Accurate Approximate Diagnosability of Stochastic Systems Nathalie - - PowerPoint PPT Presentation

March 17th 2016 – LATA

Accurate Approximate Diagnosability

• f Stochastic Systems

Nathalie Bertrand1, Serge Haddad2, Engel Lefaucheux1,2

1 Inria, France 2 LSV, ENS Cachan & CNRS & Inria, France LATA, March 17th 2016

Diagnosis Framework

LTS: Labelled transition system. Diagnoser: must tell whether a fault f occurred, based on observations. Convergence hypothesis: no infinite sequence of unobservable events.

q0 f1 f2 f3 q1 q2 f u a c c c b b c

Accurate Approximate Diagnosability of Stochastic Systems March 17th 2016 – LATA - 2

Diagnosis Framework

LTS: Labelled transition system. Diagnoser: must tell whether a fault f occurred, based on observations. Convergence hypothesis: no infinite sequence of unobservable events.

q0 f1 f2 f3 q1 q2 f u a c c c b b c

A run ρ = q0

u

− → q1

c

− → q2 has an observation sequence P(ρ) = c.

Accurate Approximate Diagnosability of Stochastic Systems March 17th 2016 – LATA - 2

Diagnosis Framework

LTS: Labelled transition system. Diagnoser: must tell whether a fault f occurred, based on observations. Convergence hypothesis: no infinite sequence of unobservable events.

q0 f1 f2 f3 q1 q2 f u a c c c b b c

A run ρ = q0

u

− → q1

c

− → q2 has an observation sequence P(ρ) = c.

• c

is surely correct as P−1(c) = {q0

u

− → q1

c

− → q2}.

Accurate Approximate Diagnosability of Stochastic Systems March 17th 2016 – LATA - 2

Diagnosis Framework

LTS: Labelled transition system. Diagnoser: must tell whether a fault f occurred, based on observations. Convergence hypothesis: no infinite sequence of unobservable events.

q0 f1 f2 f3 q1 q2 f u a c c c b b c

A run ρ = q0

u

− → q1

c

− → q2 has an observation sequence P(ρ) = c.

• c

is surely correct as P−1(c) = {q0

u

− → q1

c

− → q2}. ✗ ac is surely faulty as P−1(ac) = {q0

f

− → f1

a

− → f2

c

− → f3}.

Accurate Approximate Diagnosability of Stochastic Systems March 17th 2016 – LATA - 2

Diagnosis Framework

LTS: Labelled transition system. Diagnoser: must tell whether a fault f occurred, based on observations. Convergence hypothesis: no infinite sequence of unobservable events.

q0 f1 f2 f3 q1 q2 f u a c c c b b c

A run ρ = q0

u

− → q1

c

− → q2 has an observation sequence P(ρ) = c.

• c

is surely correct as P−1(c) = {q0

u

− → q1

c

− → q2}. ✗ ac is surely faulty as P−1(ac) = {q0

f

− → f1

a

− → f2

c

− → f3}. ? b is ambiguous as P−1(b) = {q0

f

− → f1

b

− → f1, q0

u

− → q1

b

− → q1}.

Accurate Approximate Diagnosability of Stochastic Systems March 17th 2016 – LATA - 2

Diagnosis Problems

Diagnoser requirements:

◮ Soundness: if a fault is claimed, a fault occurred. ◮ Reactivity: every fault will be detected.

Accurate Approximate Diagnosability of Stochastic Systems March 17th 2016 – LATA - 3

Diagnosis Problems

Diagnoser requirements:

◮ Soundness: if a fault is claimed, a fault occurred. ◮ Reactivity: every fault will be detected.

A decision problem (diagnosability): does there exist a diagnoser? A synthesis problem: how to build a diagnoser?

Accurate Approximate Diagnosability of Stochastic Systems March 17th 2016 – LATA - 3

Diagnosis Problems

Diagnoser requirements:

◮ Soundness: if a fault is claimed, a fault occurred. ◮ Reactivity: every fault will be detected.

