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Miguel E. Andrés Radboud University, The Netherlands Significant Diagnostic Counterexamples in Probabilistic Model Checking Pedro D’Argenio Famaf, Argentina.

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Presentation on theme: "Miguel E. Andrés Radboud University, The Netherlands Significant Diagnostic Counterexamples in Probabilistic Model Checking Pedro D’Argenio Famaf, Argentina."— Presentation transcript:

1 Miguel E. Andrés Radboud University, The Netherlands Significant Diagnostic Counterexamples in Probabilistic Model Checking Pedro D’Argenio Famaf, Argentina Peter van Rossum Radboud University, The Netherlands

2 Haifa Verification 2008 - October 28th IBM Haifa Research Lab - Israel Miguel E. Andres Radboud University 2 j = : R eac h Classic Model Checking (Qualitative) Classic Model Checking (Qualitative) Motivation MODEL j = Á Counterexamples Counterexamples (Not satisfaction)

3 Haifa Verification 2008 - October 28th IBM Haifa Research Lab - Israel Miguel E. Andres Radboud University 3 Quantitative Model Checking Motivation j = · p I n t hi scase t h eproper t y i sno t sa t i s ¯ e dif p < 0 ; 6. Counterexamples (MORE COMPLEX),, …,, … R eac h

4 Haifa Verification 2008 - October 28th IBM Haifa Research Lab - Israel Miguel E. Andres Radboud University 4 Motivation Problems Not aqurate evidences Similar evidences Low probability evidences Infinite evidences Proposed Solution j = · 0 : 5 How do we deal with Counterexamples (so far) R eac h

5 Haifa Verification 2008 - October 28th IBM Haifa Research Lab - Israel Miguel E. Andres Radboud University 5 Motivation j = · 0 : 5 Non Determinism is allowed The property is satisfied if for every possible way to resolve the nondeterminism the reachability probability is at most 0.5 R eac h

6 Haifa Verification 2008 - October 28th IBM Haifa Research Lab - Israel Miguel E. Andres Radboud University 6 Overview Motivation Background  Markov Chains  LTL for probabilistic systems  Counterexamples Solution Reduced Case (Reachability and deterministic)  Reduction to Acyclic (SCC analysis)  Rails and Torrents Solution General Case  From general formulas to reachability  From MDPs to MCs Implementation Conclusion Future work

7 Haifa Verification 2008 - October 28th IBM Haifa Research Lab - Israel Miguel E. Andres Radboud University 7 Overview Motivation Background  Markov Chains  LTL for probabilistic systems  Counterexamples Solution Reduced Case (Reachability and deterministic)  Reduction to Acyclic (SCC analysis)  Rails and Torrents Solution General Case  From general formulas to reachability  From MDPs to MCs Implementation Conclusion Future work

8 Haifa Verification 2008 - October 28th IBM Haifa Research Lab - Israel Miguel E. Andres Radboud University 8 Backgorund Discrete Time Markov Chains DTMC = ( S ; s 0 ; L ; P ) Finite Paths s 0 s 1 s 3 s 0 s 1 s 1 s 3 s 0 s 1 s 1 s 1 s 3 s 0 s 1 s 1 s 1 s 1 s 3 s 0 s 1 s 1 s 1 s 1 s 1 s 3 Prob 0. 2 0. 1 0. 05 0. 025 0. 0125

9 Haifa Verification 2008 - October 28th IBM Haifa Research Lab - Israel Miguel E. Andres Radboud University 9 Background Linear Temporal Logic (LTL)  Sintaxis  Probabilistic Semantic D j =./ p Á, P r D ( S a t ( Á ))./ p ²./ 2 f < ; · ; > ; ¸ g ² S a t ( Á ), f ¾ 2 P a t h s ( D ) j ¾ j = Á g Á :: = V j : Á j Á ^ Á j Á U Á _ ; ! ; § ; an d ¤ aresyn t ac t i csugar  Semantic ¾ j = D v, v 2 L ( ¾ 0 ) ¾ j = D : Á, no t ( ¾ j = D Á ) ¾ j = D Á ^ °, ¾ j = D Á an d ¾ j = D ° ¾ j = D Á U °, 9 i ¸ 0 : ¾ # i j = D °an d 8 0 · j < i : ¾ # j j = D Á

