2 Binary Decision Diagrams 2. Binary Decision Diagrams Verification - - PowerPoint PPT Presentation

• 2. Binary Decision Diagrams

1

2 Binary Decision Diagrams

Fachgebiet Rechnersysteme

• 2. Binary Decision Diagrams

Verification Technology

Content

2.1 BDD concepts 2 2 Variable orderings 2.2 Variable orderings 2.3 OBDD algorithms 2 4 FDD´s and OKFDD´s 2.4 FDD s and OKFDD s 2.5 Integer valued decision diagrams

• 2. Binary Decision Diagrams

2



The problem of logic verification: show that two circuits implement the same boolean function p a g

=1

b

1

a a

&

g a b

&

g

& &

b

&

• 2. Binary Decision Diagrams

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2.1 BDD concepts



Problem: efficient representation of Boolean functions 2.1 BDD concepts



Problem: efficient representation of Boolean functions  DNF: linear for OR of n variables, exponential for XOR XOR  Reed-Muller: linear for XOR of n variables, exponential for OR



Problem: efficient application of Boolean operations  example: p DNF  Negation  DNF, e.g.: ab + cd + ef + gh  (ab + cd + ef + gh)  ? ab + cd + ef + gh  (ab + cd + ef + gh)  ?



Possible solution in many cases: binary decision diagrams (BDD´s) d ag a s ( s)

• 2. Binary Decision Diagrams

4 2.1 BDD concepts



Investigated systematically first by R. Bryant (CMU)  Seminal paper by Bryant in ´86  Seminal paper by Bryant in 86



Early work by Shannon ~ 1940 (relais-networks)



Revolutionary impact on logic synthesis logic



Revolutionary impact on logic synthesis, logic verification, etc.



Many modern CAD-tools employ BDD´s



Many modern CAD tools employ BDD s

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5 2.1 BDD concepts

0 0 0 0 a b c d f



Idea: decompose a function into two sub- 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 1 1 functions which do not depend on a certain variable e g x 0 0 1 1 0 0 1 0 0 0 0 1 0 1 0 variable, e.g., x1 0 1 1 0 0 0 1 1 1 0 1 0 0 0 1 0 0 0 0 1 0 0 1 0 1 0 1 0 0 1 0 1 1 1 0 1 1 0 1 1 0 0 0 1 1 0 1 1 1 0 1 0 1 1 1 0 0 1 1 1 1 1

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6 2.1 BDD concepts

0 0 0 0 a b c d f



Idea: Decompose a function into two sub- 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 1 1 functions which do not depend on a certain variable e g x 0 0 1 1 0 0 1 0 0 0 0 1 0 1 0 f(0, b, c, d) variable, e.g., x1



Apply Boole´s expansion theorem in a systematic 0 1 1 0 0 0 1 1 1 0 1 0 0 0  theorem in a systematic way to all variables



Represent result 1 0 0 0 0 1 0 0 1 0 1 0 1 0 0 1 0 1 1 graphically 1 0 1 1 0 1 1 0 0 0 1 1 0 1 0 f(1, b, c, d) 1 1 1 0 0 1 1 1 1 1

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7

f *f( ) *f( )

2.1 BDD concepts

0 0 0 0 a b c d f f f = a*f(0, b, c, d) + a*f(1, b, c, d) 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 1 1 a 0 0 1 1 0 0 1 0 0 0 0 1 0 1 0 f(0, b, c, d) a 1 0 1 1 0 0 0 1 1 1 0 1 0 0 0  0 0 0 0 0 0 f(0, b, c, d) f(1, b, c, d) 1 0 0 0 0 1 0 0 1 0 1 0 1 0 0 1 0 1 1 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 1 0 0 1 0 0 1 0 1 1 0 1 1 0 0 0 1 1 0 1 0 f(1, b, c, d) 0 1 1 0 1 0 0 0 1 0 1 0 1 1 0 1 0 0 0 1 0 1 1 1 1 0 0 1 1 1 1 1 1 0 1 0 1 1 0 0 1 1 1 0 1 0 1 0 1 1 0 0 1 1 1 1

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8 2.1 BDD concepts

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The application of Boole´s expansion theorem to all variables leads to a decision tree. Example: XOR in 3 variables a b c

a b c   a b c  

a 1 0 0 0 0 0 0 1 1 b 1 b 1 0 1 0 1 0 1 1 0 1 0 0 1 c c c c 1 0 0 1 1 0 1 0 1 1 0 0 1 1 1 1 1 1 1 1 1 1 1 1 Function values

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9 2.1 BDD concepts



The application of Boole´s expansion theorem to all variables leads to a decision tree. a b c

a b c   a b c  

Example: XOR in 3 variables a 1 0 0 0 0 0 0 1 1 b 1 b 1 0 1 0 1 0 1 1 0 1 0 0 1 c 1 c 1 c 1 c 1 1 0 0 1 1 0 1 0 1 1 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 Function values

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10 2.1 BDD concepts



Variable ordering: order in which Boole's expansion theorem is applied pp

a b c  

• rder a b c

a 1

• rder a, b, c

b 1 b 1 c 1 c 1 c 1 c 1 1 1 1 1 1 1 1 1

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11 2.1 BDD concepts

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Concepts: Nodes Ed l b lli Edge labellings a 1 b b Directed edges b (Direct) successors of node a

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12 2.1 BDD concepts

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Concepts: Root node a 1 Paths b 1 b 1 c c c c 1 1 1 1 1 1 1 1 Leafs or Terminal nodes

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13 2.1 BDD concepts



Decision trees are ordered (identical variable

• rdering on all paths) or free

g p ) — Example of a free decision tree: a 1 c 1 b c 1 b 1 b 1 b 1 c 1 c 1 1 1 1 1

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14 2.1 BDD concepts

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A fully expanded decision tree has 2n leaf nodes

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Example of a (free) decision tree which is not fully

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Example of a (free) decision tree which is not fully expanded: a 1 c 1 b c 1 b 1 b 1 c 1 1 1 1

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15 2.1 BDD concepts

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Observation: there are identical sub-trees a 1 b 1 b b 1 b 1 c 1 c 1 c 1 c 1 1 1 1 1

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16 2.1 BDD concepts

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Observation: there are identical sub-trees a 1 b 1 b b 1 b 1 c 1 c 1 1 1

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17 2.1 BDD concepts

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Merging identical sub-trees results in a decision- graph a g p 1 a b c

a b c  

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

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18 2.1 BDD concepts

a 1 a b c

a b c  

b b 1 1 0 0 0 0 0 0 1 1 0 1 0 1 c c 1 1 0 1 0 1 0 1 1 0 1 0 0 1 1 0 1 1 1 1 0 1 0 1 1 0 0 1 1 1 1 1 "0-part" "1-part"