A decision problem (diagnosability): does there exist a diagnoser? A synthesis problem: how to build a diagnoser?

q0 f1 f2 f3 q1 q2 f u a c c c b b c

A sound but not reactive diagnoser : claiming a fault when a occurs.

Accurate Approximate Diagnosability of Stochastic Systems March 17th 2016 – LATA - 3

Diagnosis of Probabilistic Systems

[TT05]

q0 f1 f2 f3 q1 q2 f, 1

2

u, 1

2

a, 1

2

c, 1

2

c c b, 1

2

b, 1

2

c

[TT05] Thorsley and Teneketzis Diagnosability of stochastic discrete-event systems, IEEE TAC, 2005.

Accurate Approximate Diagnosability of Stochastic Systems March 17th 2016 – LATA - 4

Diagnosis of Probabilistic Systems

[TT05]

q0 f1 f2 f3 q1 q2 f, 1

2

u, 1

2

a, 1

2

c, 1

2

c c b, 1

2

b, 1

2

c

bn ambiguous but... [TT05] Thorsley and Teneketzis Diagnosability of stochastic discrete-event systems, IEEE TAC, 2005.

Accurate Approximate Diagnosability of Stochastic Systems March 17th 2016 – LATA - 4

Diagnosis of Probabilistic Systems

[TT05]

q0 f1 f2 f3 q1 q2 f, 1

2

u, 1

2

a, 1

2

c, 1

2

c c b, 1

2

b, 1

2

c

bn ambiguous but... lim

n→∞ P(fbn + ubn) = 0

[TT05] Thorsley and Teneketzis Diagnosability of stochastic discrete-event systems, IEEE TAC, 2005.

Accurate Approximate Diagnosability of Stochastic Systems March 17th 2016 – LATA - 4

Diagnosis of Probabilistic Systems

[TT05]

q0 f1 f2 f3 q1 q2 f, 1

2

u, 1

2

a, 1

2

c, 1

2

c c b, 1

2

b, 1

2

c

bn ambiguous but... lim

n→∞ P(fbn + ubn) = 0

How to adapt soundness and reactivity?

[TT05] Thorsley and Teneketzis Diagnosability of stochastic discrete-event systems, IEEE TAC, 2005.

Accurate Approximate Diagnosability of Stochastic Systems March 17th 2016 – LATA - 4

Exact Diagnosis

[BHL14]

An exact diagnoser fulfills

◮ Soundness: if a fault is claimed, a fault happened.

[BHL14] Bertrand, Haddad, Lefaucheux

Foundation of Diagnosis and Predictability in Probabilistic Systems, FSTTCS’14.

Accurate Approximate Diagnosability of Stochastic Systems March 17th 2016 – LATA - 5

Exact Diagnosis

[BHL14]

An exact diagnoser fulfills

◮ Soundness: if a fault is claimed, a fault happened. ◮ Reactivity: the diagnoser will provide information almost surely. q0 f1 f2 f3 q1 q2 f, 1

2

u, 1

2

a, 1

2

c, 1

2

c c b, 1

2

b, 1

2

c

Exactly diagnosable.

[BHL14] Bertrand, Haddad, Lefaucheux

Foundation of Diagnosis and Predictability in Probabilistic Systems, FSTTCS’14.

Accurate Approximate Diagnosability of Stochastic Systems March 17th 2016 – LATA - 5

Exact Diagnosis

[BHL14]

An exact diagnoser fulfills

◮ Soundness: if a fault is claimed, a fault happened. ◮ Reactivity: the diagnoser will provide information almost surely. q0 f1 f2 f3 q1 q2 f, 1

2

u, 1

2

a, 1

2

c, 1

2

c c b, 1

2

b, 1

2

c

Exactly diagnosable. Exact diagnosability is PSPACE-complete. Also studied : exact prediction and prediagnosis.

[BHL14] Bertrand, Haddad, Lefaucheux

Foundation of Diagnosis and Predictability in Probabilistic Systems, FSTTCS’14.

