Introduction to Read-Muller Logic
2002. 10. 8.
Ahn, Ki-yong
Positive Polarity Reed-Muller Form
 f(x1,x2,…xn)
= a0 ^ a1x1 ^ a2x2 ^…^
anxn ^ … ^ amx1x2x3…xn
 All variables are positive polarities
 There is only 1 PPRM
 EX)

F = 1 ^ x1 ^ x2 ^ x1x2
Fixed Polarity Reed-Muller Form
 Each
variable has positive or negative
polarity
 Polarity of variable is fixed
 There is only 2n FPRMs
 EX)

F = 1 ^ x1 ^ x2’ ^ x1x2’
Generalized Reed-Muller Forms
 Each
variable can have any polarity
 Polarity of variable is not fixed
n-1
n2
 There is only 2
GRMs
 EX)

F = 1 ^ x1 ^ x2’ ^ x1’x2’
Fundamental Expansions
 Shannon

f(x1,x2,…,xn) = x1’f0(x2,…xn) ^ x1f1(x2,…xn)
 Positive


Expansion – S
Davio Expansion – pD
f(x1,x2,…,xn) = 1 f0(x2,…xn) ^ x1f2(x2,…xn)
f2 = f0 ^ f1
S
X1’
X1
1
pD
X1
Fundamental Expansions(2)
 Negative

Davio Expansion – nD
f(x1,x2,…,xn) = 1 f0(x2,…xn) ^ x1’f2(x2,…xn)
 Generalized


Davio Expansion – D
f(x1,x2,…,xn) = 1 f0(x2,…xn) ^ x1f2(x2,…xn)
x1 means both negative and positive
nD
1
X1’
1
D
X1
Kronecker Forms - KRO
S
a
a’
pD
pD
1
nD
1
 Using
c’ 1
b
1
nD
nD
c’1
b
nD
c’ 1
c’
S, pD, nD for each variable
 The same expansion must be used for the
same variable
Pseudo-KRO Form - PDSKRO
S
a
a’
S
pD
1
nD
1
 Using
c’ c’
b
b’
S
pD
c1
b
pD
c 1
S, pD, nD for each variable
c
S/D Tree
S
a
a’
D
1
a’
 Using
S
1
b
b’
D
S
D
c c’
a’c
a’bc’
c
1
a’bc
ab’
c 1
ab’c ab
b
D
c
abc
Shannon(S) and Generalized Davio(D)
Inclusive Forms for Two Variables
N=1
S
a’
a
S N=2
a’
a
S N=2
a’
a
S N=4
a’
a
D
D
S
D
D
S
S
b 1 b
b’
b 1 b 1
b b’ b 1
b b’ b b’
a’b’ a’b ab’ ab a’b’ a’b a ab a’ a’b ab’ ab a’ a’b a ab
S
D
1
b’
b’
S
N=4
a
1
D
N=8
a
1
D
N=8
a
D N=16
1
a
D
S
D
D
S
S
D
b
1
b b’
b 1 b
b b’ b b’
b 1 b 1
b ab’ ab b’ b a ab 1 b ab’ ab 1 b a ab
 NIF
= (1+2+2+4)+(4+8+8+16) = 45