1. Encoding CNFs to Enhance Component Analysis
Mark Chavira, Adnan Darwiche
Computer Science Department
University of California, Los Angeles
Work With
Goal: ***CLASS*** and ***MODEL COUNTING AND KNOWLEDGE COMPILATION***
Do that by applying a restricted form of resolution on the input which takes the form of a CNF
Based on observation that syntax can cripple …

2. Outline Review “search with decomposition” The effect of syntax
Preliminary strategy for circuits
Generalization of strategy for Bayesian nets

3. Search with Decomposition
CNF
A  B  C A  B  C
A  D  E A  D  E 18 A
B  C
D  E A
 B  C
 D  E 9
Review SWD
Given CNF and Goal
Overall Strategy
Sometimes for Free
In this case
Different in important way
Trace
Main Point
D  E
B  C
decompose
 D  E
 B  C 3

4. Syntax Example Syntax 1 A ^ B ^ C => D A ^ B ^ C => D
Goal / Algorithm only does this when… / Syntax Critical / Demonstrate
Describe Fragments
Ought to…
Two encodings
What happens when we split on A=false
Syntax 2
A ^ B ^ C => D
A => D
B => D
C => D A
B ^ C => D
B ^ C => D
B ^ C => D
B ^ C => D A D

5. Syntax Experiments Using Cachet
Circuit
Syntax 1
Syntax 2
Improv.
R.S. 1
s1238
7.48
0.99
7.59
1.03
7.29
s713
4.66
0.5
9.37
0.4
11.61
s526n
2.28
0.18
12.73
12.32
s1196
11.89
0.93
12.81
0.97
12.29
s526
2.27
12.82
12.61
s953
5.34
0.34
15.48
0.33
16.13
s641
5.2
0.32
16.3
15.67
s1488
2.78
0.13
21.24
s1494
2.89
22.43
22.25
s832
3.15
0.1
31.21
0.11
29.19
s838.1
2.95
0.08
38.8
s1423 -
63.78
49.94 s
186.61
199.49
S35932
2.83
3.09
On this slide
Rows
What did
Columns
Main point

6. Syntactic vs. Semantic Independence
CNFs  and  are syntactically independent iff  and  do not share variables.
CNFs  and  are semantically independent iff there exist two CNFs ’ and ’ such that
’ is logically equivalent to 
’ is logically equivalent to 
’ and ’ do not share variables
Definitions
When we have…
Gap
Goal

7. Resolution Strategy 1 …  v  v X  v ¬X …  v   v ¬X
General Idea
Simple operation any time
Bullets
Effectively a restricted form of resolution.
Always make some clause smaller
We never increase the number of clauses in the CNF.
Transforms syntax 1’s encoding of AND gate into syntax 2’s encoding.

8. The Effect of Syntax Circuit Syntax 1 Syntax 2 Improv. RS 1 s1238 7.48
0.99
7.59
1.03
7.29
s713
4.66
0.5
9.37
0.4
11.61
s526n
2.28
0.18
12.73
12.32
s1196
11.89
0.93
12.81
0.97
12.29
s526
2.27
12.82
12.61
s953
5.34
0.34
15.48
0.33
16.13
s641
5.2
0.32
16.3
15.67
s1488
2.78
0.13
21.24
s1494
2.89
22.43
22.25
s832
3.15
0.1
31.21
0.11
29.19
s838.1
2.95
0.08
38.8
s1423 -
63.78
49.94 s
186.61
199.49
S35932
2.83
3.09

9. Encoding Circuit Structure
Three ways to encode:
Using no Structure
Using “global” structure
Using global and “local” structure
No Structure
Global Structure
Local Structure
Sress: all this very interesting but…everyone already encodes according to syntax 2 for circuits so what is the point of RS1?
I’ve been talking about circuits because they have very well understood semantics and provide a great example,
but the point is to go beyond circuits, to applications where the semantics are not as well understood and we don’t have templates for how to encode.
In general, we see that these ideas can be extended to many systems that are made from components that can be encoded individually

10. Inference Bayesian Networks
Inference is #P-Complete.
Decades of research.
Most techniques use global structure only and are exponential in treewidth.
Recent algorithms attempt to be sensitive to local structure.
Weighted model counting is among the most successful.
Many networks can only be handled by model counting. A B C

11. Inference Bayesian Networks
Cachet
c2d
Compile ^ v
θa2
λc2
λc1
θc2|a2 b1
θc3|a2 b1 A B C
Encode
CNF
d-DNNF

12. Encoding Variables Parameter Variables: θ4 A B C Indicator Variables:Pr a1 b1 c1
0.7
(θ1) c2
0.0
(false) c3
0.3
(θ2) b2
0.4
(θ3) a2
0.333
(θ4)
0.2
(θ5)
0.5
(θ6) A B C
Indicator Variables:
λc1
λc2
λc3
Parameter Variables: θ4

