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Advanced Applications of Technical Development in Ordered TBDs Model Checking

This document explores advanced applications and operations concerning technically developed Ordered TBDs (Transition-Based Diagrams). It covers the intricacies of manipulating Ordered TBDs, including simplified reduction techniques and abstraction processes. Through various examples illustrating negation, conjunction, and simplification, this study elucidates model checking methods involving quantified Boolean formulas. Additionally, it addresses reduction rules, semantics, and the construction of models for state representations, aiming to enhance the understanding of operations related to Ordered TBDs.

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Advanced Applications of Technical Development in Ordered TBDs Model Checking

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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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