Knowledge Compilation

1
Knowledge Compilation

• Jinbo Huang
• NICTA and ANU

Slides made by Adnan Darwiche and Jinbo Huang
2
Propositional Logic
Is A, ?C a normal device behavior?
3
Propositional Reasoning
Is A, ?C a normal device behavior?
4
Satisfiability (SAT)

• SAT Solvers Significant growth in last decade
  many solvers publicly available (source code)
  millions of clauses not uncommon.
• Applications Verification, planning, diagnosis,
  CAD, non-propositional reasoning (e.g., SMT),

5
Knowledge Compilation
6
Knowledge Compilation
7
Knowledge Compilation
..... Prime Implicates OBDD
Compiler
8
Knowledge Compilation Map

• Whats the space of possible target compilation
  languages?
• Can it be synthesized in a semantically
  systematic way?
• How do the languages compare?
• Succinctness (relative size)
• Operations they support in polytime

9
Applications

• Diagnosis
• Is this a normal behavior?
• What are the possible faults?
• Planning
• Can this goal be achieved?
• Generate plan with highest reward
• Generate plan with highest success probability
• Probabilistic reasoning
• What is the probability of X given Y
• Formal verification / CAD
• Is it possible that the design will exhibit
  behavior X?
• Are two designs equivalent?

10
Knowledge Compilation Map

• For a given application identify needed
  operations
• Choose most succinct language that supports
  desired operations
• Compile knowledge base into chosen language

11
Agenda

• Part I Languages
• Part II Operations
• Part III Compilers
• Part IV Applications

12
Part I Languages
13
A Knowledge Compilation MAP
Negation Normal Form
or
Decomposability Determinism Smoothness Flatness De
cision Ordering
and
and
or
or
or
or
and
and
and
and
and
and
and
and
?A
B
? B
A
C
? D
D
? C
14
Propositional Logic

• Literal
• Clause
• Term
• CNF Conjunctive Normal Form
• DNF Disjunctive Normal Form

15
Propositional Logic

• Truth assignment (TA)
• TA satisfies sentence (model)
• Following TA is not a model

16
A Knowledge Compilation MAP
Negation Normal Form
or
Decomposability Determinism Smoothness Flatness De
cision Ordering
and
and
or
or
or
or
and
and
and
and
and
and
and
and
?A
B
? B
A
C
? D
D
? C
17
Queries

• Consistency (CO)
• Validity (VA)
• Sentential entailment (SE)
• Clausal entailment (CE) KB implies clause
• Implicant testing (IP) term implies KB
• Equivalence testing (EQ)
• Model Counting (CT)
• Model enumeration (ME)

18
Transformations

• Projection (existential quantification)
• Conditioning
• Conjoin
• Disjoin
• Negate

19
Representation vs Compilation Languags

• Representation Language (intuitive)CNFDNF
• Target Compilation Language (tractable) Binary
  Decision Diagrams (BDDs)DNNF

20
Negation Normal Form
rooted DAG (Circuit)
21
Negation Normal Form
Decomposability Determinism Smoothness Flatness De
cision Ordering
22
Decomposability
or
and
and
A,B
C,D
or
or
or
or
and
and
and
and
and
and
and
and
?A
B
? B
A
C
? D
D
? C
23
NNF Subsets
NNF
CO, CE, ME
DNNF
24
Determinism
or
and
and
or
or
or
or
and
and
and
and
and
and
and
and
?A
B
? B
A
C
? D
D
? C
25
Smoothness
or
and
and
or
or
or
or
and
and
and
and
and
and
and
and
?A
B
? B
A
C
? D
D
? C
26
NNF Subsets
NNF
CO, CE, ME
d-NNF
s-NNF
DNNF
VA,IP,CT
d-DNNF
EQ?
sd-DNNF
27
Flatness
Nested vs Flat languages
(X?Y ?Z)?(Z??X??Y) ?(Y?Z??X) ?(?X??Y ??Z)
28
Simple Conjunction
or
and
and
and
and
?X
X
?Y
Y
?Z
Z
Simple conjunction implies decomposability
29
Simple Disjunction
and
or
or
or
?X
X
Y
Z
30
NNF Subsets
NNF
CO, CE, ME
d-NNF
s-NNF
f-NNF
DNNF
VA,IP,CT
d-DNNF
EQ?
CNF
DNF
sd-DNNF
31
Prime Implicates (PI)

• A CNF such that
• No clause subsumes another
• If a clause is implied by the CNF, it must be
  implied by a clause in the CNF
• CNF
• PI

32
Prime Implicants (IP)

• A DNF such that
• No term subsumes another
• If a term implies the DNF, it must imply a term
  in the DNF
• DNF
• IP

33
NNF Subsets
NNF
CO, CE, ME
d-NNF
s-NNF
f-NNF
DNNF
VA,IP,CT
d-DNNF
EQ?
CNF
DNF
sd-DNNF
CO,CE,MEVA,IP,SE,EQ
VA,IP, SE,EQ
IP
PI
34
Decision
or
and
and
X
?X
a
b
a, b
Are decision nodes
35
Decision
or
and
and
? X1
X1
or
or
and
and
and
and
? X2
? X2
X2
X2
or
or
and
and
and
and
X3
X3
? X3
? X3
true
false
36
Decision
or
and
and
X
?X
a
b
37
Binary Decision Diagrams(BDDs)
Decision implies determinism
38
NNF Subsets
NNF
CO, CE, ME
d-NNF
s-NNF
f-NNF
DNNF
VA,IP,CT
BDD
d-DNNF
EQ?
CNF
DNF
sd-DNNF
CO,CE,MEVA,IP,SE,EQ
VA,IP, SE,EQ
IP
PI
39
Binary Decision Diagrams(BDDs)
40
NNF Subsets
NNF
CO, CE, ME
d-NNF
s-NNF
f-NNF
DNNF
VA,IP,CT
BDD
d-DNNF
EQ?
CNF
DNF
FBDD
EQ?
sd-DNNF
CO,CE,MEVA,IP,SE,EQ
SE,EQ
VA,IP, SE,EQ
MODS
IP
PI
41
Binary Decision Diagrams(BDDs)
Decision decomposability ordering OBDD
42
NNF Subsets
NNF
CO, CE, ME
d-NNF
s-NNF
f-NNF
DNNF
VA,IP,CT
BDD
d-DNNF
EQ?
CNF
DNF
FBDD
EQ?
sd-DNNF
CO,CE,MEVA,IP,SE,EQ
SE,EQ
SE,EQ
VA,IP, SE,EQ
MODS
IP
OBDD
PI
43
OBDD Example Odd Parity
X1
X2
X2
X3
X3
X4
X4
1
0
Symmetric Functions
44
Language Succinctness
Size p(n)
Size n
45
Odd Parity Function
46
NNF
DNNF
CNF


d-DNNF
sd-DNNF
DNF
FBDD
PI
IP
OBDD
MODS
47
Tractability Succinctness
NNF
decomposability
Diagnosis, Non-mon
Probabilistic reasoning
determinism
decision
ordering
Space Efficiency (succinctness)
48
Separating Functions

