digitaalsüsteemide verifitseerimise kursus1 formal verification: bdd bdds applied in equivalence...

Digitaalsuumlsteemide verifitseerimise kursus 1

Formal verification BDD

BDDs applied in equivalence

checking


Equivalence Checking

bull Two principal approaches

bull Transform implementation and spec (reference implementation) to a canonical form

bull Search for an input assignment that would distinguish the responses of the implementation and reference implementation

bull BDD-s applied for canonical form


Canonical forms of Boolean functions

bull Minimized function Karnaugh map

bull Full normal form

bull Truth table

These forms are not compact


Decision Diagrams and Equivalence

bull Theoretically any canonical form has exponential worst case complexity

Luckily for many practical functions compact BDDs exist

bull We will cover the topic of utilizing BDDs in equivalence checking and symbolic simulation


Examples of canonical forms

bull Is f notac + bnotc + anotb equivalent to f anotc + notbc + notab

c ab 00 01 11 10

0 1 1 1

1 1 1 1



c ab 00 01 11 10

0 1 1 1

1 1 1 1



Overview of the history of BDDs

bull 1959 Lee BDD first mentioned

bull 1976 Ubar alternative graphs akastructural BDD-s

bull 1978 Akers BDD protagonist

bull 1985 Bryant ROBDD

bull ROBDD-s started a big BDD boom


Binary Decision Diagrams

bull BDD is a directed acyclic graph with a single root

node and two terminal nodes 0-node and 1-node

bull Each internal node is labeled by a Boolean variable

bull From each internal node two edges will originate

which correspond to 0 and 1 value of the variable



x1

x2

x3

0 1

01

1

1

0

0

BDD for Boolean function

(x1x2) V x3

BDD is generated by Shannon expansion

10 fxfxf ii


BDD generation example

Shannonrsquos expansion

10 fxfxf ii

f=ab+bc

b+bc bc

c c

a) Generate a decision tree b) Reduced BDD


Interpretation of BDDs

bull By traversing the paths of

a BDD

1 Calculate the value of f

2 Derive the function

f=ab+anotbc+notabc+notanotbd


Ordered BDD (OBDD)

bull Boolean variables occur in a concrete order

along the paths of the BDD

bull No path has more than 2 occurrences of the

same variable


Reduced OBDD (ROBDD)

Rules of reduction

a) Node removal

x

f f

10

b) Node sharing

x x x

f0 f1

0 11 0

f0 f1

0 1


Reduction example


ROBDD properties

+ ROBDD is a canonical representation of a Boolean function bull Equivalence of Boolean functions is checked

by isomorphicity of their respective BDDs

ndash ROBDD worst case memory consumption is exponentialbull Not applicable in many practical cases

(Eg multiplier)


BDD size depending on the ordering

f = ab + anotbc + notabc + notanotbd



bull There are functions whose BDD size grows linearly

wrt number of variables

bull There are functions whose BDD size grows

exponentially in the worst case wrt number of

variables


ALWAYS exponentially regardless of the ordering

(Eg multiplication)



bull Finding an optimal ordering for the variables is an

unsolved problem

bull Various heuristics applied


Dynamic reordering of BDD nodes


Structural BDD-s (Ubarrsquo76)

bull Definition similar to the one of BDD

bull SSBDD is generated in a recursive manner

substituting logical gates by elementary

BDDs



bull ROBDD generated for each output of the

circuit

bull SBDD generated for each fanout-free (tree-

like) subcircuit required memory linear


SBDD properties

bull Different of ROBDD-s SBDDs are planar

bull Nodes labeled by variables and inverted variables concept of 0 and 1 direction


Alternative notation

bull Moving to the right (downwards) from a node

corresponds to variable value 1 (value 0)

bull Exiting to the right (downwards) from the graph

corresponds to function value 1 (value 0)

bull Terminal nodes not shown


Circuit its BDD and SBDD

ab

cd

e

f

y

1 0

11 0

0

10


SBDD generation

ab

cd

e

f

y

1) 2)

3) 4)


