Diagnosis of Large Software Systems Based on Colored Petri Nets

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Introduction Abstract model: CPN Diagnosis Application Conclusions & perspectives

Diagnosis of Large Software Systems

Based on Colored Petri Nets

Supervised by Tarek Melliti2and Philippe Dague1

Yingmin Li1

1 LRI IASI, Univ. Paris-Sud, CNRS/INRIA Saclay, France 2 IBISC, Univ. d’Evry Val d’Essonne, CNRS, France

December 9, 2010

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Introduction Abstract model: CPN Diagnosis Application Conclusions & perspectives

Context: basic Web services

Web service:

Technology allowing applications to dialogue remotely via Internet independently of the platforms and the languages they rest on. Cheaper and simpler for connection and integration.

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Introduction Abstract model: CPN Diagnosis Application Conclusions & perspectives

Context: composite Web services

A travel agency example

An orchestrated Web service consists of a travel agency, an airline search engine, and a bank billing system

On 09/07/2010, a client, reserved a round trip Paris-Rome, 07/11/2010-07/12/2010

0 9 / 0 7 / 2 0 1 0 - 0 7 / 1 2 / 2 0 1 0 P a r i s - R o m e T h e c h e a p e s t f l i g h t s h o u l d b e a r o u n d 2 0 0 e u r o s . I h a v e 5 0 0 e u r o s i n m y a c c o u n t 3 / 101

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Introduction Abstract model: CPN Diagnosis Application Conclusions & perspectives

Context: diagnosis problems of Web services

A travel agency example

Reservation failed because of no enough credit?

Client was confused and asked why the tickets were so expensive

0 9 / 0 7 / 2 0 1 0 - 0 7 / 1 2 / 2 0 1 0 P a r i s - R o m e N o e n o u g h c r e d i t T h e c h e a p e s t f l i g h t s h o u l d b e a r o u n d 2 0 0 e u r o s . I h a v e 5 0 0 e u r o s i n m y a c c o u n t A l a r m F a u l t 4 / 101

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Introduction Abstract model: CPN Diagnosis Application Conclusions & perspectives

An abstract view

Large software systems

Consist of several components located on different sites

Dysfunctions are generated and transformed among the components through data (signals, interface data, etc)

A communicating components system

O b s e r v a t i o n C o m p l e t e l y o r d e r e d : O 1 , O 2 , . . . , O n N o n - o r d e r e d : { O 1 , O 2 , . . . , O n } P a r t i a l l y o r d e r e d : O 1 , O 3 O 2 , O 4 O n C o m p o n e n t 2 D i a g n o s i s T o d e t e c t a n d e x p l a i n t h e p o s s i b l e f a u l t ( s ) C o m p o n e n t 1 C o m p o n e n t 3 F a u l t A l a r m 5 / 101

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Introduction Abstract model: CPN Diagnosis Application Conclusions & perspectives

A possible solution: model based diagnosis

A communicating components system diagnosis

P a r t i a l l y o r d e r e d o b s C o m p o n e n t 1 C o m p o n e n t 2 C o m p o n e n t 3 O 1 , O 3 P a r t i a l l y o r d e r e d o b s O 2 , O 4 P a r t i a l l y o r d e r e d o b s O 5 , O 6 D i a g n o s e r A b s t r a c t s y s t e m m o d e l D i a g n o s i s T o d e t e c t a n d e x p l a i n t h e p o s s i b l e d a t a , c o n t r o l o r e v e n t f a u l t ( s ) m i g h t c o m e f r o m o t h e r c o m p o n e n t s F a u l t A l a r m 6 / 101

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Introduction Abstract model: CPN Diagnosis Application Conclusions & perspectives

A possible solution: model based diagnosis

To model the faulty events asunobservableevents

P a r t i a l l y o r d e r e d o b s C o m p o n e n t 1 C o m p o n e n t 2 C o m p o n e n t 3 O 1 , O 3 P a r t i a l l y o r d e r e d o b s O 2 , O 4 P a r t i a l l y o r d e r e d o b s O 5 , O 6 D i a g n o s e r A b s t r a c t s y s t e m m o d e l D i a g n o s i s T o d e t e c t a n d e x p l a i n t h e p o s s i b l e d a t a , c o n t r o l o r e v e n t f a u l t ( s ) m i g h t c o m e f r o m o t h e r c o m p o n e n t s F a u l t A l a r m 7 / 101

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Introduction Abstract model: CPN Diagnosis Application Conclusions & perspectives

Existing works

DES models

Discrete event system (DES) models: Petri net (Petri [1973]), automata (Arto and N. [1969]), process algebra (van Glabbeek [1987]), etc.

Most of them focus on the state evolution driven by the discrete events Faults are modeled as unobservable events (Sampath et al. [1995]) while faulty data is usually not modeled

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Introduction Abstract model: CPN Diagnosis Application Conclusions & perspectives

Existing works: diagnoser (Sampath et al. [1995, 1996])

Transform the discrete event model of the system to be diagnosed into a finite state automaton which has only observable events; the historical faulty events are recorded in states. Example f 1 t 1 t 2 f 2 t 3 f 1 N f 2 t 1 t 2 t 3 N observations: t2t3 Diagnosis={f2} system diagnoser 9 / 101

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Introduction Abstract model: CPN Diagnosis Application Conclusions & perspectives

Existing works: PN unfolding (Benveniste et al. [2003])

Fully describes the concurrent behaviors in a single branching structure, represents all the possible computation steps and their mutual dependencies, as well as all reachable states.

Example

system observations unfolding diagnoser

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Introduction Abstract model: CPN Diagnosis Application Conclusions & perspectives

Existing works: PN backward reasoning (Anglano and Portinale [1994];

Cardoso et al. [1995]; Jiroveanu [2006]; Srinivasan and Jafari [1994])

Starts from the final states which represents a symptom and calculates backwardly according to the backward searching rules to detect all the traces that cover it.

Example

t1

system observation diagnosis

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Introduction Abstract model: CPN Diagnosis Application Conclusions & perspectives

Assumptions

Targets

Locallycorrect and tested software system with interacting components in a stable network environment, butfaulty activities,faulty data and controls(from user, database, interface, etc) that transmitted between the components

Symptoms

Exception(s) thrown on one or more components during the execution

Method: model based diagnosis

P a r t i a l l y o r d e r e d o b s F a u l t O 1 , O 3 P a r t i a l l y o r d e r e d o b s O 2 , O 4 P a r t i a l l y o r d e r e d o b s O 5 , O 6 A b s t r a c t s y s t e m m o d e l C o m p o n e n t 2 C o m p o n e n t 1 C o m p o n e n t 3

?

