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Automated Analysis of Reli liability Architecture Fondazione Bruno Kessler Marco Bozzano, Alessandro Cimatti, and Cristian Mattarei Alpine Verification Meeting, 2013 Outline Architectural Design in Critical Systems Redundant systems

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SLIDE 1

Automated Analysis of Reli liability Architecture

Fondazione Bruno Kessler Marco Bozzano, Alessandro Cimatti, and Cristian Mattarei Alpine Verification Meeting, 2013

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SLIDE 2

Outline

  • Architectural Design in Critical Systems
  • Redundant systems
  • Reliability Analysis
  • Automated Approaches
  • EUF modeling and Fault Tree Analysis
  • Efficient Analysis via predicate abstraction
  • Conclusion

2

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SLIDE 3

Power system: …in a perfect world

Engine +

  • 3
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SLIDE 4

Power system: …in real world

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SLIDE 5

Power system: …in real world

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SLIDE 6

Power system: …in real world

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SLIDE 7

Power system: …in real world

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SLIDE 8

Outline

  • Architectural Design in Critical Systems
  • Redundant Systems
  • Reliability Analysis
  • Automated Approaches
  • EUF modeling and Fault Tree Analysis
  • Efficient Analysis via Predicate Abstraction
  • Conclusion

8

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SLIDE 9

𝑵𝟐 𝑵𝟑 𝑵𝟓 𝑵𝟒 𝑵𝟔 𝑵𝟕 Nominal architecture

Redundant systems definition: TMR

[Abraham74]

9

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SLIDE 10
  • Increase reliability for critical design
  • Usage of redundant scheme (e.g. Triple Modular Redundancy)
  • Hard to analyze and optimize system reliability

𝑵𝟐 𝑵𝟑 𝑵𝟓 𝑵𝟒 𝑵𝟔 𝑵𝟕 Nominal architecture Redundant architecture

Redundant systems definition: TMR

[Abraham74]

10

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SLIDE 11

Triple Modular Redundancy patterns

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1 voter 2 voters 3 voters

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SLIDE 12

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Reliability analysis: manual approach

[Hamamatsu10]

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SLIDE 13

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Reliability analysis: manual approach

[Hamamatsu10]

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SLIDE 14

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Reliability analysis: manual approach

[Hamamatsu10]

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SLIDE 15

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Reliability analysis: manual approach

[Hamamatsu10]

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SLIDE 16

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Triple Redundant Module comparison (1 voter)

Reliability analysis: manual approach

[Hamamatsu10]

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SLIDE 17

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Triple Redundant Module comparison (1 voter)

Reliability analysis: manual approach

[Hamamatsu10]

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SLIDE 18

Reliability analysis: manual approach

  • Time expensive and error prone reliability

computation

  • Specific approach for linear structures (not

generalizable)

  • Needs space discretization
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SLIDE 19

Outline

  • Architectural Design in Critical Systems
  • Redundant Systems
  • Reliability Analysis
  • Automated Approaches
  • EUF modeling and Fault Tree Analysis
  • Efficient Analysis via Predicate Abstraction
  • Conclusion

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SLIDE 20

Modeling of the extended system

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SLIDE 21

Modeling of the extended system

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  • Duplicate the behavior (nominal and faulty)
  • Introduce a multiplexer, triggered by the fault event
  • Model the (generic) behavior of components

using uninterpreted functions (e.g. x = y → 𝑔 𝑦 = 𝑔 𝑧 )

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SLIDE 22

Modeling of the extended system

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SLIDE 23

Fault Tree Analysis: equivalence check

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𝐺𝑈 𝐺, 𝑈𝑀𝐹 = {𝑔 ∈ 2𝐺|∃𝑗 ∈ 𝐽. 𝑈𝑀𝐹 𝑗, 𝑔 ∧ 𝑔 𝑗𝑡 𝑛𝑗𝑜𝑗𝑛𝑏𝑚} 𝑈𝑀𝐹 𝐽, 𝐺 = 𝑂𝑝𝑛𝑗𝑜𝑏𝑚 𝐽 ≠ 𝑆𝑓𝑒𝑣𝑜𝑒𝑏𝑜𝑢(𝐽, 𝐺)

