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Conditional Probabilities over Probabilistic and Nondeterministic Systems M. E. Andrés and P. van Rossum Radboud Universiteit Nijmegen, The Netherlands.

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Presentation on theme: "Conditional Probabilities over Probabilistic and Nondeterministic Systems M. E. Andrés and P. van Rossum Radboud Universiteit Nijmegen, The Netherlands."— Presentation transcript:

1 Conditional Probabilities over Probabilistic and Nondeterministic Systems M. E. Andrés and P. van Rossum Radboud Universiteit Nijmegen, The Netherlands.

2 TACAS - April 1 st Budapest, Hungary Miguel E. Andres Radboud University 2 Overview Motivation Background Markov Decision Processes and Schedulers Conditional Probabilities pCTL Our Logic (cpCTL) Model Checking issues Fully probabilistic case Probabilistic and Nondeterministic case Comparison (pCTL vs cpCTL) cpCTL Complications Model Checker Counterexamples Future work

3 TACAS - April 1 st Budapest, Hungary Miguel E. Andres Radboud University 3 Overview Motivation Background Markov Decision Processes and Schedulers Conditional Probabilities pCTL Our Logic (cpCTL) Model Checking issues Fully probabilistic case Probabilistic and Nondeterministic case Comparison (pCTL vs cpCTL) cpCTL Complications Model Checker Counterexamples Future work

4 TACAS - April 1 st Budapest, Hungary Miguel E. Andres Radboud University 4 Motivation Model Checking Model j = Temporal Logics ' P [ § D ea d L ] § D ea d L P + [ § D ea d L ] P + [ § D ea d L j ¤ S i ng U ] · 0 : 1 · 0 : 1 · 0 : 1 (+ cond prob) cpCTL (+ nondet) pCTL (+ prob) pCTL LTL – CTL NEW

5 TACAS - April 1 st Budapest, Hungary Miguel E. Andres Radboud University 5 Motivation Conditional Probabilities Anonymity Strong Anonymity Probable innocence What we do Define cpCTL Model Checker for cpCTL Present a Notion of Counterexamples Deterministic Case Nondeterministic Case Risk assessment P[dyke breaks| it rains heavily] Diagnosability P[A failed|error message E]

6 TACAS - April 1 st Budapest, Hungary Miguel E. Andres Radboud University 6 Overview Motivation Background Markov Decision Processes and Schedulers pCTL Conditional Probabilities Our Logic (cpCTL) Model Checking issues Fully probabilistic case Probabilistic and Nondeterministic case Comparison (pCTL vs cpCTL) cpCTL Complications Model Checker Counterexamples Future work

7 TACAS - April 1 st Budapest, Hungary Miguel E. Andres Radboud University 7 Probabilistic and Nondeterministic Example Background – MDPs The Model (MDP) ² S i s t h e ¯ n i t es t a t espaceo f t h esys t em ² s 0 2 S i s t h e i n i t i a l s t a t e ² L : S ! } ( P ) i sa l a b e l i ng f unc t i on ² ¿: S ! } ( D i s t r ( S )) MDP = ( S, s 0 ; L ; ¿ ), w h ere: Finite PathsPaths s 0 s 2 s 0 s 2 s 3... s 0 s 2 ( s 3 ) ! s 0 ( s 1 ) !...

8 TACAS - April 1 st Budapest, Hungary Miguel E. Andres Radboud University 8 Background – Schedulers Schedulers resolve the Nondeterminism! Schedulers : F i n i t e P a t h ! D i s t r ( S ) ² P [ s 0 s 2 s 5 ] = 1 8 ² P [ s 0 s 2 s 6 ] = 0 S 2 ! ¼ 2 S 2 ! ¼ 3 ² P [ s 0 s 2 s 5 ] = 0 ² P [ s 0 s 2 s 6 ] = 1 40 S 2 1 4 ! ¼ 2 S 2 3 4 ! ¼ 3

9 TACAS - April 1 st Budapest, Hungary Miguel E. Andres Radboud University 9 Background – pCTL Syntaxis State Path ª : = © U © j § © j ¤ © Semantic ¾ j = Á U Ã, Áh o ld sun t i l a t somepo i n t Ãh o ld s ¾ j = § Á, ¾ j = t rue U Á ¾ j = ¤ Á, ¾ j = : § : Á © : = P j © ^ © j : © j 8 ª j 9 ª j P./ a [ ª ] a 2 [ 0 ; 1 ]./ 2 f < ; · ; > ; ¸ g

