ece 667 - synthesis & verification - lecture 10 1 ece 697b (667) spring 2006 ece 697b (667)...

1ECE 667 - Synthesis & Verification - Lecture 10

ECE 697B (667)ECE 697B (667)Spring 2006Spring 2006

Synthesis and Verificationof Digital Systems

Binary Decision DiagramsBinary Decision Diagrams

(BDD)(BDD)

ECE 667 - Synthesis & Verification - Lecture 10 2

OutlineOutline

• Background– Canonical representations

• BDD’s– Reduction rules

– Construction of BDD’s

– Logic manipulation of BDD’s

– Application to verification and SAT

• Reading:

read one of the BDD tutorials available on class web site– Anderson, or

– Somenzi


Common RepresentationsCommon Representations• Boolean functions ( f : B B )

– Truth table, Karnaugh map– SoP, PoS, ESoP– Reed-Muller expansions (XOR-based)– Decision diagrams (BDD, ZDD, etc.)

• Each minimal, canonical representation is characterized by– Decomposition type

• Shannon, Davio, moment decomposition, Taylor exp., etc.– Reduction rules

• Redundant nodes, isomorphic sub-graphs, etc.– Composition method (“Apply”, compose rule)

• What they represent– Boolean functions (f : B B)

– Arithmetic functions (f : B Int )

– Algebraic expressions (f : Int Int )


Binary Decision Diagrams (Binary Decision Diagrams (BDDBDD))

• Based on recursive Shannon expansion

f = x fx + x’ fx’

• Compact data structure for Boolean logic– can represents sets of objects (states) encoded as Boolean

functions

• Canonical representation– reduced ordered BDDs (ROBDD) are canonical– essential for verification


ROBDD’sROBDD’s

• Directed acyclic graph (DAG)• one root node, two terminals 0, 1• each node, two children, and a variable• Shannon co-factoring tree, except reduced and ordered

(ROBDD)– Reduced:

• any node with two identical children is removed• two nodes with isomorphic BDD’s are merged

– Ordered: • Co-factoring variables (splitting variables) always follow

the same order along all paths

xi1 < xi2

< xi3 < … < xin


ExampleExample

Two different orderings, same function.Two different orderings, same function.

a

b b

c c

d

0 1

c+bd b

root node

c+dc

d

f = ab+a’c+bc’d a

c

d

b

0 1

c+bd

d+b

b

1

0


ROBDDROBDDOrdered BDD (OBDD) Input variables are ordered - each path from root to

sink visits nodes with labels (variables) in ascending order.

a

c c

b

0 1

orderedorder = a,c,b

Reduced Ordered BDD (Reduced Ordered BDD (ROBDDROBDD) - reduction rules:) - reduction rules:1.1. if the two children of a node are the if the two children of a node are the samesame, the node is eliminated: f , the node is eliminated: f

= vf + vf= vf + vf2.2. if two nodes have if two nodes have isomorphic isomorphic graphs, they are replaced by one of graphs, they are replaced by one of

themthemThese two rules make it so that each node represents a distinct logic These two rules make it so that each node represents a distinct logic function.function.

a

b c

c

0 1

notordered

b


Efficient Implementation of BDD’sEfficient Implementation of BDD’s• BDDs is a compressed Shannon co-factoring tree:

• f = v fv + v fv

• leafs are constants “0” and “1”

• Three components make ROBDDs canonical (Proof: Bryant 1986):

– unique nodes for constant “0” and “1”

– identical order of case splitting variables along each paths

– hash table that ensures:

• (node(fv) = node(gv)) (node(fv) = node(gv)) node(f) = node(g)

– provides recursive argument that node(f) is unique when using the unique hash-table

v0 1

f

fv fv


Onset is Given by all Paths to “1”Onset is Given by all Paths to “1”

Notes:• By tracing paths to the 1 node, we get a cover of pair wise disjoint cubes.• The power of the BDD representation is that it does not explicitly

enumerate all paths; rather it represents paths by a graph whose size is measures by its nodes and not paths.

• A DAG can represent an exponential number of paths with a linear number of nodes.

• BDDs can be used to efficiently represent sets– interpret elements of the onset as elements of the set– f is called the characteristic function of that set

F = b’+a’c’ = ab’+a’cb’+a’c’ all paths to the 1 node

a

cb

0 1

10

1

10

0

f

fa= b’fa = cb’+c’


ImplementationImplementationVariables are totally ordered: If v < w then v occurs “higher” up in the ROBDDTop variable of a function f is a variable associated with its root node.Example: f = ab + a’bc + a’bc’. Order is (a < b < c).

fa = b, fa = b

a

b

0 1

f b is top variable of f

b

0 1

f

reducedf does not depend on a,

since fa = fa .

