Symbolic Unfoldings for Networks of Timed Automata Franck Cassez 1 Thomas Chatain 2 Claude Jard 2 1 CNRS/IRCCyN 2 IRISA Nantes, France Rennes, France Automated Technology for Veri�cation and Analysis (ATVA'06) October 23–26th, 2006 Beijing, China
Unfoldings for Network of Automata Symbolic Unfoldings for NTA Conclusion Outline of the talk Unfoldings for Network of Automata ◮ Symbolic Unfoldings for Network of Timed Automata ◮ Conclusion ◮ October 2006 (ATVA’06, Beijing) Unfoldings for Networks of Timed Automata 2 / 28
Unfoldings for Network of Automata Symbolic Unfoldings for NTA Conclusion Outline of the talk Unfoldings for Network of Automata ◮ Symbolic Unfoldings for Network of Timed Automata ◮ Conclusion ◮ October 2006 (ATVA’06, Beijing) Unfoldings for Networks of Timed Automata 2 / 28
Unfoldings for Network of Automata Symbolic Unfoldings for NTA Conclusion Outline of the talk Unfoldings for Network of Automata ◮ Symbolic Unfoldings for Network of Timed Automata ◮ Conclusion ◮ October 2006 (ATVA’06, Beijing) Unfoldings for Networks of Timed Automata 2 / 28
Unfoldings for Network of Automata Symbolic Unfoldings for NTA Conclusion Outline Unfoldings for Network of Automata ◮ Symbolic Unfoldings for Network of Timed Automata ◮ Conclusion ◮ October 2006 (ATVA’06, Beijing) Unfoldings for Networks of Timed Automata 3 / 28
Unfoldings for Network of Automata Symbolic Unfoldings for NTA Conclusion Unfoldings à la McMillan For Petri Nets [McMillan, FMSD'95] For Network of Automata [Esparza & Römer, CONCUR'99] A t 2 U 0 B t 0 t 1 t 1 t 2 1 2 C V October 2006 (ATVA’06, Beijing) Unfoldings for Networks of Timed Automata 4 / 28
Unfoldings for Network of Automata Symbolic Unfoldings for NTA Conclusion Unfoldings à la McMillan For Petri Nets [McMillan, FMSD'95] For Network of Automata [Esparza & Römer, CONCUR'99] A t 2 ⊥ 0 B U 0 A U t 0 t 1 t 1 t 2 e 2 t 2 1 2 C V Finite Automata V B e 1 t 0 ⇒ e 3 t 1 = 1-safe Petri net 1 2 C October 2006 (ATVA’06, Beijing) Unfoldings for Networks of Timed Automata 4 / 28
Unfoldings for Network of Automata Symbolic Unfoldings for NTA Conclusion Features of Unfoldings ◮ Unfolding = 1-safe Petri net ◮ Finite “good” unfoldings exist finite complete prefix ◮ Preserves concurrency ⊥ size(unfolding) < synchronous product of TA 0 A U ◮ Can be constructed efficiently e 2 t 2 ◮ Can be used for checking properties: ◮ coverability or reachability properties B V e 1 t 0 ◮ deadlock detection e 3 t 1 ◮ temporal logics properties ◮ Can be used for diagnosis: 1 2 C ◮ Induces a partial order on events ◮ Event structure = explanations for set of events October 2006 (ATVA’06, Beijing) Unfoldings for Networks of Timed Automata 5 / 28
Unfoldings for Network of Automata Symbolic Unfoldings for NTA Conclusion Features of Unfoldings ◮ Unfolding = 1-safe Petri net ◮ Finite “good” unfoldings exist finite complete prefix ◮ Preserves concurrency ⊥ size(unfolding) < synchronous product of TA 0 A U ◮ Can be constructed efficiently e 2 t 2 ◮ Can be used for checking properties: ◮ coverability or reachability properties B V e 1 t 0 ◮ deadlock detection e 3 t 1 ◮ temporal logics properties ◮ Can be used for diagnosis: 1 2 C ◮ Induces a partial order on events ◮ Event structure = explanations for set of events October 2006 (ATVA’06, Beijing) Unfoldings for Networks of Timed Automata 5 / 28
Unfoldings for Network of Automata Symbolic Unfoldings for NTA Conclusion Features of Unfoldings ◮ Unfolding = 1-safe Petri net ◮ Finite “good” unfoldings exist finite complete prefix ◮ Preserves concurrency ⊥ size(unfolding) < synchronous product of TA 0 A U ◮ Can be constructed efficiently e 2 t 2 ◮ Can be used for checking properties: ◮ coverability or reachability properties B V e 1 t 0 ◮ deadlock detection e 3 t 1 ◮ temporal logics properties ◮ Can be used for diagnosis: 1 2 C ◮ Induces a partial order on events ◮ Event structure = explanations for set of events October 2006 (ATVA’06, Beijing) Unfoldings for Networks of Timed Automata 5 / 28