Accurate Approximate Diagnosability of Stochastic Systems March 17th 2016 – LATA - 5

Exact Diagnosis versus Approximate Diagnosis

q0 qf qc f, 1

2

u, 1

2

a, 1

4

b, 3

4

a, 3

4

b, 1

4

Accurate Approximate Diagnosability of Stochastic Systems March 17th 2016 – LATA - 6

Exact Diagnosis versus Approximate Diagnosis

q0 qf qc f, 1

2

u, 1

2

a, 1

4

b, 3

4

a, 3

4

b, 1

4

Not exactly diagnosable

Accurate Approximate Diagnosability of Stochastic Systems March 17th 2016 – LATA - 6

Exact Diagnosis versus Approximate Diagnosis

q0 qf qc f, 1

2

u, 1

2

a, 1

4

b, 3

4

a, 3

4

b, 1

4

Not exactly diagnosable However a high proportion of b implies a highly probable faulty run.

Accurate Approximate Diagnosability of Stochastic Systems March 17th 2016 – LATA - 6

Exact Diagnosis versus Approximate Diagnosis

q0 qf qc f, 1

2

u, 1

2

a, 1

4

b, 3

4

a, 3

4

b, 1

4

Not exactly diagnosable However a high proportion of b implies a highly probable faulty run. Relaxed Soundness: if a fault is claimed the probability of error is small.

Accurate Approximate Diagnosability of Stochastic Systems March 17th 2016 – LATA - 6

Outline

Specification of Approximate Diagnosis AA-diagnosis is Easy Other Approximate Diagnoses are Hard

Accurate Approximate Diagnosability of Stochastic Systems March 17th 2016 – LATA - 7

Outline

Specification of Approximate Diagnosis AA-diagnosis is Easy Other Approximate Diagnoses are Hard

Accurate Approximate Diagnosability of Stochastic Systems March 17th 2016 – LATA - 8

Proportion of Correct Runs

Given an observation sequence σ ∈ Σ∗

• ,

CorP(σ) = P({π−1(σ) ∩ correct}) P({π−1(σ)})

Accurate Approximate Diagnosability of Stochastic Systems March 17th 2016 – LATA - 9

Proportion of Correct Runs

Given an observation sequence σ ∈ Σ∗

• ,

CorP(σ) = P({π−1(σ) ∩ correct}) P({π−1(σ)}) q0 qf qc f, 1

2

u, 1

2

a, 1

4

b, 3

4

a, 3

4

b, 1

4

CorP(a) = 3/4,

Accurate Approximate Diagnosability of Stochastic Systems March 17th 2016 – LATA - 9

Proportion of Correct Runs

Given an observation sequence σ ∈ Σ∗

• ,

CorP(σ) = P({π−1(σ) ∩ correct}) P({π−1(σ)}) q0 qf qc f, 1

2

u, 1

2

a, 1

4

b, 3

4

a, 3

4

b, 1

4

CorP(a) = 3/4, CorP(ab) = 1/2,

Accurate Approximate Diagnosability of Stochastic Systems March 17th 2016 – LATA - 9

Proportion of Correct Runs

Given an observation sequence σ ∈ Σ∗

• ,

CorP(σ) = P({π−1(σ) ∩ correct}) P({π−1(σ)}) q0 qf qc f, 1

2

u, 1

2

a, 1

4

b, 3

4

a, 3

4

b, 1

4

CorP(a) = 3/4, CorP(ab) = 1/2, CorP(abb) = 1/4, CorP(abbb) = 1/10.

Accurate Approximate Diagnosability of Stochastic Systems March 17th 2016 – LATA - 9

Relaxing Soundness

Given ε ≥ 0, an ε-diagnoser fulfills

◮ Soundness: If a fault is claimed after an observation sequence σ,

then CorP(σ) ≤ ε.

Accurate Approximate Diagnosability of Stochastic Systems March 17th 2016 – LATA - 10

Relaxing Soundness

Given ε ≥ 0, an ε-diagnoser fulfills

◮ Soundness: If a fault is claimed after an observation sequence σ,

then CorP(σ) ≤ ε.

◮ Reactivity: Given a faulty run ρ, the measure of undetected runs

extending ρ converges to 0.