10 TACAS - April 1 st Budapest, Hungary Miguel E. Andres Radboud University 10 Example 6 j = Background – computing satisfaction 3 4 + 1 40 = 0 ; 775 3 4 + 1 4 ( 1 2 ¡ ® ) + 1 4 ® = 0 ; 875

11 TACAS - April 1 st Budapest, Hungary Miguel E. Andres Radboud University 11 Background – Conditional Probabilities Standard Conditional Probabilities P ( A j B ) = P ( A \ B ) P ( B ) Max and Min Conditional Probabilities P + ( ¢ 1 j ¢ 2 ) = sup ´ 2 S c h > 0 ¢ 2 P ´ ( ¢ 1 j ¢ 2 ) P ¡ ( ¢ 1 j ¢ 2 ) = i n f ´ 2 S c h > 0 ¢ 2 P ´ ( ¢ 1 j ¢ 2 ) Conditional Probabilities over MDPs P ´ ( ¢ 1 j ¢ 2 ) = P ´ ( ¢ 1 \ ¢ 2 ) P ´ ( ¢ 2 ) ² ( ­ s ; B s ; P ´ ) i s t h epro b a b i l i t yspace ² ¢ 1 ; ¢ 2 2 B s are t wose t so f pa t h s ² P ´ ( ¢ 2 ) > 0 ² ( ­ ; F ; P ) i sapro b a b i l i t yspace ² A ; B 2 F are t woeven t s ² P ( B ) > 0

12 TACAS - April 1 st Budapest, Hungary Miguel E. Andres Radboud University 12 Overview Motivation Background Markov Decision Processes and Schedulers pCTL Conditional Probabilities Our Logic (cpCTL) Model Checking issues Fully probabilistic case Probabilistic and Nondeterministic case Comparison (pCTL vs cpCTL) cpCTL Complications Model Checker Counterexamples Future work

13 TACAS - April 1 st Budapest, Hungary Miguel E. Andres Radboud University 13 s j = P · a [ Á j à ] P s [ Á ^ à ] P s [ à ] · a Our Logic – cpCTL pCTL cpCTL ª : = © U © j § © j ¤ © j P./ a [ ª j ª ] © : = P j © ^ © j : © j 8 ª j 9 ª j P./ a [ ª ] Interpretation P + s [ Á j à ] s j = P · a [ Á j à ] max ´ 2 S c h > 0 P s ; ´ [ Á ^ à ] P s ; ´ [ à ] · a P [ A j B ] = P [ A \ B ] P [ B ]

14 TACAS - April 1 st Budapest, Hungary Miguel E. Andres Radboud University 14 max ´ P s 0 ; ´ [ § B ^ ¤ P ] P s 0 ; ´ [ ¤ P ] · 0 ; 99 cpCTL - Example S 0 j = P · 0 ; 99 [ § B j ¤ P ] ² P s 0 ; ´ ¼ 2 [ § B j ¤ P ] = P [ s 0 s 1 ] + P [ s 0 s 2 s 3 ] P [ s 0 s 1 ] + P [ s 0 s 2 s 3 ] + P [ s 0 s 2 s 4 ] = 1 ¡ 2 ® 7 max ( 1 ¡ 2 ® 7 ; 30 31 ) · 0 ; 99 ² P s 0 ; ´ ¼ 3 [ § B j ¤ P ] = P [ s 0 s 1 ] P [ s 0 s 1 ] + P [ s 0 s 2 s 6 ] = 30 31

15 TACAS - April 1 st Budapest, Hungary Miguel E. Andres Radboud University 15 Overview Motivation Background Markov Decision Processes and Schedulers pCTL Conditional Probabilities Our Logic (cpCTL) Model Checking issues Fully probabilistic case Probabilistic and Nondeterministic case Comparison (pCTL vs cpCTL) cpCTL Complications Model Checker Counterexamples Future work

16 TACAS - April 1 st Budapest, Hungary Miguel E. Andres Radboud University 16 Model Checking Issues Fully probabilistic case Can be reduced to a pCTL* problem, using P + s [ Á j à ] 6 = P + s [ Á ^ à ] P + s [ à ] Observation Probabilistic and Nondeterministic case pCTLcpCTL Deterministic Schedulers History Independent Schedulers Semi History Independent Schedulers Bellman EquationsNO Bellman Equations P + s [ Á j à ] = max ´ P s ; ´ [ Á ^ à ] P s ; ´ [ à ]

17 TACAS - April 1 st Budapest, Hungary Miguel E. Andres Radboud University 17 Model Checking Issues – Nondeterministic case cpCTL case Deterministic Schedulers (Not trivial) Semi History Independent Schedulers No Bellman equations Theorem: Deterministic Schedulers P ´ [ Á j à ] = P + [ Á j à ] an d P ´ 0 [ Á j à ] = P ¡ [ Á j à ] T h ereex i s t s D e t erm i n i s t i c sc h e d u l ers´an d ´ 0 suc h t h a t Coming…

