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Verification & Control Discrete Games Timed Games Symbolic Algorithms Conclusion ACI Scurit Informatique CORTOS CORTOS = Control and Observation of Real-Time Open Systems Participants: LSV + VERIMAG + IRCCyN Web:

  1. Verification & Control Discrete Games Timed Games Symbolic Algorithms Conclusion Control and Game a e ℓ 0 ℓ 1 Bad c u u b ℓ 2 ℓ 3 d Open System = 2-player game, Controller (C) vs Environment (E) ◮ Controller does Act c moves, Environment does Act u moves ◮ Control Objective = Winning condition on the game “Avoid bad states” (safety) or “Enforce good states” (reachability) ◮ Control Problem: find a strategy for the controller to win the game ◮ Various types of game models for C and E ◮ Finite or pushdown or counter automata . . . ◮ Timed or hybrid automata MSR’05 (Autrans, France) Control of Timed Systems 7 / 32

  2. Verification & Control Discrete Games Timed Games Symbolic Algorithms Conclusion Control and Game a e ℓ 0 ℓ 1 Bad c u u b ℓ 2 ℓ 3 d Open System = 2-player game, Controller (C) vs Environment (E) ◮ Controller does Act c moves, Environment does Act u moves ◮ Control Objective = Winning condition on the game “Avoid bad states” (safety) or “Enforce good states” (reachability) ◮ Control Problem: find a strategy for the controller to win the game ◮ Various types of game models for C and E ◮ Finite or pushdown or counter automata . . . ◮ Timed or hybrid automata MSR’05 (Autrans, France) Control of Timed Systems 7 / 32

  3. Verification & Control Discrete Games Timed Games Symbolic Algorithms Conclusion Control and Game a e ℓ 0 ℓ 1 Bad c u u b ℓ 2 ℓ 3 d Open System = 2-player game, Controller (C) vs Environment (E) ◮ Controller does Act c moves, Environment does Act u moves ◮ Control Objective = Winning condition on the game “Avoid bad states” (safety) or “Enforce good states” (reachability) ◮ Control Problem: find a strategy for the controller to win the game ◮ Various types of game models for C and E ◮ Finite or pushdown or counter automata . . . ◮ Timed or hybrid automata MSR’05 (Autrans, France) Control of Timed Systems 7 / 32

  4. Verification & Control Discrete Games Timed Games Symbolic Algorithms Conclusion Problems of Interest Verification Problem (or Model Checking Problem) Input: a model of the closed system S and a property ϕ Problem: Does S satisfy ϕ ? Control Problem (CP) Input: a model of the open system (game) G and a property ϕ Problem: Is there a controller (strategy) C s.t. ( C � G ) satisfy ϕ ? Control Synthesis Problem (CSP) Input: a model of the open system (game) G and a property ϕ Problem: If the answer to the CP ( G , ϕ ) is “yes”, can we effectively compute a witness controller ? MSR’05 (Autrans, France) Control of Timed Systems 8 / 32

  5. Verification & Control Discrete Games Timed Games Symbolic Algorithms Conclusion Problems of Interest Verification Problem (or Model Checking Problem) Input: a model of the closed system S and a property ϕ Problem: Does S satisfy ϕ ? Control Problem (CP) Input: a model of the open system (game) G and a property ϕ Problem: Is there a controller (strategy) C s.t. ( C � G ) satisfy ϕ ? Control Synthesis Problem (CSP) Input: a model of the open system (game) G and a property ϕ Problem: If the answer to the CP ( G , ϕ ) is “yes”, can we effectively compute a witness controller ? MSR’05 (Autrans, France) Control of Timed Systems 8 / 32

  6. Verification & Control Discrete Games Timed Games Symbolic Algorithms Conclusion Problems of Interest Verification Problem (or Model Checking Problem) Input: a model of the closed system S and a property ϕ Problem: Does S satisfy ϕ ? Control Problem (CP) Input: a model of the open system (game) G and a property ϕ Problem: Is there a controller (strategy) C s.t. ( C � G ) satisfy ϕ ? Control Synthesis Problem (CSP) Input: a model of the open system (game) G and a property ϕ Problem: If the answer to the CP ( G , ϕ ) is “yes”, can we effectively compute a witness controller ? MSR’05 (Autrans, France) Control of Timed Systems 8 / 32

