COSC 3330/6308Solutions to

First Problem Set

Jehan-François PârisSeptember 2012

First problem (I)

A program consists of two parts, namely– One part is purely sequential and takes 64 s to

complete – Another part takes 1,024 s when run on a

uniprocessor architectureCan be easily decomposed into two or more

parallel tasksSpeedup would be proportional to the number

of tasks executing in parallel.

First problem (II)

What would be the speedups that the program would achieve if it was to run on computers with 2, 4, 8, 16, and 32 processors?

What would be the maximum speedup that the program would achieve if it was to run on a computer with an unlimited number of processors?

Answer (I)

Will use Amdahl's law

in

on

TTTT

Speedup

Answer (II)

Sequential Part (s)

Number of Proc.

Other Part (s)

Total (s)

Speedup

64 1 1,024 1,088 NA

64 2 512 576 1.889

64 4 256 320 3.400

64 8 128 192 5.667

64 16 64 128 8.500

64 32 32 96 11.333

64 infinity 0 64 17.000

Second problem

Simplify the two following logical expressions:

– ABC'D' + AB'D' + ACD + CD

– (A BC) (B + A)

using both algebra and Karnaugh maps.

First expression (I)

ABC'D' + AB'D' + ACD + CD =

ABC'D' + AB'D' + CD =

ABC'D' + AB'D' + AB'C'D' + CD =

AC'D' + AB'D' + CD

First expression (II)

C'D' C'D CD CD'

A’B’ 0 0 1 0

A’B 0 0 1 0

AB 1 0 1 0

AB’ 1 0 1 1

AC'D' + AB'D' +CD

Second expression (I)

(A BC) (B + A) =

(A(BC)' + A'BC) (A + B) =

(AB' + AC' + A'BC) (A + B) =

AB' + AC' + A'BC + ABC' =

AB' +AC' +A'BC

Second expression (II)

A B C BC A BC A+ B PRODUCT

F F F F F F F

F F T F F F F

F T F F F T F

F T T T T T T A'BC

T F F F T T T AB'C'

T F T F T T T AB'C

T T F F T T T ABC'

T T T T F T F

Second expression (III)

B'C' B'C BC BC'

A’ 0 0 1 0

A 1 1 0 1

AB' + AC' +A'BC

Second expression (IV)

A'B' A'B AB AB'

C’ 0 0 1 1

C 0 1 0 1

AB' + AC' +A'BC

Third problem

Simplify the expression

– (ABC)' (A + DE)' (A BC) (B + E)

and convert it to a form that can be represented using a programmable logic array.

Answer (I)

(ABC)' (A + DE)' (A BC) (B + E) Let us do it in two parts

– (ABC)' (A + DE)' =(A' + B' + C')(A'(DE)') =(A' + B' + C')(A'(D' +E') =A'(D' +E') + A'B'(D' +E') +A’C’(D’ +E’) =A'D' +A'E'

Answer (II)

The second part is – (A BC) (B + E) =

(A(BC)' + A'BC)(B + E) =(AB' + AC' + A'BC)(B + E) =ABC' + A'BC + AB'E + AC'E + A'BCE

Answer (III)

The product is– (A'D' +A'E')

(ABC' + A'BC + AB'E + AC'E + A'BCE) =A'BCD' + A'BCE' + A'BCD'E = A'BCD' + A'BCE'

Sum of products can be represented by a PLA

I checked the answer on a spreadsheetwith 25 rows and 14 columns

Fourth problem

Implement the double implication operation:– A B = AB +A’B’

using only NAND gates.

NAND gates

A nand B = (AB)' = A' + B'

nand (A) nand nand(B) = (A'B')' = A + B

nand(A nand B) = (AB)'' = AB

nand(nand(A) nand nand(B)) =(A' B')'' = A'B'

Answer

AB

A'+B'

A

B

A+B((A + B)(A' + B')' =(A + B)' + (A' + B')' =A'B' +AB

I started with a solution havingmore NANDs and simplified it

Fifth problem

Build a regular D flip-flop using an R’S’ latch, that is, an RS latch with inverted values for R and S, and as few NOR gates as possible.

