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## Theoretical Computer Science - Dec 2013

### Computer Engineering (Semester 4)

TOTAL MARKS: 80

TOTAL TIME: 3 HOURS
(1) Question 1 is compulsory.

(2) Attempt any **three** from the remaining questions.

(3) Assume data if required.

(4) Figures to the right indicate full marks.
**1(a)** Define with example Moore and Mealy machine.(5 marks)
**1(b)** Find the equivalent DFA accepting the regular language defined by right linear grammar given as:

S$\rightarrow$aA|bB

A$\rightarrow$aA|bC|a

B$\rightarrow$aB|b

C$\rightarrow$bB(5 marks)
**1(c)** State and prove pumping lemma theorem for regular languages.(5 marks)
**1(d)** Differentiate between Deterministic PDA and Non-deterministic PDA.(5 marks)
**2(a)** Design a finite state machine to determine whether a ternary number base 3 is divisible by 5.[Hint:$\sum$={0,1,2}]
(10 marks)
**2(b)** Design a Mealy machine for the languages (0+1)* (00+11)and convert it to a Moore machine.(10 marks)
**3(a)** Convert the following NFA with ε moves to DFA:

(10 marks)
**3(b)**

Let G be the grammar G={(S,X),{a,b},P,S}where production are

S$\rightarrow$aSX|b

X$\rightarrow$Xb|a

Find:

(i)Leftmost derivation (ii) Rightmost derivation and

Parse tree for the string "aababa".

**4(a)**Design Turing machine for the language L={a

^{n}bn|n>=1} (10 marks)

**4(b)**Design a Turing machine to compare the binary numbers m and n such that if (m>n) output is G (m=n) output is E. (10 marks)

**5(a)**List and explain decision properties of regular language.

Explain the test for checking emptiness of a regular languages.(10 marks)

**5(b)**Construct left linear and right linear grammar for the regular expression:

(((01+10)

*11)00)*(10 marks)

**6(a)**Construct a PDA equivalent to following grammar:

S$\rightarrow$OBB

B$\rightarrow$OS|1S|O

and show acceptance of 010

^{4}

by the PDA.(10 marks)

**6(b)**Reduce the following grammar to Greibach Normal form.

(i)S$\rightarrow$AB

A$\rightarrow$BSB|BB|b

B$\rightarrow$a

(ii)S$\rightarrow$01S|01

S$\rightarrow$10S|10

S$\rightarrow$00|^ (^ means Null (E)) (10 marks)

### Any Four

**7(a)** Write short notes on(any four)

Post Correspondence problem(5 marks)
**7(b)** Chomsky Hierachy(5 marks)
**7(c)** Universal turing machine(5 marks)
**7(d)** Recursive and Recursively enumerable languages(5 marks)
**7(e)** Classes of complexity.(5 marks)