vldl satisfiability and model checking via tree automata
play

VLDL Satisfiability and Model Checking via Tree Automata Alexander - PowerPoint PPT Presentation

Mar 04, 2023 •154 likes •1.91k views

VLDL Satisfiability and Model Checking via Tree Automata Alexander Weinert Saarland University December 12th, 2017 FSTTCS 2017 - Kanpur, India Alexander Weinert Saarland University VLDL Satisfiability and Model Checking 1/26 VLDL VLDL

  1. Visibly Linear Dynamic Logic (VLDL) 1 Syntax of VLDL : guarded eventually ϕ = p | ¬ ϕ | ϕ ∧ ϕ | ϕ ∨ ϕ | � A � ϕ | [ A ] ϕ guarded always � A � ϕ a a b c a a b c b b a b a b c a a . . . ϕ ϕ ϕ or or . . . or [ A ] ϕ a a b c a a b c b b a b a b c a a . . . ϕ . . . ϕ ϕ and and and 1 (W. and Zimmermann, 2016) Alexander Weinert Saarland University VLDL Satisfiability and Model Checking 4/26

  2. Visibly Linear Dynamic Logic (VLDL) 1 Syntax of VLDL : guarded eventually ϕ = p | ¬ ϕ | ϕ ∧ ϕ | ϕ ∨ ϕ | � A � ϕ | [ A ] ϕ guarded always � A � ϕ a a b c a a b c b b a b a b c a a . . . ϕ ϕ ϕ or or . . . or [ A ] ϕ a a b c a a b c b b a b a b c a a . . . ϕ . . . ϕ ϕ and and and 1 (W. and Zimmermann, 2016) Alexander Weinert Saarland University VLDL Satisfiability and Model Checking 4/26

  3. Visibly Linear Dynamic Logic (VLDL) 1 Syntax of VLDL : guarded eventually ϕ = p | ¬ ϕ | ϕ ∧ ϕ | ϕ ∨ ϕ | � A � ϕ | [ A ] ϕ guarded always � A � ϕ a a b c a a b c b b a b a b c a a . . . ϕ ϕ ϕ or or . . . or [ A ] ϕ a a b c a a b c b b a b a b c a a . . . ϕ . . . ϕ ϕ and and and 1 (W. and Zimmermann, 2016) Alexander Weinert Saarland University VLDL Satisfiability and Model Checking 4/26

  4. Visibly Pushdown Automata 2 Visibly Pushdown Automata are restricted Pushdown Automata Σ When reading call, automaton has to push onto the stack When reading return, automaton has to pop off the stack When reading local action, automaton has to ignore the stack ⇒ Closed under intersection! 2 (Alur and Madhusudan, 2005) Alexander Weinert Saarland University VLDL Satisfiability and Model Checking 5/26

  5. Visibly Pushdown Automata 2 Visibly Pushdown Automata are restricted Pushdown Automata Σ When reading call, automaton has to push onto the stack When reading return, automaton has to pop off the stack When reading local action, automaton has to ignore the stack ⇒ Closed under intersection! 2 (Alur and Madhusudan, 2005) Alexander Weinert Saarland University VLDL Satisfiability and Model Checking 5/26

  6. Visibly Pushdown Automata 2 Visibly Pushdown Automata are restricted Pushdown Automata Σ Local Calls Returns Actions When reading call, automaton has to push onto the stack When reading return, automaton has to pop off the stack When reading local action, automaton has to ignore the stack ⇒ Closed under intersection! 2 (Alur and Madhusudan, 2005) Alexander Weinert Saarland University VLDL Satisfiability and Model Checking 5/26

  7. Visibly Pushdown Automata 2 Visibly Pushdown Automata are restricted Pushdown Automata Σ Local Calls Returns Actions When reading call, automaton has to push onto the stack When reading return, automaton has to pop off the stack When reading local action, automaton has to ignore the stack ⇒ Closed under intersection! 2 (Alur and Madhusudan, 2005) Alexander Weinert Saarland University VLDL Satisfiability and Model Checking 5/26

  8. Visibly Pushdown Automata 2 Visibly Pushdown Automata are restricted Pushdown Automata Σ Local Calls Returns Actions When reading call, automaton has to push onto the stack When reading return, automaton has to pop off the stack When reading local action, automaton has to ignore the stack ⇒ Closed under intersection! 2 (Alur and Madhusudan, 2005) Alexander Weinert Saarland University VLDL Satisfiability and Model Checking 5/26

  9. Visibly Pushdown Automata 2 Visibly Pushdown Automata are restricted Pushdown Automata Σ Local Calls Returns Actions When reading call, automaton has to push onto the stack When reading return, automaton has to pop off the stack When reading local action, automaton has to ignore the stack ⇒ Closed under intersection! 2 (Alur and Madhusudan, 2005) Alexander Weinert Saarland University VLDL Satisfiability and Model Checking 5/26

