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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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