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19 2.1 BDD concepts



Shannon: A symbolic analysis of relay and switching circuits (1938) circuits (1938)  Size of the networks grows linearly in the number of variables

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20 2.1 BDD concepts

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Some simple examples of BDD's: a 1 a 1 1 1 1 1 1 a a a b a 1 b 1 b 1 b 1 1 1

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21 2.1 BDD concepts



AND-, OR-, XOR-operation in n variables

x x

1

x1 x

1

x1 1 1

x2

1 1

x

2

x2 x2 1 1 ... 1 . ... 1

xn

1

x

n

. . xn xn 1 1 1 1 1

 #nodes grows linearly in #variables

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22

&

3 2 1

S S S

2.1 BDD concepts

— Example: SN 74181 ALU:

& 1 & & 1 & & &

1 n+4

D E B G c S

3 3

1 & & & 1 & & =1

3

E A P F

3

D

3

=1 =1 1 & & & 1 & & & & & 1

2

Q A

2 2

E D B

3

& & 1 & & &

2

F A

1

B D

2 2

Q

=1 =1 & & & & & 1 1 & & =1

1 1

A=B F

1

B A E

1

& & 1 & & & 1 =1

F E D B

1 1

Q

1 =1 1 & & &

n

M c A 0 Q

=1 =1

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23

— SN 74181 BDD ("shared BDD"):

2.1 BDD concepts

s3 g c4 p f3 f2 aeqb f1 f0 s3 s2 s1 s0 b3 a3 b2 a3 a2 b1 a2 a1 b0 a0 m c

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24 2.1 BDD concepts



One path to the 1 leaf-node corresponds to a product – an implicant of the function. Example: p p a 1 a b c

a b c  

b b 1 1 0 0 0 0 0 0 1 1

c b a

0 1 0 1 0 1 1 0 1 0 0 1 c c 1 1 1 0 0 1 1 0 1 0 1 1 0 0 1 1 1 1 1

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25 2.1 BDD concepts



2 Problems:  Given a binary decision diagram  Given a binary decision diagram. How to derive the Boolean function represented by the BDD?  Given a Boolean function. How to derive the BDD for it?  First: BDD  Boolean function

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26 2.1 BDD concepts



A node v of a BDD is characterized by a triple (x, v0 ,v1), where v0 ,v1 are the successors of v

0 , 1

v x v0 v1



The leaf nodes 0 and 1 represent the Boolean functions 0 and 1



According to Boole's expansion theorem, the Boolean function bf(v) is associated with node v as follows (where var(v) is the variable of node v): var(v) is the variable of node v): bf(v) = var(v)· bf(v0) + var(v)· bf(v1) f f f

1

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27 2.1 BDD concepts

— Example: which Boolean function is represented by the following BDD? y g a 1 b 1 1 1

 The function associated with a node can be

1 determined only if the functions associated with the successor nodes are known bf(v) = var(v)· bf(v0) + var(v)· bf(v1)

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28 2.1 BDD concepts

 "Bottom-up procedure": 1. Step a 1 b b 1 1 Functions and 1 Functions 0 and 1

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29 2.1 BDD concepts

 "Bottom-up procedure": 2. Step a 1 b bf(v) = var(v)· bf(v0) + var(v)· bf(v1) b 1 = b · 0 + b · 1 = b 1 Functions and 1 Functions 0 and 1

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30 2.1 BDD concepts

 "Bottom-up procedure": 3. Step bf( ) ( ) bf( ) ( ) bf( ) a 1 bf(v) = var(v)· bf(v0) + var(v)· bf(v1) = a · 0 + a · b = a · b 1 b bf(v) = var(v)· bf(v0) + var(v)· bf(v1) b 1 = b · 0 + b · 1 = b 1 Functions and 1 Functions 0 and 1

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31 2.1 BDD concepts



There are many variants of binary decision diagrams



Most useful and common: OBDD's (Ordered Binary



Most useful and common: OBDD s (Ordered Binary Decision Diagrams, Bryant 1986)



OBDD properties: OBDD properties:  Ordered : The variables appear in a fixed ordering on all paths — Technically, an index (a positive integer) is associated with each variable index(var(v)) — For each node v with successors v0 and v1 we have: index(var(v)) < index(var(v0)) und index(var(v)) < index(var(v )) index(var(v)) < index(var(v1))

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32 2.1 BDD concepts

a b c  

i bl a

a b c  

variable

• rder a, b, c

index(a) = 1 1 index(b) = 2 b 1 b 1 index(b) = 2 index(c) = 3 c 1 c 1 c 1 c 1 index(c) 3 1 1 1 1

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33 2.1 BDD concepts



OBDD properties (cont'd.):  Reduced:  Reduced: — The function represented by one node is different from the functions of all other nodes — The two successors of each node are distinct

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34 2.1 BDD concepts

— Reduction example: a 1 a 1 b 1 b 1 b 1 b 1 1 1 1 1 a 1 Several representations of 1 b 1 Identical 1 1 Identical successors

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35 2.1 BDD concepts



Simplified representations exist, e.g.,  1-edges to the right 0-edges to the left  1-edges to the right, 0-edges to the left  Edges to 0 omitted  etc  etc. — Example: (a  b) · (c  d) · (e  f)  or: 0 edges are dashed lines

a b b

 or: 0 edges are dashed lines

1

b b c

all

d d e

• thers

e f f

1

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36 2.1 BDD concepts



Now: Boolean function  OBDD  Example above: (a  b) · (c  d) · (e  f)  Example above: (a  b) · (c  d) · (e  f)  Let

F (a b) (c d) (e f) (a b) r         

 Variable ordering a, b, c, d, e, f  Following Boole's expansion theorem, we have the

( ) ( ) ( ) ( )

 Following Boole s expansion theorem, we have the following cofactors of F w.r.t. a a 1

F (1 b) r b r 

F (0 b) r b r 

1

F (1 b) r b r

a 

   

F (0 b) r b r

a 

   

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37 2.1 BDD concepts

 More expansions: a 1

F (1 b) r b r

a 

    F (0 b) r b r

a 

   

a 1 b b 1 b b 1

F F r

ab ab

  F

ab 

F

ab 

etc.

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38 2.1 BDD concepts



The problem of reduction:  In the example above it was easy to detect F

F r

 In the example above it was easy to detect and to merge the nodes for and  Redundant nodes have to be removed

F F r

ab ab

  F

ab

F

ab

Redundant nodes have to be removed  Redundant nodes — Either represent the same function Either represent the same function — Or have identical successors (easy to detect)



How to know that two nodes represent the same function?



How to know that two nodes represent the same function?