  7. Verification & Control Discrete Games Timed Games Symbolic Algorithms Conclusion Problems of Interest Verification Problem (or Model Checking Problem) Input: a model of the closed system S and a property ϕ Problem: Does S satisfy ϕ ? Control Problem (CP) Input: a model of the open system (game) G and a property ϕ Problem: Is there a controller (strategy) C s.t. ( C � G ) satisfy ϕ ? Control Synthesis Problem (CSP) Input: a model of the open system (game) G and a property ϕ Problem: If the answer to the CP ( G , ϕ ) is “yes”, can we effectively compute a witness controller ? MSR’05 (Autrans, France) Control of Timed Systems 8 / 32

  8. Verification & Control Discrete Games Timed Games Symbolic Algorithms Conclusion Outline ◮ Verification & Control ◮ Control of Finite Automata ◮ Timed Game Automata ◮ Symbolic Algorithms for Timed Game Automata ◮ Conclusion MSR’05 (Autrans, France) Control of Timed Systems 9 / 32

  9. Verification & Control Discrete Games Timed Games Symbolic Algorithms Conclusion Game Automata, Strategies & Winning States a e ℓ 0 ℓ 1 Bad d c u u b Game Automaton ℓ 2 ℓ 3 d Strategy ◮ A strategy f gives for each finite run the controllable action to take. We assume full observability of the system MSR’05 (Autrans, France) Control of Timed Systems 10 / 32

  10. Verification & Control Discrete Games Timed Games Symbolic Algorithms Conclusion Game Automata, Strategies & Winning States a e ℓ 0 ℓ 1 Bad d c u u b Game Automaton ℓ 2 ℓ 3 d Strategy ◮ A strategy f gives for each finite run the controllable action to take. We assume full observability of the system MSR’05 (Autrans, France) Control of Timed Systems 10 / 32

  11. Verification & Control Discrete Games Timed Games Symbolic Algorithms Conclusion Game Automata, Strategies & Winning States a e ℓ 0 ℓ 1 Bad d c u u b Game Automaton ℓ 2 ℓ 3 d Strategy ◮ A strategy f gives for each finite run the controllable action to take. We assume full observability of the system Example of Strategies: f ′ ( · · · ℓ 0 ) = a f ( ℓ 0 ) = a a f ′ ( · · · ℓ 1 ) = c f ( ℓ 0 − → ℓ 1 ) = c a u f ′ ( · · · ℓ 2 ) = b − → ℓ 1 − → ℓ 2 ) = b f ( ℓ 0 a u b a f ′ ( · · · ℓ 3 ) = d f ( ℓ 0 − → ℓ 1 − → ℓ 2 − → ℓ 0 − → ℓ 1 ) = e MSR’05 (Autrans, France) Control of Timed Systems 10 / 32

  12. Verification & Control Discrete Games Timed Games Symbolic Algorithms Conclusion Game Automata, Strategies & Winning States a e ℓ 0 ℓ 1 Bad d c u u b Game Automaton ℓ 2 ℓ 3 d Strategy ◮ A strategy f gives for each finite run the controllable action to take. We assume full observability of the system Example of Strategies: f ′ ( · · · ℓ 0 ) = a f ( ℓ 0 ) = a a f ′ ( · · · ℓ 1 ) = c f ( ℓ 0 − → ℓ 1 ) = c a u f ′ ( · · · ℓ 2 ) = b − → ℓ 1 − → ℓ 2 ) = b f ( ℓ 0 a u b a f ′ ( · · · ℓ 3 ) = d f ( ℓ 0 − → ℓ 1 − → ℓ 2 − → ℓ 0 − → ℓ 1 ) = e MSR’05 (Autrans, France) Control of Timed Systems 10 / 32

  13. Verification & Control Discrete Games Timed Games Symbolic Algorithms Conclusion Game Automata, Strategies & Winning States a e ℓ 0 ℓ 1 Bad d c u u b Game Automaton ℓ 2 ℓ 3 d Strategy ◮ A strategy f gives for each finite run the controllable action to take. We assume full observability of the system ◮ A strategy restricts the set of runs of the system. from a state s it generates of subset of the runs of the initial game MSR’05 (Autrans, France) Control of Timed Systems 10 / 32