Accurate Approximate Diagnosability of Stochastic Systems March 17th 2016 – LATA - 10

Relaxing Soundness

Given ε ≥ 0, an ε-diagnoser fulfills

◮ Soundness: If a fault is claimed after an observation sequence σ,

then CorP(σ) ≤ ε.

◮ Reactivity: Given a faulty run ρ, the measure of undetected runs

extending ρ converges to 0. A uniform ε-diagnoser ensures for reactivity a uniform convergence over the faulty runs.

Accurate Approximate Diagnosability of Stochastic Systems March 17th 2016 – LATA - 10

Relaxing Soundness

Given ε ≥ 0, an ε-diagnoser fulfills

◮ Soundness: If a fault is claimed after an observation sequence σ,

then CorP(σ) ≤ ε.

◮ Reactivity: Given a faulty run ρ, the measure of undetected runs

extending ρ converges to 0. A uniform ε-diagnoser ensures for reactivity a uniform convergence over the faulty runs. 0-diagnosers correspond to exact diagnosers.

Accurate Approximate Diagnosability of Stochastic Systems March 17th 2016 – LATA - 10

Approximate Diagnosis Problems

Reactivity

ε-diagnosability uniform ε-diagnosability

Given ε > 0, does there exist Given ε > 0, does there exist an ε-diagnoser? a uniform ε-diagnoser? Accuracy

AA-diagnosability uniform AA-diagnosability

For all ε > 0, does there exist For all ε > 0, does there exist an ε-diagnoser? a uniform ε-diagnoser? AA-diagnosability allows to select ε depending on external requirements.

Accurate Approximate Diagnosability of Stochastic Systems March 17th 2016 – LATA - 11

Illustration

q0 qf qc f, 1

2

u, 1

2

a, 1

4

b, 3

4

a, 3

4

b, 1

4

AA-diagnosable but not uniformly AA-diagnosable

Accurate Approximate Diagnosability of Stochastic Systems March 17th 2016 – LATA - 12

Illustration

q0 qf qc f, 1

2

u, 1

2

a, 1

4

b, 3

4

a, 3

4

b, 1

4

AA-diagnosable but not uniformly AA-diagnosable q0 qf qc f, 1

3

u, 1

3

a, 1

3

b a Uniformly AA-diagnosable but not exactly diagnosable

Accurate Approximate Diagnosability of Stochastic Systems March 17th 2016 – LATA - 12

Establishing relations between the Specifications

uniformly exact diagnosis exact diagnosis uniformly AA-diagnosable uniformly ε-diagnosable AA-diagnosable ε-diagnosable for all ε for all ε

Accurate Approximate Diagnosability of Stochastic Systems March 17th 2016 – LATA - 13

Complexity of the Problems

Simple Uniform ε-diagnosability undecidable undecidable AA-diagnosability PTIME undecidable

Accurate Approximate Diagnosability of Stochastic Systems March 17th 2016 – LATA - 14

Outline

Specification of Approximate Diagnosis AA-diagnosis is Easy Other Approximate Diagnoses are Hard

Accurate Approximate Diagnosability of Stochastic Systems March 17th 2016 – LATA - 15

A Simple Case

Initial fault pLTS. Initially, an unobservable split towards two subpLTS:

◮ a correct event u leads to a correct subpLTS; ◮ a faulty event f leads to an arbitrary subpLTS.

Accurate Approximate Diagnosability of Stochastic Systems March 17th 2016 – LATA - 16

A Simple Case

Initial fault pLTS. Initially, an unobservable split towards two subpLTS:

◮ a correct event u leads to a correct subpLTS; ◮ a faulty event f leads to an arbitrary subpLTS.

q0 qf qc q′

f

f, 1

2

u, 1

2

f, 1

2

a, 1

2

a, 1

4

b, 3

4

a, 3

4

b, 1

4 ◮ an initial state, q0; ◮ an arbitrary pLTS with states {qf , q′ f }; ◮ a correct pLTS with state qc.