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39 2.1 BDD concepts



Two functions

a a

f a af f  

a a

g a ag g  

are equal iff they have identical cofactors

a a

b b 1 1

fa

 1 1

ga

 a 1 1

fa

 a 1 1

ga

 c f c g a 1 a 1

fa fa ga ga

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40 2.1 BDD concepts



If we presume that all successor nodes of two nodes represent distinct functions then the two nodes represent p p identical functions iff the direct successor nodes are pairwise identical a 1 a 1 1 1

f f g g fa fa ga ga

 This results in a simple bottom-up-procedure:

redundant nodes are eliminiated in the redundant nodes are eliminiated in the bottom-level first, the in the second level, etc.

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41 2.1 BDD concepts

b 1 a 1 b 1 c 1 c 1 c 1 c 1 1 1 1 1 1 1 1 1

• 1. Step:
• 1. Step:

0/1 leafs

a 1 b 1 b 1

all

c c 1 c c 1

all

• thers

1

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42

• 2. Step:

d

a 1

2.1 BDD concepts

c nodes

b 1 b 1 c c 1 c c 1

all

• thers

1 1 a b 1 b 1 c c

all

• thers

1 1

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43 2.1 BDD concepts

• 3. Step

b nodes

1 a

b nodes ll

b 1 b 1

all

• thers

c c 1 1 1

 We can decide that the two b nodes do not  We can decide that the two b-nodes do not

represent the same function by means of the c- nodes

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44 2.1 BDD concepts



Example: derive the OBDD for the following function, variable order r e g variable order r,e,g r e g p p = eg + rg + reg 0 0 0 0 0 1 1 0 1 0 1 pr = eg + g = g 0 1 0 1 0 1 1 1 0 0 1 1 0 1 23 = 8 cases pr = eg + eg 1 0 1 1 1 0 1 1 1 1

Traffic- r Light Checker e g p

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45 2.1 BDD concepts

r pr = eg + g = g + p = eg + rg + reg r

1

pr = eg + eg p g g g e g g g 1 1

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46 2.1 BDD concepts



Reduction was necessary in the original concept by R. Bryant (1986), but can be avoided completely (s. Sect. 2.3) y ( ), p y ( )

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47 2.1 BDD concepts



OBDD´s can be implemented easily by means of 2:1-Multiplexors p

1

x 1 x

1 1

fx fx _

& &

fx fx _ fx _ fx

x

x x

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48 2.1 BDD concepts

— Example: a

1

a 1

1

b 1 b

1

c c c 1 c 1

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49 2.1 BDD concepts



Given a certain variable ordering, OBDD´s are canonical representations of Boolean functions, i.e., there exists p , , exactly one OBBD-representation for each Boolean function



Two circuits implementing the same function have identical OBDD's a

1

a b b b

1 1

=1

=

1 a a a b b

1 1 1

a b a

& & &

1 b b

&

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50

OBDD´ Ci it 1

2.1 BDD concepts

OBDD´s

& 1 & & 1 & & & & &

3 2 1 n+4

D E S B G c S S S

3 3 3

=1

Circuit 1:

& & 1 1 & & & & 1 & & & & & & & 1 =1

2 2 3

Q A P F F A

3 2 2

E D B

3 3 2

=1 1 =1 & & 1 & & & & 1 & & 1 & 1 & & & & & 1 =1

1 1

A=B F

1 1

B A D E E D B

2 1 2 1

Q Q

=1 =1 =1 &

3 2

S S S

Circuit 2:

=

1 & & & &

n

M c F E A 0 Q

=1 =1 & 1 & & & & 1 1 & 1 & & & & & & & =1

1 3 n+4

D E Q B A G c P F S S

3 3 2 3

D

3

=1 =1 & 1 & & & & 1 & 1 & & & 1 & & & 1 1 & & &

2 1 2

Q F A=B A

2

E

1 1

B B A D E

3 2 2

Q

=1 =1 & & 1 1 & & & & 1 & & & 1 & =1

1 n

M c F F A E D B A 0

1 1

Q Q

=1 =1 =1

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51

2.2 Variable Orderings 2.2 Variable Orderings



The variable ordering has a critical impact on the size of the OBDD (= #nodes)



There are static and d namic proced res to determine



There are static and dynamic procedures to determine "good" orderings

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52 2.2 Variable orderings



The number of nodes of a OBDD depends critically on the variable ordering  Classical example (Bryant 1986): f = x1x2 + x3x4 + x5x6 x1 x1

1

x2

1

x3

1

x3

1 1 1

x3

1

x5 x5 x5 x5 x

1

x

1 1 1 1

x5 x4

1

x2 x2 x2 x2

1 1

x4 x4

1 1 0

x5 x6

1 1

x6

1

x4 x4 1

1 0

1

1

6

1

1

• 2. Binary Decision Diagrams

53 2.2 Variable orderings

 Example: n-bit adder: — Order R1: an, bn, an-1, bn-1,..., a0, b0 — Order R2: an, an-1,..., a0, bn, bn-1,..., b0 n= 8 16 32 64 time 0 02 0 03 0 11 0 19 time 0.02 0.03 0.11 0.19 #nodes 35 75 155 315 R1: time 0.39 16.34 #nodes 750 196574 R2:

 

• 2. Binary Decision Diagrams

54 2.2 Variable orderings



Calculating the best order may result in exponential run time



For a given circuit, "good" orderings can be heuristically determined — Example: Distribution of a "weight" 1/4 z x 1/2 1/4 1/2 1/ 4

& &

y 1 1/2 1/4 1/2 1/4

& & &

x 1/ 4

&

Sum of weights: x=1/2, y=1/4, z=1/4,  first use x for expansion 4

• 2. Binary Decision Diagrams

55 2.2 Variable orderings

— Delete selected variable and distribute weight again weight again z 1/4 z 1 1/2 1/4 3/4

& & &

y 1 1/2 1/2 3/4 3/4

& &

Sum of weights : y=3/4, z=1/4,  next use y for expansion — Order: x, y, z

• 2. Binary Decision Diagrams

56 2.2 Variable orderings



Sifting: dynamic ordering procedure (Rudell ICCAD´93)  Basic step: exchange two adjacent variables (Fujita  Basic step: exchange two adjacent variables (Fujita et al. EDAC´91) a 1 b b 1 1 c c 1 1 1

• 2. Binary Decision Diagrams

57 2.2 Variable orderings

 Principle: exchange 0-1 and 1-0 path b 1 c c 1 1 g0 g1 g2 g3 c 1 b b 1 1 g0 g2 g1 g3

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58 2.2 Variable orderings

a 1 b b 1 1 1 b b 1 1 c c 1 1 1 1

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59 2.2 Variable orderings

a 1 c c 1 1 1 c c 1 1 b b 1 1 1 1

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60 2.2 Variable orderings



Sifting-procedure:  Calculate variable with max #nodes (the "thickest"  Calculate variable with max. #nodes (the thickest part of the OBDD)  Shift variable over OBDD by pairwise exchange of y p g adjacent variables  Shift variable to a position where #nodes is minimal Minimum Minimum 1