  10. VLDL Complexity 3 VLDL : Extension of LTL , temporal operators guarded by visibly pushdown automata Satisfiability ExpTime -complete Model Checking ExpTime -complete Games 3ExpTime -complete Contribution: Novel, conceptually simple algorithms 3 (W. and Zimmermann, 2016) Alexander Weinert Saarland University VLDL Satisfiability and Model Checking 6/26

  11. VLDL Complexity 3 VLDL : Extension of LTL , temporal operators guarded by visibly pushdown automata Satisfiability ExpTime -complete Model Checking ExpTime -complete Games 3ExpTime -complete Contribution: Novel, conceptually simple algorithms 3 (W. and Zimmermann, 2016) Alexander Weinert Saarland University VLDL Satisfiability and Model Checking 6/26

  12. VLDL Complexity 3 VLDL : Extension of LTL , temporal operators guarded by visibly pushdown automata Satisfiability ExpTime -complete Model Checking ExpTime -complete Games 3ExpTime -complete Contribution: Novel, conceptually simple algorithms 3 (W. and Zimmermann, 2016) Alexander Weinert Saarland University VLDL Satisfiability and Model Checking 6/26

  13. VLDL Complexity 3 VLDL : Extension of LTL , temporal operators guarded by visibly pushdown automata Satisfiability ExpTime -complete Model Checking ExpTime -complete Games 3ExpTime -complete Contribution: Novel, conceptually simple algorithms 3 (W. and Zimmermann, 2016) Alexander Weinert Saarland University VLDL Satisfiability and Model Checking 6/26

  14. VLDL Complexity 3 VLDL : Extension of LTL , temporal operators guarded by visibly pushdown automata Satisfiability ExpTime -complete ExpTime -complete Model Checking ExpTime -complete ExpTime -complete Games 3ExpTime -complete Contribution: Novel, conceptually simple algorithms 3 (W. and Zimmermann, 2016) Alexander Weinert Saarland University VLDL Satisfiability and Model Checking 6/26

  15. VLDL Satisfiability and Model Checking Theorem (W. and Zimmermann, 2016) VLDL Satisfiability is ExpTime -complete. VLDL 1 − AJA VPA ϕ A aja A vpa ✓ / ✗ 1. What are 1 − AJA s? 2. How to translate 1 − AJA s into tree automata? Alexander Weinert Saarland University VLDL Satisfiability and Model Checking 7/26

  16. VLDL Satisfiability and Model Checking Theorem (W. and Zimmermann, 2016) VLDL Satisfiability is ExpTime -complete. VLDL 1 − AJA VPA ϕ A aja A vpa ✓ / ✗ 1. What are 1 − AJA s? 2. How to translate 1 − AJA s into tree automata? Alexander Weinert Saarland University VLDL Satisfiability and Model Checking 7/26

  17. VLDL Satisfiability and Model Checking Theorem (W. and Zimmermann, 2016) VLDL Satisfiability is ExpTime -complete. VLDL 1 − AJA VPA ϕ A aja A vpa ✓ / ✗ 1. What are 1 − AJA s? 2. How to translate 1 − AJA s into tree automata? Alexander Weinert Saarland University VLDL Satisfiability and Model Checking 7/26

  18. VLDL Satisfiability and Model Checking Theorem (W. and Zimmermann, 2016) VLDL Satisfiability is ExpTime -complete. VLDL 1 − AJA VPA ϕ A aja A vpa ✓ / ✗ 1. What are 1 − AJA s? 2. How to translate 1 − AJA s into tree automata? Alexander Weinert Saarland University VLDL Satisfiability and Model Checking 7/26

  19. VLDL Satisfiability and Model Checking Theorem (W. and Zimmermann, 2016) VLDL Satisfiability is ExpTime -complete. 1 − AJA VLDL VPA ϕ A aja A vpa ✓ / ✗ 1. What are 1 − AJA s? 2. How to translate 1 − AJA s into tree automata? Alexander Weinert Saarland University VLDL Satisfiability and Model Checking 7/26

  20. VLDL Satisfiability and Model Checking Theorem (W. and Zimmermann, 2016) VLDL Satisfiability is ExpTime -complete. Tree VLDL 1 − AJA Aut. ϕ A aja ✓ / ✗ T 1. What are 1 − AJA s? 2. How to translate 1 − AJA s into tree automata? Alexander Weinert Saarland University VLDL Satisfiability and Model Checking 7/26

  21. VLDL Satisfiability and Model Checking Theorem (W. and Zimmermann, 2016) VLDL Satisfiability is ExpTime -complete. Tree VLDL 1 − AJA Aut. ϕ A aja ✓ / ✗ T 1. What are 1 − AJA s? 2. How to translate 1 − AJA s into tree automata? Alexander Weinert Saarland University VLDL Satisfiability and Model Checking 7/26

  22. VLDL Satisfiability and Model Checking Theorem (W. and Zimmermann, 2016) VLDL Satisfiability is ExpTime -complete. Tree VLDL 1 − AJA Aut. ϕ A aja ✓ / ✗ T 1. What are 1 − AJA s? 2. How to translate 1 − AJA s into tree automata? Alexander Weinert Saarland University VLDL Satisfiability and Model Checking 7/26

