1. Introduction to BDDs

2. Overview Decision trees and reduction rules
Building already reduced diagrams
Dynamic variable reordering
BDD operators
Implementations issues
(Dis)advantages of BDDs
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3. Truth Table, DNF, and CNF DNF (sum-of-products) CNF (product-of-sums)
F = x1’x2x3 + x1x2’x3 + x1x2x3
CNF (product-of-sums)
F = (x1+x2+x3) & (x1+x2+x3’) &
(x1+x2’+x3) & (x1’+x2+x3) &
(x1’+x2’+x3) x1 x2 x3 F 1
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4. Truth Table and Decision Treex1 1 x2 x3 F x1 1 x2 x2 1 1 x3 x3 x3 x3 1 1 1 1 1 1 1
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5. Examples F = a F = ab F=a1b1+ a2b2 F=a1b1+a2+b2 F = a F = a+b a a1 a1 b1 1
F = a a2
F = ab a2 a b2 b2 a b 1 1 1 1
F=a1b1+ a2b2
F=a1b1+a2+b2
F = a
F = a+b
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6. Reduction of Decision Tree
Rule 2: Elimination Rule: Redundant tests should not be present
Rule 1: Merging Rule: Nodes must be unique a a a a b b b b b b
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7. Example of Decision Tree Reduction1 1 1
reduction
Decision tree
BDD
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8. BDD Construction Typically done using APPLY operator Reduction rules
remove duplicate terminals
merge duplicate nodes
(isomorphic subgraphs)
remove redundant nodes
Redundant nodes:
nodes with identical children a b f 1 b c
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9. BDD Construction – your first BDD
Construction of a Reduced Ordered BDD
1 edge
0 edge
f = ac + bc 1 a b c f
a b c f
Truth table
Decision tree
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10. BDD Construction – cont’d1 a b c f f 1 a b c
f = (a+b)c 1 a b c
1. Remove duplicate terminals
2. Merge duplicate nodes
3. Remove redundant nodes
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11. Application to Verification
Equivalence of combinational circuits
Canonicity property of BDDs:
if F and G are equivalent, their BDDs are identical (for the same ordering of variables)
F = a’bc + abc +ab’c 1 a b c
G = ac +bc 1 a b c 
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12. Application to Verification, cont’d
Functional test generation
SAT, Boolean satisfiability analysis
to test for H = 1 (0), find a path in the BDD to terminal 1 (0)
the path, expressed in function variables, gives a satisfying solution (test vector) H 1 a b c ab
ab’c
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13. Logic Manipulation using BDDs
Useful operators 1 F F’
Complement ¬ F = F’
(switch the terminal nodes) 1
F(x,y)
x=b
F(y)
Restrict
Restrict: F|x=b = F(x=b) where b = const
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14. Useful BDD Operators - cont’d
Apply: F G
where stands for any Boolean operator (AND, OR, XOR, )  =  F G 1
Any logic operation can be expressed using only Restrict and Apply
Efficient algorithms, work directly on BDDs
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15. Apply Operation
Basic operator for efficient BDD manipulation (structural)
Based on recursive Shannon expansion
F OP G = x (Fx OP Gx) + x’(Fx’ OP Gx’)
where OP = OR, AND, XOR, etc
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16. Apply Operation - AND = = a AND c ac a c 1 a 2 1 a c 2.3 c 1 3 1.3 0.3a 2 1 a c
2.3 = = c 1 3 c
1.3
AND
0.3
1.0
1.1
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17. Apply Operation - OR = = f = ac+bc ac bc a b c 1 a c 4 5 1 a b c 1 b ca c 4 5 1 a b c bc 1 b c 6 7 c
4+6
0+0 a
7+5 1
0+6 b
6+5
0+5
0+7 OR = =
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18. Building Already Reduced Tree
assuming variable ordering
x1 < x2 < x3 < … < xn
function Build ( F, i )
if ( i > n )
if (F == 0) return Node0
else return Node1;
else
v0 = Build ( F(xi=0), i+1 );
v1 = Build ( F(xi=1), i+1 );
return CreateNode ( i, v0, v1 );
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19. Creating a Unique Node assuming there is a node table and
functions CheckExists(), Insert(), and Hash()
function CreateNode( Var, LowF, HighF )
if ( LowF == HighF )
return LowF;
else if ( CheckExists( Var, LowF, HighF ) )
return the existing node;
else Insert( Var, LowF, HighF );
return the new node;
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20. Variable Ordering F(x1, x2, y1, y2) = (x1 = y1) & (x2 = y2)
x1 < y1 < x2 < y x1 < x2 < y1 < y2 x1 x1 y1 y1 x2 x2 x2 y1 y1 y1 y1 y2 y2 y2 y2 1 1
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21. Dynamic Variable ReorderingF F F0 F1 x1 x2 x2 x2 F0 F1 x2 x2 x1 x1 x3 x3 x3 x3 x3 x3 x3 x3
Variable reordering is a local operation
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22. Characteristic Functions: Example
Problem: Given the set {p1, p2, p3, p4, p5, p6 }, create the characteristic function of the subset {p1, p3, p4} and represent it using BDDs
Step 1: Introduce an encoding of the set
p1 – p2 – p3 –
p4 – p5 – p6 –
Step 2: Define a function over the encoding variables (x1, x2, x3) such that it will be equal to 1 for minterms encoding the subset {p1, p3, p4}.
