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PG – 492

*PG492*

I Semester M.Sc. Degree Examination, February 2013 (Y2K5 Scheme) COMPUTER SCIENCE MCS 101 : Theory of Computation Time : 3 Hours

Max. Marks : 80

Instruction : Answer any five full questions choosing atleast 2 full questions from each Part. PART – A 1. a) Define language and the operations performed on languages. b) Define NFA and the language accepted by NFA. c) Describe the notations to represent automata.

(6+5+5)

2. a) Show that if D = (Q D, ∑, δD, {q0 ), FD ) is the DFA constructed from NFA N = (Q N, ∑, δN, q0, FN ) by subset construction, then L (D) = L (N). b) Design an algorithm to simulate the behavior of DFA on any string. c) Construct a DFA equivalent to the following NFA : 0

1

− > q0

{ q0, q1}

q0

q1

−

q2

* q2

−

−

(5+4+7)

3. a) Define Pumping lemma for regular languages and show that L = {wwR|w ∈{a, b}} is not regular. b) Construct regular expression for the following DFA : 0

1

→p

q

r

q

p

r

*r

q

p

c) Show that regular expressions are closed under union and intersection.

(6+6+4) P.T.O.

PG – 492

-2-

*PG492*

4. a) Define equivalent states. Use table-filling algorithm to find the equivalent states and minimize the following DFA : 0

1

→A

B

A

B

A

C

C

D

B

*D

D

A

E

D

F

F

G

E

G

F

G

H

G

D

b) Define context free grammar and the context free language. Construct a context free grammar to generate the set of all palindromes. Generate any two palindromes. c) Define DPDA.

(8+6+2) PART – B

5. a) Show that the grammar G = (V, T, S, P) with P : {S – > aSb| ∈ } is sound and complete. b) Consider the grammar G = ({E}, {+, *, num, –}, {E → E + E|E*E|(E) | –E| num}, E). Is the grammar ambiguous ? Justify. c) Describe any one application of context free grammar.

(6+6+4)

6. a) Define PDA and the language accepted by PDA. Construct a PDA to accept the language L = {wwR|w is in {0 +, 1} *}. b) Compare and contrast Turing machines and computers. c) Define ID. Illustrate the working of the PDA.

(8+4+4)

*PG492*

-3-

PG – 492

7. a) Define Turing machine and the language acceptance of a Turing machine. b) Design a Turing machine M to recognize the language {1n 2n 3n| n ≥ 1} and show the behavior of the TM on a string of length greater than 6. c) Explain the terms “undecidability” and “intractability”. 8. Write short notes on any two of the following : a) Halting problem b) Universal Turing machine c) Moore and Mealy machines. _________________

(4+8+4) (2×8=16)