F a u l t A l a r m 12 / 101

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Introduction Abstract model: CPN Diagnosis Application Conclusions & perspectives

Challenges

Existing works

focus onstateevolution

diagnose by trajectories reconstruction

We need an abstract model

to represent the correct and fault of

data

and

controls

in a unique

way

to represent the correct and fault

behaviors

of the activities

to represent the

concurrency

and

partially ordered observations

allows to handle the

loops

in an elegant way to avoid unfolding

the trajectories

to diagnose the

orchestrated

software systems

Our choice

Colored Petri net

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Introduction Abstract model: CPN Diagnosis Application Conclusions & perspectives

Outline

1 _Introduction

Context: Web service diagnosis problem Model-based diagnosis for Web services Assumptions

2 Abstract model: CPN

CPN definition

Partially ordered observation

3 Diagnosis

CPN as a fault model Diagnosis algorithm

4 _Application

Architecture of BPEL Monitoring and Diagnosis Decentralized diagnosis architecture

Example: travel agency diagnosis problem

5 _{Conclusions & perspectives}

Contributions Perspectives

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Introduction Abstract model: CPN Diagnosis Application Conclusions & perspectives

CPN has same structure as PN

Example · · · p1 t · p2 2 3

Definition (Petri net)

N=hP, T , Pre, Posti

P: a set of labeled places T : a set of labeled transitions

Pre:P×T →N, a backward matrix of consumed

token number

Post:P×T → N, a forward matrix of produced

token number

Example (Incidence matrix C=Post-Pre)

C t p1 -2 p2 3 = Post t p1 p2 3 - Pre t p1 2 p2

Example (State equation)

M′ ₌ _M

0 + C × T

p1 1 3 -2

p2 4 1 3 1

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Introduction Abstract model: CPN Diagnosis Application Conclusions & perspectives

CPN has same structure as PN

Example · p1 t · p2 2 3

token number

M′ ₌ _M

0 + C × T

p1 1 3 -2

p2 4 1 3 1

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Introduction Abstract model: CPN Diagnosis Application Conclusions & perspectives

CPN has same structure as PN

Example · p1 t · ··· p2 2 3

token number

M′ ₌ _M

0 + C × T

p1 1 3 -2

p2 4 1 3 1

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Introduction Abstract model: CPN Diagnosis Application Conclusions & perspectives

CPN definition: Structure & dynamic (Li et al. [2009a])

Example p1: Π₁ t: Γ₁ p2: Π₂ F: Γ₁→Ψ(Π₁) F′: Γ₁→Ψ(Π₂) PN to CPN

Placesare typed (p1: Π1) Transitionare typed (t: Γ1) Markingsare multi-sets of place types

(m(p1) =

X

i∈|Π₁|

n8iei,ni>0,ei∈Π1)

Weightsof an edges are multi-set

expressions (Pre(p1,t) : Γ1→Ψ(Π1))

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Introduction Abstract model: CPN Diagnosis Application Conclusions & perspectives

CPN definition: Structure & dynamic (Li et al. [2009a])

(m(p1) =

X

i∈|Π₁|

expressions (Pre(p1,t) : Γ1→Ψ(Π1)) Example (Bindingβ) 1‘a+ 2‘b p1: Π1={a,b} t: Γ₁={m1,_m2} 1‘c _p2: Π₂={a,b,c} Pre(t,_p1) ={(_m1:1‘b+1‘x) ,(_m2:1‘a)} β 1m1:{x=a} or β 2m1:{x=b} Post(t,_p2) ={(_m1:2‘x), (_m2:2‘c)}

Mode firing rule: M[tmiβM′

M′=M+C(.,t)(m1)β_with_β_:_x₌_a

Marking: M′isreachablefrom M underβand m

can be extended to sequence firing

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Introduction Abstract model: CPN Diagnosis Application Conclusions & perspectives

CPN definition: Structure & dynamic (Li et al. [2009a])

(m(p1) =

X

i∈|Π₁|

n8_iei,ni>0,ei∈Π1)

expressions (Pre(p1,t) : Γ1→Ψ(Π1)) Example (Bindingβ) 1‘b _p1: Π₁={a,b} t: Γ₁={m1,_m2} 1‘c _p2: Π₂={a,b,c} Pre(t,_p1) ={(_m1:1‘b+1‘x) ,(_m2:1‘a)} β 1m1:{x=a} or β 2m1:{x=b} Post(t,_p2) ={(_m1:2‘x), (_m2:2‘c)}

M′=M+C(.,t)(m1)βwithβ:x=a

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Introduction Abstract model: CPN Diagnosis Application Conclusions & perspectives

CPN definition: Structure & dynamic (Li et al. [2009a])

Example p1: Π1 t: Γ₁ p2: Π₂ F: Γ₁→Ψ(Π₁) F′: Γ₁→Ψ(Π₂) PN to CPN

(m(p1) =

X

i∈|Π₁|

expressions (Pre(p1,t) : Γ1→Ψ(Π1)) Example (Bindingβ) 1‘b _p1: Π₁={a,b} t: Γ₁={_m1,_m2} 1‘c+2‘a _p2: Π₂={a,b,c} Pre(t,_p1) ={(_m1:1‘b+1‘x) ,(_m2:1‘a)} β 1m1:{x=a} or β 2m1:{x=b} Post(t,_p2) ={(_m1:2‘x), (_m2:2‘c)}

M′=M+C(.,t)(m1)β_with_β_:_x₌_a

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Introduction Abstract model: CPN Diagnosis Application Conclusions & perspectives

CPN definition: Dynamic

Definition (Characteristic Vector)

Sequence of modes:δ∈Mod∗

Characteristic vector ofδ:

−

⇀

δ :Mod→N

Transition characteristic vector

− ⇀ δT :T →N Example δ=t1.m1t2.m3t1.m2t1.m1t1.m2 − ⇀ δ(m1) =2 − ⇀ δT(t1) =4 State equation

Given S=hN,Mia CPN-S andmodessequence,δ∈Mod∗with M[δithen the reached marking M′after the firing ofδis :

M

′

=

M

+

C

×

−

⇀

δ

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Introduction Abstract model: CPN Diagnosis Application Conclusions & perspectives

CPN definition: example (1/2)

Dining philosophers (of size four) No blocking version (taking both forks concurrently)

Forks are identified by the serial number 1-4

One of them (num 4) is

well-organized:

• to take the forks, and before eat,

he/she checks the id of the forks

Three of them (num 1, 2 and 3) are

Unorganized:

• don’t verify the forks’id and might

exchange forks in hands before restore them.