Each node is written as a triple: f = (v,g,h) where g = fv and h = fv . We read this triple as:

f = if v then g else h = ite (v,g,h) = vg+v ’ h

vf

0 1

h g

1 0f

v

g h

mux

v is top variable of f


BDD ConstructionBDD Construction• Reduced Ordered BDDReduced Ordered BDD

1 edge

0 edgea b c f

0 0 0 00 0 1 00 1 0 00 1 1 11 0 0 01 0 1 11 1 0 01 1 1 1

Truth table

f = ac + bc

Decision tree

10 0 0 1 0 10

a

b

c

b

c c c

f


BDD Reduction Rules -1BDD Reduction Rules -1• Eliminate redundant nodes Eliminate redundant nodes

(with both edges pointing to same node)(with both edges pointing to same node)

f = fa g(b) + fa’ g(b) = g(b)

(fa + fa’ = 1)bg

a

b

f

g


BDD Reduction Rules -2BDD Reduction Rules -2• Merge duplicate nodes (Merge duplicate nodes (isomorphicisomorphic subgraphs) subgraphs)

• Nodes must be unique

f1 = fa’ g(b) + fa h(c) = f2 f = f1 = f2

a a

b chg

f1 f2

a

b cg h

f


BDD Construction – cont’dBDD Construction – cont’d

10

a

b

c

b

c c c

f f

10

a

b

c

b

c

10

a

b

c

f = (a+b)c

2. Merge duplicate nodes

1. Merge terminal nodes 3. Remove redundant nodes


APPLYAPPLY Operator Operator

• Useful in constructing BDD for arbitrary Boolean logicUseful in constructing BDD for arbitrary Boolean logic

• Any logic operation can be expressed using Any logic operation can be expressed using Apply (ITE)Apply (ITE)

• Efficient algorithms, work directly on BDD graphsEfficient algorithms, work directly on BDD graphs

• Apply: F G, Apply: F G, any Boolean operationany Boolean operation((AND, OR, XOR, AND, OR, XOR, ))

=

F G

0 1 0 10 1

F G


Logic Manipulation using BDDsLogic Manipulation using BDDs

• Useful operatorsUseful operators

– Complement ¬ F = F’

(switch the terminal nodes)

– Restrict: F|x=b = F(x=b) where b = const

¬

1 00 1

F F’

0 1

F(x,y)

x=b 0 1

F(y)Restrict


Useful BDD Operators – Apply OperationUseful BDD Operators – Apply Operation• Basic operator for efficient BDD manipulation (structural)Basic operator for efficient BDD manipulation (structural)• Based on recursive Shannon expansionBased on recursive Shannon expansion

F F OPOP G = x (F G = x (Fxx OPOP G Gxx) + x’(F) + x’(Fx’x’ OPOP G Gx’x’))

wherewhere OPOP = OR, AND, XOR, etc = OR, AND, XOR, etc


Apply Operation (cont’d)Apply Operation (cont’d)

• Apply: F GApply: F Gwhere stands for any Boolean operator (AND, OR, XOR, where stands for any Boolean operator (AND, OR, XOR, ))

=

F G

0 1 0 10 1

F G

• Any logic operation can be expressed using only Restrict and

Apply

• Efficient algorithms, work directly on BDDs


Apply Operation - Apply Operation - ANDAND

10

a

c

aca AND c

10

a 2

c

10

3

0.3

2.3a

c1.3

1.11.0

AND

= =


Apply Operation - Apply Operation - OROR

OR

ac

10

a

c

4

5

bc

10

b

c

6

7 ==

10

a

b

c

f = ac+bc

c

4+6

0+0

a

7+5

1

0+6 b6+5

0+5

0

0+7


Application to VerificationApplication to Verification

• Equivalence Checking of Equivalence Checking of combinationalcombinational circuits circuits• CanonicityCanonicity property of BDDs: property of BDDs:

– if F and G are equivalent, their BDDs are identical (for the same if F and G are equivalent, their BDDs are identical (for the same ordering of variablesordering of variables))

10

a

b

c

F = a’bc + abc +ab’c G = ac +bc

10

a

b

c


Application to SATApplication to SAT

• Functional test generationFunctional test generation– SAT, Boolean SAT, Boolean satisfiability satisfiability

analysisanalysis

– to test for H = 1 (0), find a path to test for H = 1 (0), find a path in the BDD to terminal in the BDD to terminal 11 (0) (0)

– the path, expressed inthe path, expressed in function function variables, gives a satisfying variables, gives a satisfying solution (solution (test vectortest vector)) ab

ab’c

H

0 1

a

b

c

• Problem: size explosionProblem: size explosion


Efficient Implementation of BDD’sEfficient Implementation of BDD’s

Unique Table:• avoids duplication of existing nodes

– Hash-Table: hash-function(key) = value– identical to the use of a hash-table in AND/INVERTER circuits

Computed Table:• avoids re-computation of existing results

hash valueof key

collisionchain

hash valueof key

No collision chain

ece 667 - synthesis & verification - lecture 10 1 ece 697b (667) spring 2006 ece 697b (667)...

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