2020 question paper

Formal Language & Automata Theory

27 questions

1. Q1a. Definition of a language L with alphabet \{a\} is given as L = \{a^{nk} \mid k > 0, \text{ and } n \text{ is a positive integer constant}\} What is the minimum number of states needed in a DFA to recognize L? (i) k + 1 (ii) n + 1 (iii) 2^{n + 1} (iv) 2^{k + 1}20202m

   Module 1: Introduction, Regular languages and finite automata

   Definition of a language $L$ with alphabet $\{a\}$ is given as

   $$L = \{a^{nk} \mid k > 0, \text{ and } n \text{ is a positive integer constant}\}$$

   What is the minimum number of states needed in a DFA to recognize $L$?
   (i) $k + 1$
   (ii) $n + 1$
   (iii) $2^{n + 1}$
   (iv) $2^{k + 1}$

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2. Q1b. Which of the following versions of Unix came up with YACC first? (i) V3 (ii) V5 (iii) CB UNIX (iv) UNIX-RT20202m

   Module 2: Context-free languages and pushdown automata

   Which of the following versions of Unix came up with YACC first?

   (i) V3
   (ii) V5
   (iii) CB UNIX
   (iv) UNIX-RT

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3. Q1c. A pushdown automata can be represented as \text{PDA} = \epsilon\text{-NFA} + [\text{stack}]. (i) True (ii) False20202m

   Module 2: Context-free languages and pushdown automata

   A pushdown automata can be represented as $\text{PDA} = \epsilon\text{-NFA} + [\text{stack}]$.

   (i) True
   (ii) False

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4. Q1d. A language accepted by deterministic pushdown automata is closed under which of the following? (i) Complement (ii) Union (iii) Both (i) and (ii) (iv) None of the above20202m

   Module 2: Context-free languages and pushdown automata

   A language accepted by deterministic pushdown automata is closed under which of the following?

   (i) Complement
   (ii) Union
   (iii) Both (i) and (ii)
   (iv) None of the above

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5. Q1e. A _____ is context free grammar with atmost one non-terminal in the right handside of the production. (i) linear grammar (ii) linear bounded grammar (iii) regular grammar (iv) None of the above20202m

   Module 1: Introduction, Regular languages and finite automata

   A _____ is context free grammar with atmost one non-terminal in the right handside of the production.

   (i) linear grammar
   (ii) linear bounded grammar
   (iii) regular grammar
   (iv) None of the above

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6. Q1f. The lexical analysis for a modern language such as Java needs the power of which one of the following machine models in a necessary and sufficient sense? (i) Finite state automata (ii) Deterministic pushdown automata (iii) Non-deterministic pushdown automata (iv) Turing machine20202m

   Module 1: Introduction, Regular languages and finite automata

   The lexical analysis for a modern language such as Java needs the power of which one of the following machine models in a necessary and sufficient sense?

   (i) Finite state automata
   (ii) Deterministic pushdown automata
   (iii) Non-deterministic pushdown automata
   (iv) Turing machine

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7. Q1g. Let L = \{w \mid w \in \{0+1\}^*, w \text{ has even number of 1s}\}. Which one of the regular expression below represents L? (i) (0^* 10^* 1)^* (ii) 0^*(10^* 10^*)^* (iii) 0^* (10^* 1^*)^* 0^* (iv) 0^* 1(10^* 1)^* 10^*20202m

   Module 1: Introduction, Regular languages and finite automata

   Let $L = \{w \mid w \in \{0+1\}^*, w \text{ has even number of 1s}\}$. Which one of the regular expression below represents $L$?

   (i) $(0^* 10^* 1)^*$
   (ii) $0^*(10^* 10^*)^*$
   (iii) $0^* (10^* 1^*)^* 0^*$
   (iv) $0^* 1(10^* 1)^* 10^*$

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8. Q1h. What is the minimum number of states in deterministic finite automata (DFA) for string starting with ba^2 and ending with a over alphabet \{a, b\}? (i) Ten (ii) Nine (iii) Eight (iv) Six20202m

   Module 1: Introduction, Regular languages and finite automata

   What is the minimum number of states in deterministic finite automata (DFA) for string starting with $ba^2$ and ending with $a$ over alphabet $\{a, b\}$?

   (i) Ten
   (ii) Nine
   (iii) Eight
   (iv) Six

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9. Q1i. The decision problem is the function from string to _____. (i) char (ii) int (iii) boolean (iv) None of the above20202m

   Module 5: Undecidability

   The decision problem is the function from string to _____.