𝐽 𝑈𝑀𝐹

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SLIDE 24

Fault Tree Analysis: equivalence check

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𝐺𝑈 = 𝐺𝑁11 ∧ 𝐺𝑁12 ∨ 𝐺𝑁11 ∧ 𝐺𝑁13 ∨ ⋯ ∨ (𝐺𝑁23 ∧ 𝐺𝑊

2)

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SLIDE 25

Reliability Function Extraction

𝑮𝒏𝟐 𝑮𝒘 𝑮𝒏𝟒 𝑮𝒏𝟑 ⊤ ⊥ 𝑮𝒏𝟑

25

𝐺

𝑡𝑧𝑡 ∶ ℝ 0,1 × ⋯ × ℝ 0,1 ⟼ ℝ 0,1

BDD representation of the Fault Tree

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SLIDE 26

Reliability Function Extraction

𝑮𝒏𝟐 𝑮𝒘 𝑮𝒏𝟒 𝑮𝒏𝟑 ⊤ ⊥ 𝑮𝒏𝟑

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𝑮𝒘 + 𝐺

𝑡𝑧𝑡 𝐺 𝑤, 𝐺𝑛1, 𝐺𝑛2, 𝐺𝑛3 =

𝐺

𝑡𝑧𝑡 ∶ ℝ 0,1 × ⋯ × ℝ 0,1 ⟼ ℝ 0,1

BDD representation of the Fault Tree

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SLIDE 27

Reliability Function Extraction

𝑮𝒏𝟐 𝑮𝒘 𝑮𝒏𝟒 𝑮𝒏𝟑 ⊤ ⊥ 𝑮𝒏𝟑

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𝐺

𝑤 +

𝐺

𝑡𝑧𝑡 𝐺 𝑤, 𝐺𝑛1, 𝐺𝑛2, 𝐺𝑛3 =

+(𝟐 − 𝑮𝒘) ∗ 𝑮𝒏𝟐 ∗ 𝑮𝒏𝟑 +

𝐺

𝑡𝑧𝑡 ∶ ℝ 0,1 × ⋯ × ℝ 0,1 ⟼ ℝ 0,1

BDD representation of the Fault Tree

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SLIDE 28

Reliability Function Extraction

𝑮𝒏𝟐 𝑮𝒘 𝑮𝒏𝟒 𝑮𝒏𝟑 ⊤ ⊥ 𝑮𝒏𝟑

28

𝐺

𝑤 +

𝐺

𝑡𝑧𝑡 𝐺 𝑤, 𝐺𝑛1, 𝐺𝑛2, 𝐺𝑛3 =

+(1 − 𝐺

𝑤) ∗ 𝐺𝑛1 ∗ 𝐺𝑛2 +

+(𝟐 − 𝑮𝒘) ∗ 𝑮𝒏𝟐 ∗ (𝟐 − 𝑮𝒏𝟑) ∗ 𝑮𝒏𝟒 +

𝐺

𝑡𝑧𝑡 ∶ ℝ 0,1 × ⋯ × ℝ 0,1 ⟼ ℝ 0,1

BDD representation of the Fault Tree

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SLIDE 29

Reliability Function Extraction

𝑮𝒏𝟐 𝑮𝒘 𝑮𝒏𝟒 𝑮𝒏𝟑 ⊤ ⊥ 𝑮𝒏𝟑

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𝐺

𝑤 +

𝐺

𝑡𝑧𝑡 𝐺 𝑤, 𝐺𝑛1, 𝐺𝑛2, 𝐺𝑛3 =

+(1 − 𝐺

𝑤) ∗ 𝐺𝑛1 ∗ 𝐺𝑛2 +

+(𝟐 − 𝑮𝒘) ∗ (𝟐 − 𝑮𝒏𝟐) ∗ 𝑮𝒏𝟑 ∗ 𝑮𝒏𝟒 +(1 − 𝐺

𝑤) ∗ 𝐺𝑛1 ∗ (1 − 𝐺𝑛2) ∗ 𝐺𝑛3 +

𝐺

𝑡𝑧𝑡 ∶ ℝ 0,1 × ⋯ × ℝ 0,1 ⟼ ℝ 0,1

BDD representation of the Fault Tree

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SLIDE 30

Reliability Function Extraction

𝑮𝒏𝟐 𝑮𝒘 𝑮𝒏𝟒 𝑮𝒏𝟑 ⊤ ⊥ 𝑮𝒏𝟑

30

𝐺

𝑤 +

𝐺

𝑡𝑧𝑡 𝐺 𝑤, 𝐺𝑛1, 𝐺𝑛2, 𝐺𝑛3 =

+(1 − 𝐺

𝑤) ∗ 𝐺𝑛1 ∗ 𝐺𝑛2 +

+(1 − 𝐺

𝑤) ∗ (1 − 𝐺𝑛1) ∗ 𝐺𝑛2 ∗ 𝐺𝑛3

+(1 − 𝐺

𝑤) ∗ 𝐺𝑛1 ∗ (1 − 𝐺𝑛2) ∗ 𝐺𝑛3 +

𝐺

𝑡𝑧𝑡 ∶ ℝ 0,1 × ⋯ × ℝ 0,1 ⟼ ℝ 0,1

BDD representation of the Fault Tree

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SLIDE 31

Automated Analysis of Reliability Architecture

  • 1. Model the extended system with uninterpreted

functions

