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Parallel LTL-X Model Checking of High-Level Petri Nets Based on Unfoldings

UNIVERSITY OF STUTTGART. Parallel LTL-X Model Checking of High-Level Petri Nets Based on Unfoldings. Claus Schröter* and Victor Khomenko** *University of Stuttgart, Germany **University of Newcastle upon Tyne, UK. Basis for our work. Esparza and Heljanko (ICALP 2000, SPIN 2001):

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Parallel LTL-X Model Checking of High-Level Petri Nets Based on Unfoldings

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  1. UNIVERSITY OF STUTTGART Parallel LTL-X Model Checking of High-Level Petri Nets Based on Unfoldings Claus Schröter* and Victor Khomenko** *University of Stuttgart, Germany **University of Newcastle upon Tyne, UK

  2. Basis for our work Esparza and Heljanko (ICALP 2000, SPIN 2001): A New Unfolding Approach to LTL Model-Checking • Net system is constructed as the product of • the original net system and • an Büchi automaton accepting  • Model-checking problem is reduced to detection of • illegal ω-traces and • illegal livelocks • by exploiting finite complete prefixes

  3. Basis for our work • Simplicity of this approach • Partial order semantics of Petri nets • Alleviates the state space explosion problem  Input are low level Petri nets  Low level Petri nets are not convenient for modelling

  4. ColouredPNs a good intermediate formalism High-level descriptions: • Verification is hard  Convenient for modelling Low-level PNs: • Can be efficiently verified • Not convenient for modelling Gap

  5. Coloured PNs {1,2} {1,2} 1 2 u v w<u+v w {1..4}

  6. Coloured PNs {1,2} {1,2} 1 2 u v w<u+v w {1..4}

  7. Coloured PNs {1,2} {1,2} u v w<u+v w {1..4} 1

  8. Coloured PNs {1,2} {1,2} u v w<u+v w {1..4} 2

  9. {1,2} {1,2}  1 2  u v w<u+v w {1..4} Expansion

  10. {1,2} {1,2} 1 2 u v w<u+v w {1..4} Expansion  

  11. {1,2} {1,2} 1 2 u v w<u+v w {1..4} Expansion 

  12. {1,2} {1,2} 1 2 u v w<u+v w {1..4} Expansion  

  13. {1,2} {1,2} 1 2 u v w<u+v w {1..4} Expansion 

  14.  Expansion {1,2} {1,2} 1 2 • The expansion faithfully models the original net u v w<u+v w {1..4}  Blow up in size

  15. Finite complete prefix • Introduced by McMillan in 1992 • Relies on the partial order view of concurrent computation • Represents system states implicitly, using an acyclic net • Satisfies two key properties: • Completeness: Each reachable marking of the original net is represented by at least one reachable marking in the prefix • Finiteness: The prefix is finite and thus can be used as an input to model-checking algorithms

  16. Relationship diagram expansion Low-level PNs Coloured PNs unfolding unfolding ? Low-level prefix Coloured prefix

  17. Relationship diagram expansion Low-level PNs Coloured PNs unfolding unfolding ~ Low-level prefix Coloured prefix Khomenko and Koutny proved isomorphism (TACAS’03)

  18.  1 2 u=1 v=2 w=1 u=1 v=2 w=2 1 2 Relationship diagram {1,2} {1,2} 1 2 u v w<u+v w {1..4}

  19. p1:{0,1} p3:{0,1} 0 1 a a a a t1 t2 t3 a a b b p2:{0,1} p4:{0,1} Example: Buffer of capacity 2