• OBDD/FBDD
• Hidden weighted bit function hwb(x1,..,xn)
• DNNF/DNF
• odd parity function parity(x1,..,xn)
• DNNF/OBDD
• Distinct integers function distinct(x1,..,xn)

49
Agenda

• Part I Languages
• Part II Operations
• Part III Compilers
• Part IV Applications

50
Part II Operations
51
Knowledge Compilation
KB
52
Queries

• Consistency (CO)
• Validity (VA)
• Sentential entailment (SE)
• Clausal entailment (CE) KB implies clause
• Implicant testing (IP) term implies KB
• Equivalence testing (EQ)
• Model Counting (CT)
• Model enumeration (ME)

53
Transformations

• Projection (existential quantification)
• Conditioning
• Conjoin
• Disjoin
• Negate

54
Decomposability
55
Decomposability
or
and
and
A,B
C,D
or
or
or
or
and
and
and
and
and
and
and
and
?A
B
? B
A
C
? D
D
? C
56
Example Knowledge Base
A ? okX ? ?B ?A ? okX ? B B ? okY ? ?C ?B ? okY
? C
57
Decomposable
58
Decomposable
or
B
A, okX
and
and
C, okY
or
or
or
or
B
B
A
C
C
A
okX
okY
59
Decomposable
or
and
and
or
or
or
or
B
B
A
C
C
A
okX
okY
60
Satisfiability
y

• SAT(A or B) iff SAT(A) or SAT(B)
• SAT(A and B) iff SAT(A) and SAT(B)
• SAT(X) is true
• SAT(X) is true
• SAT(True) is true
• SAT(False) is false

61
Satisfiability
62
Satisfiability
n
63
Partial Decomposability
Decomposable except on okZ
64
Clausal Entailment

• KB entails L1 v L2 v v Ln ?
• KB ? ?L1 ? ?L2 ? ? ?Ln SAT ?

65
Literal Conjoin
A
66
Literal Conjoin
A
67
Literal Conjoin
Conditioning
or
A
and
and
or
or
or
or
B
B
C
C

true
false
okX
okY
68
Literal Conjoin
or
A
and
and
or
or
or
or
B
B
C
C

true
false
okX
okY
69
(No Transcript)
70
A
C
okX
okY
or
and
and
or
or
or
or
B
B
false
false
true
true
false
false
71
n
n
A
C
okX
okY
or
and
and
or
or
or
or
B
B
false
false
true
true
false
false
72
Partial Decomposability
Decomposable except on okZ
Clausal entailment test works as long as clause
mentioned all variables on which we dont have
decomposability!
73
(No Transcript)
74
and
okZ
75
and
okZ
76
Projection Existential Quantification
Knowledge Base
77
Projection Existential Quantification
Formal Definition

• If Knowledge base is a CNF
• Close under resolution
• Remove all clauses that mention X

78
Projection Existential Quantification
79
Projection Existential Quantification
A ? okX ? ?B ?A ? okX ? B
or
and
and
or
or
or
or
B
B
A
A
true
true
okX
true
80
I1
I2
I3
O
I4
I5
I6
D
O
81
Minimum Cardinality
A ? okX ? ?B ?A ? okX ? B B ? okY ? ?C ?B ?
okY ? C
82
Minimum Cardinality
1
83
Minimizing Requires Smoothness
1
or
and
and
or
or
or
or
and
and
and
and
and
and
and
and
?A
B
? B
A
C
? D
D
? C
84
Minimizing
1
or
and
and
or
or
or
or
and
and
and
and
and
and
and
and
?A
B
? B
A
C
? D
D
? C
85
Minimizing
1
or
and
and
or
or
or
or
and
and
and
and
and
and
and
and
?A
B
? B
A
C
? D
D
? C
86
Minimizing
1
or
and
and
or
or
or
or
and
and
and
and
and
and
?A
B
? B
A
C
? D
D
? C
87
Minimizing
?A,B,C,D
or
and
and
or
or
or
or
and
and
and
and
and
and
?A
B
? B
A
C
? D
D
? C
88
Minimizing
A, ?B,C,D
or
and
and
or
or
or
or
and
and
and
and
and
and
?A
B
? B
A
C
? D
D
? C
89
Minimizing
A, B,C, ?D
or
and
and
or
or
or
or
and
and
and
and
and
and
?A
B
? B
A
C
? D
D
? C
90
Minimizing
A, B, ?C,D
or
and
and
or
or
or
or
and
and
and
and
and
and
?A
B
? B
A
C
? D
D
? C
91
Decomposability

Query DNNF
CO Consistency Yes
VA Validity
CE Clausal entailment Yes
SE Sentential entailment
IP Implicant testing
EQ Equivalence testing
MC Model Counting
ME Model enumeration Yes
92
Decomposability

Transformation DNNF
CD Conditioning Yes
SFO Single variable Yes
FO Multiple variable Yes
Conjoin
B Bounded Conjoin
Disjoin Yes
B Bounded Disjoin Yes
Negate
93
Determinism
94
Determinism
or
and
and
or
or
or
or
and
and
and
and
and
and
and
and
?A
B
? B
A
C
? D
D
? C
95
Satisfiability
Is there a satisfying assignment?
A B C okX okY
T T T T T
T T T T F
...


F F F F F
96
Model Counting
How many satisfying assignments?
A B C okX okY
T T T T T
T T T T F
...


