AND/OR Search for

1

• AND/OR Search for
• Mixed Networks
• CSP

Robert Mateescu
ICS280 Spring 2004 - Current Topics in Graphical
Models Professor Rina Dechter
2
Mixed Networks

• Belief Networks

Constraint Networks
A
A
F
F
B
C
B
C
D
E
D
E
3
Mixed Networks
A
A
F
F
B
C
B
C
A
D
E
D
E
F
B
C
D
E
4
Tasks for Mixed Networks

• Tasks for Belief Networks
• Belief updating evaluating the posterior
  probability of each singleton proposition given
  some evidence
• Most probable explanation (MPE) finding a
  complete assignment to all variables having
  maximum probability given the evidence
• Maximum a posteriori hypothesis (MAP) finding
  the most likely assignment to a subset of
  hypothesis variables given the evidence
• Tasks for Constraint Networks
• Consistent Decide if network is consistent
• Find solutions Find one, some or all solutions
• Tasks for Mixed Networks
• Belief updating, MPE, MAP
• Constraint Probability Evaluation (CPE) Find the
  probability of the constraint query

5
Auxiliary Network
A
A
F
F
B
C
B
C
A
X
F
D
E
D
E
B
C
Y
Q
D
E
Z
6
Mixed Graph

• The mixed graph is the union of the belief
  network graph and the constraint network graph
• The moral mixed graph is the union of the moral
  graph of the belief network and the graph of the
  constraint network
• Given a mixed graph GM (GB,GR) of a mixed
  network M(B,R) where GB is the directed graph of
  B, and GR is the undirected constraint graph of
  R, the ancestral graph of Y \in X in GM is the
  union of GR and the ancestral graph of Y in GB

7
dm-separation

• Definition Given a mixed graph, GM and given
  three subsets of variables W, Y and Z which are
  disjoint, we say that W and Y are dm-separated
  given Z in the mixed graph GM, denoted ltW,Z,Ygtdm,
  iff in the ancestral mixed graph of WUYUZ, all
  the paths between W and Y are intercepted by
  variables in Z.
• Theorem Given a mixed network M M(B,R) and its
  mixed graph GM, then GM is a minimal I-map
  relative to dm-separation

8
AND/OR Search for Mixed Networks
A
A
F
B
B
C
C
E
D
E
D
F
Mixed network
DFS (legal) tree
9
AND/OR Search Space
OR
A
AND
0
1
OR
B
B
AND
0
1
0
1
OR
E
C
E
C
E
C
E
C
AND
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
OR
D
F
D
F
D
F
D
F
D
F
D
F
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
10
Forward Expanding an AND node
OR
A
AND
0
1
OR
B
B
AND
0
1
0
1
OR
E
C
E
C
E
C
E
C
AND
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
OR
D
F
D
F
D
F
D
F
D
F
D
F
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
g-value(ANDnode) ? Pi (or 1, if product is
empty) e.g. g-value(ltB,1gt) P(B1A0)
11
Forward Expanding an OR node
OR
A
AND
0
1
OR
B
B
AND
0
1
0
1
OR
E
C
E
C
E
C
E
C
AND
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
OR
D
F
D
F
D
F
D
F
D
F
D
F
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
g-value(ORnode) 0
12
Backward Propagating g-values
OR
A
AND
0
1
OR
B
B
AND
0
1
0
1
OR
E
C
E
C
E
C
E
C
AND
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
OR
D
F
D
F
D
F
D
F
D
F
D
F
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 nodes perform summation AND nodes perform
product
13
The main idea

• Search algorithms can exploit a wide range of
  constraint propagation techniques
• When generating AND nodes, consistency is checked
  according to the desired level of constraint
  propagation