Structural paths lt=gt BDD nodesbull 11 correspondence between SBDD nodes and

structural paths in the fanout-free circuit

=gt saumlilib info skeemi struktuuri kohta

ab

cd

e

f

y


ROBDD vs SBDD

ROBDD SBDD

Canonical representation

+ -

Structural representation

- +

Speed of generation

linear linear

Memory consumption

exponential linear


SBDD summary

bull SBDD model has a number of advantageous properties for modeling logic circuits

1) Preserve structural information

2) Suitable for fault modeling and simulation

3) Linear memory requirements






PowerPoint Presentation



Slide 9



Ordered BDD (OBDD)


Reduction example

ROBDD properties


Slide 17

Slide 18



Slide 21

SBDD properties



SBDD generation

Structural paths lt=gt BDD nodes

ROBDD vs SBDD

SBDD summary



bull Two principal approaches

bull Transform implementation and spec (reference implementation) to a canonical form

bull Search for an input assignment that would distinguish the responses of the implementation and reference implementation

bull BDD-s applied for canonical form





bull Truth table










c ab 00 01 11 10

0 1 1 1

1 1 1 1



c ab 00 01 11 10

0 1 1 1

1 1 1 1


















x1

x2

x3

0 1

01

1

1

0

0


(x1x2) V x3


10 fxfxf ii




10 fxfxf ii

f=ab+bc

b+bc bc

c c





a BDD





Ordered BDD (OBDD)




same variable



Rules of reduction

a) Node removal

x

f f

10

b) Node sharing

x x x

f0 f1

0 11 0

f0 f1

0 1


Reduction example


ROBDD properties




(Eg multiplier)










variables



(Eg multiplication)




unsolved problem









BDDs




circuit




SBDD properties












ab

cd

e

f

y

1 0

11 0

0

10


SBDD generation

ab

cd

e

f

y

1) 2)

3) 4)





ab

cd

e

f

y


ROBDD vs SBDD

ROBDD SBDD


+ -


- +

Speed of generation

linear linear

Memory consumption

exponential linear


SBDD summary













Slide 9



Ordered BDD (OBDD)


Reduction example

ROBDD properties


Slide 17

Slide 18



Slide 21

SBDD properties



SBDD generation


ROBDD vs SBDD

SBDD summary





bull Truth table










c ab 00 01 11 10

0 1 1 1

1 1 1 1



c ab 00 01 11 10

0 1 1 1

1 1 1 1


















x1

x2

x3

0 1

01

1

1

0

0


(x1x2) V x3


10 fxfxf ii




10 fxfxf ii

f=ab+bc

b+bc bc

c c





a BDD





Ordered BDD (OBDD)




same variable



Rules of reduction

a) Node removal

x

f f

10

b) Node sharing

x x x

f0 f1

0 11 0

f0 f1

0 1


Reduction example


ROBDD properties




(Eg multiplier)










variables



(Eg multiplication)




unsolved problem









BDDs




circuit




SBDD properties












ab

cd

e

f

y

1 0

11 0

0

10


SBDD generation

ab

cd

e

f

y

1) 2)

3) 4)





ab

cd

e

f

y


ROBDD vs SBDD

ROBDD SBDD


+ -


- +

Speed of generation

linear linear

Memory consumption

exponential linear


SBDD summary













Slide 9



Ordered BDD (OBDD)


Reduction example

ROBDD properties


Slide 17

Slide 18



Slide 21

SBDD properties



SBDD generation


ROBDD vs SBDD

SBDD summary









c ab 00 01 11 10

0 1 1 1

1 1 1 1



c ab 00 01 11 10

0 1 1 1

1 1 1 1


















x1

x2

x3

0 1

01

1

1

0

0


(x1x2) V x3


10 fxfxf ii




10 fxfxf ii

f=ab+bc

b+bc bc

c c





a BDD





Ordered BDD (OBDD)




same variable



Rules of reduction

a) Node removal

x

f f

10

b) Node sharing

x x x

f0 f1

0 11 0

f0 f1

0 1


Reduction example


ROBDD properties




(Eg multiplier)










variables



(Eg multiplication)




unsolved problem









BDDs




circuit




SBDD properties












ab

cd

e

f

y

1 0

11 0

0

10


SBDD generation

ab

cd

e

f

y

1) 2)

