a

S1 eS|

C b

The grammar is NOT LL(1) because:

1) It is left recursive

2) it is right recursive

3) it is ambiguous

4) it is not context-free


30.Consider the following two statements : P : Every regular grammar is LL (1)

Q : Every regular set has a LR

(1) grammar

Which of the

following is TRUE ?

1) Both P and Q are true.

2) P is true and Q is false.

3) P is false and Q is true.

4) Both P and Q are false.



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Consider the following Finite State Automaton :

31.The language accepted by this automaton is given by the regular expression

1) b * ab * ab * ab *

2) (a + b) *

3) b * a (a + b) *

4) b * ab * ab *


32.The minimum state automaton equivalent to the above FSA has the following number

of states

1) 1 2) 2 3) 3 4) 4


33.Which of the following is true for the language {ap | p

is a prime} ?

1) It is not accepted by a Turing Machine

2) It is regular but not context-free

3) It is context-free but not regular

4) It is neither regular nor context-free, but accepted by a Turing machine


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34.Which of the following are decidable?

I. Whether the intersection of two regular languages is infinite

II. Whether a given context-free language is regular

III. Whether two push-down automata accept the same language

IV. Whether a given grammar is context-free

1) I and II

2) I and IV

3) II and III

4) II and IV


35.If L and are recursively enumerable then L is

1) regular

2) context-free

3) context-sensitive

4) recursive


36.Which of the following statements is false?

1) Every NFA can be converted to an equivalent DFA

2) Every non-deterministic Turing machine can be converted to an

equivalent deterministic Turing machine

3) Every regular language is also a context-free language

4) Every subset of a recursively enumerable set is recursive


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Solution:4

[Gate-2008]

37.Which of the following statements are true?

I. Every left-recursive grammar can be converted to a right-recursive grammar and

vice- versa

II. All -productions can be removed from any context-free grammar by

suitable transformations

III. The language generated by a context-free grammar all of whose productions are of

the form X w or X wY (where, w is a string of terminals and Y is a non-terminal),

is always regular

IV. The derivation trees of strings generated by a context-free grammar in

Chomsky

Normal Form are always binary

trees

1) I, II, III and IV

2) II, III and IV only

3) I, III and IV only

4) I, II and IV only


38.. Match the following

(E)

Checking that identifiers are

declared before their use (P) L = (anbmcndm | n 1, m

(F)

Number of formal parameters i

the declaration of a function agree

with the number of actual parameter

in a use of that function

(Q) X X b X | X c X | d X f | g


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(G)

Arithmetic expression with matche

pairs of parentheses (R) L = {wcw | w (a | b) *}

(H) Palindromes (S) X b X b | c X c |

1) E P, F R, G Q, H S

2) E R, F P, G S, H Q

3) E R, F P, G Q, H S

4) E P, F R, G S, H Q


39.Match the following NFAs with the regular expressions they correspond to.

1. + 0(01*1 + 00)*01*

2. + 0(10*1 + 00)*0

3. + 0(10*1 + 10)*1

4. + 0(10*1 + 10)*10*


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1) P 2, Q 1, R 3, S 4

2) P 1, Q 3, R 2, S 4

3) P 1, Q 2, R 3, S 4

4) P 3, Q 2, R 1, S 4


40.Which of the following are regular sets?

I. {anb2m | n 0, m 0}

II. {an

bm

| n = 2m}

III. {anbm | n m}

IV. {xcy | x, y {a, b}*}

1) I and IV only

2) I and III only

3) I only

4) IV only


41.S aSa | bSb | a | b; The language generated by the above grammar over the

alphabet

{a, b} is the set of

1) All palindromes.

2) All odd length palindromes.

3) Strings that begin and end with the same symbol.


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4) All even length palindromes.


42.Which one of the following languages over the alphabet {0, 1} is described by

the regular expression:

(0 + 1)* 0(0 + 1)* 0(0 + 1)*?

1) The set of all strings containing the substring 00.

2) The set of all strings containing at most two 0s.

3) The set of all strings containing at least two 0s.

4) The set of all strings that begin and end with either 0 or 1.


43.Which one of the following is FALSE?

1) There is unique minimal DFA for every regular language.

2) Every NFA can be converted to an equivalent PDA.

3) Complement of every context-free language is recursive.

4) Every nondeterministic PDA can be converted to an equivalent deterministic PDA.


44.L = L1

L2

, where L1

and L2

are languages as defined below:


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L1 = {am bm c an bm | m,

n 0} L2 = {aibjck | i, j, k

0}

Then L is

1) Not recursive

2) Regular

3) Context free but not regular

4) Recursively enumerable but not context free


45.

