These slides, presented at RCRA 2009, summarises my work for the MSc thesis. They introduce some fundamentals on LPADs, the standard inference mechanisms, two approximate algorithms and the graphs documenting our findings.

1. Stefano Bragaglia
DEIS – University of Bologna
stefano.bragaglia@unibo.it
Fabrizio Riguzzi
ENDIF – University of Ferrara
fabrizio.riguzzi@unife.it

2. 

Increasing interest in combining
logic and probability
•
C1: sneezing(X): 0.7 v
not_sneezing(X): 0.3 ←
flu(X).
C2: sneezing(X): 0.6 v
not_sneezing(X): 0.4 ←
hayfever(X).
C3: flu(andrew).
C4: flu(david).
C5: hayfever(david).
C6: hayfever(robert).
Many formalism:
 Markov Logic Networks
 ProbLog
 Logic Programs with Annotated
Disjunctions (LPADs)
 etc.

LPADs can easily express:
•
Queries:
?- sneezing(andrew).
?- sneezing(robert).
?- sneezing(david).
 cause-effect relationships among
events
 possible effects of a single cause
 contemporary contribution of
more causes to the same effect
Example: simple medical diagnosis
•
0.7
0.6
0.88
Probability:
Instances where query is true: 3 out of 4
P = 0.7×0.6 + 0.7×0.4 + 0.3×0.6 =
0.88

3. 
Syntax





Program: set of disjunctive clauses
Head: set of mutually exclusive and
exhaustive logical atoms annotated with
probability values between 0 and 1
whose sum is 1
Body: event whose effects are
represented by the atoms of the
corresponding head
1.
2.
Semantic



Inference: in 2 steps (by cplint)
Explanations: a meta-interpreter
performs resolution keeping current
set of choices
Probability of a query:
a dynamic algorithm converts
explanations into a Binary Decision
Diagram (BDD) and traverse it; this
makes explanations mutually
exclusive
?- sneezing(david).
(C1;{X1/david};0)
(C2;{X2/david};0)
:- flu(david).
:- hayfever(david).
Instance: normal logic program obtained
by choosing a logical atom from the
head of each grounding of every clause
Probability of an instance: product of
the probability values of all the atoms
chosen for that instance
Probability of a query: sum of the
probabilities of each instance where the
query is true


0.4
0.3
0
XC2
XC1
0.7
0.6
1
P = 0.3×0.6 + 0.7 = 0.88

Not possible in some domains

4.  Best K
▪ Deterministic algorithm
based on branch and bound
technique
▪ Explanations built
incrementally by keeping
only the k most probable
ones (k = 64)
▪ Probability of the query
computed on chosen
explanations trough BDD
(lower bound)
▪ Note: limiting explanations
allows better control on
complexity
 Monte Carlo
▪ Stochastic algorithm based
on Monte Carlo approach
▪ Instances sampled
repeatedly from LPAD by
considering only relevant
clauses
▪ Head atoms of resolving
clauses chosen stochastically
by a meta-interpreter
▪ Note: the probability of the
query is the fraction of
sampled instances where the
query is true (no BDD is
needed)

5. 
Real datasets: 10 incremental
samples of a graph describing
biological entities responsible of
Alzheimer’s disease
Biological Graph
HGNC_1505
PubMed_12653
0.643
0.493
0.665
EntrezGene_11803
PubMed_15529
0.523


Artificial datasets: 3
procedurally built graphs of
increasing complexity
Queries: compute the probability
that a path exists between two
given nodes of each graph
0.567
EntrezProtein_7891032
Tests: performed on Linux
machines equipped with Intel
Core 2 Duo E6550 (2333 MHz) and
4 Gb RAM with a 24 hours time
limit
PubMed_1741124
0.602
0.621
PubMed_157782
0.713
EntrezProtein_8147603
Artificial Graphs
0.3
0.3
0.3
0.3
0.3
0.3
0.3
0.3
0.3
0.3
0.3
0.3
“lanes”
0.3
0.3
0.3
0.3

HGNC_620
0.3
0.3
0.3
0.3
0.3
0.3
0.3
“branches”
“parachutes”

6. 12
10
8
6
4
2
0
Biological Graphs:
CPU times averaged on successes
100000
Standard
Best K
Monte Carlo
Size (edges)
Time (log s)
Answers
Biological Graphs:
number of successes
1000
Standard
10
Best K
Monte Carlo
0,1
0,001
Size (edges)

7. Lanes Graphs: CPU times
Parachutes Graphs: CPU times
100
Standard
Best K
Monte Carlo
0,01
Monte Carlo
0,000001
Size (steps)
Size (steps)
Branches Graphs: CPU times
Time (log s)
100
1
1
3
5
7
9
11 13 15 17 19
0,01
Standard
Best K
Monte Carlo
0,0001
0,000001
Size (steps)
Standard
Best K
0,0001
0,0001
0,000001
1
1
23
45
67
89
111
133
155
177
199
221
243
265
287
0,01
Time (log s)
1
1
23
45
67
89
111
133
155
177
199
221
243
265
287
Time (log s)
100