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From Monotonic Transition Systems to Monotonic Games

From Monotonic Transition Systems to Monotonic Games. Parosh Aziz Abdulla Uppsala University. Outline. Model Checking Infinite-State Systems Methodology: Monotonicity Well Quasi-Orderings Models Petri Nets Lossy Channel Systems Timed Petri Nets Extension to Games.

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From Monotonic Transition Systems to Monotonic Games

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  1. From Monotonic Transition Systems to Monotonic Games Parosh Aziz Abdulla Uppsala University

  2. Outline • Model Checking • Infinite-State Systems • Methodology: • Monotonicity • Well Quasi-Orderings • Models • Petri Nets • Lossy Channel Systems • Timed Petri Nets • Extension to Games

  3. Model Checking Tsatf? transition system specification

  4. Transition System Model Checking Tsatf? transition system specification

  5. Transition System Reachability Init Fin InitreachesFin?

  6. Transition Systems Reachability Init Saftey Properties = Reachability Fin InitreachesFin?

  7. Forward Reachability Analysis

  8. Forward Reachability Analysis Post

  9. Forward Reachability Analysis Post Forward Reachability Analysis = computing Post Init Fin

  10. Backward Reachability Analysis

  11. Backward Reachability Analysis Pre

  12. Backward Reachability Analysis Pre Backward Reachability Analysis = computing Pre Init Fin

  13. Forward Reachability Analysis Init Fin Backward Reachability Analysis Init Fin

  14. Infinite-State Systems 1. Unbounded Data Structures • stacks • queues • clocks • counters, etc. 2. Unbounded Control Structures • Parameterized Systems • Dynamic Systems

  15. Backward Reachability Analysis Init Fin infinite

  16. Backward Reachability Analysis Init Fin infinite effective symbolic representation

  17. Petri Nets

  18. States = Markings

  19. Transitions

  20. Transitions t Firing t

  21. Transitions t t is disabled

  22. Monotonicity

  23. Monotonicity

  24. Monotonicity

  25. Petri Nets: infinite state

  26. Petri Nets: infinite state

  27. Petri Nets: infinite state

  28. Petri Nets: infinite state

  29. Petri Nets: infinite state

  30. Mutual Exclusion W R=1? R:=1 R:=0 C

  31. Mutual Exclusion W R=1? R:=1 R:=0 C R=1? R=1? R=1? R:=1 R:=1 R:=1 R:=0 R:=0 R:=0

  32. Mutual Exclusion R=1? R=1? R=1? R:=1 R:=1 R:=1 R:=0 R:=0 R:=0

  33. Mutual Exclusion R=1? R=1? R=1? R:=1 R:=1 R:=1 R:=0 R:=0 R:=0 • Initial states: • R=1 • All processes in Infinitely many

  34. Mutual Exclusion R=1? R=1? R=1? R:=1 R:=1 R:=1 R:=0 R:=0 R:=0 • Initial states: • R=1 • All processes in Infinitely many Bad states: Two or more processes in

  35. Mutual Exclusion R=1? R=1? R=1? R:=1 R:=1 R:=1 R:=0 R:=0 R:=0 R=1 C W

  36. R=1 C W Mutual Exclusion Set of initial states : infinite

  37. R=1 C W Mutual Exclusion

  38. R=1 C W Mutual Exclusion R=1 C W

  39. Mutual Exclusion R=1 C W

  40. R=1 C W Mutual Exclusion R=1 C W

  41. Safety Properties • mutual exclusion: • #tokens in critical section > 1 critical section

  42. Safety Properties • mutual exclusion: • #tokens in critical section > 1 Ideal = Upward closed set of markings critical section

  43. Safety Properties • mutual exclusion: • #tokens in critical section > 1 Ideal = Upward closed set of markings critical section safety = reachability of ideals

  44. Petri Nets • Concurrent systems • Infinite-state: symbolic representation • Monotonic behaviour • Safety properties: reachability of ideals

  45. Petri Nets • Concurrent systems • Infinite-state: symbolic representation • Monotonic behaviour • Safety properties: reachability of ideals

  46. Monotonicity ideals closed under computing Pre

  47. Monotonicity ideals closed under computing Pre I

  48. Monotonicity ideals closed under computing Pre I

  49. Monotonicity ideals closed under computing Pre I

  50. Monotonicity ideals closed under computing Pre Pre(I) I

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