Accurate Approximate Diagnosability of Stochastic Systems March 17th 2016 – LATA - 16

Solving AA-diagnosability for Initial-Fault pLTS

• Transform the correct and arbitrary subpLTS in labelled Markov chains

by merging the unobservable transitions. qf qc q′

f 1 2 1 2

a, 3

8

b, 1

8

a, 1

2

a, 1

4

b, 3

4

a, 3

4

b, 1

4

Mc Mf

Accurate Approximate Diagnosability of Stochastic Systems March 17th 2016 – LATA - 17

Solving AA-diagnosability for Initial-Fault pLTS

• Transform the correct and arbitrary subpLTS in labelled Markov chains

by merging the unobservable transitions. qf qc q′

f 1 2 1 2

a, 3

8

b, 1

8

a, 1

2

a, 1

4

b, 3

4

a, 3

4

b, 1

4

Mc Mf

• PM(E) = measure of infinite runs of M with observation in E.

Distance 1 problem: ∃E (measurable) ⊆ Σω

• , PMc(E) − PMf (E) = 1?
• Illustration: E = {σ | lim supn→∞

|σ↓n|b |σ↓n|a > 1}

[CK14] Chen and Kiefer On the Total Variation Distance of Labelled Markov Chains, CSL-LICS’14.

Accurate Approximate Diagnosability of Stochastic Systems March 17th 2016 – LATA - 17

Solving AA-diagnosability for Initial-Fault pLTS

• Transform the correct and arbitrary subpLTS in labelled Markov chains

by merging the unobservable transitions. qf qc q′

f 1 2 1 2

a, 3

8

b, 1

8

a, 1

2

a, 1

4

b, 3

4

a, 3

4

b, 1

4

Mc Mf

• PM(E) = measure of infinite runs of M with observation in E.

Distance 1 problem: ∃E (measurable) ⊆ Σω

• , PMc(E) − PMf (E) = 1?
• Illustration: E = {σ | lim supn→∞

|σ↓n|b |σ↓n|a > 1}

The distance 1 problem is decidable in PTIME.

[CK14] Chen and Kiefer On the Total Variation Distance of Labelled Markov Chains, CSL-LICS’14.

Accurate Approximate Diagnosability of Stochastic Systems March 17th 2016 – LATA - 17

Solving AA-diagnosability

• Identifying relevant pairs of states by reachability analysis in the

synchronised self-product.

q0 f q ρf ρc

P(ρc) = P(ρf )

Accurate Approximate Diagnosability of Stochastic Systems March 17th 2016 – LATA - 18

Solving AA-diagnosability

• Identifying relevant pairs of states by reachability analysis in the

synchronised self-product.

q0 f q ρf ρc

P(ρc) = P(ρf )

• Checking distance 1 for all relevant pairs.

f q Mf Mc

Accurate Approximate Diagnosability of Stochastic Systems March 17th 2016 – LATA - 18

Solving AA-diagnosability

• Identifying relevant pairs of states by reachability analysis in the

synchronised self-product.

q0 f q ρf ρc

P(ρc) = P(ρf )

• Checking distance 1 for all relevant pairs.

f q Mf Mc

AA-diagnosability is decidable in PTIME.

Accurate Approximate Diagnosability of Stochastic Systems March 17th 2016 – LATA - 18

Diagnoser Synthesis

An ε-diagnoser may need infinite memory.

Accurate Approximate Diagnosability of Stochastic Systems March 17th 2016 – LATA - 19

Diagnoser Synthesis

An ε-diagnoser may need infinite memory. q0 qf qc f, 1

2

u, 1

2

a, 1

4

b, 3

4

a, 3

4

b, 1

4

Accurate Approximate Diagnosability of Stochastic Systems March 17th 2016 – LATA - 19

Diagnoser Synthesis

An ε-diagnoser may need infinite memory. q0 qf qc f, 1

2

u, 1

2

a, 1

4

b, 3

4

a, 3

4

b, 1

4

For all k < n, ak and an lead to different states of the diagnoser.

