Download BDDs and MDDs for State-space Generation and Symbolic Model Checking | CS 246 and more Study notes Computer Science in PDF only on Docsity!

PART:

FILE:

.

BDDs and MDDs

for state-space generation

and symbolic model checking

PART:

FILE:

BDD/binary-decision-diagramsMOD.tex

(Reduced ordered) binary decision diagrams (BDDs)

“Graph-based algorithms for boolean function manipulation”

Randy Bryant (Carnegie Mellon University)

IEEE Transactions on Computers, 1986

CiteSeer most cited document!

0

1

x 1 x 2 + x 1

x

x

x

x

x

x

x

x

x

x

x

x

1 + ( x 4 + x 3

x

x

x )

1

x 2 x 3 x 4 x 2

0 -edge

1 -edge

BDDs are a

canonical

representation of boolean functions

f : B L → B

For the

root

node,

f (^) ( x

4 (^) = 0

(^) x

3 (^) = 1

(^) x

2 (^) = 1

(^) x

1 (^) = 0

f (^) ( x

4 (^) = 0

x

3

= 1

x

2

= 1

x

1

= 1

PART:

FILE:

BDD/bdd-def.tex

Canonical versions of BDDs

For

canonical

BDDs, we further require that

There are no

duplicates

: if

p.lvl

q.lvl

and

p [0] =

q [0]

and

p [1] =

q [1]

, then

p

=

q

Then, if the BDD is

quasi-reduced

, there is

no level skipping

:

The only

root

nodes with no incoming arcs are at level

L

The children

p [0]

and

p [1]

of a node

p

are at level

p.lvl

Or, if the BDD is

fully-reduced

, there is

maximum level skipping

:

There are no

redundant

nodes

p

satisfying

p [0] =

p [1]

Both versions are

canonical

, if functions

f

and

g

are encoded using BDDs:

Satisfiability,

f

, or equivalence,

f

g

O

Conjunction,

f

g , disjunction,

f

g , relational product:

O

f (^) || × ||

g || ) , if fully-reduced

. ∑ L ≥ k ≥ 1 O (

f (^) ||

k

× ||

g || k ) , if quasi-reduced

f (^) ||

= number of nodes in the BDD encoding

f

f (^) ||

k

= number of nodes at level

k

in the BDD encoding

f

PART:

FILE:

BDD/bdd-def.tex

Quasi-reduced vs. fully-reduced BDDs

x 4 x 3 (^) x

2 (^) x

1

• (

x 4

x 3 ) (

x 2 + x 1 ) 0 1

0

1

0

1

0

1

0

1

0

1

x 3 x 2 (^) x

1

x 3 (^) ( x 2 + x 1 )

x 2 x 1 x 2 + x 1

x 1

1

0

0

1

0

1

0

1

0

1

x 2 + x 1 x 1

x 2

x 3

x 4 x 4 x 3

(^) x

2 (^) x

1

• (

x 4

x 3 ) (

x 2 + x 1 ) 0 1

0

1

0

1

0

1

0

1

0

1

x 3 x 2 (^) x

1

x 3 (^) ( x 2 + x 1 )

x 2 x 1 x 2 + x 1

x 1

1

0

Fully-reduced BDDs: each node

in the BDD

encodes a different function

Quasi-reduced BDDs: each node

at a given level of the BDD

encodes a different function

PART:

FILE:

BDD/ite.tex

ITE

: the if-then-else operator

Any function

f : B L → B

of

L

variables

x L

,... ,

(^) x

1

can be expressed as

a

boolean vector

of size

L , or a

full binary tree

of height

L

an

expression

over

x

L ,... ,

(^) x

1

using the operator set

, {¬

, or

a

disjunctive normal form (DNF)

expression over

x

L ,... ,

(^) x

1

d D

C d

c

t d,c

where each

t d,c

is either

x k

or its complement

x

k , also written as

x k

a

conjunctive normal form (CNF)

expression over

x

L ,... ,

x 1

c C

D

c

d

t c,d

where each

t c,d

is either

x k

or its complement

x

k , also written as

x k

an

expression

over

x

L ,... ,

(^) x

1

using only the

ITE

operator and the constants

and

ITE

f, g, h

) = ( f ∧ g ) ∨ ( ¬ f ∧ h )

an

if-then-else normal form (INF)

expression over

x L

,... ,

x 1

where variables appear only in

tests, i.e., as the first parameter of

ITE

f

ITE

f,

f

g

=

ITE

f,

ITE

g,

f

g

=

ITE

f,

ITE

g,

(^1)

,

f

g

=

ITE

f,

ITE

g,

f

g

=

ITE

f,

ITE

g,

ITE

g,

PART:

FILE:

BDD/ite.tex

ITE

, Shannon expansion, and BDDs

Let

f (^) [ c/x

]

be the expression obtained from

f

by substituting variable

x

with the constant

c

∈

B

Then,

f

ITE

x, f

[

/x

]