Unfoldings for Network of Automata Symbolic Unfoldings for NTA Conclusion Features of Unfoldings ◮ Unfolding = 1-safe Petri net ◮ Finite “good” unfoldings exist finite complete prefix ◮ Preserves concurrency ⊥ size(unfolding) < synchronous product of TA 0 A U ◮ Can be constructed efficiently e 2 t 2 ◮ Can be used for checking properties: ◮ coverability or reachability properties B V e 1 t 0 ◮ deadlock detection e 3 t 1 ◮ temporal logics properties ◮ Can be used for diagnosis: 1 2 C ◮ Induces a partial order on events ◮ Event structure = explanations for set of events October 2006 (ATVA’06, Beijing) Unfoldings for Networks of Timed Automata 5 / 28
Unfoldings for Network of Automata Symbolic Unfoldings for NTA Conclusion Features of Unfoldings ◮ Unfolding = 1-safe Petri net ◮ Finite “good” unfoldings exist finite complete prefix ◮ Preserves concurrency ⊥ size(unfolding) < synchronous product of TA 0 A U ◮ Can be constructed efficiently e 2 t 2 ◮ Can be used for checking properties: ◮ coverability or reachability properties B V e 1 t 0 ◮ deadlock detection e 3 t 1 ◮ temporal logics properties ◮ Can be used for diagnosis: 1 2 C ◮ Induces a partial order on events ◮ Event structure = explanations for set of events October 2006 (ATVA’06, Beijing) Unfoldings for Networks of Timed Automata 5 / 28
Unfoldings for Network of Automata Symbolic Unfoldings for NTA Conclusion Features of Unfoldings ◮ Unfolding = 1-safe Petri net ◮ Finite “good” unfoldings exist finite complete prefix ◮ Preserves concurrency ⊥ size(unfolding) < synchronous product of TA 0 A U ◮ Can be constructed efficiently e 2 t 2 ◮ Can be used for checking properties: ◮ coverability or reachability properties B V e 1 t 0 ◮ deadlock detection e 3 t 1 ◮ temporal logics properties ◮ Can be used for diagnosis: 1 2 C ◮ Induces a partial order on events ◮ Event structure = explanations for set of events October 2006 (ATVA’06, Beijing) Unfoldings for Networks of Timed Automata 5 / 28
Unfoldings for Network of Automata Symbolic Unfoldings for NTA Conclusion Features of Unfoldings ◮ Unfolding = 1-safe Petri net ◮ Finite “good” unfoldings exist finite complete prefix ◮ Preserves concurrency ⊥ size(unfolding) < synchronous product of TA 0 A U ◮ Can be constructed efficiently e 2 t 2 ◮ Can be used for checking properties: ◮ coverability or reachability properties B V e 1 t 0 ◮ deadlock detection e 3 t 1 ◮ temporal logics properties ◮ Can be used for diagnosis: 1 2 C ◮ Induces a partial order on events ◮ Event structure = explanations for set of events October 2006 (ATVA’06, Beijing) Unfoldings for Networks of Timed Automata 5 / 28
Unfoldings for Network of Automata Symbolic Unfoldings for NTA Conclusion Network of Timed Automata A x ≤ 10 Def. of NTA Semantics of NTA t 2 ; x : = 0 0 B x ≤ 2 U y ≤ 3 t 0 ; z > 5 t 1 t 1 ; x ≤ 2 t 2 ; y ≤ 3 2 C V 1 Clocks are NOT shared October 2006 (ATVA’06, Beijing) Unfoldings for Networks of Timed Automata 6 / 28
Unfoldings for Network of Automata Symbolic Unfoldings for NTA Conclusion Network of Timed Automata A x ≤ 10 Def. of NTA Semantics of NTA t 2 ; x : = 0 0 B x ≤ 2 U y ≤ 3 t 0 ; z > 5 t 1 t 1 ; x ≤ 2 t 2 ; y ≤ 3 2 C V 1 State of a NTA: ((1 , A, U ) , x = 1 , y = 1 , z = 1) Symbolic state: ((1 , A, U ) , x = y = z ∧ y ≤ 3) Clocks are NOT shared October 2006 (ATVA’06, Beijing) Unfoldings for Networks of Timed Automata 6 / 28
Unfoldings for Network of Automata Symbolic Unfoldings for NTA Conclusion Network of Timed Automata A x ≤ 10 Def. of NTA Semantics of NTA t 2 ; x : = 0 0 B x ≤ 2 U y ≤ 3 t 0 ; z > 5 t 1 t 1 ; x ≤ 2 t 2 ; y ≤ 3 2 C V 1 State of a NTA: ((1 , A, U ) , x = 1 , y = 1 , z = 1) Symbolic state: ((1 , A, U ) , x = y = z ∧ y ≤ 3) Clocks are NOT shared October 2006 (ATVA’06, Beijing) Unfoldings for Networks of Timed Automata 6 / 28
Unfoldings for Network of Automata Symbolic Unfoldings for NTA Conclusion Unfoldings for Network of Timed Automata ? A x ≤ 10 t 2 ; x : = 0 0 B x ≤ 2 U y ≤ 3 t 0 ; z > 5 t 1 t 1 ; x ≤ 2 t 2 ; y ≤ 3 ⊥ δ ⊥ = 0 1 2 C V 0 A U e 2 t 2 , δ e 2 ≤ 3 B V e 1 t 0 , δ e 1 > 5 e 3 t 1 , δ e 3 – δ e 2 ≤ 2 1 2 C October 2006 (ATVA’06, Beijing) Unfoldings for Networks of Timed Automata 7 / 28
Unfoldings for Network of Automata Symbolic Unfoldings for NTA Conclusion Unfoldings for Network of Timed Automata ? A x ≤ 10 t 2 ; x : = 0 0 B x ≤ 2 U y ≤ 3 t 0 ; z > 5 t 1 t 1 ; x ≤ 2 t 2 ; y ≤ 3 ⊥ δ ⊥ = 0 1 2 C V 0 A U e 2 t 2 , δ e 2 ≤ 3 B V e 1 t 0 , δ e 1 > 5 e 3 t 1 , δ e 3 – δ e 2 ≤ 2 1 2 C October 2006 (ATVA’06, Beijing) Unfoldings for Networks of Timed Automata 7 / 28
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