3) 4)





ab

cd

e

f

y


ROBDD vs SBDD

ROBDD SBDD


+ -


- +

Speed of generation

linear linear

Memory consumption

exponential linear


SBDD summary













Slide 9



Ordered BDD (OBDD)


Reduction example

ROBDD properties


Slide 17

Slide 18



Slide 21

SBDD properties



SBDD generation


ROBDD vs SBDD

SBDD summary



c ab 00 01 11 10

0 1 1 1

1 1 1 1


















x1

x2

x3

0 1

01

1

1

0

0


(x1x2) V x3


10 fxfxf ii




10 fxfxf ii

f=ab+bc

b+bc bc

c c





a BDD





Ordered BDD (OBDD)




same variable



Rules of reduction

a) Node removal

x

f f

10

b) Node sharing

x x x

f0 f1

0 11 0

f0 f1

0 1


Reduction example


ROBDD properties




(Eg multiplier)










variables



(Eg multiplication)




unsolved problem









BDDs




circuit




SBDD properties












ab

cd

e

f

y

1 0

11 0

0

10


SBDD generation

ab

cd

e

f

y

1) 2)

3) 4)





ab

cd

e

f

y


ROBDD vs SBDD

ROBDD SBDD


+ -


- +

Speed of generation

linear linear

Memory consumption

exponential linear


SBDD summary













Slide 9



Ordered BDD (OBDD)


Reduction example

ROBDD properties


Slide 17

Slide 18



Slide 21

SBDD properties



SBDD generation


ROBDD vs SBDD

SBDD summary

















x1

x2

x3

0 1

01

1

1

0

0


(x1x2) V x3


10 fxfxf ii




10 fxfxf ii

f=ab+bc

b+bc bc

c c





a BDD





Ordered BDD (OBDD)




same variable



Rules of reduction

a) Node removal

x

f f

10

b) Node sharing

x x x

f0 f1

0 11 0

f0 f1

0 1


Reduction example


ROBDD properties




(Eg multiplier)










variables



(Eg multiplication)




unsolved problem









BDDs




circuit




SBDD properties












ab

cd

e

f

y

1 0

11 0

0

10


SBDD generation

ab

cd

e

f

y

1) 2)

3) 4)





ab

cd

e

f

y


ROBDD vs SBDD

ROBDD SBDD


+ -


- +

Speed of generation

linear linear

Memory consumption

exponential linear


SBDD summary













Slide 9



Ordered BDD (OBDD)


Reduction example

ROBDD properties


Slide 17

Slide 18



Slide 21

SBDD properties



SBDD generation


ROBDD vs SBDD

SBDD summary




10 fxfxf ii

f=ab+bc

b+bc bc

c c





a BDD





Ordered BDD (OBDD)




same variable



Rules of reduction

a) Node removal

x

f f

10

b) Node sharing

x x x

f0 f1

0 11 0

f0 f1

0 1


Reduction example


ROBDD properties




(Eg multiplier)










variables



(Eg multiplication)




unsolved problem









BDDs




circuit




SBDD properties












ab

cd

e

f

y

1 0

11 0

0

10


SBDD generation

ab

cd

e

f

y

1) 2)

3) 4)





ab

cd

e

f

y


ROBDD vs SBDD

ROBDD SBDD


+ -


- +

Speed of generation

linear linear

Memory consumption

exponential linear


SBDD summary













Slide 9



Ordered BDD (OBDD)


Reduction example

ROBDD properties


Slide 17

Slide 18



Slide 21

SBDD properties



SBDD generation


ROBDD vs SBDD

SBDD summary




a BDD





Ordered BDD (OBDD)




same variable



Rules of reduction

a) Node removal

x

f f

10

b) Node sharing

x x x

f0 f1

0 11 0

f0 f1

0 1


Reduction example


ROBDD properties




(Eg multiplier)










variables



(Eg multiplication)




unsolved problem









BDDs




circuit




SBDD properties












ab

cd

e

f

y

1 0

11 0

0

10


SBDD generation

ab

cd

e

f

y

1) 2)

3) 4)





ab

cd

e

f

y


ROBDD vs SBDD

ROBDD SBDD


+ -


- +

Speed of generation

linear linear

Memory consumption

exponential linear


SBDD summary













Slide 9



Ordered BDD (OBDD)