• 2. Binary Decision Diagrams

61 2.2 Variable orderings

 Movie "Sifting" by Stefan Höreth:

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62 2.2 Variable orderings

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63 2.2 Variable orderings

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64 2.2 Variable orderings

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65 2.2 Variable orderings

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66 2.2 Variable orderings

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67 2.2 Variable orderings

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68 2.2 Variable orderings

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69 2.2 Variable orderings

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70

 im Detail:

2.2 Variable orderings

V3 V4

1

V4 V4 V3 V3 V5 V5 V5 V5 1 1

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71 2.2 Variable orderings

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72 2.2 Variable orderings

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73 2.2 Variable orderings

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74 2.2 Variable orderings

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75 2.2 Variable orderings

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76 2.2 Variable orderings

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77 2.2 Variable orderings

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78 2.2 Variable orderings

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79 2.2 Variable orderings

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80 2.2 Variable orderings

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81 2.2 Variable orderings

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82 2.2 Variable orderings

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83 2.2 Variable orderings

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84 2.2 Variable orderings

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85 2.2 Variable orderings

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86 2.2 Variable orderings

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87 2.2 Variable orderings

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88 2.2 Variable orderings

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89 2.2 Variable orderings

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90

2.3 OBDD Construction 2.3 OBDD Construction a 1 b 1 a ? c 1 c & a b 1 1 1 1

• 2. Binary Decision Diagrams

91 2.3 OBDD construction



Principle: build OBDD while traversing the circuit from inputs to outputs p p a OBDD-Package C-Program Traverser 1 Traverser a 1 C-Program C-Program 1 c & a b 1 g C-Program & b 1

*

1

• 2. Binary Decision Diagrams

92 2.3 OBDD construction

a a OBDD-Package C-Program Traverser 1 1 b 1 Traverser a 1 1 C-Program c & a b 1 C-Programm 1 C-program 1 C-Program 1 b 1 1 c g C-Programm & p g g 1 1

• 2. Binary Decision Diagrams

93 2.3 OBDD construction

a a OBDD-Package C-Program Traverser 1 1 b 1 a Traverser a 1 1 1 b C-Program C-Program 1 c & a b 1 b 1 g C-Programm & b 1 1 c c 1 C-Program & 1 1 1

• 2. Binary Decision Diagrams

94 2.3 OBDD construction



Orthogonality of Boole's expansion f+g = x*(fx + gx) + x*(fx + gx), f*g = x*(fx * gx) + x*(fx * gx), f = x*fx + x*fx * x g x f x x 1 1 * * fx fx gx gx

• 2. Binary Decision Diagrams

95 2.3 OBDD construction



AND-operation of two OBDD´s  Assumption: nodes are represented as triples  Assumption: nodes are represented as triples (x,v0,v1) var low high f ti access-functions

• 2. Binary Decision Diagrams

96 2.3 OBDD construction

function AND(bdd1, bdd2): IF bdd1 0 OR bdd2 0 THEN t IF bdd1=0 OR bdd2=0 THEN return 0; ELSEIF bdd1=1 THEN return bdd2; ELSEIF bdd2=1 THEN return bdd1; ELSE var1:=var(bdd1);var2:=var(bdd2); ( ); ( ); IF var1=var2 THEN x:=var1; v0:= AND(low(bdd1), low(bdd2)), v1:= AND(high(bdd1),high(bdd2)); ( g ( ), g ( )); ELSEIF index(var1) < index(var2) THEN x:=var1; v0:= AND(low(bdd1), bdd2), ( ( ), ), v1:= AND(high(bdd1), bdd2); ELSEIF ... IF v0 = v1 THEN return v0 ELSE return (x v0 v1); IF v0 = v1 THEN return v0 ELSE return (x,v0,v1); ...

• 2. Binary Decision Diagrams

97 2.3 OBDD construction

3 5 c & a b 1

a

*

4 c

1 b 1

*

c 1 1 3 1 4

bdd1 bdd2 var1=a var2=c => index(var1) < index(var2)

• 2. Binary Decision Diagrams

98 2.3 OBDD construction

3 5 & a b 1

a

*

4 c

1 b 1

*

c 1 1 1 3 1 4 3 4

bdd1 bdd2 var1=a var2=c => index(var1) < index(var2) x:=var1 := a x:=var1 := a v0:= and(low(bdd1),bdd2), v1:= and(high(bdd1),bdd2)

• 2. Binary Decision Diagrams

99 2.3 OBDD construction

3 5 & a b 1

a

*

4 c

1 b 1

*

c 1 1 3 1 4 3 4

bdd1 bdd2 var1=b var2=c => index(var1) < index(var2)

• 2. Binary Decision Diagrams

100 2.3 OBDD construction

a 1

3 5 & a b 1

a

*

1 b 1

4 c

1 b 1

*

c 1 c 1 1 1 3 1 4 1 5 3 4 5

bdd1 bdd2 var1=b var2=c => index(var1) < index(var2) x:=var1 := b x:=var1 := b v0:= and(low(bdd1),bdd2), v1:= and(high(bdd1),bdd2)

• 2. Binary Decision Diagrams

101 2.3 OBDD construction

a 1

3 5 & a b 1

*

1 b 1

4 c

a

*

c 1 c 1 1 b 1 3 1 4 1 5 1 1 3 4 5

bdd1 bdd2 var2=c => index(var1) < index(var2) x:=var1 := b var1=b x:=var1 := b v0:= and(low(bdd1),bdd2), v1:= and(high(bdd1),bdd2)

• 2. Binary Decision Diagrams

102 2.3 OBDD construction

a 1

3 5 & a b 1

a

*

1 b 1

4 c

1 b 1

*

c 1 c 1 1 1 3 1 4 1 5 3 4

bdd1 bdd2

5

bdd1 bdd2 var2=c => index(var1) < index(var2) x:=var1 := b var2=c var1=b x:=var1 := b v0:= and(low(bdd1),bdd2), v1:= and(high(bdd1),bdd2)  

• 2. Binary Decision Diagrams

103 2.3 OBDD construction

a 1

3 5 & a b 1

*

1 b 1

4 c

a

*

c 1 c 1 1 b 1 1 4 1 5 1 1 3 4

bdd1 bdd2

5 3

var1=a var2=c => index(var1) < index(var2) x:=var1 := a x:=var1 := a v0:= and(low(bdd1),bdd2), v1:= and(high(bdd1),bdd2) 

• 2. Binary Decision Diagrams

104 2.3 OBDD construction

a 1

3 5 & a b 1

a

*

1 b 1

4 c

1 b 1

*

c 1 c 1 1 1 3 1 4 1 5 3 4

bdd1 bdd2

5

var2=c => index(var1) < index(var2) x:=var1 := a var1=a x:=var1 := a v0:= and(low(bdd1),bdd2), v1:= and(high(bdd1),bdd2)

