Knowledge Compilation Properties of TreeofBDDs

1
Knowledge Compilation Properties of Tree-of-BDDs
Sathiamoorthy Subbarayan
Lucas Bordeaux, Youssef Hamadi
IT University of Copenhagen, Denmark
Microsoft Research, Cambridge, UK
Tree
-
of
-
BDDs
(
ToB
)
Example
CNF to
ToB
Propagation in
ToB
CNF to BDD
Binary
Decision Diagrams
(

a

e
)
C1
C1
C4
C3
(
ab
)
C1
C2

Use
a
rooted
BFS
tree
C1
C2
C5
C2
C3
CNF
C4
C4
C5
C2
(
b

c
)
C3
Tree
clauses
c1
c2
C5
C6
C3
C6
C1
C2
Tree
Decomposition
(
c

d
)
Decomposition
C4

Propagate
up
C7
(

b

d
)
c4
c6
C4
C5
C5
CNF
AND

Makes
each
parent
bdd
C7
clauses
consistent
with
children
C3
C6
c5
c3
C7

Propagate
down
BDD
c7
CNF

Makes
each
child
clause
consistent
with
parent
A
compressed
representation
of
solutions
clauses
BDDs
ANDPropagate
Group
clause
-
BDDs
ToB
clause
Answers
many
queries
in
polytime
ToB
BDDs
4
5
3
1
2
Some
Interesting
Queries
Knowledge
Compilation
Example
Comparison
Conditioning
ToB
Clausal Entailment,
Implicant
Check

Consistency
(CO)
(
ToB
, term)
Condition

Validity
(VA)

Compile
a system model in to a
form
(
ToB
,
clause
)
(
ToB
, term)
IsCE
IsIM
//Restrict
ToB
by term

Clausal
Entailment
(CE)
each BDD with term
Restrict
//Does
ToB
entails the clause
//Does the term entails
ToB

Implicant
Check (IM)

Quickly
answer
interesting
queries
(
ToB
,

clause
)
(
ToB
, term)
Condition
Condition
Propagate

Equality
(EQ)
if
(
ToBfalse
)
if
( term
entails
each
BDD )
return
true

Sentential
Entailment
(SE)
return
true

Applications

configuration
,
fault
-
trees
,
else
else
Bayesian
networks
,
model
-
based

Model
Counting
(CT)
Exponential
in
tree
-
width
!
return
false
return
false
diagnosis
,
verification
, etc.,
d
-
DNNF

Model
Enumeration
(ME)
Exploits
15
edges
tree
-
width
!
8
7
6
9
10
Equivalence Checking
Complexity
of
Queries
Compilation
Overview
Experiments
Projection in
ToB

91
realistic
CNF
instances
(ToB
, ToB
)
IsEQ
(
ToB
, K)
Project
1
2

35 Mercedes
car
configuration
problems
//Project
ToB
over K variables

7 ISCAS85
circuits
ToB
over
vars
in
each
node
of
ToB
Project
1
2
Conjoin
BDDs
from
bottom
to top

17 ISCAS89
circuits
-
Check
whether
the
two
BDDs
agree
Early
quantification
of
non
K
variables
-

14 ISCAS93
Addendum
circuits
ToB
over
vars
in
each
node
of
ToB
Project
2
1

13 ISCAS99
circuits
for successful compilations
-
Check
whether
the
two
BDDs
agree

5
Bounded
model
checking
circuits
Exponential
in (
wK
)
Exponential in
w
Polytime
ToB
tool
compiles
all
the
used

Compare
with
c2d
, a
d
-
DNNF
compiler
instances
in just 926
seconds
Exponential
in (w
w
)

Simulate
CE and IM
queries
on
ToBs
1
2
Exponential in
Not unless
(
w
w
)
PNP
1
2
12
15
14
13
11
Compilation
Space

ToB
vs.
d
-
DNNF
Conclusion
Compilation
Time
ToB
vs.
d
-
DNNF
CE and IM Simulation
Details
Cases
hard
for
d
-
DNNF
ToB
d
-
DNNF (mf)
d
-
DNNF (hg)
Instance

ToBs
often
use
very
small time and
Size
Time
Size
Time
Size
Time
w
C
V
Name
space
0.02
0.1
TO
26.1
269
46
2053
751
c1908
0.22
1.02
TO
28.9
371
44
8588
3185
c7552
0.05
0.18
TO
10
119
40
1616
623
s1269

ToBs
still
answer
queries
for
many
0.15
0.55
TO
TO
35
4269
1714
s3271
1.31
5.09
FSE
FSE
33
6434
2495
s4863
realistic
instances
0.16
0.44
57.53
348
TO
26
8423
3392
s6669
0.09
1.86
0.01
168
TO
23
30678
9207
cnt09
0.21
5.41
TO
TO
26
68561
20470
cnt10

Future
work

0.21
8.05
5.54
296
TO
110
7477
1909
C168FW
0.58
15
5.9
365
TO
203
10610
1874
C170FR

Succinctness
of
ToB
vs
d
-
DNNF
0.41
9.58
TO
TO
88
8883
1990
C202FS

Transformations
supported
by
ToB
0.94
65
TO
TO
99
11342
2038
C202FW

Adopting
ToBs
for
multi
-
core
CPUs
0.93
18
9.56
243
TO
194
7518
1922
C208FC
0.91
12
51.34
2370
TO
84
7982
1990
C210FS
3.64
663
71.63
3352
TO
98
9705
2024
C210FW
19
20
18
17
16