Accurate Approximate Diagnosability of Stochastic Systems March 17th 2016 – LATA - 19

Diagnoser Synthesis

An ε-diagnoser may need infinite memory. q0 qf qc f, 1

2

u, 1

2

a, 1

4

b, 3

4

a, 3

4

b, 1

4

For all k < n, ak and an lead to different states of the diagnoser. Otherwise for all i ∈ N, ak+i(n−k) lead to the same state.

Accurate Approximate Diagnosability of Stochastic Systems March 17th 2016 – LATA - 19

Diagnoser Synthesis

An ε-diagnoser may need infinite memory. q0 qf qc f, 1

2

u, 1

2

a, 1

4

b, 3

4

a, 3

4

b, 1

4

For all k < n, ak and an lead to different states of the diagnoser. Otherwise for all i ∈ N, ak+i(n−k) lead to the same state. By reactivity for some σ, the diagnoser must claim a fault after akσ and thus after all ak+i(n−k)σ.

Accurate Approximate Diagnosability of Stochastic Systems March 17th 2016 – LATA - 19

Diagnoser Synthesis

An ε-diagnoser may need infinite memory. q0 qf qc f, 1

2

u, 1

2

a, 1

4

b, 3

4

a, 3

4

b, 1

4

For all k < n, ak and an lead to different states of the diagnoser. Otherwise for all i ∈ N, ak+i(n−k) lead to the same state. By reactivity for some σ, the diagnoser must claim a fault after akσ and thus after all ak+i(n−k)σ. But limi→∞ CorP(ak+i(n−k)σ) = 1.

Accurate Approximate Diagnosability of Stochastic Systems March 17th 2016 – LATA - 19

Diagnoser Synthesis

An ε-diagnoser may need infinite memory. q0 qf qc f, 1

2

u, 1

2

a, 1

4

b, 3

4

a, 3

4

b, 1

4

For all k < n, ak and an lead to different states of the diagnoser. Otherwise for all i ∈ N, ak+i(n−k) lead to the same state. By reactivity for some σ, the diagnoser must claim a fault after akσ and thus after all ak+i(n−k)σ. But limi→∞ CorP(ak+i(n−k)σ) = 1. For exact diagnosis, one can build a diagnoser exponential in the size of the pLTS [BHL14].

Accurate Approximate Diagnosability of Stochastic Systems March 17th 2016 – LATA - 19

Outline

Specification of Approximate Diagnosis AA-diagnosis is Easy Other Approximate Diagnoses are Hard

Accurate Approximate Diagnosability of Stochastic Systems March 17th 2016 – LATA - 20

The Emptiness Problem for Probabilistic Automata (PA)

a, 1

2

a, 1

2

a a, 1

2

a c b b a, 1

2

a

P(b) = 0, P(baa) = 1

4, P(baaa) = 7 8

Accurate Approximate Diagnosability of Stochastic Systems March 17th 2016 – LATA - 21

The Emptiness Problem for Probabilistic Automata (PA)

a, 1

2

a, 1

2

a a, 1

2

a c b b a, 1

2

a

P(b) = 0, P(baa) = 1

4, P(baaa) = 7 8

Emptiness problem: Given a PA A, ∃w ∈ Σ∗, PA(w) > 1

2?

Accurate Approximate Diagnosability of Stochastic Systems March 17th 2016 – LATA - 21

The Emptiness Problem for Probabilistic Automata (PA)

a, 1

2

a, 1

2

a a, 1

2

a c b b a, 1

2

a

P(b) = 0, P(baa) = 1

4, P(baaa) = 7 8

Emptiness problem: Given a PA A, ∃w ∈ Σ∗, PA(w) > 1

2?

The emptiness problem for PA is undecidable even when for all w,

1 4 ≤ PA(w) ≤ 3 4.

[P71] Paz, Introduction to Probabilistic Automata, Academic Press 1971.