The above DFA accepts the set of all strings over {0, 1} that

1) begin either with 0 or 1

2) end with 0

3) end with 00

4) contain the substring 00


46. Let L = { (0 + 1) * | has even number of 1s}, i.e. L is the set of all

bit strings with even number of 1s. Which one of the regular expressions

below represents L ?


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1) (0 * 10 * 1) *

2) 0 * (10 * 10 * ) *

3) 0 * (10 * 1 *) * 0 *

4) 0 * 1(10 * 1) * 10 *


47.Consider the languages

L1 = {0i1j | i j}. L2 = {0i1j | i =j}. L3 = {0i1j | i = 2j + 1}. L4 = {0i1j | i 2j}.

Which one of the following statements is true ?

1) Only L2 is context free

2) Only L2 and L3 are context free

3) Only L1 and L2 are context free

4) All are context free


48. Let be any string of length n in {0, 1}*. Let L be the set of all

substrings of. What is the minimum number of states in a non-deterministic

finite automaton that accepts L ?

1) n - 1

2) n

3) n + 1

4) 2n - 1


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49.What is the complement of the language acceptedby the NFA show below?

Assume ={a} and is the empty string.

(1) (2) {} (3) a G (4) {a, }


50.

Solution:B [Gate-2012]

51. The following grammar

G = (N, T, P, S)

N = {S, A, B}

T = {a, b, c}


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P : S aSa

S aAa

A bB

B bB

B c is

a. is type 3b. is type 2 but not type 3c. is type 1 but not type 2d. is type 0 but not type 1


G = (N, T, P, S)

N = {S, A, B, C, D, E}

T = {a, b, c}

P : S aAB

AB CD

CD CE

C aC

C bbE bc is



G = (N, T, P, S)

N = {S, A, B, C}


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T = {a, b, c}

P : S aS

A bB

B cC

C a is



G = (N, T, P, S)

N = {S, A, B, C, D, E}

T = (a, b, c}

P : S ABCD

BCD DE

D aD

D a

E bEE c is


55. Consider the following CFG

S aB S bAB b A a

B bS A aS


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S aB S bA

B aBB A bAA

Consider the following derivation

S aB

aaBB

aaBb

aabSb

aabbAb

aabbab

This derivation is

a. a leftmost derivationb. a rightmost derivationc. both leftmost and rightmost derivationd. neither leftmost nor rightmost derivation

56. Consider the following language

L = {anbncndn|n 1}

L is

a. CFL but not regularb. CSL but not CFLc. regulard. type 0 language but not type 1


L = {anbn|n 1}

L is

a. CFL but not regular


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b. CSL but not CFLc. regulard. type 0 language but not type 1


L = {anbmcpdq|n, m, p, q 1}

L is

a. CFL but not regularb. CSL but not CFLc. regulard. type 0 language but not type 1

59. The following CFG is in

S AB

B CD

B AD

B b

D AD

D d

A a

C a

a. Chomsky normal form but not strong Chomsky normal formb. Weak Chomsky normal form but not Chomsky normal formc. Strong Chomsky normal formd. Greibach normal form

60. The following CFG is in

S aBB

B bAA


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A a

B b

a. Chomsky normal form but not strong Chomsky normal formb.

Weak Chomsky normal form but not Chomsky normal formc. Strong Chomsky normal form

d. Greibach normal form61. Which of the following CF language is inherently ambiguous?

a. {anbncmdm|n, m 1}b. {anbmcpdq|n = p or m = q, n, m, p, q 1}c. {anbmcpdq|n m p q}d. {anbmcpdq|n m p q}

62. Which string is not accepted by the following FSA?

a. 00111b. 01010c. 00110d. 11010

63. Which string is accepted by the following FSA?


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a. 00111b. 11011c. 01101d. 0101

64. Can a DFSA simulate a NFSA

a. Nob. Yesc. sometimesd. depends on NFA

65. Which of the following is true for an arbitrary language L.

a.b. L*= L+{}c. L*= L+d. L*= L+ {}

66. The concept of FSA is much used in this part of the compiler

a. lexical analysisb. parserc. code generationd. code optimization

67. The concept of grammar is much used in this part of the compiler

a. lexical analysisb. parserc. code generationd. code optimization

68. (a + b)(cd)*(a + b) denotes the following set

a. {a(cd)nb|n 1}b. {a(cd)na|n 1} {b(cd)nb/n 1}c. {a(cd)na|n 0} {a(cd)nb/n 0} {b(cd)na/n 0} {b(cd)nb/n 0}d. {acndnb|n 1}