• 2. Binary Decision Diagrams

105 2.3 OBDD construction

a 1

3 5 & a b 1

a

*

1 b 1

4 c

1 b 1

*

c 1 c 1 1 1 3 1 4 1 5 3 4

bdd1 bdd2

5

var2=c => index(var1) < index(var2) x:=var1 := a var1=a x:=var1 := a v0:= and(low(bdd1),bdd2), v1:= and(high(bdd1),bdd2)  

• 2. Binary Decision Diagrams

106 2.3 OBDD construction



"OBDD-Packages" manage two tables:  The unique table (ut) with entries:  The unique table (ut) with entries: x v0 v1

 For uniqueness of OBDD's  For uniqueness of OBDD s

• 2. Binary Decision Diagrams

107 2.3 OBDD construction

 The computed table (ct) with entries Operation bdd1 bdd2 Result bdd

 Stores previously calculated results  Stores previously calculated results

• 2. Binary Decision Diagrams

108 2.3 OBDD construction



Reduction was needed in the original OBDD procedures



OBDD uniquess is guaranteed by



OBDD uniquess is guaranteed by  Checking in the unique-table (ut) if the OBDD was calculated before calculated before  Testing for identical successor nodes



In addition, it is checked in the computed table (ct) if the result was calculated before

 Many steps of recursion may be saved

• 2. Binary Decision Diagrams

109 2.3 OBDD construction

function AND(bdd1, bdd2): IF (AND,bdd1,bdd2,x)  ct THEN return x; ( ) IF bdd1=0 OR bdd2=0 THEN return 0; ELSEIF bdd1=1 THEN return bdd2; ELSEIF bdd1=1 THEN return bdd2; ELSEIF bdd2=1 THEN return bdd1; ELSE var1:=var(bdd1);var2:=var(bdd2); IF var1=var2 THEN x:=var1; v0:= AND(low(bdd1), low(bdd2)), v1:= AND(high(bdd1),high(bdd2)); ELSEIF index(var1) < index(var2) THEN x:=var1; v0:= AND(low(bdd1), bdd2), v1:= AND(high(bdd1), bdd2); ELSEIF ELSEIF ... IF v0 = v1 THEN return v0 ELSEIF (x,v0,v1) ut THEN put in ut; ELSE return (x,v0,v1); ...

• 2. Binary Decision Diagrams

110 2.3 OBDD construction



The computed table is essential for the efficiency of the algorithms: g  In principle, two additional steps of recursion may result at each step

 the number of steps may grow exponentially

in the number of variables  Using the computed table with entries Operation bdd1 bdd2 Result bdd th b f i i d d t | 1|*| 2| h Operation bdd1 bdd2 Result bdd the number of recursions is reduced to |n1|*|n2| where |n1| and |n2| are the number of nodes of bdd1 and bdd2, respectively , p y

• 2. Binary Decision Diagrams

111

*

2.3 OBDD construction

1 1 1 1 1 1 1 1 1 1

a b c d e f g      

1 1 1 1

a b c d e f g      

1 1 1 1

g

1 1 1 1 1 1

1

1 1

1

1 1

• 2. Binary Decision Diagrams

112

*

2.3 OBDD construction

1 1 1 1 1 1 1 1 1 1

• * exponential in

the number of variables?

1 1 1 1

• O(n1*n2) using the

computed table !

1 1 1 1 1 1 1 1 1 1 1 1

*

1

1 1

1

1 1

* &

• 2. Binary Decision Diagrams

113 2.3 OBDD construction



General result:  If two OBDD´s with m and n nodes are logically  If two OBDD s with m and n nodes are logically combined then the resulting OBDD has  m*n nodes

 This is due to the fact that not more than

m*n distinct functions are generated!

• 2. Binary Decision Diagrams

114 2.3 OBDD construction



Negated edges:  The OBDD of a function f and the OBDD of the  The OBDD of a function f and the OBDD of the negated function are very similar: exchange the terminal nodes 0 and 1!  Orthogonality of negation: negate a function by negating it's cofactors =

Means ti 1

x

1

x =

negation

 Problem: non-canonical representation!

• 2. Binary Decision Diagrams

115 2.3 OBDD construction

 Solution: — Only the 0-edge can be a negated edge — 1 terminal leaf only (or the dual version ...) = =

1

x

1

x

1

x

1

x x x = x x =

1 1 1 1

• 2. Binary Decision Diagrams

116 2.3 OBDD construction

 Examples: variable and negated variable 1

1 1

1 1

1

1

1 1 1 1 1 1

1 1 1 1 1 1

• 2. Binary Decision Diagrams

117 2.3 OBDD construction

 Example: XOR function

1 1

XOR function

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

1

1 1

1

1

• 2. Binary Decision Diagrams

118 2.3 OBDD construction



Cofactor calculation using OBDD's



cof(x pol OBDD): x variable pol polarity 1 or 0



cof(x, pol, OBDD): x variable, pol polarity 1 or 0



Easy if variable = top-variable, e.g., cof(a, 0, OBDD):

a 1 b b c 1 c 1 1 1 1 1 1 1

• 2. Binary Decision Diagrams

119 2.3 OBDD construction



Generally: Replace pointers to the variable by the pointer to the 1-(0-)successor: ( ) f = c f = ac +abc f = ac

a a

fb = c f ac +abc

a

fb = ac

1 b 1 1 1 c 1 c 1 c 1 1 1 1 1 1 1 1

• 2. Binary Decision Diagrams

120 2.3 OBDD construction

— Example: determine the 0-cofactor for variable d:

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

1

1 1

1

1 1

• 2. Binary Decision Diagrams

121 2.3 OBDD construction



Functional substitution: substitute function g vor variable x  The paper-and-pencil method is: replace all  The paper-and-pencil method is: replace all

• ccurrences of x textually by g

 How to do that with a OBDD-representation? How to do that with a OBDD representation? f[x g] = gfx + gfx Rationale: fx f = xfx + xfx fx f = gfx + gfx

1

fx

1

fx x g

• 2. Binary Decision Diagrams

122 2.3 OBDD construction



Functional substitution: substitute function g vor variable x f[x g] = gfx + gfx Note:  x : (f(x  g)) = [fx(0  g)] + [fx(1  g)] = fxg + fxg = f[x g] f[x g]

 Functional substitution can be reduced to the

application of the  -operator

• 2. Binary Decision Diagrams

123 2.3 OBDD construction

 Using Boolean operations plus cofactor-calculation

more advanced Boolean operations like the - and -quantifier and functional substitutions can be implemented implemented

• 2. Binary Decision Diagrams

124 2.3 OBDD construction



OBDD´s are used in many CAD-tools for synthesis, verification and simulation



Many public domain OBDD-packages



Many are based on the ite(p, f, g)-operator (if p then f else g) Many are based on the ite(p, f, g) operator (if p then f else g)



CUDD package (Boulder Univ.)

