From AND/OR Search to
AND/OR BDDs
Rina Dechter
Information and Computer Science, UC-Irvine,
and
Radcliffe Institue of Advanced Study, Cambridge
Joint work with Robert Mateescu and John Cobb
Verification/constraints workshop,
2006
Combining Decision Diagrams
f = A*E + B*F
A
B
g = C*G +D*H
C
B
D
E
D
E
G
F
G
H
0
1
0
1
A
B
B
C
D
C
D
E
OBDD f * g =
F
C
D
E
D
E
F
D
E
E
F
G
G
H
0
D
1
E
Combining AND/OR BDDs
f = A*E + B*F
A
B
g = C*G +D*H
C
B
D
E
D
E
G
F
G
H
0
1
0
AND
A
B
B
AOBDD f * g =
E
C
D
D
E
G
F
0
1
G
H
1
0
1
OBDDs vs AOBDD
AND
A
AOBDD
B
C
B
E
f *g=
D
D
E
G
F
G
H
0
1
0
1
A
B
B
OBDD
C
D
C
D
E
D
E
F
C
D
E
F
D
D
E
E
F
G
G
H
0
1
E
Outline






Background in Graphical models
AND/OR search trees and Graphs
Minimal AND/OR graphs
From AND/OR graphs to AOMDDs
Compilation of AOMDDs
AOMDDs and earlier BDDs
Constraint Networks
A
Example: map coloring
Variables - countries (A,B,C,etc.)
Values - colors (red, green, blue)
Constraints:
A  B, A  D, D  E, etc.
Constraint graph
A
B
red
red
green
green
yellow
yellow
green
yellow
red
yellow
green
red
A
E
A
E
D
B
F
C
Semantics: set of all solutions
Primary task: find a solution
D
B
F
G
G
C
Propositional Satisfiability
 = {(¬C), (A v B v C), (¬A v B v E), (¬B v C v D)}.
Graphical models

A graphical model (X,D,C):



X = {X1,…Xn}
variables
D = {D1, … Dn} domains
C = {F1,…,Ft}
functions
(constraints, CPTS, cnfs)
Fi : P( F | A, C )
Fi : F  A  C
A
F
B
C
Examples:
•Constraint networks
•Belief networks
•Cost networks
•Markov random fields
•Influence diagrams
E



D
All these tasks are NP-hard
 identify special cases
 approximate
Tree-solving is easy
CSP – consistency
(projection-join)
Belief updating
(sum-prod)
P(X)
mXY (X )
mXZ (X )
mYX (X )
mZX (X )
P(Y|X)
P(Z|X)
mYT (Y )
mYR (Y ) mZL (Z )
mZM (Z )
mTY (Y )
mRY (Y ) mLZ (Z )
mMZ (Z )
P(T|Y)
MPE (max-prod)
P(R|Y)
P(L|Z)
P(M|Z)
#CSP (sum-prod)
Trees are processed in linear time and memory
Transforming into a Tree

By Inference (thinking)


Transform into a single, equivalent tree of
sub-problems
By Conditioning (guessing)

Transform into many tree-like subproblems.
Inference and Treewidth
ABC
G
D
A
DGF
B
BDEF
F
C
EFH
E
M
K
H
L
FHK
J
HJ
KLM
Inference algorithm:
Time: exp(tree-width)
Space: exp(tree-width) treewidth = 4 - 1 = 3
treewidth = (maximum cluster size) - 1
Conditioning and Cycle cutset
H
G
D
F
O
E
C
K
M
J
A
O
E
C
P
B
K
L
Cycle cutset = {A,B,C}
H
F
B
O
E
P
K
L
C
N
D
F
M
J
H
G
N
D
P
B
L
G
N
D
F
M
A
J
H
G
N
M
J
O
E
C
P
K
L
Search over the Cutset
E
Graph
Coloring
problem
M
D
• Inference may require too much memory
L
C
A
B
• Condition (guessing) on some of the variables
K
H
F
G
J
A=yellow
E
D
A=green
M
E
D
M
L
C
L
C
B
K
B
K
H
F
H
F
G
J
G
J
Search over the Cutset (cont)
E
Graph
Coloring
problem
M
D
• Inference may require too much memory
L
C
A
B
• Condition on some of the variables
K
H
F
G
J
A=yellow
B=red
E
D
B=blue
M
E
D
B=green
M
E
L
C
E
J
E
E
L
K
K
H
F
J
M
C
H
G
D
L
K
F
J
M
C
H
G
D
B=yellow
L
K
F
G
M
C
H
F
D
B=blue
L
K
H
J
M
C
K
G
D
B=red
L
C
F
A=green
H
F
G
J
G
J
Inference vs. Conditioning