13. Encoding Clauses Parameter Clause: λa2 ^ λb1 ^ λc1 => θ4 A B CPr a1 b1 c1
0.7
(θ1) c2
0.0
(false) c3
0.3
(θ2) b2
0.4
(θ3) a2
0.333
(θ4)
0.2
(θ5)
0.5
(θ6) A B C
Indicator Clauses:
λc1 v λc2 v λc3
¬λc1 v ¬λc2
¬λc1 v ¬λc3
¬λc2 v ¬λc3
Parameter Clause:
λa2 ^ λb1 ^ λc1 => θ4

14. Encoding Groups A B C A B C Pr a1 b1 c1 0.7 (θ1) c2 0.0 (false) c3 0.3
(θ2) b2
0.4
(θ3) a2
0.333
(θ4)
0.2
(θ5)
0.5
(θ6) A B C

15. Encoding a Single Group
a1 b1 c3
a1 b2 c2
a1 b2 c3
a2 b2 c2
λa1 ^ λb1 ^ λc3 => θ2
λa1 ^ λb2 ^ λc2 => θ2
λa1 ^ λb2 ^ λc3 => θ2
λa2 ^ λb2 ^ λc2 => θ2
encode
Prime
Implicants
a1 c3
b2 c2 PI
encode
λa1 ^ λc3 => θ2
λb2 ^ λc2 => θ2

16. Old Encoding vs. New

17. Resolution Strategy 2 Use global structure
Encode each table in isolation
Use local structure as before
Suppress generation of some CNF variables
Use local structure to reveal semantic independence
For each table, partition rows into encoding groups.
For each encoding group, find prime implicants.
For each prime implicant, assert a clause

18. Observations Resolution Strategy 2 applies structured resolution.
Finding prime implicants here is efficient.
Similarity between two strategies:
Remove occurrences of literals
Seeks to make clauses more minimal
Reveals semantic independence
Main idea can be applied more generally
Encoding functions over finitely-valued variables
e.g., Truth tables, Bayesian networks, Markov networks, influence diagrams.

19. Resolution Strategy 2 Experiments
Previous
RS 2
Compile
Network TW
Time (s)
Improv.
Size gr 24
137.25
43.95
3.12
14,692,963
5,739,854
2.56 gr 25 -
292.42
35,473,955 gr
65.03
40.45
1.61
7,755,318
5,280,027
1.47 gr
407.6
46.83
8.7
35,950,912
6,128,859
5.87 gr
26.7
3,431,139 gr
44.4
22.99
1.93
4,598,373
3,159,007
1.46 gr
51.68
2.99
17.28
6,413,897
421,060
15.23 gr
86.19
32.29
2.67
10,341,755
4,280,261
2.42 gr
60.55
7,360,872 gr
133.7
287.08
0.47
15,144,602
33,561,672
0.45 gr 27
411.45
48.36
8.51
39,272,847
6,451,916
6.09 gr 28
172.13
19,037,468 gr
362.9
29.18
12.44
32,120,267
2,507,215
12.81 gr
139.81
15,933,651 gr
158.13
18,291,116 gr
403.96
52.55
7.69
37,411,619
7,111,893
5.26 gr
79.97
9,439,318 gr
42.17
5,036,670 gr
68.51
7,890,645 gr
188.66
22,387,841

20. Resolution Strategy 2 Experiments
Previous
RS 2
Compile
Network TW
Time (s)
Improv.
Size
bm-05-03 19
0.29
0.2
1.45
19,190
10,957
1.75
bm-10-03 51
19.57
4.97
3.94
938,371
275,089
3.41
bm-15-03 62
254.96
44.07
5.79
7,351,823
1,460,842
5.03
bm-20-03 90
1,505.24
388.65
3.87
37,916,087
6,195,000
6.12
bm-22-03
107
4,869.64
748.13
6.51
72,169,022
14,405,730
5.01 or 24
338.48
54.47
6.21
6,968,339
7,777,867
0.90 or 25
1.4
0.77
1.82
104,275
119,779
0.87 or 27
728.36
118.17
6.16
17,358,747
14,986,497
1.16 or 26
250.72
97.13
2.58
11,296,613
12,510,488 or
19.58
7.17
2.73
1,011,193
1,060,217
0.95
water 21
2.81
1.73
1.62
101,009
103,631
0.97
pathfinder 15
12.45
2.86
4.35
36,024
33,614
1.07
diabetes 18
6,281.23
3,391.18
1.85
15,426,793
15,751,044
0.98
mildew
6,245.45
1,869.92
3.34
1,693,750
1,696,139
1.00
barley 23 -
14,722.19
37,321,497

21. Conclusion Syntax of CNF is critical
Gap between syntactic and semantic independence
Resolution Strategy 1: a resolution strategy
Resolution Strategy 2: a generalization of strategy for Bayesian nets
Experiments using Cachet and c2d
Achieve large time and space improvements

22. Encoding Times

23. Implied Variables Syntax 1 A ^ B ^ C => D A v B v C
A => D
B => D
C => D
Syntax 1
A ^ B ^ C => D
A ^ B ^ C => D
A ^ B ^ C => D
A ^ B ^ C => D
A ^ B ^ C => D
A ^ B ^ C => D
A ^ B ^ C => D