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Introduction Abstract model: CPN Diagnosis Application Conclusions & perspectives

CPN definition: example (2/2)

Notations

pi: the forks on the table

tiand ri:

philosopher i takes and released the fork

pii: the fork in the

right hand of philosopher i

pji: the fork in the

left hand of philosopher i 1 p1 p2 2 3 p3 4 p4 p11 p21 p22 p32 p33 p43 p44 p14 t1 r1 t2 r2 t3 r3 t4 r4 χp1 χp2 χp1 χp2 χp11 χp21 N ormal:χp11 Exchange:χp21 N ormal:χp21 Exchange:χp11 χp2 χp3 χp2 χp3 χp22 χp32 N ormal:χp22 Exchange:χp32 N ormal:χp32 Exchange:χp22 χp3 χp4 χp3 χp4 χp33 χp43 N ormal:χp33 Exchange:χp43 N ormal:χp43 Exchange:χp33 χp4 χp1 χp4 χp1 χp44 χp14 N ormal:χp44 N ormal:χp14 24 / 101

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Introduction Abstract model: CPN Diagnosis Application Conclusions & perspectives

CPN definition: example (2/2)

Notations

tiand ri:

left hand of philosopher i p1 p2 p3 p4 p11 p21 p22 p32 p33 p43 p44 p14 t1 r1 t2 r2 t3 r3 t4 r4 χp1 χp2 χp1 χp2 χp11 χp21 N ormal:χp11 Exchange:χp21 N ormal:χp21 Exchange:χp11 χp2 χp3 χp2 χp3 χp22 χp32 N ormal:χp22 Exchange:χp32 N ormal:χp32 Exchange:χp22 χp3 χp4 χp3 χp4 χp33 χp43 N ormal:χp33 Exchange:χp43 N ormal:χp43 Exchange:χp33 χp4 χp1 χp4 χp1 χp44 χp14 N ormal:χp44 N ormal:χp14 25 / 101

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Introduction Abstract model: CPN Diagnosis Application Conclusions & perspectives

CPN definition: example (2/2)

Notations

tiand ri:

left hand of philosopher i p1 p2 p3 p4 p11 p21 2 p22 3 p32 p33 p43 4 p44 1 p14 t1 r1 t2 r2 t3 r3 t4 r4 χp1 χp2 χp1 χp2 χp11 χp21 N ormal:χp11 Exchange:χp21 N ormal:χp21 Exchange:χp11 χp2 χp3 χp2 χp3 χp22 χp32 N ormal:χp22 Exchange:χp32 N ormal:χp32 Exchange:χp22 χp3 χp4 χp3 χp4 χp33 χp43 N ormal:χp33 Exchange:χp43 N ormal:χp43 Exchange:χp33 χp4 χp1 χp4 χp1 χp44 χp14 N ormal:χp44 N ormal:χp14 26 / 101

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Introduction Abstract model: CPN Diagnosis Application Conclusions & perspectives

CPN definition: example (2/2)

Notations

tiand ri:

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Introduction Abstract model: CPN Diagnosis Application Conclusions & perspectives

CPN definition: example (2/2)

Notations

tiand ri:

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Introduction Abstract model: CPN Diagnosis Application Conclusions & perspectives

CPN definition: example (2/2)

Notations

tiand ri:

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Introduction Abstract model: CPN Diagnosis Application Conclusions & perspectives

CPN definition: example (2/2)

Notations

tiand ri:

left hand of philosopher i p1 p2 p3 p4 1 p11 3 p21 p22 p32 2 p33 4 p43 p44 p14 t1 r1 t2 r2 t3 r3 t4 r4 χp1 χp2 χp1 χp2 χp11 χp21 N ormal:χp11 Exchange:χp21 N ormal:χp21 Exchange:χp11 χp2 χp3 χp2 χp3 χp22 χp32 N ormal:χp22 Exchange:χp32 N ormal:χp32 Exchange:χp22 χp3 χp4 χp3 χp4 χp33 χp43 N ormal:χp33 Exchange:χp43 N ormal:χp43 Exchange:χp33 χp4 χp1 χp4 χp1 χp44 χp14 N ormal:χp44 N ormal:χp14 30 / 101

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Introduction Abstract model: CPN Diagnosis Application Conclusions & perspectives

CPN definition: example (2/2)

Notations

tiand ri:

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Introduction Abstract model: CPN Diagnosis Application Conclusions & perspectives

CPN definition: example (2/2)

Notations

tiand ri:

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Introduction Abstract model: CPN Diagnosis Application Conclusions & perspectives

CPN definition: example (2/2)

Notations

tiand ri:

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Introduction Abstract model: CPN Diagnosis Application Conclusions & perspectives

CPN definition: example (2/2)

Notations

tiand ri:

left hand of philosopher i p1 p2 p3 p4 3 p11 1 p21 p22 p32 4 p33 2 p43 p44 p14 t1 r1 t2 r2 t3 r3 t4 r4 χp1 χp2 χp1 χp2 χp11 χp21 N ormal:χp11 Exchange:χp21 N ormal:χp21 Exchange:χp11 χp2 χp3 χp2 χp3 χp22 χp32 N ormal:χp22 Exchange:χp32 N ormal:χp32 Exchange:χp22 χp3 χp4 χp3 χp4 χp33 χp43 N ormal:χp33 Exchange:χp43 N ormal:χp43 Exchange:χp33 χp4 χp1 χp4 χp1 χp44 χp14 N ormal:χp44 N ormal:χp14 34 / 101

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Introduction Abstract model: CPN Diagnosis Application Conclusions & perspectives

CPN definition: example (2/2)

Notations

tiand ri:

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Introduction Abstract model: CPN Diagnosis Application Conclusions & perspectives

CPN definition: example (2/2)