NOR gates

A nor B = (A + B)' = A' B'

nor (A) nor nor(B) = (A' + B')' = AB

nor(A nor B) = (A+B)'' = A + B

nor(nor(A) nor nor(B)) =(A' + B')'' = A' + B'

From S'R' latch to D flip-flop

S'R' has three inputs– S' sets latch when S' = 0

and clock = 1– R' resets latch when R' = 0

and clock = 1 D flip-flop

– Stores a 1 at clock transition when input is 1– Stores a 0 at clock transition when input is 0

S'

R'

Q'

Q

Clock

D flip-flop using an S-R latch

X

Clock

S

R

Q

Q'

Clock

D flip-flop using an S'-R' latch

X

Clock

S'

R'

Q

Q'

Clock

Introducing NORs

X

Clock

S'

R'

Q

Q'

Clock

(Clock+Clock')'=ClockClock'

Note

The solution is fairly simple because we assumed that the R'S' latch had a clock entry

Solution for R'S' latch without clock input is more complex

Q

Q’R’

S’

Sixth problem (I)

Build a synchronous sequential circuit with– Two inputs, respectively named P—for

plus—and M—for minus– Two outputs respectively named O—for

overflow—and U—for underflow

Sixth problem (II)

The transitions are:

Answer

Eight states– Three flip-flops X Y Z

Will consider separately– How P inputs affect X, Y, Z– How M inputs affect X, Y, Z– When counters outputs O and U

How P inputs affect Z

Y'Z' Y'Z YZ YZ'

P'X' 0 1 1 0

P'X 0 1 1 0

PX 1 0 0 1

PX’ 1 0 0 1

Solution

Z = P'Z + PZ' – Will use a T flip-flop triggered by P

How P inputs affect Y

Y'Z' Y'Z YZ YZ'

P'X' 0 0 1 1

P'X 0 0 1 1

PX 0 1 0 1

PX’ 0 1 0 1

Solution

Y = P'Y + PYZ' + PY'Z – Will use a T flip-flop triggered by PZ

How P inputs affect X

Y'Z' Y'Z YZ YZ'

P'X' 0 0 0 0

P'X 1 1 1 1

PX 1 1 0 1

PX’ 0 0 1 0

Solution

X = P'X + PXY' + PXZ' + PX'YZ – Will use a T flip-flop triggered by PYZ

How M inputs affect Z

Y'Z' Y'Z YZ YZ'

M'X' 0 1 1 0

M'X 0 1 1 0

MX 1 0 0 1

MX’ 1 0 0 1

Solution

Z = M'Z + MZ' – Will use a T flip-flop triggered by M

How M inputs affect Y

Y'Z' Y'Z YZ YZ'

M'X' 0 0 1 1

M'X 0 0 1 1

MX 1 0 1 0

MX’ 1 0 1 0

Solution

Y = M'Y + MYZ + M Y'Z' – Will use a T flip-flop triggered by MZ'

How M inputs affect X

Y'Z' Y'Z YZ YZ'

M'X' 0 0 0 0

M'X 1 1 1 1

MX 0 1 1 1

MX’ 1 0 0 0

Solution

X = M'X + MXY + MXZ +MX'Y'Z' – Will use a T flip-flop triggered by MY'Z'

How P inputs affect O

Y'Z' Y'Z YZ YZ'

P'X' 0 0 0 0

P'X 0 0 0 0

PX 0 0 1 0

PX’ 1 0 0 0

Solution

O = PXYZ

How M inputs affect U

Y'Z' Y'Z YZ YZ'

M'X' 0 0 0 0

M'X 0 0 0 0

MX 0 0 0 0

MX’ 1 0 0 0

Solution

U = MX'Y'Z' – Will use a T flip-flop triggered by MY'Z'

Summary

Flip-flops– Z: T flip-flop triggered by P + M– Y: T flip-flop triggered by PZ + MZ'– X: T flip-flop triggered by PYZ + MY'Z'

Outputs– O = PXYZ– U = MX'Y'Z'