18 TACAS - April 1 st Budapest, Hungary Miguel E. Andres Radboud University 18 Model Checking Issues – Nondeterministic case Semi History Independent Schedulers Why? I f P + s 0 [ § B j § P ] = P s 0 ; ´ [ § B j § P ] t h en´sa t i s ¯ es ´ ( ¾ ) = 8 < : ¼ 3 i f ¾ = s 0 ¼ 5 i f ¾ = s 0 s 3 ¼ 1 i f ¾ = s 0 s 3 s 0 Definition ´ i s'-sem i H i s t ory I n d epen d en t i f ² ´ t a k esa l ways t h esame d ec i s i on b e f ore t h esys t emreac h es' ² ´ t a k esa l ways t h esame d ec i s i on a f t er t h esys t emreac h es' P ´ [ Á j à ] = P + [ Á j à ] an d P ´ 0 [ Á j à ] = P ¡ [ Á j à ] T h ereex i s t s d e t erm i n i s t i c an d s HI sc h e d u l ers ´an d ´ 0 suc h t h a t Theorem: sHI Schedulers Stopping condition

19 TACAS - April 1 st Budapest, Hungary Miguel E. Andres Radboud University 19 Local Bellman equation Model Checking Issues – Nondeterministic case P + s 2 [ § P ] = ¼ 2 ¼ 3 P + s [ Á ] = max ¼ 2 ¿ ( s ) 0 @ X t 2 succ ( s ) ¼ ( t ) ¢ P + t [ Á ] 1 A Bellman Equations 1 10 ¢ P + s 6 [ § P ] + 9 10 ¢ P + s 7 [ § P ] ( 1 2 ¡ ® ) ¢ P + s 3 [ § P ] + ® ¢ P + s 4 [ § P ] + 1 2 ¢ P + s 5 [ § P ] P + s 2 [ § P ] = max 8 < : ( 1 2 ¡ ® ) ¢ P + s 3 [ § P ] + ® ¢ P + s 4 [ § P ] + 1 2 ¢ P + s 5 [ § P ] 1 10 ¢ P + s 6 [ § P ] + 9 10 ¢ P + s 7 [ § P ] M ax i mumovera ll ou t go i ng di s t r ib u t i ons ¼ o f s R ecurs i ve C ompu t a t i on

20 TACAS - April 1 st Budapest, Hungary Miguel E. Andres Radboud University 20 Model Checking Issues – Nondeterministic case Why Not Bellman equations? Bellman equation on cpCTL case… P + s 0 [ Á j à ] = max ¼ 2 ¿ ( s ) 0 @ X t 2 succ ( s ) ¼ ( t ) ¢ P + t [ Á j à ] 1 A P + s 0 [ § B j ¤ P ] · 0 ; 99 max ( 1 ¡ 2 ® 7 ; 30 31 ) · 0 ; 99 P + s 0 [ § B j ¤ P ] = P s 0 ; ´ ¼ 3 [ § B j ¤ P ] I f ® ¸ 7 62 t h en …but P + s 2 [ § B j ¤ P ] = P s 2 ; ´ ¼ 2 [ § B j ¤ P ] = 1 ¡ 2 ¢ ®

21 TACAS - April 1 st Budapest, Hungary Miguel E. Andres Radboud University 21 Overview Motivation Background Markov Decision Processes and Schedulers pCTL Conditional Probabilities Our Logic (cpCTL) Model Checking issues Fully probabilistic case Probabilistic and Nondeterministic case Comparison (pCTL vs cpCTL) cpCTL Complications Model Checker Counterexamples Future work

22 TACAS - April 1 st Budapest, Hungary Miguel E. Andres Radboud University 22 Idea Model Checker - Idea P + s [ Á j à ] = max ´ µ P s ; ´ [ Á ^ à ] P s ; ´ [ à ] ¶ {By deterministic and sHI Theorem} P + s [ Á j à ] = max µ P s ; ´ 1 [ Á ^ à ] P s ; ´ 1 [ à ] ; ¢¢¢ ; P s ; ´ k [ Á ^ à ] P s ; ´ k [ à ] ¶ w h ere f ´ 1 ; ´ 2 ;:::; ´ k g i s t h ese t o f a lld e t erm i n i s t i can d s HI sc h e d u l ers What we actually compute f ( s ; Á ; à ) = © ( P s ; ´ 1 [ Á ^ à ] ; P s ; ´ 1 [ à ]) ; ¢¢¢ ; ( P s ; ´ k [ Á ^ à ] ; P s ; ´ k [ à ]) ª P + s [ Á j à ] = max ³n a b j ( a ; b ) 2 f ( s ; Á ; à ) ^ b 6 = 0 o [ f 0 g ´