Notations

tiand ri:

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Introduction Abstract model: CPN Diagnosis Application Conclusions & perspectives

CPN definition: example (2/2)

Notations

tiand ri:

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Introduction Abstract model: CPN Diagnosis Application Conclusions & perspectives

CPN definition: example (2/2)

Notations

tiand ri:

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Introduction Abstract model: CPN Diagnosis Application Conclusions & perspectives

CPN definition: example (2/2)

Notations

tiand ri:

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Introduction Abstract model: CPN Diagnosis Application Conclusions & perspectives

CPN definition: example (2/2)

Notations

tiand ri:

left hand of philosopher i p1 p2 2 1 p3 p4 p11 p21 p22 p32 p33 p43 4 p44 3 p14 t1 r1 t2 r2 t3 r3 t4 r4 χp1 χp2 χp1 χp2 χp11 χp21 N ormal:χp11 Exchange:χp21 N ormal:χp21 Exchange:χp11 χp2 χp3 χp2 χp3 χp22 χp32 N ormal:χp22 Exchange:χp32 N ormal:χp32 Exchange:χp22 χp3 χp4 χp3 χp4 χp33 χp43 N ormal:χp33 Exchange:χp43 N ormal:χp43 Exchange:χp33 χp4 χp1 χp4 χp1 χp44 χp14 N ormal:χp44 N ormal:χp14 40 / 101

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Introduction Abstract model: CPN Diagnosis Application Conclusions & perspectives

CPN definition: example (2/2)

Notations

tiand ri:

left hand of philosopher i

Observations each philosopher ate twice except philosopher 4 tiis fired before ri some activities were partially ordered p1 p2 2 1 p3 p4 p11 p21 p22 p32 p33 p43 4 p44 3 p14 t1 r1 t2 r2 t3 r3 t4 r4 χp1 χp2 χp1 χp2 χp11 χp21 N ormal:χp11 Exchange:χp21 N ormal:χp21 Exchange:χp11 χp2 χp3 χp2 χp3 χp22 χp32 N ormal:χp22 Exchange:χp32 N ormal:χp32 Exchange:χp22 χp3 χp4 χp3 χp4 χp33 χp43 N ormal:χp33 Exchange:χp43 N ormal:χp43 Exchange:χp33 χp4 χp1 χp4 χp1 χp44 χp14 N ormal:χp44 N ormal:χp14 41 / 101

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Introduction Abstract model: CPN Diagnosis Application Conclusions & perspectives

Partially ordered observation

Definition (Partially ordered observation(T,))

T ={t_ik|16k6|T(ti)|}: repeated occurred transitions

⊆(S(T)×S(T)): partially ordered relation

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Introduction Abstract model: CPN Diagnosis Application Conclusions & perspectives

Partially ordered observation

Example (less specific) i={(ti(k),r (k) i ),(r (k) i ,t (k+1) i )}

1isless specificthan1∪2

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Introduction Abstract model: CPN Diagnosis Application Conclusions & perspectives

Partially ordered observation

Example (less specific) i={(ti(k),r (k) i ),(r (k) i ,t (k+1) i )}

1isless specificthan1∪2

Definition (Minimal partially ordered relationM

T →M′ min ) T=                      t1: 2 r1: 2 t2: 2 r2: 2 t3: 2 r3: 2 t4: 2 r4: 1

⇒

S(T) = {t₁(1),t₁(2),r₁(1),r₁(2) . . . ,t₄(1),r₄(1)} _i= {(t_i(k),r_i(k)), (r_i(k),t_i(k+1))} M M ’ (ti,tj)∈M T →M′ min , iff∀δ, M[δ >M ′_∧−⇀ δT=T ⇒δtitj 44 / 101

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Introduction Abstract model: CPN Diagnosis Application Conclusions & perspectives

Adapt the CPN as a fault model

An abstract fault model

Define data correctness status

{

b

,

r

,

∗}

Map all the transition modes into diagnostic transition modes

{

OK

,

KO

}

Abstract transition functionalities as data dependency functions

{

FW

,

SRC

,

EL

}

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Introduction Abstract model: CPN Diagnosis Application Conclusions & perspectives

Data dependencies in transition modes

p1: Π₁ _p1: Ψ_D=status={b,r,∗} F: Γ₁→Ψ(Π₁) χp:status t: Γ₁ Fault: Γ₁→ {OK,KO} F′: Γ₁→Ψ(Π₂)⇒ ∀χp∈Π₁∩Π₂, χ′_p∈Π₂,Π′ 1⊆Π1 ∀c,c′ ∈status,∀C ⊆status, p2: Π₂ _p2:status 46 / 101

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Introduction Abstract model: CPN Diagnosis Application Conclusions & perspectives

Data dependencies in transition modes

p1: Π₁ _p1: Ψ_D=status={b,r,∗} F: Γ₁→Ψ(Π₁) χp:status t: Γ₁ Fault: Γ₁→ {OK,KO} F′: Γ₁→Ψ(Π₂)⇒ ∀χp∈Π₁∩Π₂, χ′_p∈Π₂,Π′ 1⊆Π1 ∀c,c′ ∈status,∀C ⊆status, D=      FW:Π₁∩Π₂→Ψ(Π₂), FW(χp) =χp SRC:∅ →Ψ(Π₂), SRC=χp EL:(Π₁∩Π₂)n→Ψ(Π₂), EL(Π′₁) =χ′_p ⇒ Dc=                    FW:status→status,FW c(c) =c SRC:∅ →status,SRCc=∗

EL :statusn→status,ELc(C) =c′, with c′=    b, iff∀c∈ C,c=b r, iff∃c∈ C,c=r ∗,iff∃c∈ C,c=∗ ∧∄c′′ ∈ C,c′′=r p2: Π₂ _p2:status 47 / 101

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Introduction Abstract model: CPN Diagnosis Application Conclusions & perspectives

Data dependencies in transition modes

p1: Π₁ _p1: Ψ_D=status={b,r,∗} F: Γ₁→Ψ(Π₁) χp:status t: Γ₁ Fault: Γ₁→ {OK,KO} F′: Γ₁→Ψ(Π₂)⇒ ∀χp∈Π₁∩Π₂, χ′_p∈Π₂,Π′ 1⊆Π1 ∀c,c′ ∈status,∀C ⊆status, D=      FW:Π₁∩Π₂→Ψ(Π₂), FW(χp) =χp SRC:∅ →Ψ(Π₂), SRC=χp EL:(Π₁∩Π₂)n→Ψ(Π₂), EL(Π′₁) =χ′_p ⇒ Dc=                    FW:status→status,FW c(c) =c SRC:∅ →status,SRCc=∗