  14. Verification & Control Discrete Games Timed Games Symbolic Algorithms Conclusion Game Automata, Strategies & Winning States a e ℓ 0 ℓ 1 Bad d c u u b Game Automaton ℓ 2 ℓ 3 d Strategy ◮ A strategy f gives for each finite run the controllable action to take. We assume full observability of the system ◮ A strategy restricts the set of runs of the system. from a state s it generates of subset of the runs of the initial game ◮ A strategy is winning if it generates only good runs. MSR’05 (Autrans, France) Control of Timed Systems 10 / 32

  15. Verification & Control Discrete Games Timed Games Symbolic Algorithms Conclusion Game Automata, Strategies & Winning States a e ℓ 0 ℓ 1 Bad d c u u b Game Automaton ℓ 2 ℓ 3 d Strategy ◮ A strategy f gives for each finite run the controllable action to take. We assume full observability of the system ◮ A strategy restricts the set of runs of the system. from a state s it generates of subset of the runs of the initial game ◮ A strategy is winning if it generates only good runs. Winning States A state s is winning if there exists a winning strategy from s . MSR’05 (Autrans, France) Control of Timed Systems 10 / 32

  16. Verification & Control Discrete Games Timed Games Symbolic Algorithms Conclusion Controllable Predecessors a e ℓ 0 ℓ 1 Bad d c u u b ℓ 2 ℓ 3 d MSR’05 (Autrans, France) Control of Timed Systems 11 / 32

  17. Verification & Control Discrete Games Timed Games Symbolic Algorithms Conclusion Controllable Predecessors a e ℓ 0 ℓ 1 Bad d c u u b ℓ 2 ℓ 3 d π ( X ) = states from which one can enforce X with a controllable action π ( X ) = Pred Act c ( X ) \ Pred Act u ( X ) MSR’05 (Autrans, France) Control of Timed Systems 11 / 32

  18. Verification & Control Discrete Games Timed Games Symbolic Algorithms Conclusion Controllable Predecessors a e ℓ 0 ℓ 1 Bad d c u u b ℓ 2 ℓ 3 d π ( X ) = states from which one can enforce X with a controllable action π ( X ) = Pred Act c ( X ) \ Pred Act u ( X ) ∃ c ∈ Act c X • π ( X ) • X • not ( ∃ u ∈ Act u ) MSR’05 (Autrans, France) Control of Timed Systems 11 / 32

  19. Verification & Control Discrete Games Timed Games Symbolic Algorithms Conclusion Controllable Predecessors a e ℓ 0 ℓ 1 Bad d c u u b ℓ 2 ℓ 3 d π ( X ) = states from which one can enforce X with a controllable action π ( X ) = Pred Act c ( X ) \ Pred Act u ( X ) Some Values of the π Operator ◮ π ( { ℓ 3 } ) = ∅ ◮ π ( { ℓ 1 } ) = { ℓ 0 } ◮ π ( { ℓ 0 , ℓ 1 } ) = { ℓ 0 , ℓ 2 } ◮ π ( { ℓ 0 , ℓ 1 , ℓ 2 } ) = { ℓ 0 , ℓ 1 , ℓ 2 } MSR’05 (Autrans, France) Control of Timed Systems 11 / 32

  20. Verification & Control Discrete Games Timed Games Symbolic Algorithms Conclusion Controllable Predecessors a e ℓ 0 ℓ 1 Bad d c u u b ℓ 2 ℓ 3 d π ( X ) = states from which one can enforce X with a controllable action A Fixpoint Characterization of Winning States: 1 let ϕ be a set of safe (good) states and G a game 2 let W ∗ be the greatest fixpoint of h ( X ) = ϕ ∩ π ( X ) 3 W ∗ is the set of winning states for ( G , ϕ ) MSR’05 (Autrans, France) Control of Timed Systems 11 / 32

