GATE 1998 PROBLEMS

THEORY OF COMPUTATION

SECTION A

Q1.9 If the regular set A is represented by A=(01+1)* ant the regular set 'B' is represented by B= ((01)*1*)*, which of the following is true?

(a) A is a proper subset of B

(b) B is a proper subset of  A

(c) A and B are incomparable

(d) A=B

Q1.10 Which of the following sets can be recognized by a deterministic finite automaton?

(a) The numbers 1,2,4,8,----------.2^n,-------------written in binary.

(b) The umbrs 1.2.4.-------,2^n,--------------written in unary.

(c) The set of binary strings in which the number of zeros is the same as the number of ones.

(d) The st {1, 101, 11011, 1110111,--------}

Q1.11 Regarding the power of recognition of languages, which of the following statements is false?

(a) The non-deterministic finite-state automata are equivalent to deterministic finite-state automata.

(b) Non-deterministic Push-down automata are equivalent to deterministic push-down automata.

(c) Non-deterministic Turing machines are equivalent to deterministic Turing machines.

(d) Multi-tape Turing machines are equivalent to single-tape Turing machines.

Q1.12 The string 1101 does not belong to the set represented by

(a) 110*(0+1)                    (b) 1(0+1)*101

(c) (10)*(01)*(00+11)*      (d) (00+(11)*0)*

Q2.5 Let L be the set of all binary strings whose last two symbols are the same. The number of states in the minimum state deterministic finite state automaton accepting L is

(a) 2         (b) 5        (c) 8        (d) 3

Q2.6 Which of the following statements is false?

(a) Every finite subset of a non-regular set is regular.

(b) Every subset of a regular set is regular.

(c) Every finite subset of a regular set is regular.

(d) The intersection of two regular sets is regular.

Q4. Design a deterministic finite state automaton (using the minimum number of states) that recognizes the following language:

L = {w in {0,1}*| w interpreted as a binary number (ignoring the leading 0's) is divisible by five}

SECTION B

Q13. Let M = ({q0,q1},{0,1},{z0,x},f,z0,0) be a pushdown automaton where f is given by

f(q0,1.z0) = {(q0,x,z0)}

f(q0,e,z0)  = {(q0,e)}

f(q0,1,X)  =  {(q0,XX)}

f(q1,1,X) = {(q1,e)}

f(q0,0,X) = {(q1,X)}
f(q1,0,z0) = {(q0,z0)}

(a) What is the language accepted by this PDA by empty store?

(b) Describe informally the working of the PDA?

Q14. Let G1 =(N,T,P,S1) be a CFG where,

N = {S1,A,B}            T= {a,b}

and P is given by

S1---------->a S1 b

S1-------->a Ab

A--------->aA

A-------->a

S1---->aBb

B---------->bB

B---------->b

What is L(G1)?

(b) Use the grammar in Part(a) to give a CFG for L2 =(a^ib^ja^kb^l|i,j,k,l>=1, i=j or k = l} by adding not more than 5 production rules.

(c) Is L2 inherently ambiguous?

ANS
1987	1988	1989
1990	1991	1992
1993	1994	1995
1997	1998	1999
2000	2001	2002
2003	2004	1987
QUES
1987	1988	1989
1990	1991	1992
1993	1994	1995
1997	1998	1999
2000	2001	2002
2003	2004	1987

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2005, GATE THEORY OF COMPUTATION