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Application of TBDs

Application of TBDs. Technical development Ordered TBDs Operations on ordered TBDs ( ,,) Reduced ordered TBDs. M odel checking == Manipulation of TBDs. Ordered TBDs. p 1. p 2. p 3. p n. p n+1. Ordered TBDs. p n+1. - p n+1. u. x. y. z. Example. A. - B. - B. - C.

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Application of TBDs

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  1. Application of TBDs Technical development Ordered TBDs Operations on ordered TBDs (,,) Reduced ordered TBDs Model checking == Manipulation of TBDs

  2. Ordered TBDs p1 p2 p3 pn pn+1

  3. Ordered TBDs pn+1 -pn+1 u x y z

  4. Example A - B - B - C A - C - D - D - D D - A

  5. Example A - B - B - C A - C - D - D - D D - A

  6. Example A - B - D - C D C - D - D - D - D D - D

  7. Operations Negation Conjunction Abstraction s s t • x s

  8. Negation u - u x y z x y z

  9. Conjunction pn+1 u u

  10. Conjunction - pn+1 - pn+1 u

  11. Conjunction a a x y z x y z x’ y’ z’ a x’ y’ z’

  12. Conjunction - a - a x y z pn+1 x y z z x’ y’ - a z’ z’ x’ y’ z’

  13. Conjunction a - a x y z x y z x’ y’ z’ z’ - a x’ y’ z’

  14. Conjunction a a x y z x y z b/-b b/-b x’ y’ z’

  15. Conjunction - a a x y z x y z b/-b b/-b b/-b b/-b x’ y’ z’

  16. Abstraction An abstraction of a TBD on a label u = Conjunction of a simplication on –u and a simplication on u

  17. Simplification on a Label u/-u A - B - B - C A - C - D - D - D D - A Select all non-terminal nodes labeled with singed/unsignedu Replace the selected nodes with a simpler one according to given rules

  18. Simplification for a node with label u u - u u x y z x y z z

  19. Simplification for a node with label -u - u - u u x y z x y z z

  20. Abstraction on u Given a TBD. (1) Make a simplification on –u and a simplification on u (2) Make a conjunction of the two simplifications u z

  21. Existential Abstraction on u u z

  22. Properties s1 s2 t1 t2 s1 s2 s1 s2 t1 t2 • u • u s1 s2

  23. Observation: comp(s) p1 : : pn s pn+1

  24. Quantified Boolean Formulas Consider formulas with variables p1,p2,…,pn pi φ φΨ x. φ pi s s • x t s - pn+1 pn+1 pn+1 φ is valid comp( ) holds s

  25. Reduced Ordered TBDs u x y - pn+1 Not allowed x - pn+1 pn+1 x pn+1 - pn+1 y pn+1 y pn+1 x x y Non-terminal x y y y pn+1 y x y x>0

  26. Reduction Rules for u u T T’ - z - z - z - z T - z z z T T T T z T

  27. Reduction Rules for u u u - z • T z - z z T T • - z z - z z T T • T’ T’ T z T’ T • T’ T z T’ T

  28. Reduction Rules for -u - u T T’ - z z - z - z T z z z T - T T T z - T

  29. Reduction Rules for -u - u - u - z • T z - z z T T • - z z - z z T T • T’ T’ T z T’ T • T’ T z T’ T - u u z • T z - z - T z T • z z - T - z z

  30. Explanation on Some Rules (Semantics) u - u u ~x ~y - z - z - z ~x ~y z ~x ~y

  31. Explanation on Some Rules (1) u - u u ~x ~y - z - z - z - x - y z - x - y x - y z x - y - x y z - x y x y z x y

  32. Explanation on Some Rules u - u u ~x ~y - z - z - z - x - y z - x - y x y z x y

  33. Explanation on Some Rules u -u/u ~x ~y - z - z - x - x z - x x x z x

  34. Explanation on Some Rules u T T’ - z - z T T z T

  35. Explanation on Some Rules (2) u - u u ~x ~y - z - z - z - x - y z - x - y x - y z x - y - x y z - x y x y z x y

  36. Explanation on Some Rules u - u u ~x ~y - z - z - z - x - y z - x - y x y z x y

  37. Explanation on Some Rules u -u/u ~x ~y - z - z - x - x z - x x x z x

  38. Explanation on Some Rules u T T’ - z - z - z - z T - z z z T T T T z T

  39. Boolean Diagram Model Checking m variables for representing states 2m variables for representing transitions Let n=2m Construct a TBD for the formula representing the initial states Construct a TBD for the formula representing the transition relation The rest follows from the CTL model checking techniques

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