EL :statusn→status,ELc(C) =c′, with c′=    b, iff∀c∈ C,c=b r, iff∃c∈ C,c=r ∗,iff∃c∈ C,c=∗ ∧∄c′′ ∈ C,c′′=r p2: Π₂ _p2:status

Diagnosis properties of data dependency (Ardissono et al. [2005]) EL mt c•_{i t}c•_{j t}ct•mt c•_{i t}c•_{j t}ct• OK b b b KO b b r OK r b r KO r b ∗ OK ∗ b ∗KO ∗ b ∗ OK ∗ ∗ ∗KO ∗ ∗ ∗ OK ∗ r r KO ∗ r ∗ FW mt c•t ct• OK/KO b b OK/KO r r OK/KO∗ ∗ SRC mt ct• OK b KO r 48 / 101

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Introduction Abstract model: CPN Diagnosis Application Conclusions & perspectives

CPN for diagnosis: fault model

Fault model: F: [ γ∈Γ γ→ {OK,KO} F(Normal) =OK , F(Exchange) =KO Example (philosopher 1) 1 p1 4 p2 p11 p21 t1 r1 χp1 χp2 χp1 χp2 χp11 χp21 normal:χp11 exchange:χp21 normal:χp21 exchange:χp11 =⇒ • p1 p2• p11 p21 t1 r1 χp1 χp2 F W(χp1) F W(χp2) χp11 χp21 OK:F W(χp11) KO:r OK:F W(χp21) KO:r 49 / 101

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Introduction Abstract model: CPN Diagnosis Application Conclusions & perspectives

CPN for diagnosis: fault model

Fault model: F: [ γ∈Γ γ→ {OK,KO} F(Normal) =OK , F(Exchange) =KO Example (philosopher 1) p1 p2 p11 p21 t1 r1 χp1 χp2 χp1 χp2 χp11 χp21 normal:χp11 exchange:r normal:χp21 exchange:r =⇒ p1 p2 p11 p21 t1 r1 χp1 χp2 F W(χp1) F W(χp2) χp11 χp21 OK:F W(χp11) KO:r OK:F W(χp21) KO:r 50 / 101

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Introduction Abstract model: CPN Diagnosis Application Conclusions & perspectives

CPN for diagnosis: fault model

Fault model: F: [ γ∈Γ γ→ {OK,KO} F(Normal) =OK , F(Exchange) =KO Example (philosopher 1) p1 p2 1 p11 4p21 t1 r1 χp1 χp2 χp1 χp2 χp11 χp21 normal:χp11 exchange:χp21 normal:χp21 exchange:χp11 =⇒ p1 p2 • p11 •p21 t1 r1 χp1 χp2 F W(χp1) F W(χp2) χp11 χp21 OK:F W(χp11) KO:r OK:F W(χp21) KO:r 51 / 101

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Introduction Abstract model: CPN Diagnosis Application Conclusions & perspectives

CPN for diagnosis: fault model

Fault model: F: [ γ∈Γ γ→ {OK,KO} F(Normal) =OK , F(Exchange) =KO Example (philosopher 1) p1 p2 p11 p21 t1 r1 χp1 χp2 χp1 χp2 χp11 χp21 normal:χp11 exchange:χp21 normal:χp21 exchange:χp11 =⇒ p1 p2 p11 p21 t1 r1 χp1 χp2 F W(χp1) F W(χp2) χp11 χp21 OK:F W(χp11) KO:r OK:F W(χp21) KO:r 52 / 101

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Introduction Abstract model: CPN Diagnosis Application Conclusions & perspectives

CPN for diagnosis: fault model

Fault model: F: [ γ∈Γ γ→ {OK,KO} F(Normal) =OK , F(Exchange) =KO Example (philosopher 1) 4 p1 1p2 p11 p21 t1 r1 χp1 χp2 χp1 χp2 χp11 χp21 normal:χp11 exchange:χp21 normal:χp21 exchange:χp11 =⇒ • p1 •p2 p11 p21 t1 r1 χp1 χp2 F W(χp1) F W(χp2) χp11 χp21 OK:F W(χp11) KO:r OK:F W(χp21) KO:r 53 / 101

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Introduction Abstract model: CPN Diagnosis Application Conclusions & perspectives

Diagnosis problem:

D

=

h

M

0

,

(

S

(

T

)

,

)

,

M

ˆ

i

(Li et al. [2009a])

Initial marking M0 p1p2p3p4p11p21p22p32p33p43p44p14 h ∗,∗,∗,∗, 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0i SymptomMˆ p1p2p3p4p11p21p22p32p33p43p44p14 h0 ,∗,∗, 0 , 0 , 0 , 0 , 0 , 0 , 0 , b , r i (S(T),) t₁(1)r₁(1)t₁(2)r₁(2), t₂(1)r₂(1)t₂(2)r₂(2), t3(1)r (1) 3 t (2) 3 r (2) 3 , t4(1)r (1) 4 t (2) 4 54 / 101

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Introduction Abstract model: CPN Diagnosis Application Conclusions & perspectives

Diagnosis problem:

D

=

h

M

0

,

(

S

(

T

)

,

)

,

M

ˆ

i

(Li et al. [2009a])

Initial marking M0 p1p2p3p4p11p21p22p32p33p43p44p14 h ∗,∗,∗,∗, 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0i SymptomMˆ p1p2p3p4p11p21p22p32p33p43p44p14 h0 ,∗,∗, 0 , 0 , 0 , 0 , 0 , 0 , 0 , b , r i (S(T),) t₁(1)r₁(1)t₁(2)r₁(2), t₂(1)r₂(1)t₂(2)r₂(2), t3(1)r (1) 3 t (2) 3 r (2) 3 , t4(1)r (1) 4 t (2) 4

A solution for a CPN diagnosis problem

− ⇀ δT t1:2 r1:2 t2:2 r2:2 t3:2 r3:2 t4:2 r4:1 − ⇀ δ t1: 2 r1.OK:_n1 r1.KO:_n2 t2: 2 r2.OK:_n3 r2.KO:_n4 t3: 2 r3.OK:n5 r3.KO:_n6 t4: 2 r4: 1 55 / 101