F F F F F
97
Counting Models
or
and
and
or
or
or
or
and
and
and
and
and
and
and
and
?A
B
? B
A
C
? D
D
? C
98
Counting Graph
8















?A
B
? B
A
C
? D
D
? C
99
Counting Models
SA, ? B
or
and
and
or
or
or
or
and
and
and
and
and
and
and
and
?A
B
? B
A
C
? D
D
? C
100
Counting Models
SA, ? B
or
and
and
or
or
or
or
and
and
and
and
and
and
and
and
false
false
true
true
C
? D
D
? C
101
Counting Graph
2
SA, ? B















false
false
true
true
C
? D
D
? C
102
Counting Graph















?A
B
? B
A
C
? D
D
? C
103
Counting Graph















V?A
VB
V? B
VA
VC
V? D
VD
V? C
104
Counting Graph
2
SA, ? B















V?A
VB
V? B
VA
VC
V? D
VD
V? C
105
Counting Graph















V?A
VB
V? B
VA
VC
V? D
VD
V? C
106
Counting Graph
1
SA, ? B,C















V?A
VB
V? B
VA
VC
V? D
VD
V? C
107
Asserting Literals
1
SA, ? B,C
? ? D















V?A
VB
V? B
VA
VC
V? D
VD
V? C
108
Retracting Literals
1
SA, ? B,C
\ ? B















V?A
VB
V? B
VA
VC
V? D
VD
V? C
109
Flipping Literals
1

SA, ? B,C
\ ? B ? B














V?A
VB
V? B
VA
VC
V? D
VD
V? C
110
Testing Equivalence
A ? okX ? ?B ?A ? okX ? B B ? okY ? ?C ?B ?
okY ? C
111
Testing Equivalence
p
KB1(A,B,C,okX,okY)
q
KB2(A,B,C,okX,okY)
If p ltgt q, then KB1 and KB2 not equivalent
If p q, then Pr(KB1 and KB2 are equivalent)gt 1/2
Run test 100 times ? error is lt 10-30
112
.50
113
Probabilistic Equivalence Testing

• Given propositional theories F G
• Compute p(F), p(G)
• If p(F) ? p(G), F G are not equivalent
• Otherwise, equivalent with probability gt½

114
Equivalence Testing
10
(1-2) (1-3) 5
x y z F(xyz)
0 0 1 1
0 1 0 1
0 1 1 1
1 0 1 1

• Map x y z to 1,2,3,4,5,6
• x?2, y ?3, z?5

12
(1-2) 3 (1 - 5)
-15
(1-2) 3 5
-20
2 (1-3) 5
Add p(F) -13

• p(F) p(G) ? F G are equivalent with
  probability (m 1)n / mn ( gt ½ for m 2n)

115
Projection under determinism
or
and
and
or
or
or
or
B
B
A
C
C
A
okX
okY
116
Projection under determinism
or
and
and
or
or
or
or
true
true
A
C
C
A
okX
okY
117
Determinism

Query d-DNNF
CO Consistency Yes
VA Validity Yes
CE Clausal entailment Yes
SE Sentential entailment
IP Implicant testing Yes
EQ Equivalence testing ?
MC Model Counting Yes
ME Model enumeration Yes
118
Determinism

Transformation d-DNNF
CD Conditioning Yes
SFO Single variable
FO Multiple variable
Conjoin
B Bounded Conjoin
Disjoin
B Bounded Disjoin
Negate ?
119
Decision
120
Decision
121
Decision

Query FBDD
CO Consistency Yes
VA Validity Yes
CE Clausal entailment Yes
SE Sentential entailment
IP Implicant testing Yes
EQ Equivalence testing ?
MC Model Counting Yes
ME Model enumeration Yes
122
Decison

Transformation FBDD
CD Conditioning Yes
SFO Single variable
FO Multiple variable
Conjoin
B Bounded Conjoin
Disjoin
B Bounded Disjoin
Negate Yes
123
Ordering
124
Ordering

Query OBDD
CO Consistency Yes
VA Validity Yes
CE Clausal entailment Yes
SE Sentential entailment Yes
IP Implicant testing Yes
EQ Equivalence testing Yes
MC Model Counting Yes
ME Model enumeration Yes
125
Ordering

Transformation OBDD
CD Conditioning Yes
SFO Single variable Yes
FO Multiple variable
Conjoin
B Bounded Conjoin Yes
Disjoin
B Bounded Disjoin Yes
Negate Yes
126
Agenda

• Part I Languages
• Part II Operations
• Part III Compilers
• Part IV Applications

127
A Knowledge Compilation MAP
Negation Normal Form
or
Decomposability Determinism Smoothness Flatness De
cision Ordering
and
and
or
or
or
or
and
and
and
and
and
and
and
and
?A
B
? B
A
C
? D
D
? C
128
NNF Subsets
NNF
CO, CE, ME
d-NNF
s-NNF
f-NNF
DNNF
VA,IP,CT
BDD
d-DNNF
EQ?
CNF
DNF
FBDD
EQ?
sd-DNNF
CO,CE,MEVA,IP,SE,EQ
SE,EQ
SE,EQ
VA,IP, SE,EQ
MODS
IP
OBDD
PI
129
Part III Compilers
130
Building Compilers

• To-down approaches
• Based on exhaustive search
• Bottom-up approaches
• Based on transformations

131
SAT by DPLL Search
x ? y x ? z w ? z ? v v ? w ? z
SAT?

• Unit resolution
• Conflict-directed backtracking
• Clause learning
• Branching heuristics
• Restarts

Terminating condition for recursion empty set
(satisfied), or empty clause (contradiction)
132
Recent Trend Exhaustive DPLL

• Count number of models
• Model counters, e.g., relsat, cachet
• Generate all/subset of models
• Image computation in model checking
• SMT (non-propositional reasoning)
• Variations on DPLL Search

