CSE 498M598M, Fall 2002 Digital Systems Testing

1
CSE 498M/598M, Fall 2002 Digital Systems Testing

• Instructor Maria K. Michael
• CSE Dept., University of Notre Dame
• LECTURE 7
• Fundamentals of
• Binary Decision Diagrams

2
Overview

• Representations of boolean functions
• Canonical and non-canonical forms
• BDDs and reduction rules
• Node Decomposition
• BDD Variants (ZBDDs, Hybrid)
• BDDs with complement edges
• Variable ordering/reordering
• BDD basic operators
• Some basic BDD routines

3
Boolean Function Representations

• Truth table
• SoP (DNF) and PoS (CNF)
• Factored Form
• Decision tree
• Reduced decision tree
• BDD
• Zero-Suppressed BDD
• BDD with complement edges

4
Truth Table, SoP, and PoS

• Sum-of-products
• F x1x2x3 x1x2x3 x1x2x3
• Product-of-sums
• F (x1x2x3) (x1x2x3)
• (x1x2x3) (x1x2x3)
• (x1x2x3)

F
x3
x2
x1
0
0
0
0
0
1
0
0
0
0
1
0
1
1
1
0
0
0
0
1
1
1
0
1
0
0
1
1
1
1
1
1
5
Whats a good representation of Boolean functions?

• Let f V? B ? 0,1, Vset of boolean variables,
  and ? be a truth assignment for f, s.t. f(?)1
• Satisfiability Exists some ? such that f(?)1?
• Tautology For all ? , is f(?)1?
• Cooks Theorem
• Satisfiability of Boolean expressions is
  NP-Complete
• Tautology is co-NP-complete
• Perfect representations are hopeless. Good
  representations are compact and efficient on
  real-life examples.

6
Truth Table and Ordered BDD
1
1
1
1
0
0
0
0
x1
1
1
0
0
1
1
0
0
x2
1
0
1
0
1
0
1
0
x3
x1
1
0
1
0
1
0
1
0
0
0
F
x2
x2
0
1
1
0
x3
x3
x3
x3
1
0
0
1
0
1
0
1
1
1
0
1
0
0
0
0
7
Reduction of OBDDs
Rule 2 Elimination Redundant nodes should not
be present
Rule 1 Merging Nodes must be unique
a
a
a
a
a
b
b
b
b
b
b
b
b
8
Example of OBDD Reduction
x1
x1
x1
x2
x2
x2
x2
x2
x3
x3
x3
x3
x3
x3
x3
0
1
0
1
0
1
0
1
0
0
0
1
Reduced OBDD (ROBDD)
OBDD
reduction
9
More BDD Examples
1 edge
0 edge
a
10
Properties of BDDs

• Canonicity
• Compactness (with some exceptions)
• Fast computation (with some exceptions)
• Represent a variety of discrete objects
• Boolean functions and relations
• Combinational sets and subsets (ZBDDs)
• Partitions of states
• Multivalued functions
• Facilitate symbolic methods
• Two-level minimization
• State traversal of FSMs, etc.
• Verification and Test

11
Applicability of BDDs

• Combinational Sequential circuits ? Facilitate
  symbolic methods, ex.
• Two-level minimization
• State traversal of FSMs, etc.
• Verification and Test
• Automata
• Combinatorial problems (ZBDDs)
• Temporal logic model checking
• Program analysis

12
Variable Ordering
Fa1b1 a2b2
a1
a1
a2
a2
b1
a2
b1
b1
b2
b2
0
1
0
1
a1lta2ltb1lt b2
a1ltb1lta2lt b2
13
Dynamic Variable Reordering
F
F0
F
F1
x1
x2
x2
x2
F1
F0
x1
x2
x2
x1
x3
x3
x3
x3
x3
x3
x3
x3
Variable reordering is a local operation
14
Complement Edges

• BDD edges can carry attributes
• Using complementation as an attribute for edges
  saves BDD nodes and makes NOT a constant time
  operation
• To maintain canonicity, the use of attribute
  edges should be constrained

15
Equivalent Pairs of Functions
x
x
x
x
x
x
x
16
BDD node decomposition

• Shannon Expansion
• A boolean function can be expanded with respect
  to any variable
• F( x,y,z ) x Fx x Fx
• where Fx and Fx are positive (negative)
  cofactors
• Fx F( 1,y,z ), Fx F( 0,y,z )

17
Node decomposition based on Shannon expansion
F x.Fx x.Fx
Fx F(x1)
Fx F(x0)
18
BDD node decomposition

• XOR-based expansion
• Negative Davio/Reed-Muller ? ZBDDs
• F( x,y,z ) Fx ? x.F?x , where F?x Fx?Fx
• Positive Davio
• F( x,y,z ) Fx ? x.F?x , where F?x Fx?Fx