  21. Verification & Control Discrete Games Timed Games Symbolic Algorithms Conclusion Controllable Predecessors a e ℓ 0 ℓ 1 Bad d c u u b ℓ 2 ℓ 3 d π ( X ) = states from which one can enforce X with a controllable action A Fixpoint Characterization of Winning States: 1 let ϕ be a set of safe (good) states and G a game 2 let W ∗ be the greatest fixpoint of h ( X ) = ϕ ∩ π ( X ) 3 W ∗ is the set of winning states for ( G , ϕ ) MSR’05 (Autrans, France) Control of Timed Systems 11 / 32

  22. Verification & Control Discrete Games Timed Games Symbolic Algorithms Conclusion Controllable Predecessors a e ℓ 0 ℓ 1 Bad d c u u b ℓ 2 ℓ 3 d π ( X ) = states from which one can enforce X with a controllable action A Fixpoint Characterization of Winning States: 1 let ϕ be a set of safe (good) states and G a game 2 let W ∗ be the greatest fixpoint of h ( X ) = ϕ ∩ π ( X ) 3 W ∗ is the set of winning states for ( G , ϕ ) MSR’05 (Autrans, France) Control of Timed Systems 11 / 32

  23. Verification & Control Discrete Games Timed Games Symbolic Algorithms Conclusion Controllable Predecessors a e ℓ 0 ℓ 1 Bad d c u u b ℓ 2 ℓ 3 d π ( X ) = states from which one can enforce X with a controllable action A Fixpoint Characterization of Winning States: 1 let ϕ be a set of safe (good) states and G a game 2 let W ∗ be the greatest fixpoint of h ( X ) = ϕ ∩ π ( X ) 3 W ∗ is the set of winning states for ( G , ϕ ) MSR’05 (Autrans, France) Control of Timed Systems 11 / 32

  24. Verification & Control Discrete Games Timed Games Symbolic Algorithms Conclusion Controllable Predecessors a e ℓ 0 ℓ 1 Bad d c u u b ℓ 2 ℓ 3 d π ( X ) = states from which one can enforce X with a controllable action A Fixpoint Characterization of Winning States: 1 let ϕ be a set of safe (good) states and G a game 2 let W ∗ be the greatest fixpoint of h ( X ) = ϕ ∩ π ( X ) 3 W ∗ is the set of winning states for ( G , ϕ ) ◮ CP: check that ℓ 0 ∈ W ∗ MSR’05 (Autrans, France) Control of Timed Systems 11 / 32

  25. Verification & Control Discrete Games Timed Games Symbolic Algorithms Conclusion Controllable Predecessors a e ℓ 0 ℓ 1 Bad d c u u b ℓ 2 ℓ 3 d π ( X ) = states from which one can enforce X with a controllable action A Fixpoint Characterization of Winning States: 1 let ϕ be a set of safe (good) states and G a game 2 let W ∗ be the greatest fixpoint of h ( X ) = ϕ ∩ π ( X ) 3 W ∗ is the set of winning states for ( G , ϕ ) ◮ CP: check that ℓ 0 ∈ W ∗ ◮ CSP: Given W ∗ and G , we can build a winning strategy MSR’05 (Autrans, France) Control of Timed Systems 11 / 32

  26. Verification & Control Discrete Games Timed Games Symbolic Algorithms Conclusion Results for Finite Games Given G a finite game, ϕ a control objective Theorem (Positional Strategies are Sufficient) Positional (or memoryless) strategies suffice to win ω -regular games. The number of states of C is ≤ number of states of G. MSR’05 (Autrans, France) Control of Timed Systems 12 / 32

  27. Verification & Control Discrete Games Timed Games Symbolic Algorithms Conclusion Results for Finite Games Given G a finite game, ϕ a control objective The fixpoint computation of W ∗ terminates Theorem (Positional Strategies are Sufficient) Positional (or memoryless) strategies suffice to win ω -regular games. The number of states of C is ≤ number of states of G. MSR’05 (Autrans, France) Control of Timed Systems 12 / 32

  28. Verification & Control Discrete Games Timed Games Symbolic Algorithms Conclusion Results for Finite Games Given G a finite game, ϕ a control objective Theorem (CP is Decidable) CP is decidable for ω -regular objectives. Theorem (Positional Strategies are Sufficient) Positional (or memoryless) strategies suffice to win ω -regular games. The number of states of C is ≤ number of states of G. MSR’05 (Autrans, France) Control of Timed Systems 12 / 32