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23. Characteristic Functions: Example (continued)
{p1, p3, p4} ( x0, x1, x2 ) = =
Step 3: Represent
this function as a BDD x2 x1 x0 1
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24. Set Manipulation
Operations on combinatorial sets can be reduced to boolean operations on characteristic functions
Empty set: = 0
Union of sets: S  T= S + T
Intersection of sets: S  T= S & T
Difference of sets: S - T= S & T’
Subset relation (S  T): S - T = S & T’ = 0
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25. Relations for Logic Blocks
Suppose variables (x1,x2,x3,…) and (y1,y2,y3,) are inputs and outputs of a logic block.
Then, we can define a relation over variables (x1,x2,x3,…) and (y1,y2,y3,…). Suppose variables are related in the following way. Assignments (y1,y2,y3,…) correspond to values, which outputs take when values (x1,x2,x3,…) are applied at the inputs.
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26. Relations for Logic Blocks: Exampley1 y2 1 x1 y1 x2 y2 x3
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27. Example (continued) x1 x2 x3 y1 y2 1 other x1 x2 x3 y1 y2 F 11 x1 x2 x3 y1 y2 1
other
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28. Relations for FSMs: Example
Ins CS
CSC NS
NSC A 00 B 10
0,1 1 C 01 C 1 B A 1
0,1
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29. Example (continued)
Relation = i'a1‘a2‘b1b2‘+ a1‘a2'b1'b2' + i'a1a2'b1b2' + ia1a2'b1'b2' + i'a1a2'b1b2' + ia1a2'b1'b2' i a1 b1 a2 b2 1
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30. Efficient Operations on BDDs
Apply – NOT, AND, OR, EXOR, etc.
Quantification (existential, universal)
Replace
Compose
Specialized operators
Generalized cofactor (constrain, restrict)
Compatible projection, etc.
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31. Components of BDDs and Their Use
Nodes (represent functions; complexity)
Terminal nodes (constant functions)
Edges (relationship between functions)
Paths (true and false var. assignments)
Cuts (variable partitions and subsets)
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32. Properties of BDDs Canonicity Compactness (with some exceptions)
Represent a variety of discrete objects
Boolean functions
Compositional sets
Encodings and labelings
Facilitate symbolic methods
Two-level minimization
State traversal of FSMs …
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33. (Dis)Advantages of BDDs
Universal (for discrete data only)
Save memory (not always)
Speed-up computation (not always)
Attractive coding style (it depends…)
Implicit computation (what about good old classical methods?)
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34. Overview ITE operator APPLY operator RESTRICT operator
Derived operators
The effect of variable ordering
Deriving the Upper Bound on BDD Size for Boolean Functions
Dynamic variable reordering
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35. 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) )
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36. 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 )
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37. Pseudocode for APPLY operator
function Apply( F, G )
if ( AlreadyComputed( F, G ) ) return the result;
else if ( F=={0,1} && G=={0,1} ) return oper( F, G );
else if ( Var( F ) == Var( G ) )
u = CreateNode( Var(F), Apply(Fx’,Gx’), Apply(Fx,Gx));
else if ( Var( F ) < Var( G ) )
u = CreateNode( Var(F) , Apply(Fx’,G ), Apply(Fx,G ));
else /* if ( Var( F ) > Var( G ) ) */
u = CreateNode( Var(F) , Apply(F,Gx’ ), Apply(F,Gx ));
InsertComputed( F,G,u );
return u;
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38. F=ac+bc+d G=ac’+d F+G = ?