19
Zero-Suppressed BDDs
Rule 2 Elimination Redundant nodes should not be
present
Rule 2 Merging Remove nodes with 1-edge
pointing to 0-terminal
a
b
b
0
20
BDD Operators

• ITE operator
• APPLY operator
• RESTRICT operator
• Derived operators
• Compose
• Existential quantification
• Universal quantification

21
IF-THEN-ELSE (ITE) Operator

• Boolean operations over 2 arguments can be
  expressed as ITE of F, G, and constants
• ITE( F, G, H ) F G F H
• Example AND( F, G ) ITE( F, G, 0 )
• Computation of boolean operations is based on the
  Shannon expansion
• ITE(F,G,H) ITE(x, ITE(Fx,Gx,Hx),
  ITE(Fx,Gx,Gx) )

22
APPLY operator

• APPLY( F, G ) operator is a shorthand for any
  two-variable boolean operator
• APPLY is reducible to ITE
• It follows that APPLY can be computed recursively
  just like ITE
• APPLY(F,G) x APPLY(Fx,Gx)
• x APPLY(Fx ,Gx )

23
Pseudocode for APPLY operator

• function Apply( F, G )
• if ( AlreadyComputed( F, G ) ) return the result
• else if ( F0,1 G0,1 ) return oper( F,
  G )
• else if ( Var( F ) Var( G ) )
• u CreateNode( Var(F), Apply(Fx,Gx),
  Apply(Fx,Gx))
• else if ( Var( F ) lt Var( G ) )
• u CreateNode( Var(F) , Apply(Fx,G ),
  Apply(Fx,G ))
• else / if ( Var( F ) gt Var( G ) ) /
• u CreateNode( Var(F) , Apply(F,Gx ),
  Apply(F,Gx ))
• InsertComputed( F,G,u )
• return u

24
Example of APPLY(F,G) with oper OR Facbcd
Gacd FG ?
A1,B1
A1
B1
a
a
A2,B2
A2
b
A6,B2
A6,B5

A6
B5
c
c
B2
A3,B2
A5,B2
A3
A3,B4
d
d
A4,B3
A5,B4
0
1
0
1
B3
B4
A4
A5
25
Example of APPLY(F,G) with oper ORFacbcd
Gacd FG abcd
A1,B1
a
a
A2,B2
b
b
A6,B2
A6,B5
c
c
c
A3,B2
A5,B2
A3,B4
d
d
1
1
A4,B3
A5,B4
0
0
1
1
26
Pseudocode for RESTRICT operator

• function Restrict( F, var, value )
• if ( AlreadyComputed( F, var, value ) ) return
  result
• else if ( Var( F ) gt var ) return F
• else if ( Var( F ) lt var )
• u CreateNode( Var(F), Restrict(Fx, var,
  value), Restrict(Fx, var, value) )
• InsertComputed(F, var, value, u ) return u
• else / ( Var( F ) var / if ( value 0 )
• return Restrict(Fx, var, value)
• else / ( Var( F ) var value 1 ) /
• return Restrict(Fx , var, value)

27
Example of RESTRICT(F,b,1) Fbcabc F(b1)
?
a
a
b
b
b
b
c
c
c
c
c
0
0
0
1
1
1
28
Derived Operations COMPOSE

• Given F(x) and G(y), find F(G(y))
• Using Shannon Expansion
• F(x) x Fx x Fx
• F(G(y)) G(y) Fx G(y) Fx
• COMPOSE is reduced to two operations RESTRICT and
  three operations APPLY

29
Derived Operations Quantification

• Given a function F(x1, x2, x3)
• Existential quantification of F w.r.t. x1 is
• ?x1 F(x1, x2, x3) F(0, x2, x3) F(1, x2,
  x3)
• Also called Smoothing
• Universal quantification of F w.r.t. x1 is
• ?x1 F(x1, x2, x3) F(0, x2, x3) F(1, x2,
  x3)
• Also called Consensus

30
Summary of Operations on BDDs

• Apply NOT, AND, OR, EXOR, etc.
• Restrict cofactoring
• Compose variable/function substitution
• Quantification (existential, universal)
• All operators are based on the ITE( ) operator,
  which is used to create a new node (CreateNode( )
  in pheudocodes)

31
Some basic BDD routines

• Tautology constant time
• Complementation constant time when using
  complemented edges
• Satisfiability linear to BDD size
• Cube/Minterm Counting linear to BDD size
• Both 3 4 are reduced to graph traversal
  problems