Exploiting the Temporal Logic Hierarchy and the Non-Confluence - PowerPoint PPT Presentation

Problem Det. via Automata Hierarchy Det. Non-Confluent Automata Conclusion Exploiting the Temporal Logic Hierarchy and the Non-Confluence Property for Efficient LTL Synthesis Andreas Morgenstern GandALF 2010 Andreas Morgenstern Symbolic LTL Synthesis via Hierarchy and Non-Confluence 1

Problem Det. via Automata Hierarchy Det. Non-Confluent Automata Conclusion Overview Motivation: What is LTL Synthesis 1 Symbolic Determinisation via the Automata Hierarchy 2 Symbolic Determinisation via Non-Confluent Automata 3 Experiments and Conclusion 4 Andreas Morgenstern Symbolic LTL Synthesis via Hierarchy and Non-Confluence 2

Problem Det. via Automata Hierarchy Det. Non-Confluent Automata Conclusion Model Checking ? | I O = System S LTL formula Φ Specification: Formula Φ in Temporal-Logic LTL Question: S | = Φ ? Andreas Morgenstern Symbolic LTL Synthesis via Hierarchy and Non-Confluence 3

Problem Det. via Automata Hierarchy Det. Non-Confluent Automata Conclusion LTL Synthesis | I O = ? LTL formula Φ Specification: Formula Φ in Temporal-Logic LTL Question: ∃ System S . S | = Φ ? Andreas Morgenstern Symbolic LTL Synthesis via Hierarchy and Non-Confluence 3

Problem Det. via Automata Hierarchy Det. Non-Confluent Automata Conclusion Model Checking ? | I O = System LTL formula Φ Question: S | = Φ ? Andreas Morgenstern Symbolic LTL Synthesis via Hierarchy and Non-Confluence 4

Problem Det. via Automata Hierarchy Det. Non-Confluent Automata Conclusion Model Checking ? �| I O = System LTL formula ¬ Φ ( S | = Φ) ↔ ( S �| = ¬ Φ) ? Andreas Morgenstern Symbolic LTL Synthesis via Hierarchy and Non-Confluence 4

Problem Det. via Automata Hierarchy Det. Non-Confluent Automata Conclusion Automata based Model Checking a c b b ξ 1 a × c 1 2 3 ξ 3 b a b d ξ 2 a c b Non-terminating systems ! B¨ uchi-Automata Automata read infinite words Automata accept, whenever a F state is visited ∞ often ! Graphsearch for one Non-Accepting run ! ( S | = Φ) ↔ ( S �| = ¬ Φ) ↔ L ( S × A ¬ Φ ) = ∅ ) ? Andreas Morgenstern Symbolic LTL Synthesis via Hierarchy and Non-Confluence 4

Problem Det. via Automata Hierarchy Det. Non-Confluent Automata Conclusion Symbolic Model Checking p 0 ∧ a ∨ r 0 ↔ ( a ∨ ¬ b ) ∧ ∧ R = p 1 ∧ ¬ b R = r 1 ↔ ( c ∨ d ) ∧ . . . . . . Using propositional logic to represent System and B¨ uchi Automata Advantages: Represent large state spaces Efficient methods like BDD / SAT Industry-sized problems managable: Verification at Intel Andreas Morgenstern Symbolic LTL Synthesis via Hierarchy and Non-Confluence 4

Problem Det. via Automata Hierarchy Det. Non-Confluent Automata Conclusion Automata based LTL Synthesis O | ¬ a I b = ? 1 2 3 b a ¬ c b ( ∃S . S | = Φ) ↔ ∃S . L ( S ) ⊆ L ( A Φ ) Idea: Search for valid sub-automaton for each input on B¨ uchi automaton! Infinite Game between Environment and System ! Andreas Morgenstern Symbolic LTL Synthesis via Hierarchy and Non-Confluence 5

Problem Det. via Automata Hierarchy Det. Non-Confluent Automata Conclusion Automata based LTL Synthesis p 1 a a p 2 p 3 c b Idea: Search for satisfying automaton for each input on specification automaton! Problem: nondeterminism intuitively: a priori not known whether b or c comes deterministic system from nondeterministic B¨ uchi automaton � Andreas Morgenstern Symbolic LTL Synthesis via Hierarchy and Non-Confluence 5

Problem Det. via Automata Hierarchy Det. Non-Confluent Automata Conclusion Determinisation of B¨ uchi Automata: Facts Rabin-Scott Subset construction not sufficient ! Safra (1988): Determinisation of B¨ uchi automata First Implementation : 2006 State space: Trees of sets of states No fully symbolic implementation known Only small examples managable Andreas Morgenstern Symbolic LTL Synthesis via Hierarchy and Non-Confluence 6

Problem Det. via Automata Hierarchy Det. Non-Confluent Automata Conclusion Core of this Work Minimizing Minimizing LTL NDet Det symbolic: Determinisation : exists symbolic symbolic translation LTL → NDet √ symbolic Algorithms for infinite games √ minimizing automata symbolically √ symbolic determinization (shown in [MoSc08,MoSc08a]) How well does it work in practice ? Andreas Morgenstern Symbolic LTL Synthesis via Hierarchy and Non-Confluence 7

Problem Det. via Automata Hierarchy Det. Non-Confluent Automata Conclusion Overview Motivation: What is LTL Synthesis 1 Symbolic Determinisation via the Automata Hierarchy 2 Symbolic Determinisation via Non-Confluent Automata 3 Experiments and Conclusion 4 Andreas Morgenstern Symbolic LTL Synthesis via Hierarchy and Non-Confluence 8