  20. Example: Buffer of capacity 2 p1:{0,1} p3:{0,1} 0 1 a a a a t1 t2 t3 a a b b p2:{0,1} p4:{0,1}

  21. Example: Buffer of capacity 2 p1:{0,1} p3:{0,1} 0 a a a a t1 t2 t3 a a b b 1 p2:{0,1} p4:{0,1}

  22. Example: Buffer of capacity 2 p1:{0,1} p3:{0,1} 0 a a a a t1 t2 t3 a a b b 1 p2:{0,1} p4:{0,1}

  23. Example: Buffer of capacity 2 p1:{0,1} p3:{0,1} 1 0 a a a a t1 t2 t3 a a b b p2:{0,1} p4:{0,1}

  24. Example: Buffer of capacity 2 p1:{0,1} p3:{0,1} 0 a a a a t1 t2 t3 a a b b 1 p2:{0,1} p4:{0,1}

  25. Example: Buffer of capacity 2 p1:{0,1} p3:{0,1} a a a a t1 t2 t3 a a b b 1 0 p2:{0,1} p4:{0,1}

  26. p1:{0,1} p3:{0,1} (p2≠0) 0 1 true a a a a true q0 q1 t1 t2 t3 (p2≠0) a a b b I0 q1:{} q0:{} p2:{0,1} p4:{0,1} u0 u1 Example: Buffer of capacity 2 Property: φ = ◊□(p2≠0) Büchi automaton Aφ

  27. Synchronisation • Standard technique: Synchronisation on all transitions  Synchronisation sequentialises the system  Not suitable for unfolding based verification • Solution: Synchronisation just on those transitions which ‘touch’ the atomic propositions of the formula  Concurrency can be exploited

  28. p1:{0,1} p3:{0,1} 0 1 a a a a t1 t2 t3 a a b b q1:{} q0:{} p2:{0,1} p4:{0,1} u0 u1 I0 (p2≠0) Synchronisation

  29. p1:{0,1} p3:{0,1} 0 1 a a a a t1 t2 t3 a a b b q1:{} q0:{} p2:{0,1} p4:{0,1} u0 u1 I0 (p2≠0) Synchronisation p2 p2

  30. p1:{0,1} p3:{0,1} 0 1 a a a a t1 t2 t3 a a b b q1:{} q0:{} p2:{0,1} p4:{0,1} u0 u1 I0 (p2≠0) Synchronisation S:{} B:{} p2 p2

  31. S:{} p1:{0,1} p3:{0,1} 0 1 a a a a t1 t2 t3 a a b b q1:{} q0:{} p2:{0,1} p4:{0,1} u0 u1 I0 (p2≠0) Synchronisation B:{} p2 p2

  32. S:{} p1:{0,1} p3:{0,1} 0 1 a a a a t1 t2 t3 a a b b q1:{} q0:{} p2:{0,1} p4:{0,1} u0 u1 I0 (p2≠0) Synchronisation B:{} p2 p2

  33. p1:{0,1} p3:{0,1} 0 1 a a a a t1 t2 t3 B:{} a a b b q1:{} q0:{} p2:{0,1} p4:{0,1} u0 u1 I0 (p2≠0) Synchronisation S:{} p2 p2

  34. p1:{0,1} p3:{0,1} 0 1 a a a a t1 t2 t3 B:{} a a b b q1:{} q0:{} p2:{0,1} p4:{0,1} u0 u1 I0 (p2≠0) Synchronisation S:{} p2 p2

  35. S:{} p1:{0,1} p3:{0,1} 0 1 a a a a t1 t2 t3 B:{} a a b b q1:{} q0:{} p2:{0,1} p4:{0,1} u0 u1 I0 (p2≠0) Synchronisation p2 p2

  36. Illegal ω-traces • Infinite transition sequence that touches q1 infinitely often violatesφ • To detect such runs we introduce a set I off all transitions putting a token into an accepting Büchi place • An infinite transition sequence of the synchronised net which is fireable from the initial marking and contains infinitely many occurrences of I-transitions violates φ (illegal ω-trace)

  37. Prefix p10 p31 p31 t3 p41 S q0 I0 S u0 p10 q0 q0 B

  38. Prefix p10 p31 p31 t3 p41 S q0 I0 S u0 p10 q0 q0 B

  39. Prefix p10 p31 p31 t3 p41 S q0 I0 S u0 p10 q0 q0 B

  40. Prefix p10 p31 p31 t3 p41 S q0 I0 S u0 p10 q0 q0 B

  41. Prefix p10 p31 p31 t3 p41 S q0 I0 S u0 p10 q0 q0 B

  42. Experimental Results

  43. More Results

  44. More Results

  45. Results for Parallel Mode

  46. Conclusions • Efficient parallel LTL-X model-checker for high level Petri nets • Based on partial order techniques (unfoldings) • Alleviates the state space explosion problem • Experimental results showed a good performance of our checker for several examples

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