Introduction to Binary Decision Diagrams

1
Introduction to Binary Decision Diagrams

• Slawomir Pilarski

2
Textbooks

• Logic Synthesis and Verification Algorithms by
  G. D. Hachtel and F. Somenzi (Chapter 6)
• Algorithms and Data Structures in VLSI Design
  by C. Meinel and T. Theobald (Chapter 6)

3
Topics

• BDDs by example
• How to build a BDD
• Reduction rules
• Variable order
• Binary operations

4

• Binary Decision Diagrams are used to represent
  and manipulate Boolean Functions
• Manipulation of Boolean functions is a
  fundamental part of computer science

5
BDD

• Graph representation of a Boolean function
• Canonical representation
• Very effective function manipulation algorithms

6
A Bit of History

• Old concept 1959 (?)
• Renewed interest Bryant, 1986
• Currently used in every sub-area of logic
  synthesis and verification

7
Constant

• A node represents 0 or 1

0
1
8
One-variable functions
F(a) a
F(a) not(a)
a
a
0
1
0
1
0
1
1
0
9
More one-variable functions? ?
10
More one-variable functions ?
F(a) 0
F(a) 1
a
a
0
1
0
1
0
0
1
1
11
Merge all identical subgraphs
F(a) 0
F(a) 0
a
a
0
1
0
1
0
0
0
12
Node elimination
F(a) 0
F(a) 0
a
0
1
0
0
13
AND
F(a,b) ab
a
0
1
b
0
1
0
0
1
14
AND - merge
a
a
0
1
0
1
b
b
0
1
0
0
1
0
1
0
1
15
AND another graph ?
b
a
0
1
0
1
a
b
0
1
0
1
0
1
0
1
16
XOR
a
0
1
b
b
0
1
0
1
1
0
0
1
17
F(a,b,c) abc
a
0
1
a
0
1
0
1
b
0
1
AND
b
0
1
0
1
0
1
18
F(a,b,c) abc
c
0
1
c
0
1
0
1
1
a
0
1
OR
a
0
1
b
0
1
b
0
1
0
1
0
1
19
F(a,b,c) abc
c
c
0
1
0
1
1
a
a
0
1
0
1
b
b
0
1
0
1
0
1
0
1
20
F(a,b,c) abc
a
0
1
b
0
1
c
0
1
0
1
21
F(a,b,c) abc bad var. order
a
0
1
c
c
0
1
0
1
b
0
1
1
0
22
Binary Operations

• Shannons expansion f g
• v ( f(v0) g(v0) ) v ( f(v1) g(v1))

v
v
v
0
1
0
1
0
1
d
bd
ac
b
c
a
23
(ab)(bc)
a
0
1
a
b
0
1
0
1
b
0
0
1
0
1


0
1
b
b
0
1
0
1
b
0
1
c
c
0
1
0
1
c
0
1
0
1
0
1
0
1
24
(ab)(bc)
a
0
1
0


b
0
1
b
0
1
c
0
1
c
0
1
0
1
0
1
25

a
0
1
b
0
1
0
1


b
0
1
c
0
1
0
1
26

b
b
0
1
0
1
0
1
0
1



b
0
1
0
c
0
1
c
0
1
0
1
0
1
27

a
0
1
b
0
1
b
0
1
0
1
0
1


b
0
1
c
0
1
0
1
28
(ab)(bc)
a
0
1
b
0
1
0
1
b
0
1
c
0
1
0
1
29
Problems

• Find a good variable order for f (abe)(dc).
  Discuss your reasoning.
• For functions f abc and g bc d
• compute fg draw the BDD

30
Next topic

• Design considerations for a BDD package
• Data structures for BDD nodes
• Unique table
• Computed table
• ITE algorithm