By Inference (thinking)
ABC
A
C
B
BDEF
F
E
H
J
L
FHK
KLM
Variable-elimination
Directional resolution
Join-tree clustering
By Conditioning (guessing)
A=yellow
E D M
L
C
A
B
K
H
F
G J
Exponential in treewidth
Time and memory
EFH
M
K
HJ

DGF
G
D
A=green
B=blue
B=green
B=red
B=blue
E D M
E D M
E D M
E D M
L C
L C
L C
L
C
K
K
K
K
H
H
H
H
F
F
F
F
G J
G J
G J
G J
Exponential in cycle-cutset
Time-wise, linear memory
Backtracking
Branch and bound
Depth-first search
Outline






Background in Graphical models
AND/OR search trees and Graphs
Minimal AND/OR graphs
From AND/OR graphs to AOMDDs
Compilation of AOMDDs
AOMDDs and earlier BDDs
Classic OR Search Space
A
F
B
E
A
1
0
E
1
0
C
F
D
0
B
D
Ordering: A B E C D F
C
1
0
0
1
1
0
0
0
1
0
1
1
0
0
1
0
1
0
1
1
0
0
0
1
0
1
1
1
0
1
0
1
0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1
0
1
1
0
0
0
1
0
1
1
0
1
0
1
0
1
1
0
0
1
0
1
1
0
1
0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1
AND/OR Search Space
A A
A
F
B
F
B
C
B
C
C
E
E
D
E
Primal graph
D D
F
DFS tree
OR
A
AND
0
1
OR
B
B
AND
0
OR
AND
E
0
OR
AND
C
1
E
0
1
D
0
F
1
0
1
0
0
D
1
0
F
1
0
0
0
F
1
0
1
D
1
E
C
1
0
D
1
1
0
1
0
1
0
E
0
1
D
F
1
C
F
1
0
0
D
1
0
1
F
1
0
C
0
1
D
1
0
F
1
0
D
1
0
F
1
0
1
A
A
F
B
AND/OR vs. OR
OR
B
C
E
E
C
D
D
F
A
AND
0
1
OR
B
B
AND
0
OR
AND
0
C
1
OR
0
E
0
1
D
AND
F
1
0
1
E
0
0
D
1
0
F
1
0
0
1
0
F
1
0
D
1
0
C
0
1
D
1
E
C
1
AND/OR
1
1
0
0
1
D
F
0
1
E
F
1
0
0
D
1
0
1
F
1
0
C
0
1
D
1
F
0
1
0
D
1
0
F
1
0
1
AND/OR size: exp(4),
OR size exp(6)
A
0
B
0
E
F
1
0
C
D
1
1
0
0
1
1
0
0
0
1
0
1
1
0
0
1
0
1
0
1
1
0
0
0
1
0
1
1
0
1
0
1
0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1
0
1
1
0
0
0
1
OR
1
0
1
1
0
1
0
1
0
1
1
0
0
1
0
1
1
0
1
0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1
AND/OR vs. OR
with Constraints
OR
No-goods
(A=1,B=1)
(B=0,C=0)
F
B
E
C
D
D
F
A
0
1
OR
B
B
AND
0
OR
0
C
1
OR
1
D
AND
0
E
0
F
1
0
0
D
1
0
F
1
0
1
A
0
0
F
1
0
D
1
0
C
0
1
D
1
E
C
1
AND/OR
0
1
E
AND
1
1
0
0
1
D
F
0
1
E
F
1
0
0
D
1
0
1
0
1
0
C
1
0
0
1
1
0
1
0
1
1
0
1
0
1
1
D
1
F
0
1
0
D
1
0
0
0
0
0
F
1
0
1
1
0
E
C
1
F
0
B
F
B
C
E
AND
D
A
A
0
1
1
0
0
0
1
0
1
1
0
1
0
1
0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1
0
1
1
0
0
0
1
OR
1
0
1
1
0
1
0
1
0
1
1
0
0
1
0
1
1
0
1
0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1
AND/OR vs. OR
with Constraints
OR
F
B
1
OR
B
B
AND
0
OR
E
0
C
E
1
1
0
D
AND
0
1
0
0
0
F
1
0
1
D
1
E
C
1
F
A
C
D
F
AND/OR
0
D
1
0
C
1
1
D
F
1
0
0
1
F
1
0
1
0
B
1
0
0
1
1
C
F
E
0
1
OR
D
D
B
A
0
E
C
E
AND
AND
A
A
No-goods
(A=1,B=1)
(B=0,C=0
0
1
0
1
1
0 1 0 1
0
1
0
1
0
1
1
0
0
0
1
OR
0
0
1
1
1
0
1
1
0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1
0
1
1
0 1 0 1
0
1
0 1 0 1
DFS algorithm (#CSP example)
A
A
F
B
B
C
solution
E
E
C
D
D
OR
F
11
A
AND
0
5
1
6
OR
B
5
B
6
AND
0
OR
AND
0
1
0
D
2
2
F
1
D
1
0
4
1
2
F
0
D
1
F
1
D
1
F
0
0
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
1
1
1
0
1
1
0
0
1
0
1
0
0
1
0
0
D
0
1
1
F
1
D
2
1
0
1
0
1
0
1
1
1
0
1
1
1
0
Value of node = number of solutions below it
0
1
1
F
D
0
1
2
C
1
0
2
2
1
E 1
C 4
E 1 Marginalization
C 1
E
OR node:
operator
(summation)
1 0
0
1operator
0 2
1 2
0
1 node:
0 1
1 0
0
AND
Combination
(product)
1
0
2
C
1
OR
AND
1
2
E
1
4
1
1
1
0
1
1
F
0
1
1
0
1
D
0
0
1
1
F
1
0
1
1
1
0
Pseudo-Trees
(Freuder 85, Bayardo 95, Bodlaender and Gilbert, 91)
4
1
1
6
2
3
m <= w* log n
2
7
5
(a) Graph
4
1
2
3
7
1
7
3
5
3
5
4
2
7
5
6
6
4
6
(b) DFS tree
depth=3
(c) pseudo- tree
depth=2
(d) Chain
depth=6
Tasks and value of nodes