   (i) char
   (ii) int
   (iii) boolean
   (iv) None of the above

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10. Q1j. A language L may not be accepted by a turing machine if (i) it is recursively enumerable (ii) it is recursive (iii) L can be enumerated by some turing machine (iv) None of the above20202m

    Module 4: Turing machines

    A language $L$ may not be accepted by a turing machine if

    (i) it is recursively enumerable
    (ii) it is recursive
    (iii) $L$ can be enumerated by some turing machine
    (iv) None of the above

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12. Q2a. Write the context-free grammar to create palindrome over \{a, b\}.20207m

    Module 2: Context-free languages and pushdown automata

    Write the context-free grammar to create palindrome over $\{a, b\}$.

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13. Q2b. Construct a DFA which accepts the set of all binary strings that interpreted as binary representation of an unsigned decimal integer, is divisible by 5.20207m

    Module 1: Introduction, Regular languages and finite automata

    Construct a DFA which accepts the set of all binary strings that interpreted as binary representation of an unsigned decimal integer, is divisible by 5.

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14. Q3a. Design a turing machine to compute the sum of two positive integers m and n.20207m

    Module 4: Turing machines

    Design a turing machine to compute the sum of two positive integers $m$ and $n$.

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15. Q3b. Design ANPDA for accepting the string L = \{\text{set of all palindrome over } \{a, b\}\} by the empty stack and by final state.20207m

    Module 2: Context-free languages and pushdown automata

    Design ANPDA for accepting the string $L = \{\text{set of all palindrome over } \{a, b\}\}$ by the empty stack and by final state.

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16. Q4a. Construct finite automaton corresponding to the regular expression: (a+b)^* c d^* e20207m

    Module 1: Introduction, Regular languages and finite automata

    Construct finite automaton corresponding to the regular expression:

    $$(a+b)^* c d^* e$$

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17. Q4b. Explain the multi-tape version of turing machine and its significance.20207m

    Module 4: Turing machines

    Explain the multi-tape version of turing machine and its significance.

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18. Q5a. Show given grammar over alphabet \{a, b\} verify whether it is ambiguous or unambiguous: S \to a \mid abSb \mid aAb A \to bS \mid aAAb20207m

    Module 2: Context-free languages and pushdown automata

    Show given grammar over alphabet $\{a, b\}$ verify whether it is ambiguous or unambiguous:

    $$S \to a \mid abSb \mid aAb$$
    $$A \to bS \mid aAAb$$

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19. Q6a. Prove that if L is generated by a CFG, then L is accepted by a non-deterministic PDA by empty stack.20207m

    Module 2: Context-free languages and pushdown automata

    Prove that if $L$ is generated by a CFG, then $L$ is accepted by a non-deterministic PDA by empty stack.

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20. Q6b. Design a pushdown automaton for the following context-free grammar: S \to aB \mid bA A \to aS \mid bAA \mid a B \to bS \mid aBB \mid b20207m

    Module 2: Context-free languages and pushdown automata

    Design a pushdown automaton for the following context-free grammar:

    $$S \to aB \mid bA$$
    $$A \to aS \mid bAA \mid a$$
    $$B \to bS \mid aBB \mid b$$

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21. Q7a. Design a turing machine that accepts all palindromes over \Sigma = \{a, b\}.20207m

    Module 4: Turing machines

    Design a turing machine that accepts all palindromes over $\Sigma = \{a, b\}$.

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23. Q7b. Explain Myhill-Nerode theorem for minimization of automata with suitable example.20207m

    Module 1: Introduction, Regular languages and finite automata

    Explain Myhill-Nerode theorem for minimization of automata with suitable example.

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24. Q8a. Prove that if L is the language generated by an unrestricted grammar G = (N, T, P, S), then L is recognized by a turing machine.20207m

    Module 4: Turing machines

    Prove that if $L$ is the language generated by an unrestricted grammar $G = (N, T, P, S)$, then $L$ is recognized by a turing machine.

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25. Q8b. Design a pushdown automata for accepting the string for the language L = \{WW^R \mid W \in \{a, b\}^*\} by the empty stack as well as final state.20207m

    Module 2: Context-free languages and pushdown automata

    Design a pushdown automata for accepting the string for the language $L = \{WW^R \mid W \in \{a, b\}^*\}$ by the empty stack as well as final state.

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26. Q9a. Write a short note on: Minimization of automata20203.5m

    Module 1: Introduction, Regular languages and finite automata

    Write a short note on: Minimization of automata

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27. Q9b. Write a short note on: Type 2 grammar (context free)20203.5m

    Module 2: Context-free languages and pushdown automata

    Write a short note on: Type 2 grammar (context free)

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28. Q9c. Write a short note on: NP-hard problem20203.5m

    Module 5: Undecidability

    Write a short note on: NP-hard problem

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29. Q9d. Write a short note on: Pumping lemma for CFL20203.5m

    Module 2: Context-free languages and pushdown automata

    Write a short note on: Pumping lemma for CFL

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