  • 2. Perform Fault Tree Analysis
  • 3. Extract Reliability Function, from BDD

representation of Fault Tree

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SLIDE 32

Automated Analysis of Reliability Architecture

  • 1. Model the extended system with uninterpreted

functions

  • 2. Perform Fault Tree Analysis
  • 3. Extract Reliability Function, from BDD

representation of Fault Tree

  • 4. Evaluate the results with analytical tools

(Octave/Matlab)

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SLIDE 33

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Automated Analysis of Reliability Architecture

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SLIDE 34

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Automated Analysis of Reliability Architecture

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SLIDE 35

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𝐺

𝑡𝑧𝑡 = 𝐺 𝑤 + 3 ∗ 𝐺 𝑛 2 − 3 ∗ 𝐺 𝑤 ∗ 𝐺 𝑛 2 …

Automated Analysis of Reliability Architecture

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SLIDE 36

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Triple Redundant Module comparison (1 voter)

e d c b a

1-Rv 1-Rm

𝐺

𝑡𝑧𝑡 = 𝐺 𝑤 + 3 ∗ 𝐺 𝑛 2 − 3 ∗ 𝐺 𝑤 ∗ 𝐺 𝑛 2 …

Automated Analysis of Reliability Architecture

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SLIDE 37

Uniform probability analysis

1 voter patterns comparison (2D) 1 voter patterns comparison (3D)

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SLIDE 38

Uniform probability analysis

1 vs 2 voters comparison (2D) 1 vs 2 voters comparison (3D)

38

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SLIDE 39

Not uniform probability analysis

Varying 𝐺

𝑛 for 𝑁1 (1 voter)

Varying 𝐺

𝑤 for 𝑊 1 (2 voters)

39

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SLIDE 40

Automated Analysis of Reliability Architectures

  • Full automated technique for the Analysis of

Reliability Architecture

  • Symbolic technique (it generates the closed form of

Reliability function)

  • Allows for the reusability of analysis results (i.e.

generation of Reliability Functions Libraries)

  • AllSMT approach: Hard to deal with big system

definition (> 10 stages)