V( n) is the value of the tree T(n) for the task:





Consistency: v(n) is 0 if T(n) inconsistent, 1 othewise.
Counting: v(n) is number of solutions in T(n)
Optimization: v(n) is the optimal solution in T(n)
Belief updating: v(n), probability of evidence in T(n).
Partition function: v(n) is the total probability in T(n).

Goal: compute the value of the root node recursively using dfs

AND/OR search tree and algorithms are
search of the AND/OR tree.
([Freuder & Quinn85], [Collin, Dechter & Katz91], [Bayardo & Miranker95],[Darwiche 2001],
[Bacchus et. Al, 2003])

Space: O(n)

Time: O(exp(m)), where m is the depth of the pseudo-tree

Time: O(exp(w* log n))

BFS is time and space O(exp(w* log n)
F
From AND/OR Tree
B
B
C
E
K
C
H
A
AND
0
1
OR
B
B
AND
0
OR
AND
E
D
G
OR
A
J
A
E
0
C
1
OR
0
1
E
0
1
D
F
0
D
1
F
0
0
1
D
F
D
C
1
G
J
H
K
E
0
1
D
F
F
1
E
C
D
F
C
0
D
1
F
0
1
D
F
D
F
AND
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
OR
G
G
J
J
G
G
J
J
G
G
J
J
G
G
J
J
G
G
J
J
G
G
J
J
G
G
J
J
G
G
J
J
AND
0 1 0
OR
H H H H K K K K H H H H K K K K H H H H K K K K H H H H K K K K H H H H K K K K H H H H K K K K H H H H K K K K H H H H K K K K
AND
01010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101
1 0 1 0
1 0 1 0
1 0 1 0
1 0 1 0
1 0 1 0
1 0 1 0
1 0 1 0
1 0 1 0
1 0 1 0
1 0 1 0
1 0 1 0
1 0 1 0
1 0 1 0
1 0 1 0
1 0 1 0
1
To an AND/OR
Graph
F
B
B
C
E
K
A
0
1
OR
B
B
AND
0
OR
OR
C
H
AND
AND
E
D
G
OR
A
J
A
E
0
C
1
D
0
1
E
0
1
F
D
0
F
1
D
0
0
1
F
D
C
1
D
F
AND
0
1
0
1
OR
G
G
J
J
AND
0 1 0
OR
H H H H K K K K
AND
0101010101010101
1 0 1 0
1
F
G
J
H
K
1
E
C
D
E
0
1
F
D
0
F
C
1
0
D
1
F
D
F
AND/OR Search Graphs