A1,B1 A1 B1 a a
A2,B2 A2 b +
A6,B2
A6,B5 A6 c B5 c A3 B2
A3,B2
A5,B2
A3,B4 d d
A4,B3 1 1
A5,B4 A4 A5 B3 B4
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39. F=ac+bc+d G=ac’+d F+G=a+bc+d1 1
A4,B3
A5,B4 1 1
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40. Pseudocode for RESTRICT operator
function Restrict( F, var, value )
if ( AlreadyComputed( F, var, value ) ) return result;
else if ( Var( F ) > var ) return F;
else if ( Var( F ) < 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);
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41. F=bc+ab’c’ F(b=1) = ? a a b b b b c c c c c 1 1 1
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42. 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
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43. 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)
Universal quantification of F w.r.t. x1 is
x1 F(x1, x2, x3) = F(0, x2, x3) & F(1, x2, x3)
Unique quantification of F w.r.t. x1 is
!x1 F(x1, x2, x3) = F(0, x2, x3)  F(1, x2, x3)
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44. Summery of Operations on BDDs
Apply – NOT, AND, OR, EXOR, etc.
Restrict
Compose (Replace)
Quantification (existential, universal)
Specialized operators
Generalized cofactor (constrain, restrict)
Compatible projection, etc.
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45. Worst-Case Complexity
CreateNode - O(1)
Build - O(2n)
APPLY - O(|F|*|G|)
RESTRICT - O(|F|)
COMPOSE - O(|F|2 *|G|2)
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46. Variable Ordering for F=a1b1+ a2b21 1
a1<b1<a2< b2
a1<a2<b1< b2
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47. Dynamic Variable ReorderingF F F0 F1 x1 x2 x2 x2 F0 F1 x2 x2 x1 x1 x3 x3 x3 x3 x3 x3 x3 x3
Variable reordering is a local operation
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48. Introduction to BDDs: References
Henrik Reif Andersen. An Introduction to Binary Decision Diagrams. Dept. of Information Technology, Technical University of Denmark, 1997.
R. E. Bryant. Symbolic Boolean Manipulation with Ordered Binary Decision Diagrams. ACM Computing Surveys, Vol. 24, No. 3 (September, 1992), pp
R. E. Bryant. Graph-Based Algorithms for Boolean Function Manipulation. IEEE Transactions on Computers, Vol. C-35, No. 8 (August, 1986), pp
Reprinted in M. Yoeli, Formal Verification of Hardware Design, IEEE Computer Society Press, 1990, pp
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49. Existential Quantification
Existential quantification (abstraction)
x f = f |x=0 + f |x=1
Example: x (x y + z) = y + z
Note: x f does not depend on x (smoothing)
Useful in symbolic image computation (sets of states)
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50. Existential Quantification - cont’d
Function can be existentially quantified w.r.to a vector: X = x1x2…
X f = x1x2... f = x1 x2 ... f
Can be done efficiently directly on a BDD
Very useful in computing sets of states
Image computation: next states
Pre-Image computation: previous states
from a given set of initial states
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51. Image Computation
Computing set of next states from a given initial state (or set of states)
Img( S,R ) = u S(u) • R(u,v)
Img(v)
R(u,v)
S(u)
FSM: when transitions are labeled with input predicates x, quantify w.r.to all inputs (primary inputs and state var)
Img( S,R ) = x u S(u) • R(x,u,v)
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52. Image Computation - example
Compute a set of next states from state s1
Encode the states: s1=00, s2=01, s3=10, s4=11
Write transition relations for the encoded states: R = (ax’y’X’Y + a’x’y’XY’ + xy’XY + ….) s1 s2 s3 s4 a a’ 00 01 10 11
a xy XY
……….
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53. Example - cont’d Compute Image from s1 under R
Img( s1,R ) = a xy s1(x,y) • R(a,x,y,X,Y)
= a xy (x’y’) • (ax’y’X’Y + a’x’y’XY’ + xy’XY + ….)