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Introduction Abstract model: CPN Diagnosis Application Conclusions & perspectives

Diagnosis problem:

D

=

h

M

0

,

(

S

(

T

)

,

)

,

M

ˆ

i

(Li et al. [2009a])

Initial marking M0 p1p2p3p4p11p21p22p32p33p43p44p14 h ∗,∗,∗,∗, 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0i SymptomMˆ p1p2p3p4p11p21p22p32p33p43p44p14 h0 ,∗,∗, 0 , 0 , 0 , 0 , 0 , 0 , 0 , b , r i (S(T),) t₁(1)r₁(1)t₁(2)r₁(2), t₂(1)r₂(1)t₂(2)r₂(2), t3(1)r (1) 3 t (2) 3 r (2) 3 , t4(1)r (1) 4 t (2) 4

A solution for a CPN diagnosis problem

− ⇀ δT t1:2 r1:2 t2:2 r2:2 t3:2 r3:2 t4:2 r4:1 − ⇀ δ t1: 2 r1.OK:_n1 r1.KO:_n2 t2: 2 r2.OK:_n3 r2.KO:_n4 t3: 2 r3.OK:n5 r3.KO:_n6 t4: 2 r4: 1 Covering relation:∗<b,∗<r ˆ M=M0+C× − ⇀ δ =⇒_Mˆ<X_M0+C×X−⇀ δ 56 / 101

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Introduction Abstract model: CPN Diagnosis Application Conclusions & perspectives

Inequations system

ˆ M 0 ∗ ∗ 0 0 0 0 0 0 0 b r < M0 ∗ ∗ ∗ ∗ 0 0 0 0 0 0 0 0 + C _t1 _t2 _t3 _t4 r1 r2 r3 _r4 OK KO OK KO OK KO p1 −χ_p1 −χ_{p1 FW}(χ_p11) r FW(χ_p13) p2 −χ_p2 −χ_p2 FW(χ_p21) r FW(χ_p22) r FW(χ_p33) p3 −χ_p3 −χ_p3 FW(χ_p32) r FW(χ_p33) r p4 −χ_p4 −χ_p4 FW(χ_p43) r FW(χ_p44) p11 FW(χ_p1) −χ_p11 −χ_p11 p21 FW(χ_p2) −χ_p21 −χ_p21 p22 FW(χ_p2) −χ_p22 −χ_p22 p32 FW(χ_p3) −χ_p32 −χ_p32 p33 FW(χ_p3) −χ_p33 −χ_p33 p43 FW(χ_p4) −χ_p43 −χ_p43 p44 FW(χ_p4) −χ_p44 p14 FW(χ_p1) −χ_p14 × t1:2 r1.OK:_n1 r1.KO:_n2 t2:2 r2.OK:_n3 r2.KO:_n4 t3:2 r3.OK:n5 r3.KO:_n6 t4:2 r4:1 n1+_{n2 =2} n3+_{n4 =2} n5+_{n6 =2}

Example (Inequations system)

                                 n1+_n2=2 _n3+_n4=2 n5+_n6=1 p1:0<∗ −28χ_p1−28χ_p1+n8 1FW(χp11) +n 8 2r+1 8_FW₍_χ p14) p2:∗<∗ −28χ_p2−28χ_p2+n8 1FW(χp21) +n 8 2r+n 8 3FW(χp22) +n 8 4r p3:∗<∗ −28χ_p3−28χ_p3+n₃8FW(χ_p32) +n8₄r+n₅8FW(χ_p33) +n8₆r p4:0<∗ −28χ_p4−28χ_p4+n₅8FW(χ_p43) +n₆8r+18FW(χ_p44) p11:0<0+28FW(χ_p1)−n8₁χ_p11−n8₂χ_p11 p21:0<0+28FW(χ_p2)−n8₁χ_p21−n8₂χ_p21 p22:0<0+28FW(χ_p2)−n8₃χ_p22−n8₄χ_p22 p32:0<0+28FW(χ_p3)−n8₃χ_p32−n8₄χ_p32 p33:0<0+28FW(χ_p3)−n8 5χp33−n86χp33 p43:0<0+28FW(χ_p4)−n8₅χ_p43−n8 6χp43 p44:b<0+28FW(χ_p4)−18χ_p44 p14:r<0+28FW(χ_p1)−18χ_p14 57 / 101

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Introduction Abstract model: CPN Diagnosis Application Conclusions & perspectives

Inequations system

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Introduction Abstract model: CPN Diagnosis Application Conclusions & perspectives

Diagnosis algorithm: single fault

Sketch 1 _{to propagate with b} tokens 2 _{to propagate with r} tokens 3 _{to integrate results}

                                         n1+_n2=2 _n3+_n4=2 n5+_n6=1 p1: 0<∗ −28χ_p1−28χ_p1+n8₁FW(χ_p11) +n₂8r+18FW(χ_p14) p2:∗<∗ −28χ_p2−28χ_p2+n₁8FW(χ_p21) +n8₂r+n₃8FW(χ_p22) +n8₄r p3:∗<∗ −28χ_p3−28χ_p3+n₃8FW(χ_p32) +n8₄r+n₅8FW(χ_p33) +n8₆r p4: 0<∗ −28χ_p4−28χ_p4+n8₅FW(χ_p43) +_n68r+18FW(χ_p44) p11: 0<0+28FW(χ_p1)−n8₁χ_p11−n₂8χ_p11 p21:0<0+28FW(χ_p2)−n8₁χ_p21−n₂8χ_p21 p22:0<0+28FW(χ_p2)−n8₃χ_p22−n₄8χ_p22 p32:0<0+28FW(χ_p3)−n8₃χ_p32−n₄8χ_p32 p33:0<0+28FW(χ_p3)−n8₅χ_p33−n₆8χ_p33 p43: 0<0+28FW(χ_p4)−n8₅χ_p43−n₆8χ_p43 p44: b<0+28FW(χ_p4)−18χ_p44 p14:r<0+28FW(χ_p1)−18χ_p14 59 / 101

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Introduction Abstract model: CPN Diagnosis Application Conclusions & perspectives