 OBDD are very efficient decision procedures for

propositional calculus and are integrated into propositional calculus, and are integrated into many theorem provers like PVS and ACL2

• 2. Binary Decision Diagrams

125

2.4 FDD's and OKFDD's 2.4 FDD s and OKFDD s



OBDD's are based on Boole's expansion theorem



OBDD's represent the systematic decomposition in all variables variables



Q: Are there other types of "decomposition"? How many?

• 2. Binary Decision Diagrams

126 2.4 FDD's and OKFDD's



Boole' expansion:

f x f x f 



There are more types of expansion (exactly two more):

x x

f x f x f    

 Positive Davio-expansion

) f f ( x f f    

 Negative Davio-expansion

) f f ( x f f

x x x

  

 Negative Davio expansion

) f f ( x f f

x x x

   

• 2. Binary Decision Diagrams

127 2.4 FDD's and OKFDD's



FDD´s (Functional Decision Diagrams, Kebschull et al. 92)  f = f  x*(f  f )  f = fx  x (fx  fx) — for x = 0  fx for x = 1  f  f  f = f — for x = 1  fx  fx  fx = fx  Same graph structure, but different interpretation: f x Boolean difference

• f f w.r.t. x

1 fx (fx  fx)

• 2. Binary Decision Diagrams

128 2.4 FDD's and OKFDD's



FDD´s (Functional Decision Diagrams, Kebschull et al. 92)  f = f  x*(f  f )  f = fx  x (fx  fx) — for x = 0  fx for x = 1  f  f  f = f — for x = 1  fx  fx  fx = fx  Same graph structure, but different interpretation: f Rule: x Rule: variable = 1  XOR both branches t t th l f f 1 to get the value of f fx (fx  fx)

• 2. Binary Decision Diagrams

129 2.4 FDD's and OKFDD's



FDD´s are canonical representations



FDD's obey a different rule of reduction:



FDD s obey a different rule of reduction: f f x 1 1 fx fx (fx  fx) : if the Boolean difference is 0 then f does not depend on x

• 2. Binary Decision Diagrams

130 2.4 FDD's and OKFDD's



Orthogonality of XOR and AND for FDD´s:  f  g = f  x*(f  f )  g  x*(g  g ) =  f  g = fx  x (fx  fx)  gx  x (gx  gx) = (fx  gx )  x*[(fx  fx)  (gx  gx)] Yes!  f  g = (fx  x*(fx  fx))  (gx  x*(gx  gx)) = (fx*gx)  x*[fx * (gx  gx)  (fx  fx ) *gx  (f  f )*(g  g )] No! (fx  fx)*(gx  gx)]

 All 4 combinations have to be considered

No!

 All 4 combinations have to be considered

for the AND of 2 FDD's

• 2. Binary Decision Diagrams

131 2.4 FDD's and OKFDD's



OBDD and FDD for 4-bit adder

• 2. Binary Decision Diagrams

132 2.4 FDD's and OKFDD's



OKFDD´s (Ordered Kronecker FDD´s, Drechsler et al. 94)  Allows any of the three types of decomposition for  Allows any of the three types of decomposition for each variable  The type of decomposition is stored in a decomposition type list (DTL) f = a*[(0c*(01)) b*((0c*(01))1)] + a*[0c*(01)] B l a*[0c*(01)] = a*(cbc) + a*c

f f f

a 1 b Boole p Davio

x x

f x f x f     ) f f ( x f f    

0 b 1 p.Davio 1 c p Davio

) f f ( x f f

x x x

    ) f f ( x f f

x x x

   

1 1 p.Davio

) (

x x x

DTL

• 2. Binary Decision Diagrams

133 2.4 FDD's and OKFDD's



OBDD's/FDD's/OKFDD'S in comparison  OKFDD´s have OBDD´s and FDD´s as subclasses  OKFDD s have OBDD s and FDD s as subclasses  OBDD´s: AND OR XOR of two OBDD´s of size n and m — AND, OR, XOR of two OBDD s of size n and m requires max. n*m operations  FDD´s/OKFDD´s:  FDD s/OKFDD s: — XOR requires max. n*m, but AND and OR may need exponentially many operations! y y

 However: #nodes of FDD/OKFDD may be

exponentially smaller than #nodes of the OBDD ( d i ) OBDD (and vice versa)

 Important for logic synthesis

 OKFDD´ d t i i th d iti t li t  OKFDD´s: determining the decomposition-type list (DTL) is an additional problem

• 2. Binary Decision Diagrams

134 2.4 FDD's and OKFDD's



The OBDD size grows only linearly in #variables for many circuits (AND, OR, XOR, adders, ALU's, etc.) y ( , , , , , )



Can all circuits be represented with linear (or polynomial) effort?



The theoretical answer is that there will never be any representation with this nice property for all circuits



While the OBDD's are very compact representations for many classes of circuits they fail for others ...

• 2. Binary Decision Diagrams

135 2.4 FDD's and OKFDD's



Example: multiplier circuits

A0 A1 A2 A3 B0 B1 B2 B3 P7 P6 P5 P4 P3 P2 P1 P0

Word length : 4 6 8

 Interest in other types of decision diagrams

Word length : 4 6 8 #OBDD nodes : 150 2.183 10.766

• 2. Binary Decision Diagrams

136

2.5 Integer-Valued Decision Diagrams



So far type Bn  Bm now: type Bn  Z: 2.5 Integer Valued Decision Diagrams



So far type Bn  Bm, now: type Bn  Z:  MTBDD´s (Multi Terminal Binary Decision Diagrams, Clarke et al. DAC ´93) Clarke et al. DAC 93)  BMD´s (Binary Moment Diagrams, Bryant/Chen DAC ´ 95) a 1 a rule: a 1 a 1 rule: variable = 1  sum both branches b b 1 1 b 1 1 4 5 1 4 4a + b MTBDD 4a + b BMD

• 2. Binary Decision Diagrams

137 2.5 Integer-valued decision diagrams

 MTBDD: f = (1 - x)*fx + x*fx h * th dditi bt ti d where +, -, * are the addition, subtraction and multiplication, respectively  BMD:  BMD: f = fx + x*(fx - fx)  HDD´s (Clarke/Zhao): combination of MTBDD/BMD, one decomposition types for each variable ( OKFDD´s) decomposition types for each variable (~ OKFDD s)

• 2. Binary Decision Diagrams

138 2.5 Integer-valued decision diagrams



Example of application (Fujita et a. ´96):  Vector matrix operations employing MTBDD´s  Vector-matrix operations employing MTBDD s  Idea: encode rows and columns by means of boolean — encode rows and columns by means of boolean variables — elements ~ leafs elements leafs  Example 2*2 Matrix: fxy fxy x y