23 TACAS - April 1 st Budapest, Hungary Miguel E. Andres Radboud University 23 Model Checker - Example Optimizations Reusing information Ussing pCTL algorithms after reaching the stopping condition Example ¡ C ase P + s [ Á 1 U Á 2 j à 1 U à 2 ] ¢ f ( s ; Á 1 U Á 2 ; à 1 U à 2 ) = f ( P + s [ à 1 U à 2 ] ; P + s [ à 1 U à 2 ]) g i f s j = Á 2 f ( s ; Á 1 U Á 2 ; à 1 U à 2 ) = f ( P + s [ Á 1 U Á 2 ] ; 1 ) g i f s j = : Á 2 ^ à 2 f ( s ; Á 1 U Á 2 ; à 1 U à 2 ) = f ( 0 ; P ¡ s [ à 1 U à 2 ]) g i f s j = : Á 1 ^ : Á 2 ^ : à 2 f ( s ; Á 1 U Á 2 ; à 1 U à 2 ) = f ( 0 ; 0 ) g i f s j = Á 1 ^ : Á 2 ^ : à 1 ^ : à 2 f ( s ; Á 1 U Á 2 ; à 1 U à 2 ) = S ¼ 2 ¿ ( s ) ³ L t 2 succ ( s ) ¼ ( t ) ¯ f ( t ; Á 1 U Á 2 ; à 1 U à 2 ) ´ i f s j = Á 1 ^ : Á 2 ^ à 1 ^ : à 2

24 TACAS - April 1 st Budapest, Hungary Miguel E. Andres Radboud University 24 Overview Motivation Background Markov Decision Processes and Schedulers pCTL Conditional Probabilities Our Logic (cpCTL) Model Checking issues Fully probabilistic case Probabilistic and Nondeterministic case Comparison (pCTL vs cpCTL) cpCTL Complications Model Checker Counterexamples Future work

25 TACAS - April 1 st Budapest, Hungary Miguel E. Andres Radboud University 25 Why?Counterexamples Counterexamples Model ' j = Counterexamples for cpCTL A coun t erexamp l e f or P · a [ Á j à ] i sapa i r ( ¢ 1 ; ¢ 2 ) o f measura bl ese t s o f pa t h ssa t i s f y i ng ¢ 1 µ ¢ Á ^ Ã, ¢ 2 µ ¢ : Ã, an d a < P ´ ( ¢ 1 ) 1 ¡ P ´ ( ¢ 2 ), f orsome sc h e d u l er´. s j = P · a [ Á j à ], f ora ll ´ P s ; ´ [ Á ^ à ] P s ; ´ [ à ] · a Lemma w h ere ¢ 1 µ ¢ Á ^ Ã, f ! 2 ­ j ! j = Á ^ à g an d ¢ 2 µ ¢ : Ã, f ! 2 ­ j ! j = : à g P ´ [ Á ^ à ] P ´ [ à ] > a P ´ ( ¢ 1 ) 1 ¡ P ´ ( ¢ 2 ) > a

26 TACAS - April 1 st Budapest, Hungary Miguel E. Andres Radboud University 26 Overview Motivation Backgorund Markov Decision Processes and Schedulers pCTL Conditional Probabilities Our Logic (cpCTL) Model Checking issues Fully probabilistic case Probabilistic and Nondeterministic case Comparison (pCTL vs cpCTL) cpCTL Complications Model Checker Counterexamples Future work

27 TACAS - April 1 st Budapest, Hungary Miguel E. Andres Radboud University 27 Future Work Implement our Algorithms in a probabilistic model checker. Investigate features of cpCTL (expressivness – bisimulation issues). Improve complexity. Extend cpCTL to cpCTL*. More research about counterexamples in cpCTL and cpCTL*.

28 TACAS - April 1 st Budapest, Hungary Miguel E. Andres Radboud University 28 Thanks for your attention!

29 TACAS - April 1 st Budapest, Hungary Miguel E. Andres Radboud University 29 Why Deterministic Schedulers? ÁÁÃà s 0 s 1 s 2 1 ¡ ® ® P s 0 [ Á j à ] = ® P s 1 [ Á ^ à ] + ( 1 ¡ ® ) P s 2 [ Á ^ à ] ® P s 1 [ à ] + ( 1 ¡ ® ) P s 2 [ à ]


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