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SLIDE 41

Outline

  • Architectural Design in Critical Systems
  • Redundant Systems
  • Reliability Analysis
  • Automated Approaches
  • EUF modeling and Fault Tree Analysis
  • Efficient Analysis via Predicate Abstraction
  • Conclusion

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SLIDE 42

Modular Abstraction

Boolean Data

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Modular Abstraction

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𝑄𝑝1 𝑝𝑜 = 𝑝1 𝑄𝑝2 𝑝𝑜 = 𝑝2 𝑄𝑝3 𝑝𝑜 = 𝑝3 … 𝑄𝑗2 𝑗𝑜1 = 𝑗21 𝑄𝑗3 𝑗𝑜1 = 𝑗31 … 𝑄𝑗1 𝑗𝑜1 = 𝑗11 𝑄𝑗4 𝑗𝑜2 = 𝑗12

Modular Abstraction

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… 𝐵1 𝑇1

𝐵

𝐷1 𝐺

1

= =

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SLIDE 45

𝑄𝑝1 𝑝𝑜 = 𝑝1 𝑄𝑝2 𝑝𝑜 = 𝑝2 𝑄𝑝3 𝑝𝑜 = 𝑝3 … 𝑄𝑗2 𝑗𝑜1 = 𝑗21 𝑄𝑗3 𝑗𝑜1 = 𝑗31 … 𝑄𝑗1 𝑗𝑜1 = 𝑗11 𝑄𝑗4 𝑗𝑜2 = 𝑗12

Modular Abstraction

45

… 𝐵1 𝑇1

𝐵

𝐷1 𝐺

1

= = … 𝐵2 𝑇2

𝐵

𝐷2 𝐺2

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SLIDE 46

𝑄𝑝1 𝑝𝑜 = 𝑝1 𝑄𝑝2 𝑝𝑜 = 𝑝2 𝑄𝑝3 𝑝𝑜 = 𝑝3 … 𝑄𝑗2 𝑗𝑜1 = 𝑗21 𝑄𝑗3 𝑗𝑜1 = 𝑗31 … 𝑄𝑗1 𝑗𝑜1 = 𝑗11 𝑄𝑗4 𝑗𝑜2 = 𝑗12

Modular Abstraction

46

… 𝐵1 𝑇1

𝐵

𝐷1 𝐺

1

= = … 𝐵2 𝑇2

𝐵

𝐷2 𝐺2 … 𝐵1 𝑇1

𝐵

… 𝑇2

𝐵

𝐷2 V

≠?

𝐺

1

𝐺2 = =

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SLIDE 47

Modular Abstraction

47

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SLIDE 48

Concrete vs Abstraction: linear

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SLIDE 49

DAG like example with 60 modules

49

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SLIDE 50

Concrete vs Abstraction: Tree and DAG (< 15 modules)

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SLIDE 51

Abstraction: Tree and DAG

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SLIDE 52

Outline

  • Architectural Design in Critical Systems
  • Redundant Systems
  • Reliability Analysis
  • Manual Reliability techniques
  • Automated Approaches
  • EUF modeling and Fault Tree Analysis
  • Efficient Analysis via Predicate Abstraction
  • Conclusion

52

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SLIDE 53

Conclusion

  • Automated technique for the analysis of reliability

architectures

  • Management of linear, Tree and DAG like structures
  • Efficient analysis of large systems (> 140 modules)

via predicate abstraction

Automated Analysis of Reliability Architectures Marco Bozzano, Alessandro Cimatti and Cristian Mattarei In proc. of ICECCS 2013 Efficient Analysis of Reliability Architectures via Predicate Abstraction Marco Bozzano, Alessandro Cimatti and Cristian Mattarei Under review of FMCAD 2013

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SLIDE 54

Thank you!

Cristian Mattarei Fondazione Bruno Kessler FBK ES-Group mattarei@fbk.eu