1

11 11 fxy fxy y

1

43 3 11 43 3

• 2. Binary Decision Diagrams

139 2.5 Integer-valued decision diagrams



Type Bn  Z and attributed edges  EVBDD´s (Edge Valued Binary Decision Diagrams  EVBDD s (Edge Valued Binary Decision Diagrams, Lai et al. ICCAD ´93)  *BMD´s (Multiplicative Binary Moment Diagrams, BMD s (Multiplicative Binary Moment Diagrams, Bryant/ Chen DAC ´95) a a rule: rule: a 1 a 1 rule: variable = 1 => add both branches, lti l b i ht 2 4 rule: add weight b b 1 multiply by weight 1 2 1 4 4a + 2b EVBDD 4a + 2b *BMD

• 2. Binary Decision Diagrams

140 2.5 Integer-valued decision diagrams

 EVBDD: f = a + (1 - x)*fx + x*fx h + * th dditi bt ti d where +, -, * are the addition, subtraction and multiplication, respectively  *BMD:  BMD: f = m*(fx + x*(fx - fx))  K*BMD´s (Drechsler EDTC´96): one decomposition type for each variable (~ OKFDD´s HDD´s) additive + type for each variable (~ OKFDD s, HDD s), additive + multiplicative weights  *PHDD (Chen/Bryant ICCAD´97): multiplicative power  PHDD (Chen/Bryant ICCAD 97): multiplicative power hybrid decision diagrams for floating-point circuits

• 2. Binary Decision Diagrams

141 2.5 Integer-valued decision diagrams



For *BMD´s we have for an edge without weight: f = fx + x*(fx - fx) = fx + x*fx

. 

*BMD´s are canonical representations provided that:  1. Rule: f f f x

1

w w fx fx fx fx

.

fx

• 2. Binary Decision Diagrams

142 2.5 Integer-valued decision diagrams

 2. Rule: the weight on an edge equals the gcd of the weights of the successor edges weights of the successor edges f f f t x

1

x

1 1

w0 w1

1

w0/t w1/t fx

.

fx

.

fx fx

 a leave "n" is a 1 node with weight n

• 2. Binary Decision Diagrams

143 2.5 Integer-valued decision diagrams

— Example: *BMD for f = 4*x + 2 f = f + x*(f - f ) f = fx + x (fx - fx) = 2 + x*(6 - 2), gcd(2, 4) = 2 f f 2 x x

1

1 1

1

1 1 2 4 1 2

• 2. Binary Decision Diagrams

144 2.5 Integer-valued decision diagrams

— Next example: *BMD for f = 3*y + 4*x + 2 f = fy + y*(fy - fy)

y

y ( y

y)

= (4*x + 2) + y*((4*x + 5) - (4*x + 2)) = (4*x + 2) + y*3 f y 2

1

x

1

1 2 3

• 2. Binary Decision Diagrams

145 2.5 Integer-valued decision diagrams

 3. Rule: sign of t is sign of left branch f f f t x

1

x

1 1

w0 w1

1

w0/t w1/t fx

.

fx

.

fx fx

• 2. Binary Decision Diagrams

146 2.5 Integer-valued decision diagrams



For *BMD´s with range {0, 1}, the boolean operations can be reduced to integer addition, subtraction and g , multiplication: f 1 f f 1 - f f and g f*g f or g f + g - f*g f xor g f + g - 2*f*g

• 2. Binary Decision Diagrams

147 2.5 Integer-valued decision diagrams



Some example *BMD´s:  4 variable AND (a boolean function)  4 variable AND (a boolean function)

1

x1

1

x1

1

x2

1

x2

1

x3

1

x3

3

x

1

3

x

1

x4

1

x4

1

1 OBDD 1 *BMD

• 2. Binary Decision Diagrams

148 2.5 Integer-valued decision diagrams

 4 variable OR (a boolean function) x1 x1

1 1

x2 x2 x2

1

• 1

1 1

x3 x3 x3

1

• 1

1 1

x4

1

x4 x4

1 1

x4

1 1

1 OBDD *BMD 1

• 1

• 2. Binary Decision Diagrams

149 2.5 Integer-valued decision diagrams

 4 variable OR (a boolean function) x4= 0, x3 = 0, x2 = 1, x1 = 0 1 x1

1

x1

1

x2 x2 x2

1

• 1

1

1

1

x3 x3 x3

• 1

1

1

1

x4

1

x4 x4

1

1

1

1

1

x4 x4 x4

1 1

1 1 OBDD *BMD 1

• 1

• 2. Binary Decision Diagrams

150 2.5 Integer-valued decision diagrams

 Example: 2-Bit multiplication x *(y *21 +y *20)+ x x y y

1

x0 2 x0*(y1*21 +y0*20)+ 2*x1*(y1*21 +y0*20) Result: x1 , x0 y1 , y0 x1

1

x *(y *21 +y *20) (x1*21 +x0*20)*(y1*21 +y0*20) = 20*x *(y *21 +y *20)+ y0

1

x1*(y1*21 +y0*20) 2 x0 (y1 2 +y0 2 )+ 21*x1*(y1*21 +y0*20) y

1

2 y1*21 +y0*20 y1

1

1

• 2. Binary Decision Diagrams

151 2.5 Integer-valued decision diagrams



Classification schema of decision diagrams (based on Minato ´96): a

1

a

1 1 1

Shannon p Davio decomposition mixed FDD OBDD Shannon

• p. Davio

Bn  B p type: mixed OKFDD O  O BMD MTBDD Bn  Z HDD Bn  Z, tt ib t d *BMD EVBDD attributed edges K*BMD

• 2. Binary Decision Diagrams

152

2.6 Bit-Vector Expressions Bit t 2.6 Bit Vector Expressions

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Bit-vectors:  Used for the compact representation of complex digital hardware digital hardware  More adequate than single bits in many cases Examples: data paths arithmetic circuits — Examples: data-paths, arithmetic circuits, register-transfer-level (rtl) descriptions, storage elements, ...  Provided by many hardware description languages (HDL's) as a basic data-type

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153 2.6 Bit-vector expressions

— Example: specification of 74181 ALU Generic bit-vector function "A PLUS B"