10 Haifa Verification 2008 - October 28th IBM Haifa Research Lab - Israel Miguel E. Andres Radboud University 10 Backgorund Counterexamples Reachability property R emem b er: D j =./ p Á, P r D ( S a t ( Á ))./ p ² D j = · p Á : C µ S a t ( Á ) suc h t h a t P r ( C ) > p ² D j = ¸ p Á : C µ S a t ( : Á ) suc h t h a t P r ( C ) > 1 ¡ p C, P a t h s ( D ), C 1 [ C 2 C 1, f ½ 2 P a t h s ( D ) j 9 i ¸ 0 : ½ = s 0 ( s 1 ) i s 3 g C 2, f ½ 2 P a t h s ( D ) j 9 i ¸ 0 : ½ = s 0 ( s 2 ) i s 4 g Example D j = < 1 § ( v 1 _ v 2 )

11 Haifa Verification 2008 - October 28th IBM Haifa Research Lab - Israel Miguel E. Andres Radboud University 11 Overview Motivation Background  Markov Chains  LTL for probabilistic systems  Counterexamples Solution Reduced Case (Reachability and deterministic)  Reduction to Acyclic (SCC analysis)  Rails and Torrents Solution General Case  From general formulas to reachability  From MDPs to MCs Implementation Conclusion Future work

12 Haifa Verification 2008 - October 28th IBM Haifa Research Lab - Israel Miguel E. Andres Radboud University 12 Solution Reduced Case D j = · p § Ã C oun t erexamp l esaregenera t e df or A c ( D ) !!! Preserves reachability probabilities! D A c ( D ) Ac Torr D j scc = T orren t s P a t h s ( A c ( D )) = R a i l s P r ( ¾ ) = a P r ( T orr ( ¾ )) = a P r ( ¾ ) = a We focus on:

13 Haifa Verification 2008 - October 28th IBM Haifa Research Lab - Israel Miguel E. Andres Radboud University 13 Solution Reduced Case [SCC Analysis I] 1) Identify SCCs 2) Identify Input/Output states 3) Compute reachability probability from input to output states Reduction

14 Haifa Verification 2008 - October 28th IBM Haifa Research Lab - Israel Miguel E. Andres Radboud University 14 Solution Reduced Case [SCC Analysis II] 1) Identify SCCs 2) Identify Input/Output States 3) Compute reachability probability from input to output states Acyclic MC Example

15 Haifa Verification 2008 - October 28th IBM Haifa Research Lab - Israel Miguel E. Andres Radboud University 15 Subsequences Solution Reduced Case [Rails and Torrents] Issues  Freshness  Inertia Subsequences* (Torrents) ¾ ¹ !, ¾ v !an d F res h nessan d I ner t i a f ! ¾ s 0 s 2 s 5 s 11 s 14 6 ´ s 0 s 2 s 6 s 11 s 14 s 0 s 2 s 6 s 14 6 ´ s 0 s 2 s 6 s 11 s 14 ¾ v !, ex i s t ssuc h a f unc t i on S 6 S 0 S 2 S 14 S 5 S 8 S 6 S 0 S 2 S 6 S 14

16 Haifa Verification 2008 - October 28th IBM Haifa Research Lab - Israel Miguel E. Andres Radboud University 16 T orr ( ¾ ), f ! 2 P a t h s ( D ) j ¾ ¹ ! g R a i l s, P a t h s ( A c ( D )) Solution Reduced Case [Rails and Torrents] Torrents and Rails We Generate Counterexamples on the Acyclic Chain!!! T h eorem 1 ) S ¾ 2 P a t h s ( A c ( D )) T orr ( ¾ ) = P a t h s ( D ) 2 ) ¾ 6 = ¾ 0 ) T orr ( ¾ ) \ T orr ( ¾ 0 ) = ; 3 ) P r A c ( D ) ( ¾ ) = P r D ( T orr ( ¾ )) 4 ) A c ( D ) j = · p § Ã i f an d on l y i f D j = · p § Ã