133
The Language of Search
Knowledge Compiler
Exhaustive DPLL
Record Trace
Variations
Languages
134
Trace of DPLL
X ? Y X ? ?Y ? ?Z ?X ? Y ? ?Z
X
0
Y
0
1
Z
unsat
0
1
unsat
sat
135
Exhaustive DPLL
Run to Exhaustion
X ? Y X ? ?Y ? ?Z ?X ? Y ? ?Z
X
1
0
Y
Y
0
1
0
1
Z
Z
sat
unsat
0
1
0
1
unsat
unsat
sat
sat
136
Trace of DPLL
a Formula
or
X
and
and
?X
X
or
or
Y
Y
and
and
and
and
Z
Z
sat
unsat
?Y
Y
0
?Y
Y
1
?Z
?Z
unsat
unsat
sat
sat
137
Trace of DPLL a Formula
or
Equivalent to original CNF Tractable (e.g.,
count models)
and
and
?X
X
or
or
and
and
and
and
?Y
Y
0
?Y
Y
1
?Z
?Z
138
Dealing with Redundancy
Level One Do not record redundant portions of
trace Level Two Try not to solve equivalent
subproblems
X
Y
Y
Z
Z
unsat
sat
unsat
unsat
sat
sat
139
Dealing with Redundancy
X
Y
Y
Z
Z
unsat
unsat
unsat
sat
140
Dealing with Redundancy
Do not create
Simply point to existing node
X
Y
Y
Z
Z
unsat
sat
141
This is an OBDD!
X
Y
Y
Z
0
1
142
This is an OBDD!
or
X
and
and
Y
Y
?X
X
or
or
and
and
and
and
Z
0
1
?Z
?Y
Y
0
1
NNF decision, decomposability, ordering
143
A Non-traditional OBDD Compiler
X
Exhaustive DPLL, Fixed variable order, Unique
nodes
X ? Y X ? ?Y ? ?Z ?X ? Y ? ?Z
Compile
Y
Y
Z
0
1
New complexity guarantees
144
FBDD
X
Exhaustive DPLL, Dynamic variable order, Unique
nodes
X ? Y X ? ?Y ? ?Z ?X ? Y ? ?Z
Y
Z
Compile
Y
Z
0
1
NNF decision, decomposability
145
FBDD vs OBDD

• FBDD more succinct than OBDD (dynamic var
  ordering in SAT)
• OBDD equivalence test (canonical)
• FBDD probabilistic equivalence test
• Both allow model counting

146
Dealing with Redundancy

• Level One Unique nodes (done)
• Level Two Avoid redundant compilation (search)

147
Redundant Compilation
x5 ? x6 x4 ? ?x5 ? x6 x1 ? x3 ? x4 ? x5 x2 ?
x3 x1 ? x2 ? ?x3
X1
1
0
X2
X2
1
0
X3
X3
1
1
x5 ? x6 x4 ? ?x5 ? x6
x5 ? x6 x4 ? ?x5 ? x6
Formula Caching complexity guarantees
148
Formula Caching

• Majercik and Litmman, 1998
• Darwiche, 2002
• Bacchus et al, 2003, 2004
• Huang Darwiche, 2004
• Sang, Kautz, Beam, 2004, 2005
• Thurley, 2006

149
Caching for DPLL
v1
OBDD(?)

OBDD(?v11)
OBDD(?v10)


Recursive calls may be made on equivalent CNFs

v2
OBDD(?v11,v21)
OBDD(?v10,v20)
150
Caching for DPLL
? v5 v6 v4 ?v5 v6 v1
v3 v4 v5 v2 v3 v1 v2
?v3
v1v2v3 ?
0 0 0 contradiction
0 0 1 contradiction
0 1 0 v5 v6, v4 ?v5 v6, v4 v5
0 1 1 v5 v6, v4 ?v5 v6
1 0 0 contradiction
1 0 1 v5 v6, v4 ?v5 v6
1 1 0 v5 v6, v4 ?v5 v6
1 1 1 v5 v6, v4 ?v5 v6
OBDD(?)



151
Caching for DPLL
Cutset_3
c5 v5 v6
c4 v4 ?v5 v6
c3 v1 v3 v4 v5
c2 v2 v3
c1 v1 v2 ?v3
v1 v2 v3 v4 v5 v6

• After instantiation of v1v2v3, ? is either
  contradictory, or determined by clause c3 alone.
• c3 can only be in one of two states satisfied or
  shrunk to v4v5

152
Caching for Basic DPLL

• In general, cutset_i is set of clauses mentioning
  a variable ? vi and one gt vi
• After instantiation of v1v2vi, ? is either
  contradictory, or determined by states of clauses
  cutset_i
• Number of distinct ? is ? 2 cutset_i 1
• Maintain a cache for each i, and use the value of
  cutset_ia bit vectoras key

153
CNF to OBDD

• OBDD(?, i)
• if(contradiction) return 0-sink
• if(satisfied) return 1-sink
• key value(cutseti-1)
• lookup cachei-1key
• if(lookup ? nil) return lookup
• result getnode_node(vi, OBDD(?vi0, i1),
  OBDD(?vi1, i1))
• cachei-1key result
• return result

154
Complexity

• For each i, 2cutset_i bounds
• number of recursive calls OBDD(?, i1)
• number of entries in cachei
• number of OBDD nodes labeled with vi
• Size of largest cutset is known as cutwidth of
  variable order
• Time and space complexities of algorithm and size
  of OBDD are all linear in number of variables,
  and exponential only in cutwidth
• Variable orders with small cutwidth can help

155
Complexity Theorems

• size(OBDD) ? n2w 2
• n number of variables
• w cutwidth of variable order (size of largest
  cutset)
• Time complexity O(sn2w)
• s size of CNF
• Also hold for w pathwidth, using a slightly
  different caching scheme
• Cutwidth and pathwidth are incomparable

156
Beyond BDDs
FBDD
Plain DPLL
Fixed Variable Ordering
OBDD
157
Decomposition (Component Analysis)
Solve disjoint subproblems independently
d-DNNF
Combine as AND node
158
A ? B ? C ?A ? ?B ? C A ? D ? E ?A ? ?D ? E
Deterministic Decomposable Negation Norm
Form (d-DNNF)
A
B ? C D ? E
and
and
D ? E
B ? C
B
D
B
D
C
E
0
1
159
Deterministic Decomposable Negation Norm Form
(d-DNNF)
or
and
and
?A
A
and
and
or
or
or
or
and
and
and
and
?B
?D
B
D
C
E
160
Decomposition Methods

• Dynamically detect disjoint components
• Most effective, but very expensive
• Static structural analysis
• Constructs a decomposition tree (dtree)
• Does not detect all decompositions
• Low overhead at runtime

161
FBDD vs d-DNNF

• d-DNNF more succinct than FBDD (effectiveness of
  decomposition)
• Deterministic equivalence test open
• Probabilistic equivalence test applies
• Other queries same

162
The Language of Search
Fixed Variable Ordering
OBDD
Plain DPLL
FBDD
Allowing Decomposition
d-DNNF
Other languages deterministic DNF
163
Relation to AND/OR Search (CP)

• AND/OR graphs are deterministic and decomposable
• AND/OR search algorithms are doing enough work to
  compile networks into (multi-valued equivalent
  of) d-DNNF
• Capable of more than answering a single query
  (model counting, belief revision, etc)