S3 S2 S1 S0 M = H M = L Cn = H M = L Cn = L L L L L F = not(A) F = A F = A PLUS 1 L L L H L L H L L L H H L H L L L H L H F = not(A+B) F = not(A) B F = 0 F = not(A B) F (B) F = A + B F = A + not(B) F = MINUS 1 F = A PLUS A not(B) F (A B) PLUS A (B) F = (A + B) PLUS 1 F = (A + not(B)) PLUS 1 F = 0 F = A PLUS A not(B) PLUS 1 F (A B) PLUS AB PLUS 1 L H L H L H H L L H H H H L L L H L L H F = not(B) F = A  B F = A not(B) F = not(A)+B F t(AB) F = (A + B) PLUS A not(B) F = A MINUS B MINUS 1 F = A not(B) MINUS 1 F = A PLUS AB F = A PLUS B F = (A + B) PLUS AB PLUS 1 F = A MINUS B F = A not(B) F = A PLUS AB PLUS 1 F = A PLUS B PLUS 1 H L L H H L H L H L H H H H L L H H L H F = not(AB) F = B F = A B F = 1 F = A + not(B) F = A PLUS B F = (A + not(B)) PLUS AB F = A B MINUS 1 F = A PLUS A F = (A + B) PLUS A F = A PLUS B PLUS 1 F = (A + not(B)) PLUS AB PLUS 1 F = AB F = A PLUS A PLUS F = (A + B) PLUS A PLUS 1 H H L H H H H L H H H H F = A + not(B) F = A + B F = A F (A + B) PLUS A F = (A + not(B)) PLUS A F = A MINUS 1 F (A + B) PLUS A PLUS 1 F = (A + not(B)) PLUS A PLUS 1 F = A

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154 2.6 Bit-vector expressions

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Bit-vector functions are necessary for input/output specifications, i.e., for the abstraction from internal p , , details I/O- Specification Specification

S S

& 1 & & & & 1 & & 1 & 1 & & & & & & & & 1 =1

D E Q B A G c P F S S S

E

D B

=1 =1 & 1 & & & & 1 & & & & 1 & 1 & & & 1 & & & & & 1 =1

F A=B F A

E

B A D E D B

Q Q

=1 =1 =1 1 & & & &

M c F E A 0 Q

=1 =1

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155 2.6 Bit-vector expressions

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Typical bit-vector functions: Function Meaning Example: Function FAE(A,N) ADC(A B C) Meaning fan-out dditi Example: FAE(1,3)="111" ADC(A,B,C) ADD(A,B) INC(A) addition addition modulo increment ADC("11","01",´1´)="101" ADD("11","01")="00" INC("111")="000" INC(A) DCR(A) RSH(C,V) LSH(V C) decrement rigth-shift left shift INC( 111 ) 000 DCR("111")="110" RSH(´0´,"111")="011" LSH("111" ´0´) "110" LSH(V,C) ROL(A) ROR(A) left-shift rotate left rotate right LSH("111", 0 )="110" ROL("001")="010" ROR("010")="001" ( ) MPX1(A,S) MINT(A,N) multiplexor 1. Dim. minterm ( ) MPX1("0010","10")=´1´ MINT(A(1:2),0)= (not A(1)) and (not A(2)) GT(A,B) LESS(A,B) A greater B A less B (not A(1)) and (not A(2)) GT("100","010")=1 LESS("100","010")=0

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156 2.6 Bit-vector expressions

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Bit-vectors in VHDL  Type bit vector predefined  Type bit_vector predefined signal X: bit_vector (1 to 16) or: (16 downto 1)  Selection of single elements X(4) or slices Selection of single elements X(4) or slices X(2 to 4)  Constant-denotation (B)"1001"  Assignments X(2 to 4) <= X(8 to 10)  Overloaded boolean primitives, e.g., g "0101" AND "0011" = "0001"  Concatenation &, e.g., X(2 to 4)&X(5 to 7) = X(2 to 7)  ...

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157 2.6 Bit-vector expressions

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Verification problems:  How to demonstrate the equality of arbitrary bit-  How to demonstrate the equality of arbitrary bit- vector expressions?  Do we have to reason formally about tuples, etc.? Do we have to reason formally about tuples, etc.?

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158 2.6 Bit-vector expressions

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Decision procedure: procedure to decide the truth of a statement in some domain

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For generic expressions (may contain expressions of arbitrary length), inductive reasoning is typically used  Example: prove ADD(A, B) = ADD(B, A) for arbitrary vectors A and B of the same length  Typically a theorem prover is employed — You have to derive the proof in large part by lf yourself — The theorem prover checks if the proof is correct

 N t

t t d d i t ti

 Not automated, needs user interaction

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159 2.6 Bit-vector expressions

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For fixed-length expressions, the problem becomes much simpler p  Example: prove ADD(A, B) = ADD(B, A) for 32-bit vectors A and B

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Several decision procedures exist:  The problem can be reduced to OBDD's  The problem can be reduced to an integer-linear programming (ILP) problem  A specific decision procedure was given by Cyrluk et

• al. (CAV´97) for a restricted repertoire of bit-vector

functions functions

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160 2.6 Bit-vector expressions

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Reduction to OBDD's ("bit-blasting"):  Translate expression using bit-vector-functions into  Translate expression using bit-vector-functions into multi-level gate-networks — e.g., A PLUS B, where A and B are two 4-bit e.g., A PLUS B, where A and B are two 4 bit vectors, is transformed into the gate-network of a four-bit adder  Then as before ! vector expression 1 gate network 1 OBDD 1 ? vector expression 2 gate network 2 OBDD 2 = ? expression 2 network 2

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161 2.6 Bit-vector expressions

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Technique offers the general possibility to carry out proofs involving complex bit-vector expressions p g p p  Examples: ADD(A, B) = ADD(B, A) (ADD(A, B) > ADD(B, C))  (A>C) (A>B) = NOT(ADD(0&A, 1&NOT(B))(1)) where A B C have fixed length by declaration where A, B, C have fixed length by declaration  Typically takes < 1 sec. for 64-bit vectors

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162 2.6 Bit-vector expressions

A > B = NOT (ADD(0&A, 1&NOT(B)))(1) B N N A N ´0´ ´1´ N N A B N > N+1 ADD 1 N+1 1 N 8 16 32 64 128 CPU-time N 0.5 8 0.6 16 0.7 32 1.1 64 1.8 128

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Example: verification of ALU´s  Verification of VHDL-specification using bit-  Verification of VHDL-specification using bit- vector-operations vs. network of standard-cells CPU-time Wordlength 1.0 4 1.5 8 2.8 16 6.6 32  32-Bit ALU: 2 * 32 boolean functions in up to 77 CPU time 1.0 1.5 2.8 6.6 variables

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164 2.6 Bit-vector expressions

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Some references:  Hassoun/Sasao (Eds ): Logic Synthesis and  Hassoun/Sasao (Eds.): Logic Synthesis and Verification, Springer — Book-Chapters on BDD's, SAT, Equivalence Book Chapters on BDD s, SAT, Equivalence checking  Hachtel/Somenzi: Logic Synthesis and Verification Algorithms, Springer

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Written exam in the summer  between 18 Juli and 7 October 2011  between 18. Juli and 7. October 2011  please follow Doodle link http://www.doodle.com/y4igrfy6wcrmx24h http://www.doodle.com/y4igrfy6wcrmx24h