Any two nodes that root identical
subtrees/subgraphs (are unifiable) can be merged
Minimal AND/OR search graph: of R relative to
tree T is the closure under merge of the AND/OR
search tree of R relative to T, where inconsistent
subtrees are pruned.
Canonicity: The minimal AND/OR graph relative to
T is unique for all constraints equivalent to R.
Context based caching

Caching is possible when context is the same

context
=
=
parent-separator set in induced pseudo-graph
current variable +
ancestors connected to subtree below
A
J
A
F
B
C
E
D
G
H
context(B) = {A, B}
B
K
E
C
context(C) = {A,B,C}
D
F
G
J
H
K
context(D) = {D}
context(F) = {F}
Caching
F
B
AND
OR
0
OR
0
E
0
1
D
F
F
G
J
H
K
B
C
1
D
1
1
E
C
A
0
OR
E
H
B
AND
AND
K
D
G
context(B) = {A, B}
context(C) = {A,B,C}
context(D) = {D}
0
context(F) = {F}
B
C
E
OR
A
J
A
0
D
F
E
C
1
0
0
1
D
1
F
D
C
1
0
1
D
F
E
F
C
0
D
1
F
0
1
D
F
D
F
AND
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
OR
G
G
J
J
G
G
J
J
G
G
J
J
G
G
J
J
G
G
J
J
G
G
J
J
G
G
J
J
G
G
J
J
AND
0 1 0
OR
H H H H K K K K H H H H K K K K H H H H K K K K H H H H K K K K H H H H K K K K H H H H K K K K H H H H K K K K H H H H K K K K
AND
01010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101
1 0 1 0
1 0 1 0
1 0 1 0
1 0 1 0
1 0 1 0
1 0 1 0
1 0 1 0
1 0 1 0
1 0 1 0
1 0 1 0
1 0 1 0
1 0 1 0
1 0 1 0
1 0 1 0
1 0 1 0
1
A
J
A
F
Caching
B
E
context(D)={D}
K
A
0
1
OR
B
B
0
OR
AND
OR
0
C
1
D
0
1
E
E
0
1
F
D
0
F
C
H
AND
AND
E
D
G
context(F)={F}
OR
B
C
1
D
0
0
1
F
D
C
1
D
F
AND
0
1
0
1
OR
G
G
J
J
AND
0 1 0
OR
H H H H K K K K
AND
0101010101010101
1 0 1 0
1
F
G
J
H
K
1
E
C
D
E
0
1
F
D
0
F
C
1
0
D
1
F
D
F
C
A
D
F
B
E
All Four Search Spaces
A
B
C
D
E
F
0
0
1
0
0
1
1
0
0
0
1
A
1
0
1
1
0
1
1
0
0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1
B
1
0
1
0
0
1
C
1
0
1
0
0
D
1
0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1
0
1
0
1
0
0
0
1
E
01 01 0 1 0 101 01 0 1 0 1 0 1 0 1 01 010 1 0 1 01 01 0 1 0 1 01 010 1 0 1 01 01 01 01 0 1 0 101 01 0 1 0 1
1
1
0
0
1
0
1
1
0
0
1
1
0
0
F
0
1
1
1
Full OR search tree
Context minimal OR search graph
126 nodes
28 nodes
OR
1
A
AND
0
1
OR
B
B
AND
1
0
C
OR
C
E
1
0
C
E
C
E
E
AND
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
OR
AND
D
D
F
F
D
D
F
F
D
D
F
F
D
D
F
F
0 1 0 1 0 1 0 1
0 1 0 1 0 1 0 1
0 1 0 1 0 1 0 1
Full AND/OR search tree
54 AND nodes
OR
AND
OR
AND
OR
AND
OR
AND
A
0
1
B
B
1
0
C
C
E
1
0
C
E
0
1
0
1
0
D
D
F F
01
D
C
E
E
1
0
1
D
F
F
01
0 1 0 1 0 1 0 1
Context minimal AND/OR search graph
18 AND nodes
Properties of minimal AND/OR graphs
Complexity:

Minimal AND/OR R relative to pseudo-tree T is O(exp(w*))
where w* is the tree-width of R along T.