  29. Verification & Control Discrete Games Timed Games Symbolic Algorithms Conclusion Results for Finite Games Given G a finite game, ϕ a control objective Theorem (CP is Decidable) CP is decidable for ω -regular objectives. Theorem (Effectiveness of CSP) Strategy synthesis is effective. We can build a finite automaton (controller) C that specifies a winning strategy. Theorem (Positional Strategies are Sufficient) Positional (or memoryless) strategies suffice to win ω -regular games. The number of states of C is ≤ number of states of G. MSR’05 (Autrans, France) Control of Timed Systems 12 / 32

  30. Verification & Control Discrete Games Timed Games Symbolic Algorithms Conclusion Results for Finite Games Given G a finite game, ϕ a control objective Theorem (CP is Decidable) CP is decidable for ω -regular objectives. Theorem (Effectiveness of CSP) Strategy synthesis is effective. We can build a finite automaton (controller) C that specifies a winning strategy. Theorem (Positional Strategies are Sufficient) Positional (or memoryless) strategies suffice to win ω -regular games. The number of states of C is ≤ number of states of G. Add Dense Time ... CP and CSP ? MSR’05 (Autrans, France) Control of Timed Systems 12 / 32

  31. Verification & Control Discrete Games Timed Games Symbolic Algorithms Conclusion Outline ◮ Verification & Control ◮ Control of Finite Automata ◮ Timed Game Automata ◮ Symbolic Algorithms for Timed Game Automata ◮ Conclusion MSR’05 (Autrans, France) Control of Timed Systems 13 / 32

  32. Verification & Control Discrete Games Timed Games Symbolic Algorithms Conclusion Timed Automata [Alur & Dill’94] [ x ≤ 4 ] [ x ≤ 5 ] x ≤ 4; c 1 x > 3; u x := 0 ℓ 0 ℓ 1 Bad c 2 c 3 ; x := 0 ℓ 2 x < 2; u [ x ≤ 5 ] Runs = sequence of discrete and time steps 1 . 55 c 1 1 . 67 u ρ 1 : ( ℓ 0 , 0 ) − − − → ( ℓ 0 , 1 . 55 ) − − → ( ℓ 1 , 1 . 55 ) − − − → ( ℓ 1 , 3 . 22 ) − → ( Bad , 3 . 22 ) 1 . 1 c 1 2 . 1 c 2 ρ 2 : ( ℓ 0 , 0 ) − − → ( ℓ 0 , 1 . 1 ) − − → ( ℓ 1 , 1 . 1 ) − − → ( ℓ 1 , 3 . 2 ) − − → ( ℓ 2 , 3 . 2 ) 0 . 1 u − − → ( ℓ 2 , 3 . 3 ) − → ( ℓ 0 , 0 ) · · · · · · · · · MSR’05 (Autrans, France) Control of Timed Systems 14 / 32

  33. Verification & Control Discrete Games Timed Games Symbolic Algorithms Conclusion Timed Automata [Alur & Dill’94] [ x ≤ 4 ] [ x ≤ 5 ] x ≤ 4; c 1 x > 3; u x := 0 ℓ 0 ℓ 1 Bad c 2 c 3 ; x := 0 ℓ 2 x < 2; u [ x ≤ 5 ] Runs = sequence of discrete and time steps 1 . 55 c 1 1 . 67 u ρ 1 : ( ℓ 0 , 0 ) − − − → ( ℓ 0 , 1 . 55 ) − − → ( ℓ 1 , 1 . 55 ) − − − → ( ℓ 1 , 3 . 22 ) − → ( Bad , 3 . 22 ) 1 . 1 c 1 2 . 1 c 2 ρ 2 : ( ℓ 0 , 0 ) − − → ( ℓ 0 , 1 . 1 ) − − → ( ℓ 1 , 1 . 1 ) − − → ( ℓ 1 , 3 . 2 ) − − → ( ℓ 2 , 3 . 2 ) 0 . 1 u − − → ( ℓ 2 , 3 . 3 ) − → ( ℓ 0 , 0 ) · · · · · · · · · MSR’05 (Autrans, France) Control of Timed Systems 14 / 32