Accurate Approximate Diagnosability of Stochastic Systems March 17th 2016 – LATA - 21

From PA to Uniform AA-diagnosability

q1 q2 I[q1] I[q2] a, Pa[q1, q2] a, Pa[q2, q1]

Accurate Approximate Diagnosability of Stochastic Systems March 17th 2016 – LATA - 22

From PA to Uniform AA-diagnosability

q1 q2 I[q1] I[q2] a, Pa[q1, q2] a, Pa[q2, q1] q0 qu

1

qu

2

bu u, I[q1]

2

u, I[q2]

2

a, Pa[q1,q2]

1+|Σ|

a, Pa[q2,q1]

1+|Σ|

♯,

I[q1] 1+|Σ|

♯,

I[q2] 1+|Σ|

♭, 1

2

♯, 1

2

♭,

1 1+|Σ|

qf

1

qf

2

bf f, I[q2]

2

f, I[q1]

2

a, Pa[q1,q2]

1+|Σ|

a, Pa[q2,q1]

1+|Σ|

♯,

I[q2] 1+|Σ|

♯,

I[q1] 1+|Σ|

♭, 1 ♭,

1 1+|Σ|

Accurate Approximate Diagnosability of Stochastic Systems March 17th 2016 – LATA - 22

From PA to Uniform AA-diagnosability

q1 q2 I[q1] I[q2] a, Pa[q1, q2] a, Pa[q2, q1] q0 qu

1

qu

2

bu u, I[q1]

2

u, I[q2]

2

a, Pa[q1,q2]

1+|Σ|

a, Pa[q2,q1]

1+|Σ|

♯,

I[q1] 1+|Σ|

♯,

I[q2] 1+|Σ|

♭, 1

2

♯, 1

2

♭,

1 1+|Σ|

qf

1

qf

2

bf f, I[q2]

2

f, I[q1]

2

a, Pa[q1,q2]

1+|Σ|

a, Pa[q2,q1]

1+|Σ|

♯,

I[q2] 1+|Σ|

♯,

I[q1] 1+|Σ|

♭, 1 ♭,

1 1+|Σ|

If ∃w ∈ Σ∗

• , PA(w) > 1/2 then limn−

→∞ CorP((w♯)n♭) = 1.

Accurate Approximate Diagnosability of Stochastic Systems March 17th 2016 – LATA - 22

From PA to Uniform AA-diagnosability

q1 q2 I[q1] I[q2] a, Pa[q1, q2] a, Pa[q2, q1] q0 qu

1

qu

2

bu u, I[q1]

2

u, I[q2]

2

a, Pa[q1,q2]

1+|Σ|

a, Pa[q2,q1]

1+|Σ|

♯,

I[q1] 1+|Σ|

♯,

I[q2] 1+|Σ|

♭, 1

2

♯, 1

2

♭,

1 1+|Σ|

qf

1

qf

2

bf f, I[q2]

2

f, I[q1]

2

a, Pa[q1,q2]

1+|Σ|

a, Pa[q2,q1]

1+|Σ|

♯,

I[q2] 1+|Σ|

♯,

I[q1] 1+|Σ|

♭, 1 ♭,

1 1+|Σ|

If ∃w ∈ Σ∗

• , PA(w) > 1/2 then limn−

→∞ CorP((w♯)n♭) = 1. If ∀w ∈ Σ∗

• , PA(w) ≤ 1/2 then ∀n CorP((w♯)n♭) ≤ 3

4.

Accurate Approximate Diagnosability of Stochastic Systems March 17th 2016 – LATA - 22

Conclusion

Contributions

◮ Investigation of semantical issues ◮ Complexity of the notions of approximate diagnosis

◮ A PTIME algorithm for AA-diagnosability ◮ Undecidability of other approximate diagnosability Accurate Approximate Diagnosability of Stochastic Systems March 17th 2016 – LATA - 23

Conclusion

Contributions

◮ Investigation of semantical issues ◮ Complexity of the notions of approximate diagnosis

◮ A PTIME algorithm for AA-diagnosability ◮ Undecidability of other approximate diagnosability

Future work

◮ Approximate prediction and prediagnosis ◮ Diagnosis of infinite state stochastic systems

Accurate Approximate Diagnosability of Stochastic Systems March 17th 2016 – LATA - 23