, f

[

/x

])

is the

Shannon expansion

of

f

with respect to

x

Given an arbitrary boolean expression

f

over

x

L ,... ,

(^) x

1 , we can put it in INF as follows:

let

f 0

be

f (^) [

/ x L ] f 0

does not contain

x

L

let

f 1

be

f (^) [

/ x L ] f 1

does not contain

x

L

let

f 00

be

f 0 [

/ x L − 1 ] f

00

contains neither

x

L

nor

x L −

1

let

f 01

be

f 0 [

/ x L − 1 ] f

01

contains neither

x

L

nor

x L −

1

let

f 10

be

f 1 [

/ x L − 1 ] f

10

contains neither

x

L

nor

x L −

1

let

f 11

be

f 1 [

/ x L − 1 ] f

11

contains neither

x

L

nor

x L −

1

...

f

ITE

x

L , (^) ITE

x

L −

1 ,

f 11

f 10

f 1

ITE

x L −

1 ,

f 01

f 00

f 0

PART:

FILE:

FOILS/ut-and-cache.tex

The unique table and the operation cache

To ensure canonicity, all decision diagram operations use a

Unique Table

(a hash table):

Search key

: the node’s level and sequence of children’s

node

id

s

Return value

: a

node

id

Alternative: one UT per level, no need to store the node’s level, but more fragmentation

All (non-dead) nodes are referenced by the UT

Collisions must be

lossless

, multiple nodes with different

node

id

may have the same

hash

val

With the UT, we avoid

duplicate nodes

To achieve polynomial complexity, all operations use an

Operation Cache

(a hash table):

Search key

: OpCODE

and sequence of operands’

node

id

s

Return value

: node

id

Alternative: one OC per operation type, no need to store

OpCODE

(^) , but more fragmentation

Before computing

OpCODE

node

id

1 , node

id

2 , ...

, we search the OC

If the search is successful, we avoid recomputing a result

Collisions can be either

lossless

or

lossy

With the OC, we consider

every node combination

instead of

every path combination

PART:

FILE:

BDD/apply.tex

IT E

and the

Apply

operator for fully-reduced BDDs

11

The

if-then-else

, or

ITE

, ternary operator is defined as

ITE

f, g, h

f ∧ g ) ∨ ( ¬ f ∧ h )

Let

f (^) [ c/x

k ] be the function obtained from

f

by substituting variable

x k

with the constant

c

∈

B

Then,

f

ITE

x k , f

[

/x

k ] , f

[

/x

k ])

is the

Shannon expansion

of

f

with respect to variable

x k

For any binary boolean operator

:

ITE

x, u, v

ITE

x, y, z

ITE

x, u

y, v

z )

This is the basis for the

recursive

BDD operator

Apply

bdd Apply

( operator

⊙

, (^) bdd

p,

(^) bdd

q ) is

fully-reduced version

local

bdd

r ;

1 if

p

∈ {

0 , (^1)

}

and

q

∈ {

0 , (^1)

}

then return

p

⊙

q ;

2 if

Cache

contains entry

〈⊙

, p, q

: r 〉 then return

r ;

3 if

p.

lvl

=

q.

lvl

then

4

r

←

UniqueTableInsert

( p.

lvl

(^) , (^) Apply

( ⊙

, p

[0]

, q

[0])

, Apply

( ⊙

, p

[1]

, q

[1]))

;

5 else if

p.

lvl

q.

lvl

then

6

r

←

UniqueTableInsert

( p.

lvl

(^) , (^) Apply

( ⊙

, p

[0]

, q

) , (^) Apply

( ⊙

, p

[1]

, q

))

;

7 else since

p.

lvl

< q.

lvl

then

8

r

←

UniqueTableInsert

( q.

lvl

(^) , (^) Apply

( ⊙

, p, q

[0])

, (^) Apply

( ⊙

, p, q

[1]))

;

9 enter

〈⊙

, p, q

: r 〉 in

Cache

;

10 return

r ;

PART:

FILE:

BDD/relational-product.tex

Computing the relational product symbolically

Given an

L

-level BDD on

x L

, ..., x

1 )

rooted at

p ∗

encoding a set

Y ⊆

X

Given a

L

-level BDD on

x L

, x

L′ ..., x

1 , x

1 ′ ) rooted at

r ∗

encoding a function

N

X →

b X

RelationalProduct

p ∗ , r

∗ ) returns the root of the BDD encoding

j

: ∃ i ∈ Y ∧

j

∈ N

i ) }

bdd RelationalProduct

( bdd

p,

(^) bdd

r ) is

quasi-reduced version

local

bdd

q , q 1 , q 2 ;

1 if

p

=

0

or

r

=

0

then return

0 ;

2 if

p

=

1

and

r

=

1

then return

1 ;

3 if

Cache

contains entry

〈 RelationalProductCODE

(^) , p, r

: q 〉 then return

q ;

4

q 0

←

Union

( RelationalProduct

( p [0]

, r

[0][0])

