causal analysis of probabilistic...
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Vu Pham
Causal Analysis of
Probabilistic Counterexamples
1
Hichem Debbi [email protected]
University of MβSila
Causal Analysis of Probabilistic Counterexamples
Mustapha Bourahla [email protected]
Hichem Debbi
Vu Pham
Motivation
ARIOUA Abdallah
Inevitable complementary task to counterexample generation
Error location is the most difficult part of debugging [Vesey]
Counterexample Analysis
Multiple Paths
Probabilistic Nature
Challenges for Analysing Probabilistic Counterexamples
2
To answer the question:
Why is the probability threshold violated ?
Debugging Probabilistic Models
Hichem Debbi Causal Analysis of Probabilistic Counterexamples
Vu Pham
Probabilistic Computation Tree Logic
ARIOUA Abdallah
PCTL Logic
PCTL Property Satisfaction
State Formula
Path Formula
PCTL is an extension of CTL for specifying probabilistic properties
3 Hichem Debbi Causal Analysis of Probabilistic Counterexamples
~ β {<,β€,>,β₯}
Vu Pham
Probabilistic Counterexamples
4
A counterexample C for π·β€π π is a set of finite paths with
Pr( ) 0.01C 0.01( )or errP F
Probabilistic Counterexample
Pr( )C p
π β
Hichem Debbi Causal Analysis of Probabilistic Counterexamples
Vu Pham
Probabilistic Counterexamples
ARIOUA Abdallah
{b,e}
5 Hichem Debbi Causal Analysis of Probabilistic Counterexamples
ππ
ππ ππ ππ
ππ ππ {a} {c,d} 0.5
0.25
0.25
0.4
0.6
0.3
0.5 0.2
{a, b}
{c,d} {c,d}
π πΆπ2 = π π 0π 1, π 0π 2π 3, π 0π 2π 4π 3, π 0π 2π 4π 5,π 0π 4π 5 = 0.25 + 0.2 + 0.09 + 0.15 + 0.12 = π. ππ
Vu Pham
Probabilistic Counterexamples
ARIOUA Abdallah
{b,e}
6 Hichem Debbi Causal Analysis of Probabilistic Counterexamples
ππ
ππ ππ ππ
ππ ππ {a} {c,d} 0.5
0.25
0.25
0.4
0.6
0.3
0.5 0.2
{a, b}
{c,d} {c,d}
MINIMAL
π πΆπ2 = π π 0π 1, π 0π 2π 3, π 0π 2π 4π 3, π 0π 2π 4π 5,π 0π 4π 5 = 0.25 + 0.2 + 0.09 + 0.15 + 0.12 = π. ππ
Vu Pham
Probabilistic Counterexamples
ARIOUA Abdallah
{b,e}
7 Hichem Debbi Causal Analysis of Probabilistic Counterexamples
ππ
ππ ππ ππ
ππ ππ {a} {c,d} 0.5
0.25
0.25
0.4
0.6
0.3
0.5 0.2
{a, b}
{c,d} {c,d}
Most Indicative
π πΆπ2 = π π 0π 1, π 0π 2π 3, π 0π 2π 4π 3, π 0π 2π 4π 5,π 0π 4π 5 = 0.25 + 0.2 + 0.09 + 0.15 + 0.12 = π. ππ
Vu Pham ARIOUA Abdallah
ππΌππΆπ(π 0 β¨ π₯)
π₯ = πβ€π(π)
Find
Labeling and probability values in the counterexample that cause
the probability to exceed the given upper bound over the model
8 Hichem Debbi Causal Analysis of Probabilistic Counterexamples
π
π
0.5 0.5
Given
Most Indicative Probabilistic Counter Example (MIPCX)
Counterexample Debugging
Vu Pham
Causality and Responsibility for MIPCX
π , π = π₯ is a cause for violating MIPCX
if either(π , π = π₯)is critical
or πβπ€β² makes (π , π = π₯) critical, for variable subset π
ππ (π , π = π₯,π₯) = 1 if (π , π = π₯)iscritical
= 1/( π + 1) otherwise
(π , π = π₯) is critical
if ππΌππΆπ(π ,πβπ₯β²) π 0 β¨ π₯ is not a valid counterexample.
ππΌππΆπ(π ,πβπ₯β²) π 0 β¨ π₯ :
The set of finite paths resulting from ππΌππΆπ π 0 β¨ π₯
by switching the value π₯ of variable π in state π
Criticality
Causality (adapted from Halpern & Pearl)
Degree of Responsibility (adapted from Chockler & Halpern)
9 Hichem Debbi Causal Analysis of Probabilistic Counterexamples
Vu Pham
Causality and Responsibility for MIPCX
ARIOUA Abdallah
Probabilistic Causality Model
is a tuple < π, ππ >
π βΆ causality model and ππ : probability function defined over the states of ππΌππΆπ π 0 β¨ π₯
10 Hichem Debbi Causal Analysis of Probabilistic Counterexamples
Most Responsible Cause
Cause C is a most responsible cause for violating π₯ = πβ€π π
if ππ πΆ ππ πΆ β₯ ππ πΆβ² Pr (πΆβ²) for any cause Cβ.
Pr π = π(π)
π βπ| πβππΌππ(π 0β¨π₯)
Pr π , π = π₯ = Pr(π )
Vu Pham
Probabilistic Counterexamples Revisited
ARIOUA Abdallah
{b,e}
11 Hichem Debbi Causal Analysis of Probabilistic Counterexamples
ππ
ππ ππ ππ
ππ ππ {a} {c,d} 0.5
0.25
0.25
0.4
0.6
0.3
0.5 0.2
{a, b}
{c,d} {c,d}
Most Indicative
π πΆπ2 = π π 0π 1, π 0π 2π 3, π 0π 2π 4π 3, π 0π 2π 4π 5,π 0π 4π 5 = 0.25 + 0.2 + 0.09 + 0.15 + 0.12 = π. ππ
Vu Pham
Probabilistic Counterexamples Revisited
ARIOUA Abdallah
{b,e}
12 Hichem Debbi Causal Analysis of Probabilistic Counterexamples
ππ
ππ ππ ππ
ππ ππ {a} {c,d} 0.5
0.25
0.25
0.4
0.6
0.3
0.5 0.2
{a, b}
{c,d} {c,d}
(s2,b=1) is the
most responsible cause π πΉ ππ, π = π = π
π πΉ ππ, π = π = π/| π | + π = π. π
π πΉ ππ, π = π ππ« ππ, π = π = 0.35
π·π ππ, π = π = π. π + π. ππ = π. ππ
: highest
Vu Pham
Algorithm and Implementation
13 Hichem Debbi Causal Analysis of Probabilistic Counterexamples
Probabilistic Symbolic Model Checker
[Kwiatkowska et al.]
Probabilistic Counterexample Generator
[Aljazzar et al.]
ππΌππΆπ π 0 β¨ π₯
π₯ = πβ€π(π)
Debugging Algorithm
(Debbi-Bourahla)
Causes with Responsibilities
and Probabilities
Diagnosis
Vu Pham
Conclusion and Future Work
β’ We adapted and showed the usefulness of Causality and Responsibility
in the context of debugging probabilistic counterexamples
β’ We introduced the notion of Most Responsible Cause
as an indicator for the source of the error
β’ We developed a Debugging Algorithm, and tested it on real case studies
with good performance
Conclusion
Future Work
14 Hichem Debbi Causal Analysis of Probabilistic Counterexamples
β’ Visualization of diagnosis results