Reduction example

ROBDD properties


Slide 17

Slide 18



Slide 21

SBDD properties



SBDD generation


ROBDD vs SBDD

SBDD summary


Ordered BDD (OBDD)




same variable



Rules of reduction

a) Node removal

x

f f

10

b) Node sharing

x x x

f0 f1

0 11 0

f0 f1

0 1


Reduction example


ROBDD properties




(Eg multiplier)










variables



(Eg multiplication)




unsolved problem









BDDs




circuit




SBDD properties












ab

cd

e

f

y

1 0

11 0

0

10


SBDD generation

ab

cd

e

f

y

1) 2)

3) 4)





ab

cd

e

f

y


ROBDD vs SBDD

ROBDD SBDD


+ -


- +

Speed of generation

linear linear

Memory consumption

exponential linear


SBDD summary













Slide 9



Ordered BDD (OBDD)


Reduction example

ROBDD properties


Slide 17

Slide 18



Slide 21

SBDD properties



SBDD generation


ROBDD vs SBDD

SBDD summary



Rules of reduction

a) Node removal

x

f f

10

b) Node sharing

x x x

f0 f1

0 11 0

f0 f1

0 1


Reduction example


ROBDD properties




(Eg multiplier)










variables



(Eg multiplication)




unsolved problem









BDDs




circuit




SBDD properties












ab

cd

e

f

y

1 0

11 0

0

10


SBDD generation

ab

cd

e

f

y

1) 2)

3) 4)





ab

cd

e

f

y


ROBDD vs SBDD

ROBDD SBDD


+ -


- +

Speed of generation

linear linear

Memory consumption

exponential linear


SBDD summary













Slide 9



Ordered BDD (OBDD)


Reduction example

ROBDD properties


Slide 17

Slide 18



Slide 21

SBDD properties



SBDD generation


ROBDD vs SBDD

SBDD summary


Reduction example


ROBDD properties




(Eg multiplier)










variables



(Eg multiplication)




unsolved problem









BDDs




circuit




SBDD properties












ab

cd

e

f

y

1 0

11 0

0

10


SBDD generation

ab

cd

e

f

y

1) 2)

3) 4)





ab

cd

e

f

y


ROBDD vs SBDD

ROBDD SBDD


+ -


- +

Speed of generation

linear linear

Memory consumption

exponential linear


SBDD summary













Slide 9



Ordered BDD (OBDD)


Reduction example

ROBDD properties


Slide 17

Slide 18



Slide 21

SBDD properties



SBDD generation


ROBDD vs SBDD

SBDD summary


ROBDD properties




(Eg multiplier)










variables



(Eg multiplication)




unsolved problem









BDDs




circuit




SBDD properties












ab

cd

e

f

y

1 0

11 0

0

10


SBDD generation

ab

cd

e

f

y

1) 2)

3) 4)





ab

cd

e

f

y


ROBDD vs SBDD

ROBDD SBDD


+ -


- +

Speed of generation

linear linear

Memory consumption

exponential linear


SBDD summary













Slide 9



Ordered BDD (OBDD)


Reduction example

ROBDD properties


Slide 17

Slide 18



Slide 21

SBDD properties



SBDD generation


ROBDD vs SBDD

SBDD summary










variables



(Eg multiplication)




unsolved problem









BDDs




circuit




SBDD properties












ab

cd

e

f

y

1 0

11 0

0

10


SBDD generation

ab

cd

e

f

y

1) 2)

3) 4)





ab

cd

e

f

y


ROBDD vs SBDD

ROBDD SBDD


+ -


- +

Speed of generation

linear linear

Memory consumption

exponential linear


SBDD summary













Slide 9



Ordered BDD (OBDD)


Reduction example

ROBDD properties


Slide 17

Slide 18



Slide 21

SBDD properties



SBDD generation


ROBDD vs SBDD

SBDD summary




unsolved problem









BDDs




circuit




SBDD properties












ab

cd

e

f

y

1 0

11 0

0

10


SBDD generation

ab

cd

e

f

y

1) 2)