Minimal OR search graph is O(exp(pw*)) where pw* is pathwidth

w* ≤ pw*, pw* ≤ w*log n
Canonicity:

The minimal AND/OR search graph is unique (canonical) for all
equivalent formulas (Boolean or Constraints), consistent with its
pseudo tree.
Searching AND/OR Graphs

AO(j): searches depth-first, cache i-context
 j = the max size of a cache table (i.e.
number of variables in a context)
i=0
i=w*
j
Space: O(n)
Space: O(exp w*)
Time:
Time:
O(exp(w* log n))
Space:
O(exp(j) )
Time:
O(exp(m_j+j )
O(exp w*)
Context-Based Caching
A
B
C
E
D
F
context(A) = {A}
context(B) = {B,A}
context(C) = {C,B}
context(D) = {D}
context(E) = {E,A}
context(F) = {F}
Cache Table (C)
C Value
0
5
1
2
0
2
1
0
Space: O(exp(2))
A
D
B
F
E
Primal graph
A
5 0
7 1
5 B
7 B
6 0
3 C
B
0
0
1
1
C
5 1
4 C
E
6 0
E
4 C
4 1
E
5 0
2 1
2 0
0 1
D
D
D
D
4 C
E
Outline






Background in Graphical models
AND/OR search trees and Graphs
Minimal AND/OR graphs
From AND/OR search graphs to AOMDDs
Compilation of AOMDDs
AOMDDs and earlier BDDs
OR Search Graphs vs OBDDs
A
A
0
0
0
0
1
1
1
1
B
0
0
1
1
0
0
1
1
C
0
1
0
1
0
1
0
1
f(ABC)
0
0
0
1
0
1
0
1
0
A
B
B
0
0
C
1
0
1
C
1
0
C
1
0
1
Full AND/OR search tree
A
A
0
1
1
0
C
B
1
1
0
redundant
1
1
1
Minimal AND/OR search graph
Backtrack free AND/OR search tree
1
B
1
C
C
C
1
0
1
B
B
1
0
C
C
B
C
B
1
A
0
1
1
An OBDD
AND/OR Search Graphs; AOBDDs
A
F(A,B,C,D)= (0,1,1,1), (1,0,1,0), (1,1,1,0)
B
C
D
AOBDD(F)
A
A
0
0
1
1
B
B
redundancy
B
0
1
1
1
D
C
D
D
C
D
1
1
0
1
1
0
AND/OR Search Graphs; AOBDDs
A
F(A,B,C,D)= (0,1,1,1), (1,0,1,0), (1,1,1,0)
B
C
D
AOBDD(F)
A
A
0
0
1
1
B
B
redundancy
B
0
1
1
1
D
C
D
D
C
D
1
1
0
1
1
0
AOBDD Conventions
A
B
D
C
A
0
A
0
Point dead-ends to terminal 0
Point goods to terminal 1
1
1
B
0
1
B
C
1
0
C
D
1
D
1
0
D
1
0
1
D
1
0
0
1
AOBDD Conventions
A
B
D
C
Introduce
Meta-nodes
A
A
0
0
1
1
B
0
B
1
1
C
0
C
D
1
D
1
0
D
1
0
1
D
1
0
0
1
Combining AOBDD (apply)
A
0
0
0
0
1
1
1
1
B
0
0
1
1
0
0
1
1
C
0
1
0
1
0
1
0
1
f(ABC)
0
0
0
1
0
1
0
1
A
B
C
A
0
0
0
0
1
1
1
1
B
0
0
1
1
0
0
1
1
D
0
1
0
1
0
1
0
1
A
0
1
0
*
1
C
0
A
B
B
D
D
C
A
1
0
B
0
A
A
B
0
g(ABC)
0
0
0
1
1
0
1
0
1
1
0
=
1
D
0
0
D
1
0
B
0
1
C
0
1
1
1
0
D
1
0
D
1
0
1
1
Example:
(f+h) * (a+!h) * (a#b#g) * (f+g) * (b+f) * (a+e) * (c+e) * (c#d) * (b+c)
A
m7
A
D
m7
B
A
A
0
0
1
B
0
1
0
C
0
C
1
0
1
0
0
1
C
1
0
0
1
0
1
0
F
1
0
0
F
1
0
B
1
D
D
G
E
1
0
0
1
G
1
0
H
1
0
C
H
1
0
0
1
m6
E
A
H
C
F
D
E
G
pseudo tree
F
1
D
0
F
primal graph
B
0
B
1
F
1
C
A
B
F
C
1
1
B
B
G
E
G
H
1
m3
A
0
B
A
0
B
0
C
0
C
0
0
0 1
0
1
1
0
0
1
1
0
0
m5
D
m4
C
D
1
D
E
0
1
C8
D
C
1
0
G
1
0
0
1
F
1
0
1
B
F
1
0
H
G
1
0
1
1
0
A
A
0
0
1
H
1
0
F
1
0 1
0 1
C6
C7
0
0
1
C
G
0
1
0
1
C3
0
1
F
A
1
0
H
B
H
1
0 1
C4
H
m1
0
G
E
G
1
1
A
1
G
0
G
0
F
0
B
1
m2
1
A
1
B
1
0
C5
E
E
E
0
F
0
0 1
1
0
0
A
1
C
A
1
0
1
F
0
1
D
0
B
1
C
0
0
F
A
B
1
B
1
E
D
C9
0
C
D
1
A
1
C
1
1
0
H
1
0
1
F
0 1
0 1
G
C1
C2
F
H
H
Example (continued)
A
0
1
B
0
C
0
B
1
0
C
1
0
C
1
0
C
1
0
F
1
0
1
F
1
0
F
1
0
F
1
0
1
A
D
0
D
1
0
E
1
0
G
1
0
0
G
1
0
H
1
0
H
1
0
1
1
m7
m7
A
A
0
0
1
B
0
1
0
C
0
C
1
0
1
0
0
1
C
1
0
0
1
0
0
F
1
A
B
1
F
C
1
1
B
B
0
1
F
1
0
F
1
0
B
1
B
D
0
D
G
E
1
0
0
1
G
1
0
H
1
0
C
H
1
0
D
0
m6
1
0
m3
1
F
1
E
G
H
AOBDD vs. OBDD
D
A
G
C
B
F
E
A
H
B
C
primal graph
A
0
C
0
D
0
0
C
1
D
0
C
1
0
D
1
0
D
F
1
0
E
1
F
1
0
F
1
0
F
1
0
D
E
1
0
G
1
0
G
1
0
H
1
0
0
1
G
F
G
AOBDD
B
18 nonterminals
F
G
D
0
1
47 arcs
G
A
H
H
C
1
E
F
OBDD
27 nonterminals
G
H
F
G
A
E
D
E
F
D
D
D
H
1
B
C
C
E
F
1
G
0
C
B
1
C
1
C
1
B
0
B
H
54 arcs
0
Complexity