14
AND/OR Search (1)
g(A)0
OR
A
AND
OR
AND
OR
AND
OR
AND
15
AND/OR Search (2)
g(A)0
OR
A
g(ltA,0gt)P(A0)
AND
0
1
g(ltA,1gt)P(A1)
OR
AND
OR
AND
OR
AND
16
AND/OR Search (3)
g(A)0
OR
A
g(ltA,0gt)P(A0)
AND
0
1
g(ltA,1gt)P(A1)
g(B)0
OR
B
AND
OR
AND
OR
AND
17
AND/OR Search (4)
g(A)0
OR
A
g(ltA,0gt)P(A0)
AND
0
1
g(ltA,1gt)P(A1)
g(B)0
OR
B
g(ltB,0gt)P(B0A0)
g(ltB,1gt)P(B1A0)
AND
0
1
OR
AND
OR
AND
18
AND/OR Search (5)
g(A)0
OR
A
g(ltA,0gt)P(A0)
AND
0
1
g(ltA,1gt)P(A1)
g(B)0
OR
B
g(ltB,0gt)P(B0A0)
g(ltB,1gt)P(B1A0)
AND
0
1
g(E)0
g(C)0
OR
E
C
AND
OR
AND
19
AND/OR Search (6)
g(A)0
OR
A
g(ltA,0gt)P(A0)
AND
0
1
g(ltA,1gt)P(A1)
g(B)0
OR
B
g(ltB,0gt)P(B0A0)
g(ltB,1gt)P(B1A0)
AND
0
1
g(E)0
g(C)0
OR
E
C
AND
0
g(ltE,0gt)P(E0A0,B0)
OR
AND
20
AND/OR Search (6)
g(A)0
OR
A
g(ltA,0gt)P(A0)
AND
0
1
g(ltA,1gt)P(A1)
g(B)0
OR
B
g(ltB,0gt)P(B0A0)
g(ltB,1gt)P(B1A0)
AND
0
1
g(C)0
OR
E
C
g(E)P(E0A0,B0)
AND
0
OR
AND
21
AND/OR Search (8)
g(A)0
OR
A
g(ltA,0gt)P(A0)
AND
0
1
g(ltA,1gt)P(A1)
g(B)0
OR
B
g(ltB,1gt)P(B1A0)
AND
0
1
g(ltB,0gt) P(B0A0) P(E0A0,B0)
g(C)0
OR
E
C
AND
0
OR
AND
22
Constraint propagation
All domains are 1,2,3,4
23
Constraint checking only
A
OR
AND
1
2
3
4
OR
C
B
C
B
B
B
2
3
4
2
3
4
3
4
3
4
4
AND
E
D
D
E
D
F
F
F
D
D
F
F
D
OR
AND
3
4
3
4
4
4
4
3
4
4
OR
H
G
G
I
I
G
K
K
K
G
K
AND
4
4
4
4
24
Forward checking
A
OR
AND
1
2
3
4
OR
C
B
C
B
B
B
2
3
4
2
3
4
3
4
3
4
4
AND
E
D
D
E
D
F
F
F
D
D
F
F
D
OR
AND
3
4
3
4
4
4
4
3
4
4
OR
H
G
G
I
I
G
K
K
K
G
K
AND
4
4
4
4
25
Maintaining arc-consistency
A
OR
AND
1
2
3
4
OR
C
B
C
B
B
B
2
3
4
2
3
4
3
4
3
4
4
AND
E
D
D
E
D
F
F
F
D
D
F
F
D
OR
AND
3
4
3
4
4
4
4
3
4
4
OR
H
G
G
I
I
G
K
K
K
G
K
AND
4
4
4
4
26
AND/OR vs. OR Space
27
Linear space algorithms
28
Linear space algorithms
29
AND/OR search vs. Bucket Elimination
30
CSP

• CSP Counting the solutions of a CSP problem is
  very similar to the CPE task in mixed networks
• If the belief part in a mixed network is empty,
  we can translate the AND/OR search for mixed
  networks to an AND/OR search for CSPs.
• OR nodes are initialized with g-value 0
• AND nodes are initialized with g-value 1

31
Parent set and parent separator set
Parent set Parent separator set
A
B A AB
C AB ABC
D BC D
E AB E
F AC F
A
B
C
E
D
F
32
Caching in AND/OR search

• If space is available, parts of the search tree
  can be cached, transforming the search space into
  a search graph
• In principle, we can cache at OR level and/or at
  AND level
• Caching at AND level gt use parent separator set
• Caching at OR level gt use parent set
• size(parent separator set) size(parent set) 1

33
ORcache, N20, K3, Nc20 ,Pc4, inst20Time
34
ORcache, N20, K3, Nc20 ,Pc4, inst20 of nodes