  23. Intermediate Automata: 1 − AJA 4 Extension of alternating automata: δ ( q , a ) = q 1 ∧ ( q 2 ∨ q 3 ). Stack Height 3 2 1 0 . . . c r c c r c l l l l · · · 4 (Bozzelli, 2007) Alexander Weinert Saarland University VLDL Satisfiability and Model Checking 8/26

  24. Intermediate Automata: 1 − AJA 4 Extension of alternating automata: δ ( q , a ) = q 1 ∧ ( q 2 ∨ q 3 ). Stack Height 3 2 1 0 . . . c r c c r c l l l l · · · 4 (Bozzelli, 2007) Alexander Weinert Saarland University VLDL Satisfiability and Model Checking 8/26

  25. Intermediate Automata: 1 − AJA 4 Extension of alternating automata: δ ( q , a ) = q 1 ∧ ( q 2 ∨ q 3 ). Stack Height 3 2 1 0 . . . c r c c r c l l l l · · · 4 (Bozzelli, 2007) Alexander Weinert Saarland University VLDL Satisfiability and Model Checking 8/26

  26. Intermediate Automata: 1 − AJA 4 Extension of alternating automata: δ ( q , a ) = q 1 ∧ ( q 2 ∨ q 3 ). Stack Height 3 2 1 0 . . . c r c c r c l l l l · · · 4 (Bozzelli, 2007) Alexander Weinert Saarland University VLDL Satisfiability and Model Checking 8/26

  27. Intermediate Automata: 1 − AJA 4 Extension of alternating automata: δ ( q , a ) = q 1 ∧ ( q 2 ∨ q 3 ). Stack Height 3 2 1 0 . . . c r c c r c l l l l · · · 4 (Bozzelli, 2007) Alexander Weinert Saarland University VLDL Satisfiability and Model Checking 8/26

  28. Intermediate Automata: 1 − AJA 4 Extension of alternating automata: δ ( q , a ) = q 1 ∧ ( q 2 ∨ q 3 ). Stack Height 3 2 1 0 . . . c r c c r c l l l l · · · 4 (Bozzelli, 2007) Alexander Weinert Saarland University VLDL Satisfiability and Model Checking 8/26

  29. Intermediate Automata: 1 − AJA 4 Extension of alternating automata: δ ( q , a ) = q 1 ∧ ( q 2 ∨ q 3 ). Stack Height 3 2 1 0 . . . c r c c r c l l l l · · · 4 (Bozzelli, 2007) Alexander Weinert Saarland University VLDL Satisfiability and Model Checking 8/26

  30. Intermediate Automata: 1 − AJA 4 Extension of alternating automata: δ ( q , a ) = q 1 ∧ ( q 2 ∨ q 3 ). Stack Height 3 2 1 0 . . . c r c c r c l l l l · · · 4 (Bozzelli, 2007) Alexander Weinert Saarland University VLDL Satisfiability and Model Checking 8/26

  31. Intermediate Automata: 1 − AJA 4 Extension of alternating automata: δ ( q , a ) = q 1 ∧ ( q 2 ∨ q 3 ). Stack Height 3 2 1 0 . . . c r c c r c l l l l · · · 4 (Bozzelli, 2007) Alexander Weinert Saarland University VLDL Satisfiability and Model Checking 8/26

  32. Intermediate Automata: 1 − AJA 4 Extension of alternating automata: δ ( q , a ) = q 1 ∧ ( q 2 ∨ q 3 ). Stack Height 3 2 1 0 . . . c r c c r c l l l l · · · 4 (Bozzelli, 2007) Alexander Weinert Saarland University VLDL Satisfiability and Model Checking 8/26

  33. Intermediate Automata: 1 − AJA 4 Extension of alternating automata: δ ( q , a ) = q 1 ∧ ( q 2 ∨ q 3 ). Stack Height 3 2 1 0 . . . c r c c r c l l l l · · · 4 (Bozzelli, 2007) Alexander Weinert Saarland University VLDL Satisfiability and Model Checking 8/26

  34. Intermediate Automata: 1 − AJA 4 Extension of alternating automata: δ ( q , a ) = q 1 ∧ ( q 2 ∨ q 3 ). Stack Height 3 2 1 0 . . . c r c c r c l l l l · · · 4 (Bozzelli, 2007) Alexander Weinert Saarland University VLDL Satisfiability and Model Checking 8/26

  35. Intermediate Automata: 1 − AJA 4 Extension of alternating automata: δ ( q , a ) = q 1 ∧ ( q 2 ∨ q 3 ). Stack Height 3 2 1 0 . . . c r c c r c l l l l · · · 4 (Bozzelli, 2007) Alexander Weinert Saarland University VLDL Satisfiability and Model Checking 8/26

