GATE CS 1996

Question 51

A logic network has two data inputs A and B, and two control inputs C₀ and C₁. It implements the function F according to the following table.         . Implement the circuit using one 4 to 1 Multiplexer, one 2-input Exclusive OR gate, one 2-input AND gate, one 2-input OR gate and one Inverter.

Question 52

An 8052 based system has an output port with address 00H. Consider the following assembly language program.

ORG    0100H
MVI    A, 00H
LXI    H, 0105H
OUT    00H
INR    A
PCHL
HLT

a) What does the program do with respect to the output port 00H?
b) Show the wave forms at the three least significant bits of the port 00H.

Question 53

A demand paged virtual memory system uses 16 bit virtual address, page size of 256 bytes, and has 1 Kbyte of main memory. LRU page replacement is implemented using a list whose current status (page number in decimal) is       . For each hexadecimal address in the address sequence given below

 00FF, 010D, 10FF, 11B0

indicate i) the new status of the list ii) page faults, if any, and iii) page replacements, if any

Question 54

The Fibonacci sequence {f₁, f₂, f₃,....., f_n} is defined by the following recurrence: f_n+2 = f_n+1 + f_n, n ≥ 1; f₂ = 1 : f₁ = 1 Prove by induction that every third element of the sequence is even.

Question 55

Let be two matrices such that Express the elements of D in terms of the elements of B.

Question 56

Let G be a context-free grammar where G = ( { S, A, B, C}, { a,b, d}, P, S ) with the productions in P given below.

S → ABAC
A → aA ∣ ε
B → bB ∣ ε
C → d

(ε denotes null string). Transform the grammar G to an equivalent context-free grammar G\' that has no ε productions and no unit productions. (A unit production is of the form x → y, and x and y are non terminals.)

Question 57

Given below are the transition diagrams for two finite state machine M₁ and M₂ recognizing languages L₁ and L₂ respectively.               . a) Display the transition diagram for a machine that recognizes L₁ . L₂, obtained from transition diagrams for M₁ and M₂ by adding only ε transitions and no new states. b) Modify the transition diagram obtained in part(a) obtain a transition diagram for a machine that recognizes (L₁.L₂)^∗ by adding only ε transitions and no new states. (Final states are enclosed in double circles).

Question 58

Let Q = ( {q₁,q₂}, {a,b}, {a,b,Z}, δ, Z, ϕ) be a pushdown automaton accepting by empty stack for the language which is the set of all non empty even palindromes over the set {a,b}. Below is an incomplete specification of the transitions δ. Complete the specification. The top of the stack is assumed to be at the right end of the string representing stack contents.

δ(q1, a, Z)={(q1, Za)}
δ(q1, b, Z)={(q1, Zb)}
δ(q1, a, a)={(..... , .....)}
δ(q1, b, b)={(..... , .....)}
δ(q2, a, a)={(q2, ϵ)}
δ(q2, b, b)={(q2, ϵ)}
δ(q2, ϵ, Z)={(q2, ϵ)}

Question 59

A two dimensional array A[1...n][1...n] of integers is partially sorted if

∀i, j ∈ [1...n−1], A[i][j] < A[i][j+1] and A[i][j] < A[i+1][j]

Fill in the blanks: a) The smallest item in the array is at A[i][j] where i=..................and j=...................... b) The smallest item is deleted. Complete the following O(n) procedure to insert item x (which is guaranteed to be smaller than any item in the last row or column) still keeping A partially sorted.

procedure insert (x: integer);
var i,j: integer;
begin
    i:=1; j:=1, A[i][j]:=x;
    while (x > ...... or x > ......) do
        if A[i+1][j] < A[i][j] ......... then begin
            A[i][j]:=A[i+1][j]; i:=i+1;
        end
        else begin
            ............
        end
    A[i][j]:= .............
end
.

Question 60

Insert the characters of the string K R P C S N Y T J M into a hash table of size 10. Use the hash function

h(x) = ( ord(x) – ord("A") + 1 ) mod10

If linear probing is used to resolve collisions, then the following insertion causes collision

• Y

• C

• M

• P