Regular languages and finite automata

Question 1
Consider the languages L1 = \phi and L2 = {a}. Which one of the following represents L1 L2* U L1*
gatecs20135
A
A
B
B
C
C
D
D
GATE CS 2013    Regular languages and finite automata    
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Question 1 Explanation: 
L1 L2* U L1* Result of L1 L2* is [Tex]\phi[/Tex]. {[Tex]\phi[/Tex]} indicates an empty language. Concatenation of [Tex]\phi[/Tex] with any other language is [Tex]\phi[/Tex]. It works as 0 in multiplication. L1* = [Tex]\phi[/Tex]* which is {[Tex]\epsilon[/Tex]}. Union of [Tex]\phi[/Tex] and {[Tex]\epsilon[/Tex]} is {[Tex]\epsilon[/Tex]}
Question 2
Consider the DFA given. gatecs201313 Which of the following are FALSE?
1. Complement of L(A) is context-free.
2. L(A) = L((11*0+0)(0 + 1)*0*1*)
3. For the language accepted by A, A is the minimal DFA.
4. A accepts all strings over {0, 1} of length at least 2. 
A
1 and 3 only
B
2 and 4 only
C
2 and 3 only
D
3 and 4 only
GATE CS 2013    Regular languages and finite automata    
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Question 2 Explanation: 
1 is true. L(A) is regular, its complement would also be regular. A regular language is also context free. 2 is true. 3 is false, the DFA can be minimized to two states. Where the second state is final state and we reach second state after a 0. 4 is clearly false as the DFA accepts a single 0.
Question 3
W hat is the complement of the language accepted by the NFA shown below? gatecs2012automata
A
A
B
B
C
C
D
D
GATE CS 2012    Regular languages and finite automata    
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Question 3 Explanation: 
The given alphabet contains only one symbol {a} and the given NFA accepts all strings with any number of occurrences of ‘a’. In other words, the NFA accepts a+. Therefore complement of the language accepted by automata is empty string.
Question 4
Given the language L = {ab, aa, baa}, which of the following strings are in L*?
1) abaabaaabaa
2) aaaabaaaa
3) baaaaabaaaab
4) baaaaabaa 
A
1, 2 and 3
B
2, 3 and 4
C
1, 2 and 4
D
1, 3 and 4
GATE CS 2012    Regular languages and finite automata    
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Question 4 Explanation: 
Question 5
Consider the set of strings on {0,1} in which, every substring of 3 symbols has at most two zeros. For example, 001110 and 011001 are in the language, but 100010 is not. All strings of length less than 3 are also in the language. A partially completed DFA that accepts this language is shown below. The missing arcs in the DFA are
A
A
B
B
C
C
D
D
GATE CS 2012    Regular languages and finite automata    
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Question 6
Definition of a language L with alphabet {a} is given as following.
             L={gate2011Q42| k>0, and n is a positive integer constant}
What is the minimum number of states needed in DFA to recognize L?
A
k+1
B
n+1
C
2^(n+1)
D
2^(k+1)
GATE CS 2011    Regular languages and finite automata    
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Question 6 Explanation: 
Question 7
A deterministic finite automation (DFA)D with alphabet {a,b} is given below GATE2011AT1 Which of the following finite state machines is a valid minimal DFA which accepts the same language as D? GATE2011AT2 GATE2011AT3
A
A
B
B
C
C
D
D
GATE CS 2011    Regular languages and finite automata    
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Question 7 Explanation: 
Options (B) and (C) are invalid because they both accept ‘b’ as a string which is not accepted by give DFA. (D) is invalid because it accepts "bba" which are not accepted by given DFA.
Question 8
Let w be any string of length n is {0,1}*. Let L be the set of all substrings of w. What is the minimum number of states in a non-deterministic finite automaton that accepts L?
A
n-1
B
n
C
n+1
D
2n-1
GATE CS 2010    Regular languages and finite automata    
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Question 8 Explanation: 
We need minimum n+1 states to build NFA that accepts all substrings of a binary string. For example, following NFA accepts all substrings of “010″ and it has 4 states. NFA_FOR_SUBSTRINGS-300x90
Question 9
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)*?
A
The set of all strings containing the substring 00.
B
The set of all strings containing at most two 0’s.
C
The set of all strings containing at least two 0’s.
D
The set of all strings that begin and end with either 0 or 1.
GATE-CS-2009    Regular languages and finite automata    
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Question 9 Explanation: 
The regular expression has two 0′s surrounded by (0+1)* which means accepted strings must have at least 2 0′s.
Question 10
Which one of the following is FALSE?
A
There is unique minimal DFA for every regular language
B
Every NFA can be converted to an equivalent PDA.
C
Complement of every context-free language is recursive.
D
Every nondeterministic PDA can be converted to an equivalent deterministic PDA.
GATE-CS-2009    Regular languages and finite automata    
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Question 10 Explanation: 
Deterministic PDA cannot handle languages or grammars with ambiguity, but NDPDA can handle languages with ambiguity and any context-free grammar. So every nondeterministic PDA can not be converted to an equivalent deterministic PDA.
There are 80 questions to complete.

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