, (^) RelationalProduct

( p [1]

, r

[1][0]))

;

5

q 1

←

Union

( RelationalProduct

( p [0]

, r

[0][1])

, (^) RelationalProduct

( p [1]

, r

[1][1]))

;

6

q

←

UniqueTableInsert

( p.lvl, q

0 , q

1 ) ;

7 enter

〈 RelationalProductCODE

(^) , p, r

: q 〉 in

Cache

;

8 return

q ;

PART:

FILE:

BDD/next-state-function.tex

BDD encoding of the next-state function

Given a current set of states

S

encoded as a BDD in

L

variables (

x L

, ..., x

1 )

Gievn the

next-state function

N

encoded as a BDD in

L

variables (

x L , x

L′

..., x

1 , x

1 ′ )

We compute the set of states

N

S

reachable from

S

in one step

x

1

x

2

x

3

x

4

x

1

x

2

x

3

x

4

x

1

x

2

x

3

x

4

x

1

x

2

x

3

x

4

x

1

x

2

x

3

x

4

SET

UNION

x

1

x

2

x

3

x

4

Iterations

to generate the state space

X

reach

:

max distance

d

of any state from initial states + 1

Peak

BDD size usually occurs well before reaching the

final

BDD for

X

reach

PART:

FILE:

SYMBOLIC-GENERATION/comparing.tex

Explicit vs. symbolic state space generation

Explicit generation of the state space

X

reach

adds

one state

at a time

memory

O

(states), increases linearly, peaks at the end

i [0℄

i [1℄

i [0℄

i [1℄

i [2℄

i [3℄

i [n ℄

i [0℄

i [1℄

i [0℄

i [2℄

i [1℄

i [0℄

i [2℄

i [3℄

i [1℄

i [0℄

i [2℄

i [3℄

i [n ℄

Symbolic generation of the state space

X

reach

with decision diagrams adds

sets of states

instead

memory

O

(decision diagram nodes), grows and shrinks, usually peaks well before the end

i L [0]

i L

[0]

i 2 [0]

i 1 [0]

i [0]

PART:

FILE:

MDD/mdd-def.tex

Ordered multiway decision diagrams (MDDs)

17

Assume a

domain

X

X

L

× · · · × X

1 , where

X k = { 0 , 1

, ..., n

k (^) −

, for some

n k

∈

N

Assume the

range

X

0

=

B

An MDD is an acyclic directed edge-labeled graph where:

The only

terminal

nodes can be

and

, and are at

level

.lvl

.lvl

A

nonterminal

node

p

is at a

level

k , with

L

k

p.lvl

k

For each

i k

∈ X

k , a nonterminal node

p

at level

k

has an outgoing edge pointing to

child

p [ i k ]

The level of a child is lower than that of

p p [ i k ]

.lvl < p.lvl

A node

p

at level

k

encodes the

function

v p

:

X →

B

defined recursively by

v p ( x L

, ..., x

1 ) =

p

if

k

v p [ x k (^) ] ( x L , ..., x

1 )

if

k >

Instead of levels, we can also talk of

variables

:

The terminal nodes are associated with the

range variable

x 0

A nonterminal node is associated with a

domain variable

x

k

, with

L

k

PART:

FILE:

MDD/mdd-def.tex

Quasi-reduced vs. fully-reduced MDDs

full storage quasi-reduced

full storagefully-reduced

0

1

0

1

2

0

1

0 1 2 0 1 2 3 0 1 2

0

1

2

0

1

0

1

2

0

1

2 0

1

0

1

x 1

x 2

x 3

x 4

0

1

0 1 2 3 0 1 2

0

1

2

0

1

2 0

1

2

0

1

What if we don’t know the range of each

X

k ?

We can simply assume

X

N

L

All but a finite number of edges point to

Only nodes encoding

are redundant

sparse storage

x 1

x 2

x 3

x 4

1

1

2

3

0

1

2

0

1

2

0

1

1

0

1

0

1

2

0

0

2

Fully-reduced MDDs: each node

in the MDD

encodes a different function

Quasi-reduced MDDs: each node

at a given level of the MDD

encodes a different function

PART:

FILE:

FOILS/set-encoded-by-mdd.tex

MDDs to encode sets

An important application of decision diagrams is to encode large sets to be manipulated

symbolically

To encode a set

Y ⊆

X

(^) , we store its

indicator function

f Y

in a decision diagram rooted at node

p :

i L , ..., i

1 )

∈ Y ⇔

f Y (^) ( i L , ..., i

1 ) = 1

⇔ v p ( i L

, ..., i

1 ) = 1

X 4 = { 0 ,

X 3 = { 0 ,

X 2 = { 0 ,

X 1 = { 0 ,

1

0

1

2

3

0 1 2 0 1 2 0 1 0 1 0 1 0 1 2 0 1 2 0 1 2 0 1 2

0 1 0 1 2 3

2

0

1

2

0

1

0

1

2

0

0

1

2

0

1

1

1

0

Y