= axy (ax’y’X’Y + a’x’y’XY’ ) = (X’Y + XY’ )
= {01, 10} = {s2,s3} s1 s2 s3 s4 a a’ 00 01 10 11
Result: a set of next
states for all inputs
s1  {s2, s3}
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54. Pre-Image Computation
Computing a set of present states from a given next state (or set of states)
Pre-Img( S’,R) = v R(u,v) )• S’(v)
S’(v)
R(u,v)
Pre-Img(u)
Similar to Image computation, except that quantification is done w.r.to next state variables
The result: a set of states backward reachable from state set S’, expressed in present state variables u
Useful in computing CTL formulas: AF, EF
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55. Model Checking Example
Traffic light controller (simplified) G2 Y2 R2 G1 Y1 R1 C
Timer T
sensor
Road 2
Road 1
R1 G2
Y1 R2
G1 R2
R1 Y2
C ’+ T ’
C T
C T ’
C ’+ T
G1 R2
Y1 R2
R1 G2
R1 Y2
C = car sensor
T = timer

56. Traffic light controller - Model Checking
Model Checking task: check
safety condition
fairness conditions
Safety condition: no green lights on both roads at the same time
A G ¬ (G1  G2 )
Fairness condition: eventually one road has green light
E F (G1  G2)
R1 G2
Y1 R2
G1 R2
R1 Y2
C ’+T ’
C T
C T ’
C ’+T
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57. Explicit Model Checking - complexity
CTL model checking is linear in the size of the formula and the size of the structure M
Not a good news:
what if you have 1050 states?
Number of states grows exponentially with number of variables
Explicit model checking limited to … 109 states
Symbolic model checking can do much better
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58. Symbolic Model Checking
operates on entire sets rather than individual states
Uses BDD for efficient representation
represent Kripke structure
manipulate Boolean formulas
RESTRICT and APPLY logic operators
Quantification operators
Existential:  x f = f |x=0 + f |x=1 (smoothing)
Universal: x f = f |x=0 • f |x=1 (consensus)
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59. Symbolic Model Checking - example Traffic Light Controller
Encode the atomic propositions (G1,R1,Y1, G2,Y2,R2):
use [a b c d] for present state, [v x y z] for next state
R1 G2
Y1 R2
G1 R2
R1 Y2 s1 s3 s4 s2
11 10
10 01
01 10
10 11
a b c d G Y R G Y R
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60. Example - cont’d
Represent the set of states as Boolean formula Q: Q = abcd’ + a’bcd’ + ab’cd + ab’c’d
R1 G2
Y1 R2
G1 R2
R1 Y2 s1 s3 s4 s2
11 10
10 01
01 10
10 11
Store Q in a BDD
(It will be used to perform logic operations, such as S(G1)  S(G2)
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61. Example - cont’d
Write a characteristic function R for the transition relation R =abcd’vxyz’ + abcd’v’xyz’ + … + ab’c’dvxyz’
(6 terms)
R1 G2
Y1 R2
G1 R2
R1 Y2 s1 s3 s4 s2
11 10
10 01
01 10
10 11
abcd vxyz R
Store R in a BDD. It will be used for Pre-Image computation for EF.
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62. Example - Fairness Condition
Check fairness condition: E F (G1  G2 )
Step 1: compute S(G1), S(G2) using RESTRICT operator
S(G1): ab·Restrict Q(G1) = ab Q|ab = abcd’ = {s1}
S(G2): cd·Restrict Q(G2) = cd Q|cd = ab’cd = {s3}
Step 2: compute S(G1)  S(G2 ) using APPLY operator
Construct BDD for (abcd’ + ab’cd) = {s1, s3}, set of states labeled with G1 or G2
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63. Example – cont’d s’ {s1’,s3’} • R(s,s’) ) =
Step 3: compute S(EF (G1  G2 )) using Pre-Image computation (quanitfy w.r.to next state variables)
Recall: R = abcd’vxyz’ + abcd’v’xyz’+ … + ab’c’dvxyz’
s’ {s1’,s3’} • R(s,s’) ) =
= vxyz(vxyz’ + vx’yz) • R(a,b,c,d;v,x,y,z)
= vxyz(abcd’vxyz’ + a’bcd’vx’yz + ab’cdvx’yz + ab’c’dvxyz’)
= (abcd’ + a’bcd’ + ab’cd + ab’c’d) = {s1, s2, s3, s4}
Compare to the result of explicit algoritm 
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64. Example – Interpretation
Pre-Img(s1’,s3’,R) eliminates those transitions which do not reach {s1,s3}
R1 G2
Y1 R2
G1 R2
R1 Y2 s1 s3 s4 s2
11 10
10 01
01 10
10 11
abcd vxyz R X
Quantification w.r.to next state variables (v,x,y,z) gives the encoded present states {s1,s2,s3,s4}
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