164
Implications

• SAT techniques harnessed for knowledge
  compilation
• c2d compiler based on Rsat Solver (SAT
  Competition 2007) uses dtrees
• Language properties (succinctness/tractability)
  help characterize power and limitations of search

165
Understanding DPLL

• Take any program X that runs exhaustive
  DPLL-style search
• Examine traces, if traces ? L, then
• X can answer all queries tractable for L
• X is hopeless on any input having no
  polynomial-size representation in L

166
Power of DPLL

• Traces of several model counters (Relsat,
  Cachet, e.g.) are in d-DNNF
• Are doing enough work to
• compile formulas into d-DNNF
• solve tasks beyond model counting (e.g., minimum
  cardinality, probabilistic equivalent testing)

167
Limitation of DPLLGeneral Determinism
or
Decision nodes(d-DNNF)
or
Deterministic nodes(d-DNNF)
168
Beyond DPLLDecomposability (D) Without
Determinism (d)
or
DNNFCO, CE, ME, exist quant
and
?X3
or
and
X1
X2
169
Approximate DNNF
D
170
Approximate Compilation
171
Approximate Compilation
Sound, but not complete
172
Approximate Compilation
Complete, but not sound
173
sat D ?
Sound Incomplete
Unsound Complete
174
Bottom-up Compilation
175
Bottom-up OBDD Construction
CNF (x y) (y z) Variable order x, y, z
Final OBDD
x
x y
x
x
DEAD NODES
y z
y
y
The Apply algorithm combines two OBDDs using any
one of the 16 binary Boolean operators
z
0
1
176
Bottom-Up OBDD Construction

• OBDD packages, such as CUDD
• implement Apply (conjoin, disjoin, etc)
• garbage-collect dead nodes
• Apply is efficient quadratic in operand size
• Problem intermediate OBDDs can be much larger
  than final onemany dead nodes
• uf100-08 (32 models) OBDD has 176 nodes under
  MINCE order 30,640,582 intermediate nodes using
  CUDD taking 25 mins

177
DPLL Based Construction
x
? (x y) (y z)
?x0 y
?x1 y z
y
y
z
0
1
178
Missing Opportunities

• Bottom up construction methods for DNNF and d-DNNF

179
Agenda

• Part I Languages
• Part II Operations
• Part III Compilers
• Part IV Applications

180
Part IV Applications
181
Applications

• Model-based diagnosis
• Planning
• Probabilistic reasoning

182
Model-based Diagnosis
C
A
X
D
Y
B
System model ? okX ? (A ? ?C) okY ? (B ? C) ? D
Health variables okX, okY Observables A, B,
D Nonobservable C
183
Model-based Diagnosis
C
Abnormal observation ? ?A ? B ? ?D
A
X
D
Y
B
System model ? okX ? (A ? ?C) okY ? (B ? C) ? D
Diagnosis Values of (okX, okY) consistent with ?
? ? (0, 0), (0, 1), (1, 0)
184
Model-based Diagnosis
C
Abnormal observation ? ?A ? B ? ?D
A
X
D
Y
B
System model ? okX ? (A ? ?C) okY ? (B ? C) ? D
Minimum cardinality Diagnoses (0, 0), (0, 1),
(1, 0)
185
Model-based Diagnosis

• Methods for characterizing diagnoses
• Conflicts
• Kernel diagnoses
• Methods for manipulating diagnoses
• Find minimal diagnoses
• Find minimum-cardinality diagnoses
• Characterize lost functionality

186
Characterizing Diagnoses
187
Characterizing Diagnoses
188
Characterizing Diagnoses
low
high
high
low
189
Minimal Conflicts
An instantiation of ok1,ok5 is a diagnosis iff
it is consistent with(conjunction of) minimal
conflicts
190
Kernel Diagnoses
An instantiation of ok1,ok5 is a diagnosis iff
it is consistent with(disjunction of) kernel
diagnoses
191
Characterizing Diagnosis

• Device Model
• Observables
• Health Variables
• Others

192
OBDD
193
DNNF
194
Minimal Conflicts
CNF/PI
195
Kernel Diagnoses
DNF/IP
196
Characterizing Diagnosis

• Device Model
• Observables
• Health Variables
• Others

Compile Model
197
System Model
?
Compile
Succinctness Efficient computation of diagnoses
198
System Model
OBDD vs. DNNF
Compile
Succinctness Efficient computation of diagnoses
199
Compiling System Models

• DNNF is more succinct can lead to smaller
  compilation
• Smaller compilation of system model does not
  imply faster on-line diagnosis
• Although less succinct, OBDD is more powerful
  (DNNF is a weaker form)
• Is this extra power relevant?

200
Diagnosis Using DNNF
C
A
X
D
Y
B
T
T
Observation ?A ? B ? ?D
System model ? okX ? (A ? ?C) okY ? (B ? C) ? D
T
F
F
F
T
T
201
Diagnosis Using DNNF
C
A
X
D
Y
B
or
Observation ?A ? B ? ?D
System model ? okX ? (A ? ?C) okY ? (B ? C) ? D
?okX
?okY
202
Diagnosis Using DNNF

• Set observables
• Linear time
• Project out nonobservables
• Linear time
• Projection exponential for OBDD(can be made
  linear with particular orders)

203
Minimum Cardinality Diagnoses

• Number of diagnoses can be large
• Some may be more preferable than others
• Minimize number of faulty components in diagnosis
• Again, easy for DNNF

204
Minimum Cardinality Diagnoses
Smoothing make disjuncts mention same set of
variables O(? H)
1
1
1
or
0
0
?
0
0
1
1
?okX
?okY
205
Minimizing the DNNF
1
1
1
or
0
0
?
?
?
0
0
1
1
?okX
?okY
206
Minimum Cardinality Diagnoses
or
and
and
?okX
?okY
okX
okY
207
Minimum Cardinality Diagnoses Using OBDDs

• Create filter OBDD that asserts a given
  cardinality k
• Conjoin it with OBDD that represents set of
  diagnoses
• Repeat for k 0, 1, 2... until result is nonempty

208
Comparison
Operation OBDD DNNF
Condition(?, ?) O(?) O(?)
Project(?, H) exponential O(?)
Minimize(?d) O(mc ?d H2) O(?d H)
209
Characterizing Lost Functionality
210
Hierarchical Diagnosis
211
Scalability