69. baa*c denotes the set


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a. {bnamcp|n, m, p 1}b. {banc|n 0}c. {banc|n 1}

70. The set of all strings over the alphabet = {a, b} (including ) is denoted by

a. (a + b)*b. (a + b)+c. a+b+d. a*b*

71. Palindromes cant be recognized by any FSA because

a. FSA cannot remember arbitrarily large amount of informationb. FSA cannot deterministically fix the midpointc. Even if the mid point is known an FSA cannot find whether the second half of the string

matches the first halfd. all of the above72. Let = {a, b, c, d, e}. The number of strings in * of length 4 such that no symbol is used

more than once in a string is

a. 360b. 120c. 35d. 36

73. Which of the following denotes Chomskian hiearchy?

a. REG CFL CSL type0b. CFL REG type0 CSLc. CSL type0 REG CFLd. CSL CFL REG type0

74. A language L is accepted by a FSA iff it is

a. CFLb. CSLc.

recursived. regular

75. Which of the following regular expressions denotes a language comprising of all possible

strings over = {a, b} of length n where n is a multiple of 3.


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a. (a + b + aa + bb + aba + bba)*b. (aaa + bbb)*c. ((a + b)(a + b)(a + b))*d. (aaa + ab + a) + (bbb + bb + a)

76. A language is represented by a regular expression (a)*(a + ba). Which of the following stringdoes not belong to the regular set represented by the above expression.

a. aaab. abac. ababad. aa

77. Which of the following is not primitive recursive but partially recursive?

a. McCarthys functionb.

Riemann functionc. Ackermanns function

d. Bounded function78. Consider the following right-linear grammar G = (N, T, P, S) N = {S}

P : S aS|aA T = {a, b}

A bA|b

Which of the following regular expression denotes L(G)?

a. (a + b)*b. a(ab)*bc. aa*bb*d. a*b*

79. Which of the following strings is not generated by the following grammar? S SaSbS|

a. aabbb. ababc. aababbd.

aaabb

80. Consider the following NFSA


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The automaton accepts

a. all words of the form {(ab)na|n 1}b. all words that end with a and c. all words that end with a and not d. all words containing substring ba

81. Consider a language L for which there exists a Turing machine (TM), T, that accepts every

word in L and either rejects or loops for every word that is not in L. The language L is

a. NP hardb. NP completec. recursived. recursively enumerable

82.Consider the following statements

I. Recursive languages are closed under complementationII. Recursively enumerable languages are closed under union

III. Recursively enumerable languages are closed under complementationWhich of the above statement are TRUE?

a. I onlyb. I and IIc. I and IIId. II and III

83.Which of the following statement is wrong?

a. Any regular language can be generated by a context-free grammarb. Some non-regular languages cannot be generated by any CFGc. the intersection of a CFL and regular set is a CFLd. All non-regular languages can be generated by CFGs.

84.Recursively enumerable languages are not closed under

a. unionb. homomorphism


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c. complementationd. concatenation

85.Which of the following problem is undecidable?

a.

membership problem for CFLb. membership problem for regular setsc. membership problem for CSLd. membership problem for type 0 languages

86.Recursive languages are

a. a proper superset of CFLb. always recognized by PDAc. are also called type 0 languagesd. always recognized by FSA

87. R1and R2are regular sets. Which of the following is not true?

a. R1 R2neet not be regularb. * R1is regularc. R1R2is regulard. is regular

88. Which of the following regular expression identity is true?

a. r(*) = r*b. (r*s*)* = (r + s)*c. (r + s)* = r* + s*d. r*s* = r* + s*

89. Which one of the following statement is FALSE?

a. context-free languages are closed under unionb. context-free languages are closed under concatenationc. context-free languages are closed under intersectiond. context-free languages are closed under Kleene closure

90. Which of the following conversion is not possible (algorithmically)?

a. regular grammar to context-free grammarb. nondeterministic FSA to deterministic FSAc. nondeterministic PDA to deterministic PDA
http://my.safaribooksonline.com/9788131723562/app01lev1sec1#app01qa2a36http://my.safaribooksonline.com/9788131723562/app01lev1sec1#app01qa2a36http://my.safaribooksonline.com/9788131723562/app01lev1sec1#app01qa2a36


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d. nondeterministic TM to deterministic TM

flat it gate 2

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