Accurate Approximate Diagnosability of Stochastic Systems Nathalie - PowerPoint PPT Presentation

Accurate Approximate Diagnosability of Stochastic Systems Nathalie Bertrand 1 , Serge Haddad 2 , Engel Lefaucheux 1 , 2 1 Inria, France 2 LSV, ENS Cachan & CNRS & Inria, France LATA, March 17th 2016 March 17th 2016 – LATA

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. b c a c f 1 f 2 f 3 f q 0 u q 1 q 2 c c b 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. b c a c f 1 f 2 f 3 f q 0 u q 1 q 2 c c b u c A run ρ = q 0 − → q 1 − → q 2 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. b c a c f 1 f 2 f 3 f q 0 u q 1 q 2 c c b u c A run ρ = q 0 − → q 1 − → q 2 has an observation sequence P ( ρ ) = c . u c is surely correct as P − 1 ( c ) = { q 0 � c → q 1 − − → q 2 } . 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. b c a c f 1 f 2 f 3 f q 0 u q 1 q 2 c c b u c A run ρ = q 0 − → q 1 − → q 2 has an observation sequence P ( ρ ) = c . u c is surely correct as P − 1 ( c ) = { q 0 � c − → q 1 − → q 2 } . a c f is surely faulty as P − 1 ( ac ) = { q 0 ac − → f 1 − → f 2 − → f 3 } . ✗ 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. b c a c f 1 f 2 f 3 f q 0 u q 1 q 2 c c b u c A run ρ = q 0 − → q 1 − → q 2 has an observation sequence P ( ρ ) = c . u c is surely correct as P − 1 ( c ) = { q 0 � c − → q 1 − → q 2 } . a c f is surely faulty as P − 1 ( ac ) = { q 0 ac − → f 1 − → f 2 → f 3 } . − ✗ f b u b is ambiguous as P − 1 ( b ) = { q 0 ? b − → f 1 − → f 1 , q 0 − → q 1 − → q 1 } . 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? b c a c f 1 f 2 f 3 f q 0 u q 1 q 2 c c b 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] b , 1 c 2 a , 1 c 2 f 1 f 2 f 3 f , 1 2 q 0 u , 1 q 1 q 2 2 c , 1 2 c b , 1 2 [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] b , 1 c 2 a , 1 c 2 f 1 f 2 f 3 f , 1 b n ambiguous but... 2 q 0 u , 1 q 1 q 2 2 c , 1 2 c b , 1 2 [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] b , 1 c 2 a , 1 c 2 f 1 f 2 f 3 f , 1 b n ambiguous but... 2 q 0 n →∞ P ( f b n + ub n ) = 0 lim u , 1 q 1 q 2 2 c , 1 2 c b , 1 2 [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] b , 1 c 2 a , 1 c 2 f 1 f 2 f 3 f , 1 b n ambiguous but... 2 q 0 n →∞ P ( f b n + ub n ) = 0 lim u , 1 q 1 q 2 2 c , 1 2 c b , 1 2 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. b , 1 c 2 a , 1 c 2 f 1 f 2 f 3 f , 1 2 q 0 u , 1 q 1 q 2 2 c , 1 2 c b , 1 2 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. b , 1 c 2 a , 1 c 2 f 1 f 2 f 3 f , 1 2 q 0 u , 1 q 1 q 2 2 c , 1 2 c b , 1 2 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 a , 3 a , 1 4 4 u , 1 f , 1 2 2 q c q 0 q f b , 1 b , 3 4 4 Accurate Approximate Diagnosability of Stochastic Systems March 17th 2016 – LATA - 6

Exact Diagnosis versus Approximate Diagnosis a , 3 a , 1 4 4 u , 1 f , 1 2 2 q c q 0 q f b , 1 b , 3 4 4 Not exactly diagnosable Accurate Approximate Diagnosability of Stochastic Systems March 17th 2016 – LATA - 6

Exact Diagnosis versus Approximate Diagnosis a , 3 a , 1 4 4 u , 1 f , 1 2 2 q c q 0 q f b , 1 b , 3 4 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 a , 3 a , 1 4 4 u , 1 f , 1 2 2 q c q 0 q f b , 1 b , 3 4 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 ∗ o , 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 ∗ o , CorP ( σ ) = P ( { π − 1 ( σ ) ∩ correct } ) P ( { π − 1 ( σ ) } ) a , 3 a , 1 4 4 u , 1 f , 1 2 2 q c q 0 q f b , 1 b , 3 4 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 ∗ o , CorP ( σ ) = P ( { π − 1 ( σ ) ∩ correct } ) P ( { π − 1 ( σ ) } ) a , 3 a , 1 4 4 u , 1 f , 1 2 2 q c q 0 q f b , 1 b , 3 4 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 ∗ o , CorP ( σ ) = P ( { π − 1 ( σ ) ∩ correct } ) P ( { π − 1 ( σ ) } ) a , 3 a , 1 4 4 u , 1 f , 1 2 2 q c q 0 q f b , 1 b , 3 4 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