  36. Intermediate Automata: 1 − AJA 4 Extension of alternating automata: δ ( q , a ) = q 1 ∧ ( q 2 ∨ q 3 ). Stack Height 3 2 1 0 . . . c r c c r c l l l l · · · 4 (Bozzelli, 2007) Alexander Weinert Saarland University VLDL Satisfiability and Model Checking 8/26

  37. Intermediate Automata: 1 − AJA 4 Extension of alternating automata: δ ( q , a ) = q 1 ∧ ( q 2 ∨ q 3 ). Stack Height 3 2 1 0 . . . c r c c r c l l l l · · · 4 (Bozzelli, 2007) Alexander Weinert Saarland University VLDL Satisfiability and Model Checking 8/26

  38. Intermediate Automata: 1 − AJA · · · Acceptance: Each branch visits accepting states infinitely often. Theorem (Ext. of (W. and Zimmermann, 2016)) For each VLDL formula there exists an equivalent 1 − AJA of polynomial size. Alexander Weinert Saarland University VLDL Satisfiability and Model Checking 9/26

  39. Intermediate Automata: 1 − AJA · · · Acceptance: Each branch visits accepting states infinitely often. Theorem (Ext. of (W. and Zimmermann, 2016)) For each VLDL formula there exists an equivalent 1 − AJA of polynomial size. Alexander Weinert Saarland University VLDL Satisfiability and Model Checking 9/26

  40. Intermediate Automata: 1 − AJA · · · Acceptance: Each branch visits accepting states infinitely often. Theorem (Ext. of (W. and Zimmermann, 2016)) For each VLDL formula there exists an equivalent 1 − AJA of polynomial size. Alexander Weinert Saarland University VLDL Satisfiability and Model Checking 9/26

  41. Guiding Questions 1. What are 1 − AJA s? ✓ 2. How to translate 1 − AJA s into tree automata? Alexander Weinert Saarland University VLDL Satisfiability and Model Checking 10/26

  42. Guiding Questions 1. What are 1 − AJA s? ✓ 2. How to translate 1 − AJA s into tree automata? Alexander Weinert Saarland University VLDL Satisfiability and Model Checking 10/26

  43. Guiding Questions 1. What are 1 − AJA s? ✓ 2. How to translate 1 − AJA s into tree automata? How to translate words into trees? Alexander Weinert Saarland University VLDL Satisfiability and Model Checking 10/26

  44. Encoding Words as Trees c r c c r c c l l l l · · · ✗ ⊥ ⊥ ⊥ ⊥ ⊥ ⊥ l l · · · l ⊥ c c ⊥ c r c l · · · r c ⊥ ⊥ Adapted from (Alur and Madhusudan, 2004) Alexander Weinert Saarland University VLDL Satisfiability and Model Checking 11/26

  45. Encoding Words as Trees c r c c r c c l l l l · · · ✗ ⊥ ⊥ ⊥ ⊥ ⊥ ⊥ l l · · · l ⊥ c c ⊥ c r c l · · · r c ⊥ ⊥ Adapted from (Alur and Madhusudan, 2004) Alexander Weinert Saarland University VLDL Satisfiability and Model Checking 11/26

  46. Encoding Words as Trees c r c c r c c l l l l · · · ✗ ⊥ ⊥ ⊥ ⊥ ⊥ ⊥ l l · · · l ⊥ c c ⊥ c r c l · · · r c ⊥ ⊥ Adapted from (Alur and Madhusudan, 2004) Alexander Weinert Saarland University VLDL Satisfiability and Model Checking 11/26

  47. Encoding Words as Trees c r c c r c c l l l l · · · ✗ ⊥ ⊥ ⊥ ⊥ ⊥ ⊥ l l · · · l ⊥ c c ⊥ c r c l · · · r c ⊥ ⊥ Adapted from (Alur and Madhusudan, 2004) Alexander Weinert Saarland University VLDL Satisfiability and Model Checking 11/26

  48. Encoding Words as Trees c r c c r c c l l l l · · · ✗ ⊥ ⊥ ⊥ ⊥ ⊥ ⊥ l l · · · l ⊥ c c ⊥ c r c l · · · r c ⊥ ⊥ Adapted from (Alur and Madhusudan, 2004) Alexander Weinert Saarland University VLDL Satisfiability and Model Checking 11/26

  49. Encoding Words as Trees c r c c r c c l l l l · · · ✗ ⊥ ⊥ ⊥ ⊥ ⊥ ⊥ l l · · · l ⊥ c c ⊥ c r c l · · · r c ⊥ ⊥ Adapted from (Alur and Madhusudan, 2004) Alexander Weinert Saarland University VLDL Satisfiability and Model Checking 11/26

  50. Encoding Words as Trees c r c c r c c l l l l · · · ✗ ⊥ ⊥ ⊥ ⊥ ⊥ ⊥ l l · · · l ⊥ c c ⊥ c r c l · · · r c ⊥ ⊥ Adapted from (Alur and Madhusudan, 2004) Alexander Weinert Saarland University VLDL Satisfiability and Model Checking 11/26

