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# Saarbruecken

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Talk at TACAS 2011 by Karsten Wolf

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### Saarbruecken

1. 1. Harro Wimmel, Karsten WolfApplying CEGAR to the Petri Net State Equation
2. 2. CEGARCounterexample GuidedAbstraction Refinement 5 ? - - + 1 2 4 ? 3 - x + + Karsten Wolf: CEGAR / PN State Equation
3. 3. 1 s1 s3 t3Petri Net t1 t2 t5 s5 t4 s2 s4 1 –1 0 0 0 -1 1 0 0 0 C= 0 1 –1 0 1 0 -1 0 1 0 0 0 1 -1 -1 Incidence   Matrix Karsten Wolf: CEGAR / PN State Equation
4. 4. If m0 t t t t t 2 3 5 1 3 m then 1 m0 + t2 + t3 + t5 + t1 + t3 = m m0 + (1•t1) + (1•t2) + (2•t3) + (0•t4) + (1•t5) = m m0 + C • (1,1,2,0,1) = m Parikh-vector of t2 t3 t5 t1 t3 •If m0 w m then m0 + C • Parikh(w) = m • Marking m is reachable only if C • x = (m – m0) is feasible for natural xThe State Equation Karsten Wolf: CEGAR / PN State Equation
5. 5. 1 Example t1 t3 t5 t2 t4reachable marking: corresponding solutions (1,0,0,0,0) ( 0 , 0 , 0 , 0 , 0), ( 1 , 0 , 1 , 0 , 1), ... (0,1,0,0,1) ( 1 , 0 , 0 , 0 , 0), ... (0,0,1,1,0) ( 0 , 1 , 0 , 0 , 0), ... (0,0,0,1,1) ( 1 , 0 , 1 , 0 , 0), ( 0 , 1 , 0 , 1 , 0), ... not reachable has solution, though! :-( (0,1,1,0,0) ( 1 , 1 , 0 , 0 , 1), ... Karsten Wolf: CEGAR / PN State Equation
6. 6. 1 Solution Spaceb1, b2, …, bj: base solutions: incomparable, minimali1, i2, …, ik: increments, solutions to C • x = 0 bi + n1 i1 + … nk ik Karsten Wolf: CEGAR / PN State Equation
7. 7. 1 The role of increments t1 t2 u1 u2t1 t2 unrealizable „to lend tokens“u1 t1 t2 u2 realizable Karsten Wolf: CEGAR / PN State Equation
8. 8. CEGARCounterexample GuidedAbstraction Refinement 5 ? - - + 1 2 4 ? 3 - x + + Karsten Wolf: CEGAR / PN State Equation
9. 9. 2 Checking solutions ?Given solution x, explore state space, but fire t at most x(t) times-Apply partial order reduction technique of [Schmidt, Petri Nets `99] .... does not insert invisible transitions .... does not require invisible transitions-Skip sequences that are covered by other solutions Karsten Wolf: CEGAR / PN State Equation
10. 10. CEGARCounterexample GuidedAbstraction Refinement 5 ? - - + 1 2 4 ? 3 - x + + Karsten Wolf: CEGAR / PN State Equation
11. 11. 3 Refinement xincrement jump Karsten Wolf: CEGAR / PN State Equation
12. 12. 3Problem: Increment after Jump x b jump b increment? transform jump! Karsten Wolf: CEGAR / PN State Equation
13. 13. CEGARCounterexample GuidedAbstraction Refinement 5 ? - - + 1 2 4 ? 3 - x + + Karsten Wolf: CEGAR / PN State Equation
14. 14. Tool: Sara 4(Structures for Automated Reachability Analysis)http://service-technology.org/tools/downloadExperiments:-590 business processes (20-300 transitions) (thanks to Jana Köhler, IBM) checked for „relaxed soundness“ (510 are, 80 are not):Sara: 198 sec LoLA: failed on 17 instances 24 min on the othersHardest instance:12278 calls to lp_solve, 24 sec Karsten Wolf: CEGAR / PN State Equation
15. 15. Tool: Sara 4(Structures for Automated Reachability Analysis)http://service-technology.org/tools/downloadExperiments:-4 models with context in verification of parameterized booleanprograms (thanks to Daniel Kröning, Alexander Kaiser) checked for coverabilitySara: 0 sec LoLA: failed on 1, 0 sec on the others Karsten Wolf: CEGAR / PN State Equation
16. 16. Tool: Sara 4(Structures for Automated Reachability Analysis)http://service-technology.org/tools/downloadExperiments:-1 challenge example from Petri Net mailing list (thanks to Hubert Garavel) 776 transitions checked for quasi-livenessSara: 26 sec LoLA: 41 sec but tricks were needed on 2 instancesWitness paths:Sara: <30 LoLA: up to 6000 Karsten Wolf: CEGAR / PN State Equation
17. 17. CEGARCounterexample GuidedAbstraction Refinement 5 ? - - + 1 2 4 ? 3 - x + + Karsten Wolf: CEGAR / PN State Equation
18. 18. Diagnosing "no" 5 -i o Karsten Wolf: CEGAR / PN State Equation
19. 19. ConclusionApplied CEGAR to a structural verification technique+ performant+ short witnesses+ excellent behavior on negative instances+ applicable to infinte state systems+ diagnostic information in negative cases- inherently incomplete ( reachability for Petri nets is EXSPACE hard)Traverse solutions rather than states Karsten Wolf: CEGAR / PN State Equation