Diagnosis algorithm: single fault

                                         n1+_n2=2 _n3+_n4=2 n5+_n6=1 p1: 0<∗ −28χ_p1−28χ_p1+n8 1FW(χp11) +n 8 2r+1 8_FW₍_χ p14) p2:∗<∗ −28χ_p2−28χ_p2+n₁8FW(χ_p21) +n8₂r+n₃8FW(χ_p22) +n8₄r p3:∗<∗ −28χ_p3−28χ_p3+n₃8FW(χ_p32) +n8₄r+n₅8FW(χ_p33) +n8₆r p4: 0<∗ −28χ_p4−28χ_p4+n8₅FW(χ_p43) +_n68r+18FW(χ_p44) p11: 0<0+28FW(χ_p1)−n8 1χp11−n 8 2χp11 p21:0<0+28FW(χ_p2)−n8 1χp21−n 8 2χp21 p22:0<0+28FW(χ_p2)−n8 3χp22−n 8 4χp22 p32:0<0+28FW(χ_p3)−n8 3χp32−n 8 4χp32 p33:0<0+28FW(χ_p3)−n8₅χ_p33−n8 6χp33 p43: 0<0+28FW(χ_p4)−n8₅χ_p43−n8 6χp43 p44: b<0+28FW(χ_p4)−18χ_p44 p14:r<0+28FW(χ_p1)−18χ_p14 Example (χ_p44=b as a constraint) p44=b 60 / 101

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Introduction Abstract model: CPN Diagnosis Application Conclusions & perspectives

Diagnosis algorithm: single fault

                                         n1+_n2=2 _n3+_n4=2 n5+_n6=1 p1: 0<∗ −28χ_p1−28χ_p1+n8₁FW(χ_p11) +n₂8r+18FW(χ_p14) p2:∗<∗ −28χ_p2−28χ_p2+n₁8FW(χ_p21) +n8₂r+n₃8FW(χ_p22) +n8₄r p3:∗<∗ −28χ_p3−28χ_p3+n₃8FW(χ_p32) +n8₄r+n₅8FW(χ_p33) +n8₆r p4: 0<∗ −28χ_p4−28χ_p4+n8₅FW(χ_p43) +_n68r+18FW(χ_p44) p11: 0<0+28FW(χ_p1)−n8₁χ_p11−n₂8χ_p11 p21:0<0+28FW(χ_p2)−n8₁χ_p21−n₂8χ_p21 p22:0<0+28FW(χ_p2)−n8₃χ_p22−n₄8χ_p22 p32:0<0+28FW(χ_p3)−n8₃χ_p32−n₄8χ_p32 p33:0<0+28FW(χ_p3)−n8₅χ_p33−n₆8χ_p33 p43: 0<0+28FW(χ_p4)−n8₅χ_p43−n₆8χ_p43 p44: b<0+28FW(χ_p4)−18χ_p44 p14:r<0+28FW(χ_p1)−18χ_p14 Example (χ_p44=b as a constraint) p44=b p4=b 61 / 101

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Introduction Abstract model: CPN Diagnosis Application Conclusions & perspectives

Diagnosis algorithm: single fault

Sketch 1 _{to propagate with b} tokens 2 to propagate with r tokens 3 _{to integrate results}

                                         n1+_n2=2 _n3+_n4=2 n5+_n6=1 p1: 0<∗ −28χ_p1−28χ_p1+n8₁FW(χ_p11) +n₂8r+18FW(χ_p14) p2:∗<∗ −28χ_p2−28χ_p2+n₁8FW(χ_p21) +n8₂r+n₃8FW(χ_p22) +n8₄r p3:∗<∗ −28χ_p3−28χ_p3+n₃8FW(χ_p32) +n8₄r+n₅8FW(χ_p33) +n8₆r p4: 0<∗ −28χ_p4−28χ_p4+n8₅FW(χ_p43) +_n68_r+18FW(χ_p44) p11: 0<0+28FW(χ_p1)−n8₁χ_p11−n₂8χ_p11 p21:0<0+28FW(χ_p2)−n8₁χ_p21−n₂8χ_p21 p22:0<0+28FW(χ_p2)−n8₃χ_p22−n₄8χ_p22 p32:0<0+28FW(χ_p3)−n8₃χ_p32−n₄8χ_p32 p33:0<0+28FW(χ_p3)−n8₅χ_p33−n₆8χ_p33 p43: 0<0+28FW(χ_p4)−n8₅χ_p43−n₆8χ_p43 p44: b<0+28FW(χ_p4)−18χ_p44 p14:r<0+28FW(χ_p1)−18χ_p14 Example (χ_p44=b as a constraint) p44=b p4=b p43=b∧_n6=0 62 / 101

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Introduction Abstract model: CPN Diagnosis Application Conclusions & perspectives

Diagnosis algorithm: single fault

                                         n1+_n2=2 _n3+_n4=2 n5+_n6=1 p1: 0<∗ −28χ_p1−28χ_p1+n8₁FW(χ_p11) +n₂8r+18FW(χ_p14) p2:∗<∗ −28χ_p2−28χ_p2+n₁8FW(χ_p21) +n8₂r+n₃8FW(χ_p22) +n8₄r p3:∗<∗ −28χ_p3−28χ_p3+n₃8FW(χ_p32) +n8₄r+n₅8FW(χ_p33) +n8₆r p4: 0<∗ −28χ_p4−28χ_p4+n8₅FW(χ_p43) +_n68_r+18FW(χ_p44) p11: 0<0+28FW(χ_p1)−n8₁χ_p11−n₂8χ_p11 p21:0<0+28FW(χ_p2)−n8₁χ_p21−n₂8χ_p21 p22:0<0+28FW(χ_p2)−n8₃χ_p22−n₄8χ_p22 p32:0<0+28FW(χ_p3)−n8₃χ_p32−n₄8χ_p32 p33:0<0+28FW(χ_p3)−n8₅χ_p33−n₆8χ_p33 p43: 0<0+28FW(χ_p4)−n8₅χ_p43−n₆8χ_p43 p44: b<0+28FW(χ_p4)−18χ_p44 p14:r<0+28FW(χ_p1)−18χ_p14 Example (χ_p44=b as a constraint) p44=b p4=b p43=b∧_n6=0 63 / 101

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Introduction Abstract model: CPN Diagnosis Application Conclusions & perspectives