• Requires a health variable for each component
• c1908 has 880 gates basic encoding fails to
  compile
• New technique to reduce number of health
  variables
• Preserves soundness and completeness w.r.t.
  min-cardinality diagnoses
• Requires only 160 health variables for c1908

212
Hierarchical Diagnosis
213
Hierarchical Diagnosis
214
Hierarchical Diagnosis
215
Identifying Cones

• Gate G dominates gate X if any path from X to
  output of circuit contains G
• All gates dominated by G form a cone
• Dominators found by breath-first traversal of
  circuit
• Treat maximal cones as blackboxes

216
Abstraction of Circuit
?C T, U, V, A, B, C
217
Top-level Diagnosis
Diagnosis A, B, C
218
Diagnosis of Cone

• Need to set inputs/output of cone according to
  top-level diagnosis
• Rest is similar, but not a simple recursive call
  (to avoid redundancy)
• Once cone diagnoses found, global diagnoses
  obtained by substitution

219
Diagnosis of Cone
Top-level diagnosis A, B, C 3 diagnoses for
cone A A, D, E 3 global diagnoses by
substitution A, B, C D, B, C E, B, C
220
Soundness

• Top-level diagnoses have same cardinality.
  Substitutions do not alter cardinality (cones do
  not overlap).
• Remains to show that cardinality of these
  diagnoses, d, is smallest. Proof by
  contradiction
• Suppose there is diagnosis P lt d. Replace every
  gate in P with its highest dominator to obtain
  P.
• P is a valid top-level diagnosis, contradicting
  soundness of baseline diagnoser

221
Completeness

• Need to show every min-cardinality diagnosis is
  found
• Given diagnosis P of min cardinality d, replace
  every gate in P with its highest dominator to
  obtain P
• P has cardinality d, and only mentions gates in
  top-level abstraction, and hence will be found by
  top-level diagnosis (by completeness of baseline
  diagnoser)
• P itself will be found by substitution (by
  completeness of cone diagnosis)

222
Sequential Diagnosis
223
Sequential Diagnosis

• Set of min-cardinality diagnoses may still be
  large
• Faults not identified with certainty
• Take measurements, one at a time, until faults
  identified
• Would like to minimize number of measurement
• Use heuristic based on amount of information gain

224
Sequential Diagnosis

• Assume a probabilistic model of the system
• Failure probabilities for components
• Output behavior of faulty component (e.g., 1 and
  0 with equal probability)
• These implicitly define a joint probability
  distribution over all system variables
• Encoding and compilation described later
• Pick component with highest posterior probability
  of failure, measure variable with highest entropy
  in that component

225
and
0.5
or
and
0.5
1
and
and
P
D
A
0.475
0.025
1
1
1
and
or
or
0.05
0.5
0.95
and
and
okA
J
and
0.5
0.9
0.05
1
0.5
?okA
??J
okJ
?A
?J
?okJ
0.1
1
0
0.5
0.5
0.5
226
Hierarchical Sequential Diagnosis

• To improve scalability, previous idea of
  abstraction applies
• Treat each cone (blackbox) as a single big
  component
• Need to compute a single failure probability that
  summarizes the failure behavior of the cone
• Create copy of cone with all gates healthy, feed
  outputs of two cones into XOR gate, compute
  Pr(output 1)

227
Planning
228
Slippery Gripper

• Goal block painted and held, gripper clean

• Probabilistic action effects
• Paint paints block w. p. 1 makes gripper dirty
  w. p. 1 if it holds block, w. p. 0.1 if not
• Pick-up succeeds w. p. 0.5 if gripper wet, 0.95
  if gripper dry
• Dry dries wet gripper w. p. 0.8 doesnt affect
  dry gripper
• Probabilistic initial state
• block not painted, not held
• gripper clean, but dry with probability 0.7

229
Conformant Probabilistic Planning

• Probabilistic initial state and action effects
• Conformant action effects not observable
• Cant decide next action by observing environment
• Needs straight-line plan with max probability of
  success, for given horizon (number of steps)
• Example 2-step plan paint, pickup (succeeds
  with probability 0.7335)

230
Brute-force Approach

• Compute success probability for all plans of
  length one
• Given success probabilities of plans of length i,
  compute probability of success for plans of
  length i 1
• Iterate to planning horizon n
• Pick n-step plan with max success probability
• Exponential in planning horizon

231
Why is it hard?

• Consider decision version
• Does there exist plan of success probability gt p,
  for given horizon
• Given plan, deciding if success probability gt p
  is PP-complete
• Can be reduced to MAJ-SAT does majority of
  assignments satisfy CNF
• Alternatively Is CNF satisfiable with
  probability gt ½
• Finding plan with given property (that is free to
  test) is NP-complete, for given horizon
• Conformant probabilistic planning for given
  horizon is NPPP-complete
• NP ? PP ? NPPP

232
Propositional Encoding Initial State

• State space BP (block-painted), BH (block-held),
  GC (gripper-clean), GD (gripper-dry)
• Probabilistic initial state ?BP, ?BH, GC, p ?
  GD
• Chance variable p (labeled with 0.7)
• p 1 ?BP, ?BH, GC, GD probability 0.7
• p 0 ?BP, ?BH, GC, ?GD probability (1 -
  0.7)
• Encoded as propositional sentence some variables
  labeled with numbers (probabilities)

233
Propositional Encoding Action Effects

• Dry dries wet gripper w. p. 0.8 doesnt affect
  dry gripper
• ?dry ? GD ? (q ? GD), ?dry ? ?GD ? GD
• Frame axiom
• ?dry ? (BP ? BP), ?dry ? (BH ? BH), ?dry ? (GC
  ? GC)
• Each setting of chance variable selects one
  effect
• probability of effect is label of chance
  variable, or 1 minus that depending on value of
  variable
• multiple chance variables multiply the numbers
• Define Pr(e) where e is an event (setting of
  chance vars)

234
Propositional Encoding Goal

• Goal BP, BH, GC

235
Propositional Encoding Horizon n

• State variables S0, S1, S2, , Sn
• Chance variables P-1, P0, P1, , Pn-1
• Action variables A0, A1, A2, , An-1
• Initial state I(P-1, S0) Goal G(Sn)
• Plan step Ak ? A(Sk, Ak, Pk, Sk1)
• ?n ? I(P-1, S0) ? A0 ? A1 ? ? An-1 ? G(Sn)

