sop pos.pdf

7/28/2019 SOP POS.pdf

1/17

SOP vs. POS

A sum of products (SOP) expressioncontains:

Only OR (sum) operations at theoutermost level

Each term that is summed must be

a product of literals If any input to OR gate is true then

f is true

Aproduct of sums (POS) expressioncontains:

Only AND (product) operations atthe outermost level

Each term must be a sum of

literals If any input to the AND gate is

false then f is false

f(x,y,z) = y (x + y + z) (x + z)f(x,y,z) = y + xyz + xz


2/17

Minterms vs. Maxterms

A minterm is a special product ofliterals, in which each input variableappears exactly once. Each minterm is true for exactly

one combination of inputs:

A maxterm is a sum of literals, inwhich each input variable appearsexactly once.

Each maxterm is false for exactly onecombination of inputs:

Minterm Is true when Shorthand

xyz x=0, y=0, z=0 m0xyz x=0, y=0, z=1 m1xyz x=0, y=1, z=0 m2xyz x=0, y=1, z=1 m3xyz x=1, y=0, z=0 m4xyz x=1, y=0, z=1 m5xyz x=1, y=1, z=0 m6

xyz x=1, y=1, z=1 m7

Maxterm Is false when Shorthand

x + y + z x=0, y=0, z=0 M0x + y + z x=0, y=0, z=1 M1x + y + z x=0, y=1, z=0 M2x + y + z x=0, y=1, z=1 M3x + y + z x=1, y=0, z=0 M4x + y + z x=1, y=0, z=1 M5x + y + z x=1, y=1, z=0 M6

x + y + z x=1, y=1, z=1 M7


3/17

Billy's Burger

Billy will eat a burger with mustard or a burgerwithout mustard, but will not eat mustardwithout a burger. (trivially Billy cannot eat

without either)


4/17

Billy's Burger Truth Table


without either)

b m e

0 0 0

0 1 01 0 1

1 1 1


5/17

Billy's Burger - m/M


without either)

b m e m/M

0 0 0 M0

0 1 0 M11 0 1 m3

1 1 1 m4


6/17

Billy's Burger SOP vs. POS

E = bm'+bm

True if a burger and nomustard

E = (1*1)+(1*0)E = 1 + 0 = 1

True if a burger andmustard

E = (1*0)+(1*1)

E = 0 + 1 = 1

Otherwise false

E = (b+m).(b+m')

False if no burger andno mustard

E = (0+0)(0+1)E = (0)(1) = 0

False if no burger andmustard

E = (0+1)(0+0)

E = (1)(0) = 0

Otherwise true


7/17

Minimal Sum of Products

Minimal sum of products (MSP) form:

There are a minimal number of product terms in theexpression.

Each term has a minimal number of literals. Circuit-wise, this leads to a minimal two-level

implementation.

What is the MSP for Billy's burger?


8/17

Minimal Sum of Products

Minimal sum of products (MSP) form:

There are a minimal number of product terms in theexpression.

Each term has a minimal number of literals.

Circuit-wise, this leads to a minimal two-levelimplementation.

What is the MSP for Billy's burger?

Logically we can see that Billy will eat a burger with orwithout ketchup and we can use Boolean Algebra to easilyprove it

E = bm'+bm = b(m'+m) = b(1) = b


9/17

Billy's Burger Getting Complicated

This is just two variables. What if we want tooptimize a function to make Billy a burger, buthave more toppings?

Ketchup Pickles

Lettuce

Tomato

etc.


10/17

Karnaugh Map

The Karnaugh map (K-map for short), Maurice Karnaugh's 1953refinement of Edward Veitch's 1952 Veitch diagram, is a method tosimplify Boolean algebra expressions.

takes advantage of humans' pattern-recognition capability

In a Karnaugh map the boolean variables are transferred andordered according to the principles ofGray code

Once the table is generated and the output possibilities aretranscribed, the data is arranged into the largest possible groupscontaining 2^n cells (n=0,1,2,3...) and the MSP is generated

through the axiom laws of boolean algebra.


11/17

Billy's Toppings

We know Billy will eat a burger plain or with justmustard so we will concentrate on four toppings

Mustard(m), Ketchup(k), Special Sauce(s), and

Pickles(p)

k m p s e k m p s e

0 0 0 0 1 1 0 0 0

0 0 0 1 1 0 0 1

0 0 1 0 1 0 1 00 0 1 1 1 0 1 1

0 1 0 0 1 1 1 0 0

0 1 0 1 1 1 0 1

0 1 1 0 1 1 1 0

0 1 1 1 1 1 1 1


12/17

Billy's Toppings

We know Billy will eat a burger plain or with justmustard so we will concentrate on four toppings

Mustard(m), Ketchup(k), Special Sauce(s), and

Pickles(p)

k m p s e k m p s e

0 0 0 0 1 1 0 0 0 0

0 0 0 1 0 1 0 0 1 0

0 0 1 0 0 1 0 1 0 10 0 1 1 0 1 0 1 1 1

0 1 0 0 1 1 1 0 0 0

0 1 0 1 1 1 1 0 1 0

0 1 1 0 0 1 1 1 0 1

0 1 1 1 1 1 1 1 1 1


13/17

Billy's Topping's - m/M

It is easier to find the minterms and maxtermsof e(k,m,p,s) before transferring it to theKarnaugh Map

k m p s e m/M k m p s e m/M

0 0 0 0 1 m0 1 0 0 0 0 M8

0 0 0 1 0 M1 1 0 0 1 0 M9

0 0 1 0 0 M2 1 0 1 0 1 m10

0 0 1 1 0 M3 1 0 1 1 1 m11

0 1 0 0 1 m4 1 1 0 0 0 M12

0 1 0 1 1 m5 1 1 0 1 0 M13

0 1 1 0 0 M6 1 1 1 0 1 m14

0 1 1 1 1 m7 1 1 1 1 1 m15


14/17

K-map

Using the minterms and Maxterms we caneasily fill out the K-map

P

m0 m1 m3 m2

m4 m5 m7 m6M

Km12 m13 m15 m14

m8 m9 m11 m10

S

P

M0 M1 M3 M2

M4

M5

M7

M6 M

KM12 M13 M15 M14

M8 M9 M11 M10

S

P

M

K

S

P

M

K

S


15/17

K-map

Using the minterms or Maxterms we can easilyfill out the K-map

P

m0 m1 m3 m2

m4 m5 m7 m6M

Km12 m13 m15 m14

m8 m9 m11 m10

S

P

M0 M1 M3 M2

M4 M5 M7 M6M

KM12 M13 M15 M14

M8 M9 M11 M10

S

P

1

1 1 1M

K1 1

1 1

S

P

0 0 0

0M

K0 0

0 0

S


16/17

K-Map Group and Reduce

Now we can group terms and reduce them tofind the MSP

P

1 0 0 0

1 1 1 0M

K0 0 1 1

0 0 1 1

S


17/17

Billy's Toppings - MSP

An answer would be

But that answer isn't unique

How can we tell if something is the MSP?

Covered on Monday!

e k , m , p , s =k ' p ' s 'k ' ms ' pk

e k , m , p , s =k ' p ' s 'k ' mp 'mps pk

sop pos.pdf

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