  51. Encoding Words as Trees c r c c r c c l l l l · · · ✗ ⊥ ⊥ ⊥ ⊥ ⊥ ⊥ l l · · · l ⊥ c c ⊥ c r c l · · · r c ⊥ ⊥ Adapted from (Alur and Madhusudan, 2004) Alexander Weinert Saarland University VLDL Satisfiability and Model Checking 11/26

  52. Encoding Words as Trees c r c c r c c l l l l · · · ✗ ⊥ ⊥ ⊥ ⊥ ⊥ ⊥ l l · · · l ⊥ c c ⊥ c r c l · · · r c ⊥ ⊥ Adapted from (Alur and Madhusudan, 2004) Alexander Weinert Saarland University VLDL Satisfiability and Model Checking 11/26

  53. Encoding Words as Trees c r c c r c c l l l l · · · ✗ ⊥ ⊥ ⊥ ⊥ ⊥ ⊥ l l · · · l ⊥ c c ⊥ c r c l · · · r c ⊥ ⊥ Adapted from (Alur and Madhusudan, 2004) Alexander Weinert Saarland University VLDL Satisfiability and Model Checking 11/26

  54. Encoding Words as Trees c r c c r c c l l l l · · · ✗ ⊥ ⊥ ⊥ ⊥ ⊥ ⊥ l l · · · l ⊥ c c ⊥ c r c l · · · r c ⊥ ⊥ Adapted from (Alur and Madhusudan, 2004) Alexander Weinert Saarland University VLDL Satisfiability and Model Checking 11/26

  55. Encoding Words as Trees c r c c r c c l l l l · · · ✗ ⊥ ⊥ ⊥ ⊥ ⊥ ⊥ l l · · · l ⊥ c c ⊥ c r c l · · · r c ⊥ ⊥ Adapted from (Alur and Madhusudan, 2004) Alexander Weinert Saarland University VLDL Satisfiability and Model Checking 11/26

  56. Encoding Words as Trees c r c c r c c l l l l · · · ✗ ⊥ ⊥ ⊥ ⊥ ⊥ ⊥ l l · · · l ⊥ c c ⊥ c r c l · · · r c ⊥ ⊥ Adapted from (Alur and Madhusudan, 2004) Alexander Weinert Saarland University VLDL Satisfiability and Model Checking 11/26

  57. Encoding Words as Trees c r c c r c c l l l l · · · ✗ ⊥ ⊥ ⊥ ⊥ ⊥ ⊥ l l · · · l ⊥ c c ⊥ c r c l · · · r c ⊥ ⊥ Adapted from (Alur and Madhusudan, 2004) Alexander Weinert Saarland University VLDL Satisfiability and Model Checking 11/26

  58. Encoding Words as Trees c r c c r c c l l l l · · · ✗ ⊥ ⊥ ⊥ ⊥ ⊥ ⊥ l l · · · l ⊥ c c ⊥ c r c l · · · r c ⊥ ⊥ Adapted from (Alur and Madhusudan, 2004) Alexander Weinert Saarland University VLDL Satisfiability and Model Checking 11/26

  59. Guiding Questions 1. What are 1 − AJA s? ✓ 2. How to translate 1 − AJA s into tree automata? How to translate words into trees? ✓ Alexander Weinert Saarland University VLDL Satisfiability and Model Checking 12/26

  60. Guiding Questions 1. What are 1 − AJA s? ✓ 2. How to translate 1 − AJA s into tree automata? How to translate words into trees? ✓ Alexander Weinert Saarland University VLDL Satisfiability and Model Checking 12/26

  61. Guiding Questions 1. What are 1 − AJA s? ✓ 2. How to translate 1 − AJA s into tree automata? How to translate words into trees? ✓ Alexander Weinert Saarland University VLDL Satisfiability and Model Checking 12/26

  62. Overview Tree VLDL 1 − AJA Aut. ϕ A aja ✓ / ✗ T Alexander Weinert Saarland University VLDL Satisfiability and Model Checking 13/26

  63. Overview Tree VLDL 1 − AJA Aut. ϕ A aja ✓ / ✗ T Alexander Weinert Saarland University VLDL Satisfiability and Model Checking 13/26

  64. Tree Automata ⊥ ⊥ ⊥ ⊥ ⊥ l l · · · c c ⊥ ⊥ ⊥ c r c l · · · r r c ⊥ ⊥ Acceptance: Every branch has infinitely many accepting vertices Component Technique States ? Transitions ? Accepting States ? Alexander Weinert Saarland University VLDL Satisfiability and Model Checking 14/26

  65. Tree Automata ⊥ ⊥ ⊥ ⊥ ⊥ l l · · · c c ⊥ ⊥ ⊥ q 1 c r c l · · · r r c ⊥ ⊥ Acceptance: Every branch has infinitely many accepting vertices Component Technique States ? Transitions ? Accepting States ? Alexander Weinert Saarland University VLDL Satisfiability and Model Checking 14/26

