Basic Definitions
Let B = {0,1} Y = {0,1,2}
A logic function f in n inputs x1, x2, ...xn and
m outputs y1,y2, ...ym is a function
f : Bn
Ym
X  [ x , x ,..., x ]  B is the input
Y  [ y , y ,..., y ] Y is the output
For each component fi, i = 1,2, ...,m, define
n
1
2
n
m
1
2
m
ON_SET:the set of input values x such that fi(x) =1
OFF_SET:the set of input values x such that fi(x) = 0
DC_SET:the set of input values x such that fi(x) = 2
Completely specified function : DC_SET = f, for fi
Incompletely specified function : DC_SET  f,for some fi
m=1 => a single output function
m>1 => a multiple output function
Representations
1. Truth Table
full adder
X Y Cin Sum Cout
0 0
0
0
0
0 0
1
1
0
0 1
0
1
0
0 1
1
0
1
1 0
0
1
0
1 0
1
0
1
1 1
0
0
1
1 1
1
1
1
• a multiple output function
• Sum : on-set = {(0 0 1), (0 1 0), (1 0 0),
(1 1 1)}
off-set= {(0 0 0), (0 1 1), (1 0 1),
(1 1 0)}
• a completely specified function
Representations
2. Geometrical representation
1 variable
0
1
2 variables
00
01
10
3 variables
11
011
001
111
101
010
010
110
000
100
sum : on-set ={(0 0 1),(0 1 0),(1 0 0),(1 1 1)}
off-set ={(0 1 1),(1 0 1),(1 1 0),(0 0 0)}
Representations
3. Algebraic representations
Canonical sum of product
Cout = x’yCin + xy’Cin + xyCin’ + xyCin
Reduced sum of product
Cout = yCin + xCin + xy
= yCin + xCin + xyCin’
Multi-level representation
Cout = Cin (x + y) + xy
Representations
4. Reduced Ordered Binary Decision
Diagram (ROBDD)
Decision graph:
f = x1x2 + x3
Terminal node:
• attribute
x1
0
1
value (v) = 0
x2
x2
value (v) = 1
0 1 0 1
x3 x3 x 3 x3
0 10 10 10 1
Non-terminal node:
0 1 0 10 1 1 1
• two children
• index (v) = i
low (v)
high (v)
Evaluate an input vector
ROBDD
A BDD graph which has a vertex v as root
corresponds to the function Fv :
(1) If v is a terminal node:
a) if value (v) is 1, then Fv = 1
b) if value (v) is 0, then Fv = 0
(2) If F is a nonterminal node(with index(v) = i)
Fv(xi , ...xn) = xi’ Flow(v) (xi+1 , ...xn) +
xi Fhigh(v) (xi+1 , ...xn)
ROBDD
Reduced BDD :
1. No distinct vertices v and v’ such that subgraphs rooted by v and v’ are isomorphism.
2. No vertex v with low (v) = high (v)
F=x1x2+x1x3+x1x2x3
x1
0
x2
0
0
0
1
1
0
x3
x3
1 0
1
x3
1
1 0
x2
0
x3
1
1 0
0
1
1 1 1
0
0
x2
0
x2
1
0
0
0 3 1 0
x2
x3 1
0
1
x2
0
x3
0
0
1
0
1
1
x
x3 1
x1
x1
x1
1
1
1
0
1
x2
0
x3
0
0
1
1
1
0
1
x3
1
ROBDD
• Ordered BDD: if x < y, then all nodes
representing x precede all nodes
representing y
• Given an ordering of variables, reduced
OBDD is canonical
Example
• The size of OBDD depends on the ordering of
variables
ex : x1x2 + x3x4
x1 < x 2 < x 3 < x 4
x1
1
0
0
x2
x3 1
0
x4
0
0
1
1
1
Example
• For a good ordering, the size of an OBDD
remains reasonably small
• Multiplier is an exception
x1
1
0
x3
x3
0
0
1
x2
x2
1
0 1 0 1
x4
0
1
0
x1 < x3 < x2 < x4
1
Operations on ROBDD
Apply f1 <op> f2
(f1, f2 must have the same ordering of variables,
then operations can be operated on f1, f2 )
• f1·f2 = (x1’g1 + x1g2)(x1’g3 + x1g4)
= x1’g1g3 + x1g2g4
= x1’(g1g3) + x1(g2g4)
f1
x1
f2
f1 · f2
x1
x1
0
1
0
1
g1
g2
g3
g4
• f1 + f2
• f1  f2
• etc.

0
g1g3
1
g2g4
ROBDD
• complement : f  1
• f1
f2 : f1’ + f1f2 (implication)
• time complexity
Reduced O( G log G
Apply
O( G1
G2 )
)
Property of ROBDD
(1) Commonly encountered functions have
reasonable representations.(except
multiplier)
(2) Complementation will not blow up the
representation.
(3) Canonical form so that equivalence,
satisfiability checking can be done easily.