236
Plan Assessment

• Given propositional encoding ?n and n-step plan ?
• Whats probability of success Pr(?n, ?)?
• Plan ? is instantiation of action vars
• Pr(?n, ?) is sum of Pr(e) for e consistent with
  ?n ? ?
• Computation intractable (has to enumerate models)

237
Brute-force Algorithm

• Search through all plans ? (instantiations of
  action vars)
• For each plan ?, compute Pr(?n, ?)
• Return plan ? with max Pr(?n, ?)
• Improve in two ways
• Efficient plan assessment
• Search space pruning
• Both achieved by compiling ?n to d-DNNF

238
Compilation to d-DNNF

• Use existing compiler c2d, http//reasoning.cs.ucl
  a.edu/c2d/
• Compilation intractable in general
• For structured problems, exponential only in
  treewidth
• Treewidth does not grow with horizon in this
  encoding
• Can scale to large horizons

239
d-DNNF
and
or
and
and
?p
t
s
p
?dry
paint
?pickup
?r
?dry
?paint
pickup
240
Plan Assessment on d-DNNF

• paint, pickup

and
or
and
and
?p
t
s
p
?dry
paint
?pickup
?r
?dry
?paint
pickup
241
Plan Assessment on d-DNNF

• paint, pickup

and
or
and
and
?p
t
s
p
?dry
paint
?pickup
?r
?dry
?paint
pickup
1
1
1
1
1
1
242
Plan Assessment on d-DNNF

• paint, pickup

and
or
and
and
?p
t
s
p
?dry
paint
?pickup
?r
?dry
?paint
pickup
1
1
1
.9
1
1
1
.5
.3
.7
.95
243
Plan Assessment on d-DNNF

• paint, pickup

and
or
.665
.15
and
and
?p
t
s
p
?dry
paint
?pickup
?r
?dry
?paint
pickup
1
1
1
.9
1
1
1
.5
.3
.7
.95
244
Plan Assessment on d-DNNF

• paint, pickup

and
.815
or
.665
.15
and
and
?p
t
s
p
?dry
paint
?pickup
?r
?dry
?paint
pickup
1
1
1
.9
1
1
1
.5
.3
.7
.95
245
Plan Assessment on d-DNNF

• paint, pickup

.7335
and
.815
or
.665
.15
and
and
?p
t
s
p
?dry
paint
?pickup
?r
?dry
?paint
pickup
1
1
1
.9
1
1
1
.5
.3
.7
.95
246
Plan Assessment on d-DNNF

• Set action vars to constants according to plan ?
• Set all state vars to 1 (existential
  quantification)
• Turn chance vars and their negations into numbers
• Turn or into summation, and into multiplication
• Evaluate resulting arithmetic circuit
• Value at root is Pr(?n, ?)

247
Assessment of Partial Plan

• Partial plan ? instantiation of a subset of
  action vars
• Pr(?n, ?) success probability of best completion
  of ?
• Maximize over uninstantiated action vars
• Special d-DNNF
• Turn corresponding or node into max

or
and
and
ß ?x
x ?
248
Assessment of Partial Plan

• Summations over state vars, followed by
  maximizations over action vars
• All max must be performed after sum
• Not guaranteed in d-DNNF
• Max and sum nodes mixed in any order
• Some max performed too early Result incorrect!

249
Depth-first Branch-and-Bound

• Key observation performing max too early can
  only increase result
• Result is upper bound on true value
• Partial plan can be pruned if upper bound lt
  value of best plan already found
• Depth-first branch-and-bound will find optimal
  plan
• Tighter bounds leads to more pruning

250
Plan Search
T
C
C
B
B
B
B
tcb .023
tcb .051
tcb .031
tcb .055
tcb .053
tcb .117
tcb .079
tcb .009
251
Plan Search
Best Score 0
T
.151
C
C
B
B
B
B
tcb .023
tcb .051
tcb .031
tcb .055
tcb .053
tcb .117
tcb .079
tcb .009
252
Plan Search
Best Score 0
T
.151
C
C
.135
B
B
B
B
tcb .023
tcb .051
tcb .031
tcb .055
tcb .053
tcb .117
tcb .079
tcb .009
253
Plan Search
Best Score 0
T
.151
C
C
.135
B
B
B
B
.127
tcb .023
tcb .051
tcb .031
tcb .055
tcb .053
tcb .117
tcb .079
tcb .009
254
Plan Search
Best Score .009
T
.151
C
C
.135
B
B
B
B
.127
tcb .023
tcb .051
tcb .031
tcb .055
tcb .053
tcb .117
tcb .079
tcb .009
255
Plan Search
Best Score .079
T
.151
C
C
.135
B
B
B
B
.127
tcb .023
tcb .051
tcb .031
tcb .055
tcb .053
tcb .117
tcb .079
tcb .009
256
Plan Search
Best Score .079
T
.151
C
C
.135
B
B
B
B
.120
tcb .023
tcb .051
tcb .031
tcb .055
tcb .053
tcb .117
tcb .079
tcb .009
257
Plan Search
Best Score .117
T
.151
C
C
.135
B
B
B
B
.120
tcb .023
tcb .051
tcb .031
tcb .055
tcb .053
tcb .117
tcb .079
tcb .009
258
Plan Search
Best Score .117
T
.151
C
C
.135
B
B
B
B
.120
tcb .023
tcb .051
tcb .031
tcb .055
tcb .053
tcb .117
tcb .079
tcb .009
259
Plan Search
Best Score .117
T
.151
C
C
.101
B
B
B
B
tcb .023
tcb .051
tcb .031
tcb .055
tcb .053
tcb .117
tcb .079
tcb .009
260
Plan Search
Best Score .117
T
.151
C
C
.101
B
B
B
B
tcb .023
tcb .051
tcb .031
tcb .055
tcb .053
tcb .117
tcb .079
tcb .009
261
Probabilistic Reasoning
262
Bayesian Networks
Battery Age
Alternator
Fan Belt
Charge Delivered
Battery
Fuel Pump
Fuel Line
Starter
Distributor
Gas
Battery Power
Spark Plugs
Gas Gauge
Engine Start
Lights
Engine Turn Over
Radio
263
Bayesian Networks
Battery Age
Alternator
Fan Belt
If Battery Power OK, then Lights ON (99) .
Charge Delivered
Battery
Fuel Pump
Fuel Line
Starter
Distributor
Gas
Battery Power
Spark Plugs
Gas Gauge
Engine Start
Lights
Engine Turn Over
Radio
264
?ca
?a
?ba
A B C Pr(.)
a b c ?a ?ba ?ca
a b c ?a ?ba ?ca
a b c ?a ?ba ?ca
a b c ?a ?ba ?ca
. . .
265
Joint Distributions
B
A
Pr(.)
.03
true
true
.27
true
false
.56
false
true
false
.14
false
false
false
Pr(a)
.03
.27 .3
266
Joint Distributions
B
A
Pr(.)
.03
true
true
.27
true
false
.56
false
true
false
.14
false
false
false
Pr(b)
.27
.14 .41
267
Joint Distributions
B
A
Pr(.)
?a?b .03
.03
true
true
.27
?a?b .27
true
false
.56
?a?b .56
false
true
false
?a?b .14
.14
false
false
false
F(?a, ?b, ?a, ?b)
.03?a?b .27?a?b .56?a?b .14?a ?b
268
F(?a, ?b, ?a, ?b)
.03?a?b .27?a?b .56?a?b .14?a ?b
Pr(a,b) F(?a0, ?b1, ?a1 , ?b0) .27
Pr(a) F(?a0, ?b1, ?a1 , ?b1) .03.27
269
?ca
?a
?ba
A B C Pr(.)
a b c ?a ?ba ?ca
a b c ?a ?ba ?ca
a b c ?a ?ba ?ca
a b c ?a ?ba ?ca
. . .
270
?ca
?a
?ba
A B C Pr(.)
a b c ?a ?b ?c ?a ?ba ?ca
a b c ?a ?b ?c ?a ?ba ?ca
a b c ?a ?b ?c ?a ?ba ?ca
a b c ?a ?b ?c ?a ?ba ?ca
. . .
271
F ?a ?b ?c ?a ?ba ?ca ?a
?b ?c ?a ?ba ?ca ?a ?b ?c
?a ?ba ?ca ?a ?b ?c ?a
?ba ?ca .
272
?ca
C
?a
?dbc
?ba
F ?a ?b ?c ?d ?a ?ba ?ca ?dbc
?a ?b ?c ?d ?a ?ba ?ca ?dbc .
Each term has 2n variables (n indicators, n
parameters)
Each variable has degree one (multi-linear
function)
273
Multi-Linear Functions ?Arithmetic Circuits
A
B
274
Reduction to Logic
275
MLFs?ACsCNFs?d-DNNF
Compile
?