  66. Tree Automata ⊥ ⊥ ⊥ ⊥ ⊥ l l · · · c c ⊥ ⊥ ⊥ q 1 q 2 c r c l · · · r r c ⊥ ⊥ Acceptance: Every branch has infinitely many accepting vertices Component Technique States ? Transitions ? Accepting States ? Alexander Weinert Saarland University VLDL Satisfiability and Model Checking 14/26

  67. Tree Automata ⊥ ⊥ q 3 ⊥ ⊥ ⊥ l l · · · c c ⊥ ⊥ ⊥ q 1 q 2 c r c l · · · q 4 r r c ⊥ ⊥ Acceptance: Every branch has infinitely many accepting vertices Component Technique States ? Transitions ? Accepting States ? Alexander Weinert Saarland University VLDL Satisfiability and Model Checking 14/26

  68. Tree Automata ⊥ ⊥ q 3 ⊥ ⊥ ⊥ l l · · · c c ⊥ ⊥ ⊥ q 1 q 2 c r c l · · · q 4 r r c ⊥ ⊥ Acceptance: Every branch has infinitely many accepting vertices Component Technique States ? Transitions ? Accepting States ? Alexander Weinert Saarland University VLDL Satisfiability and Model Checking 14/26

  69. Tree Automata ⊥ ⊥ q 3 q 8 ⊥ ⊥ ⊥ l l · · · q 7 q 10 q 11 c c ⊥ ⊥ ⊥ q 1 q 2 q 9 c r c l · · · q 4 q 5 q 6 r r c ⊥ ⊥ Acceptance: Every branch has infinitely many accepting vertices Component Technique States ? Transitions ? Accepting States ? Alexander Weinert Saarland University VLDL Satisfiability and Model Checking 14/26

  70. Tree Automata ⊥ ⊥ q 3 q 8 ⊥ ⊥ ⊥ l l · · · q 7 q 10 q 11 c c ⊥ ⊥ ⊥ q 1 q 2 q 9 c r c l · · · q 4 q 5 q 6 r r c ⊥ ⊥ Acceptance: Every branch has infinitely many accepting vertices Component Technique States ? Transitions ? Accepting States ? Alexander Weinert Saarland University VLDL Satisfiability and Model Checking 14/26

  71. Tree Automata ⊥ ⊥ q 3 q 8 ⊥ ⊥ ⊥ l l · · · q 7 q 10 q 11 c c ⊥ ⊥ ⊥ q 1 q 2 q 9 c r c l · · · q 4 q 5 q 6 r r c ⊥ ⊥ Acceptance: Every branch has infinitely many accepting vertices Component Technique States ? Transitions ? Accepting States ? Alexander Weinert Saarland University VLDL Satisfiability and Model Checking 14/26

  72. Tree Automata ⊥ ⊥ q 3 q 8 ⊥ ⊥ ⊥ l l · · · q 7 q 10 q 11 c c ⊥ ⊥ ⊥ q 1 q 2 q 9 c r c l · · · q 4 q 5 q 6 r r c ⊥ ⊥ Acceptance: Every branch has infinitely many accepting vertices Component Technique States ? Transitions ? Accepting States ? Alexander Weinert Saarland University VLDL Satisfiability and Model Checking 14/26

  73. Tree Automata ⊥ ⊥ q 3 q 8 ⊥ ⊥ ⊥ l l · · · q 7 q 10 q 11 c c ⊥ ⊥ ⊥ q 1 q 2 q 9 c r c l · · · q 4 q 5 q 6 r r c ⊥ ⊥ Acceptance: Every branch has infinitely many accepting vertices Component Technique States ? Transitions ? Accepting States ? Alexander Weinert Saarland University VLDL Satisfiability and Model Checking 14/26

Recommend

Statistical Statistical Statistical Model Statistical Model Model Checking Model Checking

Statistical Statistical Statistical Model Statistical Model Model Checking Model Checking

Statistical Statistical Statistical Model Statistical Model Model Checking Model Checking Checking Checking for Timed Automata Timed Automata Coll C l C ll b llabora orators: t ors: Peter Bulychev, Alexandre David Axel Legay, Marius

725 views • 59 slides
Hoare Logic and Model Checking Model Checking Lecture 11: Model checking for Computation Tree

Hoare Logic and Model Checking Model Checking Lecture 11: Model checking for Computation Tree

Hoare Logic and Model Checking Model Checking Lecture 11: Model checking for Computation Tree Logic Dominic Mulligan Based on previous slides by Alan Mycroft and Mike Gordon Programming, Logic, and Semantics Group University of Cambridge

402 views • 25 slides
3. Satisfiability Checking 3.1 SAT-Checking Procedures Verification Technology

3. Satisfiability Checking 3.1 SAT-Checking Procedures Verification Technology

Fachgebiet RechnerSysteme Technische Universitt Verification Technology Darmstadt 3. Satisfiability Checking Computer Systems Lab 1 3. Satisfiability Checking 3 3. Satisfiability Checking 3.1 SAT-Checking Procedures Verification

278 views • 11 slides
Multiple tree automata a new model of tree automata Gwendal Collet (TU Wien), Julien David (LIPN)

