Logic Functions and

their Representation

Logic Functions and their Representation Slide 2

Combinational Networks

x1x2

xn

f

Logic Functions and their Representation Slide 3

Logic Operations

• Truth tables

x y

AND

xy

OR

xyNOT

x

NAND

xy

NOR

xy

EXOR

xy

0 0 0 0 0 1 1 0

0 1 0 1 0 1 0 1

1 0 0 1 1 1 0 1

1 1 1 1 1 0 0 0QuickTime™ and a

TIFF (LZW) decompressorare needed to see this picture.

QuickTime™ and aTIFF (LZW) decompressor

are needed to see this picture.QuickTime™ and aTIFF (LZW) decompressorare needed to see this picture.

QuickTime™ and aTIFF (LZW) decompressor

are needed to see this picture.

QuickTime™ and aTIFF (LZW) decompressor

are needed to see this picture.QuickTime™ and a

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Logic Functions and their Representation Slide 4

SOP and POS

• Definition: A variable xi has two literals xi and xi. A logical product where each variable is represented by at most one literal is a product or a product term or a term. A term can be a single literal. The number of literals in a product term is the degree. A logical sum of product terms forms a sum-of-products expression (SOP). A logical sum where each variable is represented by at most one literal is a sum term. A sum term can be a single literal. A logical product of sum terms forms a product-of-sums expression (POS).

Logic Functions and their Representation Slide 5

Minterm• A minterm is a logical product of n literals

where each variable occurs as exactly one literal

• A canonical SOP is a logical sum of minterms, where all minterms are different.

• Also called canonical disjunctive form or minterm expansion

Logic Functions and their Representation Slide 6

Maxterm

• A maxterm is a logical sum of n literals where each variable occurs as exactly one literal

• A canonical Pos is a logical product of maxterms, where all maxterms are different.

• Also called canonical conjunctive form or maxterm expansion

Show an example

Logic Functions and their Representation Slide 7

Shannon Expansion

• Theorem: An arbitrary logic function f(x1,x2,…,xn) is expanded as follows:

f(x1,x2,…,xn) = x1f(0,x2,…,xn) x1f(1,x2,…,xn)

(Proof)

When x1 = 0,

= 1f(0,x2,…,xn) 0f(1,x2,…,xn)

= f(0,x2,…,xn)

When x1 = 1,

similar

Logic Functions and their Representation Slide 8

Expansions into Minterms

• Example: Expand f(x1,x2,x3) = x1(x2 x3)

• Example: minterm expansion of an arbitrary function

• Relation to the truth table• Maxterm expansion (duality)

Logic Functions and their Representation Slide 9

Reed-Muller Expansions

• EXOR properties

(x y) z = x (y z)

x(y z) = xy xz

x y = y x

x x = 0

x 1 = x

Logic Functions and their Representation Slide 10

Reed-Muller Expansions

• Lemma xy = 0 x y = x y

(Proof)

() Let xy = 0

x y = xy xy = (xy xy) (xy xy) = x y() Let xy ≠ 0

x = y = 1. Thus x y = 0, x y = 1

Therefore, x y ≠ x y

Logic Functions and their Representation Slide 11

• An arbitrary 2-varibale function is represented by a canonical SOPf(x1,x2) = f(0,0)x1x2 f(0,1)x1x2 f(1,0)x1x2 f(1,1) x1x2

Since the product terms have no common minterms, the can be replaced with f(x1,x2) = f(0,0)x1x2 f(0,1)x1x2 f(1,0)x1x2 f(1,1) x1x2

Next, replace x1= x1 1, and x2= x2 1

Show results!

Logic Functions and their Representation Slide 12

PPRM

• An arbitrary n-variable function is uniquely represented as f(x1,x2,…,xn) = a0

a1x1 a2x2 … anxn

a12 x1x2 a13 x1x3 … an-1,nxn-1xn

… a12…nx1x2…xn