3) 4)





ab

cd

e

f

y


ROBDD vs SBDD

ROBDD SBDD


+ -


- +

Speed of generation

linear linear

Memory consumption

exponential linear


SBDD summary













Slide 9



Ordered BDD (OBDD)


Reduction example

ROBDD properties


Slide 17

Slide 18



Slide 21

SBDD properties



SBDD generation


ROBDD vs SBDD

SBDD summary








BDDs




circuit




SBDD properties












ab

cd

e

f

y

1 0

11 0

0

10


SBDD generation

ab

cd

e

f

y

1) 2)

3) 4)





ab

cd

e

f

y


ROBDD vs SBDD

ROBDD SBDD


+ -


- +

Speed of generation

linear linear

Memory consumption

exponential linear


SBDD summary













Slide 9



Ordered BDD (OBDD)


Reduction example

ROBDD properties


Slide 17

Slide 18



Slide 21

SBDD properties



SBDD generation


ROBDD vs SBDD

SBDD summary




circuit




SBDD properties












ab

cd

e

f

y

1 0

11 0

0

10


SBDD generation

ab

cd

e

f

y

1) 2)

3) 4)





ab

cd

e

f

y


ROBDD vs SBDD

ROBDD SBDD


+ -


- +

Speed of generation

linear linear

Memory consumption

exponential linear


SBDD summary













Slide 9



Ordered BDD (OBDD)


Reduction example

ROBDD properties


Slide 17

Slide 18



Slide 21

SBDD properties



SBDD generation


ROBDD vs SBDD

SBDD summary


SBDD properties












ab

cd

e

f

y

1 0

11 0

0

10


SBDD generation

ab

cd

e

f

y

1) 2)

3) 4)





ab

cd

e

f

y


ROBDD vs SBDD

ROBDD SBDD


+ -


- +

Speed of generation

linear linear

Memory consumption

exponential linear


SBDD summary













Slide 9



Ordered BDD (OBDD)


Reduction example

ROBDD properties


Slide 17

Slide 18



Slide 21

SBDD properties



SBDD generation


ROBDD vs SBDD

SBDD summary










ab

cd

e

f

y

1 0

11 0

0

10


SBDD generation

ab

cd

e

f

y

1) 2)

3) 4)





ab

cd

e

f

y


ROBDD vs SBDD

ROBDD SBDD


+ -


- +

Speed of generation

linear linear

Memory consumption

exponential linear


SBDD summary













Slide 9



Ordered BDD (OBDD)


Reduction example

ROBDD properties


Slide 17

Slide 18



Slide 21

SBDD properties



SBDD generation


ROBDD vs SBDD

SBDD summary


SBDD generation

ab

cd

e

f

y

1) 2)

3) 4)





ab

cd

e

f

y


ROBDD vs SBDD

ROBDD SBDD


+ -


- +

Speed of generation

linear linear

Memory consumption

exponential linear


SBDD summary













Slide 9



Ordered BDD (OBDD)


Reduction example

ROBDD properties


Slide 17

Slide 18



Slide 21

SBDD properties



SBDD generation


ROBDD vs SBDD

SBDD summary





ab

cd

e

f

y


ROBDD vs SBDD

ROBDD SBDD


+ -


- +

Speed of generation

linear linear

Memory consumption

exponential linear


SBDD summary













Slide 9



Ordered BDD (OBDD)


Reduction example

ROBDD properties


Slide 17

Slide 18



Slide 21

SBDD properties



SBDD generation


ROBDD vs SBDD

SBDD summary


ROBDD vs SBDD

ROBDD SBDD


+ -


- +

Speed of generation

linear linear

Memory consumption

exponential linear


SBDD summary













Slide 9



Ordered BDD (OBDD)


Reduction example

ROBDD properties


Slide 17

Slide 18



Slide 21

SBDD properties



SBDD generation


ROBDD vs SBDD

SBDD summary


SBDD summary













Slide 9



Ordered BDD (OBDD)


Reduction example

ROBDD properties


Slide 17

Slide 18



Slide 21

SBDD properties



SBDD generation


ROBDD vs SBDD

SBDD summary

digitaalsüsteemide verifitseerimise kursus1 formal verification: bdd bdds applied in equivalence...

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