Multiple tree automata a new model of tree automata Gwendal Collet (TU Wien), Julien David (LIPN)

Multiple tree automata a new model of tree automata Gwendal Collet (TU Wien), Julien David (LIPN) S eminaire CALIN, 24 mars 2015 Outline Introduction to automata: definitions and motivation 1 Description of the model: Multiple Tree Automata

612 views • 45 slides
Tree-shaped one-pass tableau systems for Linear Temporal Logic satisfiability checking Nicola

Tree-shaped one-pass tableau systems for Linear Temporal Logic satisfiability checking Nicola

Tree-shaped one-pass tableau systems for Linear Temporal Logic satisfiability checking Nicola Gigante University of Udine, Italy Joint work with Angelo Montanari, Mark Reynolds, Luca Geatti The need for formal verification Safety-critical

489 views • 47 slides
Hoare Logic and Model Checking Model Checking Lecture 10: Computation Tree Logic (CTL) Dominic

Hoare Logic and Model Checking Model Checking Lecture 10: Computation Tree Logic (CTL) Dominic

Hoare Logic and Model Checking Model Checking Lecture 10: Computation Tree Logic (CTL) Dominic Mulligan Based on previous slides by Alan Mycroft and Mike Gordon Programming, Logic, and Semantics Group University of Cambridge Academic year

1.15k views • 48 slides
OUTLINE Model Checking in a Nutshell Timed automata and TCTL Timed Automata, TCTL A

OUTLINE Model Checking in a Nutshell Timed automata and TCTL Timed Automata, TCTL A

OUTLINE Model Checking in a Nutshell Timed automata and TCTL Timed Automata, TCTL A UPPAAL Tutorial & Verification Problems Data stuctures & central algorithms UPPAAL input languages 1 2 Timed Automata: Syntax

408 views • 11 slides
Directed Model Checking (not only) for Timed Automata Sebastian Kupferschmid March, 2010 Model

Directed Model Checking (not only) for Timed Automata Sebastian Kupferschmid March, 2010 Model

Directed Model Checking (not only) for Timed Automata Sebastian Kupferschmid March, 2010 Model Checking Motivation Embedded Systems Omnipresent Safety relevant systems Pentium bug Ariane 5 Errors can be extremely harmful Correct

837 views • 67 slides
Improved Algorithms for the Automata-based Approach to Model-Checking L. Doyen (EPFL) and J.-F.

Improved Algorithms for the Automata-based Approach to Model-Checking L. Doyen (EPFL) and J.-F.

Improved Algorithms for the Automata-based Approach to Model-Checking L. Doyen (EPFL) and J.-F. Raskin (ULB) TACAS 2007 Braga - Portugal March 28, 2007 Automata-based approach to model-checking Programs and properties are formalized as

812 views • 52 slides
Limit-Deterministic Bchi Automata for Probabilistic Model Checking Jan Ketnsk Javier

Limit-Deterministic Bchi Automata for Probabilistic Model Checking Jan Ketnsk Javier

Limit-Deterministic Bchi Automata for Probabilistic Model Checking Jan Ketnsk Javier Esparza Salomon Sickert Stefan Jaax Technische Universitt Mnchen PROBABILISTIC MODEL CHECKING Markov Decision Process (MDP) . At each state,

1.31k views • 100 slides
Limit-Deterministic Bchi Automata for Probabilistic Model Checking Javier Esparza Jan K

Limit-Deterministic Bchi Automata for Probabilistic Model Checking Javier Esparza Jan K

Limit-Deterministic Bchi Automata for Probabilistic Model Checking Javier Esparza Jan K etnsk Stefan Jaax Salomon Sickert Technische Universitt Mnchen PROBABILISTIC MODEL CHECKING Markov Decision Process (MDP) . At each

430 views • 23 slides
Graph Automata Jan Leike July 2nd, 2012 Motivation We want an automata model that Motivation

Graph Automata Jan Leike July 2nd, 2012 Motivation We want an automata model that Motivation

Graph Automata Jan Leike July 2nd, 2012 Motivation We want an automata model that Motivation We want an automata model that operates on graphs, generalizes nested words and tree automata, and has some nice properties. Outline 1.

731 views • 44 slides
Live demo 2 / 27 Automata-Theoretic LTL Model Checking State-space High-level State-space

Live demo 2 / 27 Automata-Theoretic LTL Model Checking State-space High-level State-space

Contributions to LTL and -Automata for Model Checking Alexandre Duret-Lutz LRDE/EPITA 10 February 2017 Javier Esparza Technische Universitt Mnchen reviewer Radu Mateescu INRIA Grenoble reviewer Moshe Y. Vardi Rice University,

1.1k views • 89 slides
Real-time Model Checking Priced timed automata Nicolas M ARKEY Lav. Sp ecification

Real-time Model Checking Priced timed automata Nicolas M ARKEY Lav. Sp ecification

Real-time Model Checking Priced timed automata Nicolas M ARKEY Lav. Sp ecification & V erification CNRS & ENS Cachan France March 3, 2010 Time is not always sufficient Timed automata are (rather) well understood