Complexity of apply: Complexity of
apply is bounded quadratically by the
product of input AOBDDs restricted to
each branch in the output pseudo-tree.
Complexity of VE-AOBDD: is
exponential in the tree-width along the
pseudo-tree.
AOMDDs and tree-BDDs

Tree-BDDs (McMillan1994) are :
O(| g | 2

w22 w
)
AND/OR BDDS are
w
O(| g | k )
Related work

Related work in Search





Related to compilation schemes:




Backjumping + learning (Freuder 1988,Dechter 1990,Bayardo and
Mirankar 1996)
Recursive-conditioning (Darwiche 2001)
Value elimination (Bacchus et. Al. 2003)
Search over tree-decompositions (Trrioux et. Al. 2002)
Minimal AND/OR – related to tree-OBDDs (McMillan 94),
d-DNNF (Darwiche et. Al. 2002)
Case-factor diagrams (Mcallester, Collins, Pereira, 2005)
Tutorial references:
Dechter, Constraint processing, Morgan Kauffman, 2003

Dechter, Tractable Structures of Constraint Satisfaction problems,
In Handbook of constraint solving, forthcoming.

Conclusion

AND/OR search should always be used.

AND/OR BDDs are superior to OBDDs.

A search algorithm with good and no-good learning generates an
OBDD or an AND/OR Bdds.

Dynamic variable ordering can be incorporated

With caching, AND/OR search is similar to inference (variableelimination)

The real tradeoff should be rephrased:


Time vs space rather than search vs inference, or search vs model-checking.
Search methods are more sensitive to this tradeoff.