Diagnosis algorithm: single fault

                                         n1+_n2=2 _n3+_n4=2 n5+_n6=1 p1: 0<∗ −28χ_p1−28χ_p1+n8₁FW(χ_p11) +n₂8r+18FW(χ_p14) p2:∗<∗ −28χ_p2−28χ_p2+n₁8FW(χ_p21) +n8₂r+n₃8FW(χ_p22) +n8₄r p3:∗<∗ −28χ_p3−28χ_p3+n₃8FW(χ_p32) +n8₄r+n₅8FW(χ_p33) +n8₆r p4: 0<∗ −28χ_p4−28χ_p4+n8₅FW(χ_p43) +_n68_r+18FW(χ_p44) p11: 0<0+28FW(χ_p1)−n8₁χ_p11−n₂8χ_p11 p21:0<0+28FW(χ_p2)−n8₁χ_p21−n₂8χ_p21 p22:0<0+28FW(χ_p2)−n8₃χ_p22−n₄8χ_p22 p32:0<0+28FW(χ_p3)−n8₃χ_p32−n₄8χ_p32 p33:0<0+28FW(χ_p3)−n8₅χ_p33−n₆8χ_p33 p43: 0<0+28FW(χ_p4)−n8₅χ_p43−n₆8χ_p43 p44: b<0+28FW(χ_p4)−18χ_p44 p14:r<0+28FW(χ_p1)−18χ_p14 Example (χ_p44=b as a constraint) p44=b p4=b p43=b∧_n6=0 Example (χ_p14=r) p14=r 64 / 101

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Introduction Abstract model: CPN Diagnosis Application Conclusions & perspectives

Diagnosis algorithm: single fault

                                         n1+_n2=2 _n3+_n4=2 n5+_n6=1 p1: 0<∗ −28χ_p1−28χ_p1+n8₁FW(χ_p11) +n₂8r+18FW(χ_p14) p2:∗<∗ −28χ_p2−28χ_p2+n₁8FW(χ_p21) +n8₂r+n₃8FW(χ_p22) +n8₄r p3:∗<∗ −28χ_p3−28χ_p3+n₃8FW(χ_p32) +n8₄r+n₅8FW(χ_p33) +n8₆r p4: 0<∗ −28χ_p4−28χ_p4+n8₅FW(χ_p43) +_n68_r+18FW(χ_p44) p11: 0<0+28FW(χ_p1)−n8₁χ_p11−n₂8χ_p11 p21:0<0+28FW(χ_p2)−n8₁χ_p21−n₂8χ_p21 p22:0<0+28FW(χ_p2)−n8₃χ_p22−n₄8χ_p22 p32:0<0+28FW(χ_p3)−n8₃χ_p32−n₄8χ_p32 p33:0<0+28FW(χ_p3)−n8₅χ_p33−n₆8χ_p33 p43: 0<0+28FW(χ_p4)−n8₅χ_p43−n₆8χ_p43 p44: b<0+28FW(χ_p4)−18χ_p44 p14:r<0+28FW(χ_p1)−18χ_p14 Example (χ_p44=b as a constraint) p44=b p4=b p43=b∧_n6=0 Example (χ_p14=r) p14=r p1=r 65 / 101

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Introduction Abstract model: CPN Diagnosis Application Conclusions & perspectives

Diagnosis algorithm: single fault

                                         n1+_n2=2 _n3+_n4=2 n5+_n6=1 p1: 0<∗ −28χ_p1−28χ_p1+n8₁FW(χ_p11) +n₂8r+18FW(χ_p14) p2:∗<∗ −28χ_p2−28χ_p2+n₁8FW(χ_p21) +n8₂r+n₃8FW(χ_p22) +n8₄r p3:∗<∗ −28χ_p3−28χ_p3+n₃8FW(χ_p32) +n8₄r+n₅8FW(χ_p33) +n8₆r p4: 0<∗ −28χ_p4−28χ_p4+n8₅FW(χ_p43) +_n68_r+18FW(χ_p44) p11: 0<0+28FW(χ_p1)−n8₁χ_p11−n₂8χ_p11 p21:0<0+28FW(χ_p2)−n8₁χ_p21−n₂8χ_p21 p22:0<0+28FW(χ_p2)−n8₃χ_p22−n₄8χ_p22 p32:0<0+28FW(χ_p3)−n8₃χ_p32−n₄8χ_p32 p33:0<0+28FW(χ_p3)−n8₅χ_p33−n₆8χ_p33 p43: 0<0+28FW(χ_p4)−n8₅χ_p43−n₆8χ_p43 p44: b<0+28FW(χ_p4)−18χ_p44 p14:r<0+28FW(χ_p1)−18χ_p14 Example (χ_p44=b as a constraint) p44=b p4=b p43=b∧_n6=0 Example (χ_p14=r) p14=r p1=r p11=r _n2>0 66 / 101

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Introduction Abstract model: CPN Diagnosis Application Conclusions & perspectives

Diagnosis algorithm: single fault

                                         n1+_n2=2 _n3+_n4=2 n5+_n6=1 p1: 0<∗ −28χ_p1−28χ_p1+n8₁FW(χ_p11) +n₂8r+18FW(χ_p14) p2:∗<∗ −28χ_p2−28χ_p2+n₁8FW(χ_p21) +n8₂r+n₃8FW(χ_p22) +n8₄r p3:∗<∗ −28χ_p3−28χ_p3+n₃8FW(χ_p32) +n8₄r+n₅8FW(χ_p33) +n8₆r p4: 0<∗ −28χ_p4−28χ_p4+n8₅FW(χ_p43) +_n68_r+18FW(χ_p44) p11: 0<0+28FW(χ_p1)−n8₁χ_p11−n₂8χ_p11 p21:0<0+28FW(χ_p2)−n8₁χ_p21−n₂8χ_p21 p22:0<0+28FW(χ_p2)−n8₃χ_p22−n₄8χ_p22 p32:0<0+28FW(χ_p3)−n8₃χ_p32−n₄8χ_p32 p33:0<0+28FW(χ_p3)−n8₅χ_p33−n₆8χ_p33 p43: 0<0+28FW(χ_p4)−n8₅χ_p43−n₆8χ_p43 p44: b<0+28FW(χ_p4)−18χ_p44 p14:r<0+28FW(χ_p1)−18χ_p14 Example (χ_p44=b as a