1.58k views • 129 slides
2 Temporal Logic Model Checking Many designs, especially digital hardware designs, can profitably

2 Temporal Logic Model Checking Many designs, especially digital hardware designs, can profitably

Bounded Model Checking Using Satisfiability Solving Edmund Clarke 1 , Armin Biere 2 , Richard Raimi 3 , and Yunshan Zhu 4 1 Computer Science Department, CMU, 5000 Forbes Avenue Pittsburgh, PA 15213, USA, emc@cs.cmu.edu 2 Institute of Computer

600 views • 20 slides
On Variable Selection in SAT-LP-based Bounded Model Checking of Linear Hybrid Automata Marc

On Variable Selection in SAT-LP-based Bounded Model Checking of Linear Hybrid Automata Marc

On Variable Selection in SAT-LP-based Bounded Model Checking of Linear Hybrid Automata Marc Herbstritt (joint work with Bernd Becker, Erika Abrah am, Christian Herde) Institute of Computer Science Albert-Ludwigs-University Freiburg im

568 views • 33 slides
Statistical Model Checking for Distributed Probabilistic-Control Hybrid Automata with Smart Grid

Statistical Model Checking for Distributed Probabilistic-Control Hybrid Automata with Smart Grid

Statistical Model Checking for Distributed Probabilistic-Control Hybrid Automata with Smart Grid Applications ao Martins 1 , 2 e Platzer 1 ao Leite 2 Jo Andr Jo 1 Computer Science Department, Carnegie Mellon University, Pittsburgh PA 2

923 views • 57 slides
Modular Tree Automata Deriving Modular Recursion Schemes from Tree Automata Patrick Bahr

Modular Tree Automata Deriving Modular Recursion Schemes from Tree Automata Patrick Bahr

Modular Tree Automata Deriving Modular Recursion Schemes from Tree Automata Patrick Bahr University of Copenhagen, Department of Computer Science paba@diku.dk 11th International Conference on Mathematics of Program Construction Madrid,

1.29k views • 97 slides
TCTL model checking lower/upper-bound Introduction parametric timed automata without Parametric

TCTL model checking lower/upper-bound Introduction parametric timed automata without Parametric

TCTL model checking lower/upper-bound parametric timed automata without invariants TCTL model checking lower/upper-bound Introduction parametric timed automata without Parametric timed automata invariants Model checking with unknown

476 views • 33 slides
Overview Motivation 1 Quantitative Automata Models and Model Checking What are discrete-time

Overview Motivation 1 Quantitative Automata Models and Model Checking What are discrete-time

Motivation Overview Motivation 1 Quantitative Automata Models and Model Checking What are discrete-time Markov chains? 2 Reachability probabilities 3 Joost-Pieter Katoen Qualitative reachability and all that 4 RWTH Aachen University

509 views • 36 slides
MTL-Model Checking of One-Clock Parametric Timed Automata is Undecidable SynCop 2014 1st

MTL-Model Checking of One-Clock Parametric Timed Automata is Undecidable SynCop 2014 1st

MTL-Model Checking of One-Clock Parametric Timed Automata is Undecidable SynCop 2014 1st International Workshop on Synthesis of Continuous Parameters Karin Quaas University of Leipzig 6th April 2014 Outline of the Talk param param param

340 views • 23 slides
IC3 Software Model Checking on Control Flow Automata Tim Lange 1 Martin R. Neuhuer 2 Thomas

IC3 Software Model Checking on Control Flow Automata Tim Lange 1 Martin R. Neuhuer 2 Thomas

IC3 Software Model Checking on Control Flow Automata Tim Lange 1 Martin R. Neuhuer 2 Thomas Noll 1 1 Software Modeling and Verification Group, RWTH Aachen 2 Siemens AG FMCAD 2015 at Austin, TX, USA, September 29, 2015 Introduction Outline

941 views • 42 slides
Support for the Hanoi Omega-Automata Format http://adl.github.io/hoaf/support.html ltl2dstar 0.5.3

Support for the Hanoi Omega-Automata Format http://adl.github.io/hoaf/support.html ltl2dstar 0.5.3

Spot 2.0 A C++ library for model checking and -automata manipulation Alexandre D uret -L utz Alexandre L ewkowicz Amaury F auchille tienne R enault Laurent X u Thibaud M ichaud Monday, October 17th 1 / 19 A Generalized View of -Automata

500 views • 35 slides
Model Checking, Hybrid Automata, and Systems Biology Carla Piazza 1 1 Department of Mathematics

Model Checking, Hybrid Automata, and Systems Biology Carla Piazza 1 1 Department of Mathematics

Model Checking, Hybrid Automata, and Systems Biology Carla Piazza 1 1 Department of Mathematics and Computer Science, University of Udine, Udine, Italy Part of our Group Alberto Policriti Bud Mishra DIMI Udine, IGA Udine NYU New York, DMI

949 views • 53 slides

More recommend