Problem Det. via Automata Hierarchy Det. Non-Confluent Automata Conclusion ω -Automata b b 1 2 3 a a c b ω -Automata ω -Automata read infinite Worte. Different acceptance conditions: B¨ uchi : visit F states infinitely often Andreas Morgenstern Symbolic LTL Synthesis via Hierarchy and Non-Confluence 9

Problem Det. via Automata Hierarchy Det. Non-Confluent Automata Conclusion ω -Automata b b 1 2 3 a a c b ω -Automata ω -Automata read infinite Worte. Different acceptance conditions: B¨ uchi : visit F states infinitely often Co-B¨ uchi: visit ¬F states finitely often Andreas Morgenstern Symbolic LTL Synthesis via Hierarchy and Non-Confluence 9

Problem Det. via Automata Hierarchy Det. Non-Confluent Automata Conclusion ω -Automata b b 1 2 3 a a c b ω -Automata ω -Automata read infinite Worte. Different acceptance conditions: B¨ uchi : visit F states infinitely often Co-B¨ uchi: visit ¬F states finitely often Streett: boolean combination of (co)-B¨ uchi (in Normalform) Andreas Morgenstern Symbolic LTL Synthesis via Hierarchy and Non-Confluence 9

Problem Det. via Automata Hierarchy Det. Non-Confluent Automata Conclusion ω -Automata b b 1 2 3 a a c b ω -Automata ω -Automata read infinite Worte. Different acceptance conditions: B¨ uchi : visit F states infinitely often Co-B¨ uchi: visit ¬F states finitely often Streett: boolean combination of (co)-B¨ uchi (in Normalform) Safety : visit only F states Liveness : visit F states at least once Prefix : boolean combination of Safety und Liveness (in Normalform) Andreas Morgenstern Symbolic LTL Synthesis via Hierarchy and Non-Confluence 9

Problem Det. via Automata Hierarchy Det. Non-Confluent Automata Conclusion The Automata Hierarchy (Wagner, 1979) (N)Det Safety bool. comb. Det B¨ bool. comb. uchi � � � NDet Prefix NDet B¨ uchi Det Prefix (N)Det Streett � � � NDet total Liveness (N)Det Co-B¨ uchi Det Liveness bool. comb bool. comb C 1 � C 2 := automaton from C 1 can be translated to one from C 2 Andreas Morgenstern Symbolic LTL Synthesis via Hierarchy and Non-Confluence 10

Problem Det. via Automata Hierarchy Det. Non-Confluent Automata Conclusion The Temporallogic Hierarchy (Manna&Pnueli, 1987) (N)Det Safety Det B¨ uchi bool. comb. bool. comb. TL Safety TL B¨ � � uchi � NDet Prefix NDet B¨ uchi Det Prefix (N)Det Streett � TL Prefix � TL Streett � NDet total (N)Det Co-B¨ Liveness uchi Det Liveness bool. comb bool. comb TL Liveness TL Co-B¨ uchi C 1 � C 2 := automaton from C 1 can be translated to one from C 2 Andreas Morgenstern Symbolic LTL Synthesis via Hierarchy and Non-Confluence 10

Problem Det. via Automata Hierarchy Det. Non-Confluent Automata Conclusion Symbolic Determinisation via Automata Hierarchy (N)Det Safety Det B¨ Safra uchi TL Safety TL B¨ � � uchi � NDet Prefix NDet B¨ uchi Det Prefix (N)Det Streett Subset � TL Prefix � TL Streett � NDet total (N)Det Co-B¨ Liveness uchi Det Liveness TL Liveness TL Co-B¨ uchi Breakpoint BDD-represented Automata for TL Safety and TL Co-B¨ uchi Andreas Morgenstern Symbolic LTL Synthesis via Hierarchy and Non-Confluence 11

Problem Det. via Automata Hierarchy Det. Non-Confluent Automata Conclusion Symbolic Determinisation via Automata Hierarchy (N)Det Safety Det B¨ uchi bool. comb. bool. comb. TL Safety TL B¨ uchi NDet Prefix NDet B¨ uchi Det Prefix (N)Det Streett Dual Dual TL Prefix TL Streett NDet total (N)Det Co-B¨ Liveness uchi Det Liveness bool. comb bool. comb TL Liveness TL Co-B¨ uchi BDD-represented Automata for TL Liveness , TL Prefix TL B¨ uchi und TL Streett Andreas Morgenstern Symbolic LTL Synthesis via Hierarchy and Non-Confluence 11

Problem Det. via Automata Hierarchy Det. Non-Confluent Automata Conclusion Determinisation via Automata Hierarchy: Conclusion Main Idea Locate formula syntactically in Hierarchy Subset (Breakpoint) construction symbolically boolean combination of Formulas / Automata Advantages Deterministic automata never explicitely represented Efficient: due to boolean combination subautomata very small (less than < 20 ndet states) Nearly all formula belong to TL Streett Disadvantages Not every formula is in TL Streett ! Andreas Morgenstern Symbolic LTL Synthesis via Hierarchy and Non-Confluence 12

Problem Det. via Automata Hierarchy Det. Non-Confluent Automata Conclusion Overview Motivation: What is LTL Synthesis 1 Symbolic Determinisation via the Automata Hierarchy 2 Symbolic Determinisation via Non-Confluent Automata 3 Experiments and Conclusion 4 Andreas Morgenstern Symbolic LTL Synthesis via Hierarchy and Non-Confluence 13