c
?
c
Decode


?
?

?
b
b
?b
1
a
1
a
?a
Arithmetic Circuit
Smooth d-DNNF
276
Propositional Encoding of Multi-Linear Functions
a b c Encodes
T T T abc
T T F ab
T F T ac
T F F a
F T T bc
F T F b
F F T c
F F F 1

• Propositional theory
• ? c (a ? ? b)
• Encodes
• F a c a b c c

277
Encoding Network as CNF
?a ? ?a ?a ? ?a
?b ? ?b ?b ? ?b
?a ? ?a ?a ? ?a
?a ?b ? ?ba ?a ?b ? ?ba ?a
?b ? ?ba ?a ?b ? ?ba
278
Why Logic?

• Encoding local structure is easy
• Determinism encoded by adding clauses
• CSI encoded by collapsing variables

279
Global StructureTreewidth w
280
Local StructureCSI and Determinism
Battery Age
Alternator
Fan Belt
Charge Delivered
Battery
Fuel Pump
Fuel Line
Starter
Distributor
Gas
Battery Power
Spark Plugs
Gas Gauge
Engine Start
Lights
Engine Turn Over
Radio
281
Local StructureCSI and Determinism
Battery Age
Alternator
Fan Belt
Charge Delivered
Battery
Fuel Pump
Fuel Line
Starter
Distributor
Gas
Battery Power
Spark Plugs
Gas Gauge
Engine Start
Lights
Engine Turn Over
Radio
Context Specific Independence (CSI)
282
Local StructureCSI and Determinism
Battery Age
Alternator
Fan Belt
Charge Delivered
Battery
Fuel Pump
Fuel Line
Starter
Distributor
Gas
Battery Power
Spark Plugs
Gas Gauge
Engine Start
Lights
Engine Turn Over
Radio
283
Local Structure
A
B
C
Pr(SA,B,C)
0.95
a
b
c
c
a
b
0.95
a
b
c
0.20
a
b
c
0.05
qsabe
a
b
c
0.00
a
b
c
0.00
-Functional constraints -Context-specific
independence
a
b
c
0.00
a
b
c
0.00
Tabular CPT
284
Determinism
A
Pr(SA,B,E)
B
C
0.95
a
b
c
c
a
b
0.95
a
b
c
0.20
a
b
c
0.05
a
b
c
0.00
a
b
c
0.00
a
b
c
0.00
a
b
c
0.00
Tabular CPT
285
Context-Specific Independence
A
Pr(SA,B,C)
B
C
0.95
a
b
c
c
a
b
0.95
a
b
c
0.20
a
b
c
0.05
a
b
c
0.00
a
b
c
0.00
a
b
c
0.00
a
b
c
0.00
Tabular CPT
286
The Ace Systemhttp//reasoning.cs.ucla.edu/ace
Arithmetic Circuit

Smooth d-DNNF
CNF
?

• lx ? l?x
• lx ? l?x
• x ? ly ? qxy
• ...
• .
• .

287
1st International Evaluation of Exact
Probabilistic Reasoning Systems (UAI06, Boston)

• 135 problem instances from speech, coding,
  bioinformatics, circuits, medical diagnosis,
• Each team given 4 days of computation time
• UCLA Only team to solve all problems in allotted
  time (solved all problems in 1 hr)
• Failure rates of other teams 10-40

288
Inference by Compiling to d-DNNF

• Deterministic Planning
• Blai Bonet and Hector Geffner (KR 2006)
• Probabilistic Conformant Planning
• Jinbo Huang (ICAPS 2006)
• Model-based diagnosis
• Paul Elliott and Brian Williams (AAAI 2006)
• Anthony Barrett (IJCAI 2005)
• Databases (query re-write)
• Yolife Arvelo, Blai Bonet and Maria Esther Vidal
  (AAAI 2006)
• Inference in Bayesian Networks (2006 competition)
• Mark Chavira, Adnan Darwiche (IJCAI 2005)
• Inference in Probabilistic Relational Models
• Mark Chavira, Adnan Darwiche and Manfred Jaeger
  (IJAR 2006)

c2d compiler http//reasoning.cs.ucla.edu/c2d