17 Haifa Verification 2008 - October 28th IBM Haifa Research Lab - Israel Miguel E. Andres Radboud University 17 Overview Motivation Background  Markov Chains  LTL for probabilistic systems  Counterexamples Solution Reduced Case (Reachability and deterministic)  Reduction to Acyclic (SCC analysis)  Rails and Torrents Solution General Case  From general formulas to reachability  From MDPs to MCs Implementation Conclusion Future work

18 Haifa Verification 2008 - October 28th IBM Haifa Research Lab - Israel Miguel E. Andres Radboud University 18 General Case [Reduction to Reachability] Reduction to Reachability Á Probabilistic LTL Model Checker MDP LTL formula./ ; p Maximum Probabilities and Paths are related!!! M Deterministic Rabin Automota End Components M jj A Á M j =./ p Á

19 Haifa Verification 2008 - October 28th IBM Haifa Research Lab - Israel Miguel E. Andres Radboud University 19 The calculation of a maximal probability on a reachability problem can be performed by solving a linear minimization problem General Case [Reduction to Markov Chains I] Reduction to Markov Chains P t 2 S ¼ 1 ( t ) ¢ x t · x s P t 2 S ¼ 2 ( t ) ¢ x t · x s... P t 2 S ¼ n ( t ) ¢ x t · x s w h ere¿ ( s ) = f ¼ 1 ; ¼ 2 ;:::; ¼ n g F i n d f x s j s 2 S g t h a t m i n i m i ze P s 2 S x s su b j ec tt o t h ese t o f cons t ra i ns f ora ll s 2 S

20 Haifa Verification 2008 - October 28th IBM Haifa Research Lab - Israel Miguel E. Andres Radboud University 20 General Case [Reduction to Markov Chains II] Theorems: C i sacoun t erexamp l e t o M 0 j = · p § Ã + C i sacoun t erexamp l e t o M j = · p § Ã M 0 j = · p § Ã, M j = · p § Ã

21 Haifa Verification 2008 - October 28th IBM Haifa Research Lab - Israel Miguel E. Andres Radboud University 21 Overview Motivation Background  Markov Chains  LTL for probabilistic systems  Counterexamples Solution Reduced Case (Reachability and deterministic)  Reduction to Acyclic (SCC analysis)  Rails and Torrents Solution General Case  From general formulas to reachability  From MDPs to MCs Implementation Conclusion Future work

22 Haifa Verification 2008 - October 28th IBM Haifa Research Lab - Israel Miguel E. Andres Radboud University 22 Implementation [Computability] Reduce to MC problem  Using the output from the minimization problem [Bianco/de Alfaro] Reduce to acyclic MC  Tarjan or Kosaraju or Gabow Algorithm + steady state analysis Generate counterexamples on an Acyclic MC  K-SP problem [Han/Katoen]

23 Haifa Verification 2008 - October 28th IBM Haifa Research Lab - Israel Miguel E. Andres Radboud University 23 Implementation [Debugging Issues] Torrent Representative Expanding SCCs Reachability to: 1)Output States 2)Goal States EXPAND For Free! T or R ep ( T or ) = arg µ max ! 2 T or P r ( ! ) ¶

24 Haifa Verification 2008 - October 28th IBM Haifa Research Lab - Israel Miguel E. Andres Radboud University 24 Overview Motivation Background  Markov Chains  LTL for probabilistic systems  Counterexamples Solution Reduced Case (Reachability and deterministic)  Reduction to Acyclic (SCC analysis)  Rails and Torrents Solution General Case  From general formulas to reachability  From MDPs to MCs Implementation Conclusion Future work

25 Haifa Verification 2008 - October 28th IBM Haifa Research Lab - Israel Miguel E. Andres Radboud University 25 Conclusion Counterexample generation for probabilistic LTL without restrictions Show how to generalize counterexample generators on MC to MDP Defined the notion of Torrents as collections of paths behaving similarly Show how to compute Torrents-Counterexamples

26 Haifa Verification 2008 - October 28th IBM Haifa Research Lab - Israel Miguel E. Andres Radboud University 26 Future work Implementing a practical tool Visualization of Torrents (Regular Expressions) Case studies Extension to Timed Systems

27 Haifa Verification 2008 - October 28th IBM Haifa Research Lab - Israel Miguel E. Andres Radboud University 27 Questions Thanks for your attention!

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