(4) Multi-level representation
Representation
5. And-Inverter Graph (AIG)
– Simple structure
– And-Gates as nodes (shown as circles) with
two inputs as edges (shown as arrows)
– Inverters edge marked with a dot
Example
Ex: y = f( x1,x2,x3) =( (x1 ‧ x2 )’ ‧ (x2 ‧ x3)’)’
= ( x1 ‧ x2 ) ＋ (x2 ‧ x3)
AIG Construction
–
–
Start from SOP representation
Convert to AIG using DeMorgans law
x1 ＋ x2 = ( x1 ＋ x2) ’’= ( x1 ’ ‧ x2 ’) ’
2-input and
2-input nand
3-input and
2-input or
AIG Attributes
• Size is the number of AND nodes in it
• Logic level is the number of AND-gates on the
longest path from primary input to primary
output
– The inverters are ignored
6 nodes, 4 levels
AIG Canonicity
• AIGs are not canonical
– ROBDDs are canonical
– Same function represented by two
functionally equivalent AIGs with different
structures
– Different structures can still be optimal
6 nodes, 4 levels
=> area optimal
7 nodes, 3 levels
=> speed optimal
Characteristic Function
Let E be a set and A 
E
The characteristic function A is the function
XA : E -> { 0,1 }
XA(x) = 1 if x A
E
XA(x) = 0 if x 
A
A
Ex:
E = { 1,2,3,4 }
A = { 1,2 }
XA(1) = 1
XA(3) = 0
Characteristic Function
Given a Boolean function
f : Bn -> Bm ,
the mapping relation denoted as F  Bn  Bm
is defined as
F( x, y)  {( x, y)  Bn  Bm y  f ( x )}
The characteristic function of a function f is
defined for (x,y) s.t. Xf(x,y) = 1 iff (x,y) F
Characteristic Function
Ex: y = f( x1,x2 ) = x1 + x2
x1 x2 y
0
0
1
1
0
1
0
1
0
1
1
1
Fy(x1,x2,y) =
x1
x2
y
F
0
0
0
0
1
1
1
1
0
0
1
1
0
0
1
1
0
1
0
1
0
1
0
1
1
0
0
1
0
1
0
1
Operation on Logic Function
•
•
•
•
•
•
•
•
•
•
Complement
Intersection
Union
Difference
XOR
F is a tautology
Cofactor
Boolean difference
Consensus operator
Smoothing operator
Cofactor
Cofactor operation ( restriction )
fxi  fxi0 cofactor of f with respect to xi=0
f xi  f xi1 cofactor of f with respect to xi=1
f xi 0  f ( x1 , x2 ,...., xi  0,..., xn )
f xi1  f ( x1 , x2 ,...., xi  1,..., xn )
Ex :
f ( x, y , z )  xy  yz  x z
f x  0  yz  z
f x 1  y  yz
Cofator
Cofactor with respect to any cube
Ex:
f ( x, y, z, w)  xy  z w  w x
f xy  f x 0, y 0  z w  w
f xy = f x =1,y=0  z w
Shannon Expansion
f  x  f x 0  x  f x 1
 xi  y j  f xi y j  xi  y j  f xi y j  xi  y j f xi y j  xi  y j  f xi y j
Ex :
f ( x, y, z, w)  xy  zw  x w
f x  y  zw
f x  zw  w
f  x ( y  zw )  x ( zw  w )
Boolean Difference
f
x
is called Boolean difference of f with
respect to x
f
 fx  fx
x
Def :
f is sensitive to the value of x when
Ex:
f
x
f ( x , y, z , w)  xy  zw  w x
f  f ( x  0, y, z , w)  zw  w
x
f  f ( x  1, y, z , w)  y  zw
x
f  f  ( zw  w )  ( y  zw )
x
x
 yw  w y z
Consensus Operator
Def : x( f )  f x  f x
x( f ) evaluate f to be true for x=1 and
x=0
Ex:
f ( x, y, z, w)  xy  zw  w x
x( f )  f x  f x  zw  w y
Smoothing Operator
Def : x( f )  f x  f x
x ( f ) evaluate f to be true for x=1 or x=0
Ex:
f ( x, y, z, w)  xy  zw  w x
x( f )  f x  f x  zw  w  zw  y
Image Computation by Smoothing Operator
• Computation of reachable states of a sequential
finite state machine
•
Given f : Bn -> Bm and let
F ( x, y)  {( x, y)  Bn  Bm y  f ( x)}
• Use F to obtain the image of a subset A of Bn by f
– Compute the projection on Bm of the set
F∩(A× Bm)
• Smoothing operator and and are used
f ( A)( y )  S x ( F ( x, y )  A( x))
Example for Image Computation
Ex:
x1 x2
0
0
1
1
0
1
0
1
y z
0
1
1
1
A= {(0, 0), (0, 1)}
0
0
1
1
F(x1, x2, y, z) =
x1
x2
0
0
0
0
0
0
0
0
0
1
0
1
0
1
0
1
1
0
1
0
1
0
1
0
…….
y z
F
0
0
1
1
0
0
1
1
0
0
1
1
1
0
0
0
0
0
1
0
0
0
0
1
0
1
0
1
0
1
0
1
0
1
0
1
How to Build Characteristic Functions?
Ex:
x 1 x2
y z
0
0
1
1
0
1
1
1
0
1
0
1
0
0
1
1
y = x1+ x2
F1 (x1, x2, y) =
z = x1
F2 (x1, x2, z) =
F(x1, x2, y, z) =
Image computation:
S{x1, x2} (F(x1, x2, y, z)  X1 )