  34. Verification & Control Discrete Games Timed Games Symbolic Algorithms Conclusion Timed Automata [Alur & Dill’94] [ x ≤ 4 ] [ x ≤ 5 ] x ≤ 4; c 1 x > 3; u x := 0 ℓ 0 ℓ 1 Bad c 2 c 3 ; x := 0 ℓ 2 x < 2; u [ x ≤ 5 ] Runs = sequence of discrete and time steps 1 . 55 c 1 1 . 67 u ρ 1 : ( ℓ 0 , 0 ) − − − → ( ℓ 0 , 1 . 55 ) − − → ( ℓ 1 , 1 . 55 ) − − − → ( ℓ 1 , 3 . 22 ) − → ( Bad , 3 . 22 ) 1 . 1 c 1 2 . 1 c 2 ρ 2 : ( ℓ 0 , 0 ) − − → ( ℓ 0 , 1 . 1 ) − − → ( ℓ 1 , 1 . 1 ) − − → ( ℓ 1 , 3 . 2 ) − − → ( ℓ 2 , 3 . 2 ) 0 . 1 u − − → ( ℓ 2 , 3 . 3 ) − → ( ℓ 0 , 0 ) · · · · · · · · · MSR’05 (Autrans, France) Control of Timed Systems 14 / 32

  35. Verification & Control Discrete Games Timed Games Symbolic Algorithms Conclusion Timed Automata [Alur & Dill’94] [ x ≤ 4 ] [ x ≤ 5 ] x ≤ 4; c 1 x > 3; u x := 0 ℓ 0 ℓ 1 Bad c 2 c 3 ; x := 0 ℓ 2 x < 2; u [ x ≤ 5 ] Runs = sequence of discrete and time steps 1 . 55 c 1 1 . 67 u ρ 1 : ( ℓ 0 , 0 ) − − − → ( ℓ 0 , 1 . 55 ) − − → ( ℓ 1 , 1 . 55 ) − − − → ( ℓ 1 , 3 . 22 ) − → ( Bad , 3 . 22 ) 1 . 1 c 1 2 . 1 c 2 ρ 2 : ( ℓ 0 , 0 ) − − → ( ℓ 0 , 1 . 1 ) − − → ( ℓ 1 , 1 . 1 ) − − → ( ℓ 1 , 3 . 2 ) − − → ( ℓ 2 , 3 . 2 ) 0 . 1 u − − → ( ℓ 2 , 3 . 3 ) − → ( ℓ 0 , 0 ) · · · · · · · · · MSR’05 (Autrans, France) Control of Timed Systems 14 / 32

  36. Verification & Control Discrete Games Timed Games Symbolic Algorithms Conclusion Timed Automata [Alur & Dill’94] [ x ≤ 4 ] [ x ≤ 5 ] x ≤ 4; c 1 x > 3; u x := 0 ℓ 0 ℓ 1 Bad c 2 c 3 ; x := 0 ℓ 2 x < 2; u [ x ≤ 5 ] Runs = sequence of discrete and time steps 1 . 55 c 1 1 . 67 u ρ 1 : ( ℓ 0 , 0 ) − − − → ( ℓ 0 , 1 . 55 ) − − → ( ℓ 1 , 1 . 55 ) − − − → ( ℓ 1 , 3 . 22 ) − → ( Bad , 3 . 22 ) 1 . 1 c 1 2 . 1 c 2 ρ 2 : ( ℓ 0 , 0 ) − − → ( ℓ 0 , 1 . 1 ) − − → ( ℓ 1 , 1 . 1 ) − − → ( ℓ 1 , 3 . 2 ) − − → ( ℓ 2 , 3 . 2 ) 0 . 1 u − − → ( ℓ 2 , 3 . 3 ) − → ( ℓ 0 , 0 ) · · · · · · · · · MSR’05 (Autrans, France) Control of Timed Systems 14 / 32

  37. Verification & Control Discrete Games Timed Games Symbolic Algorithms Conclusion Timed Automata [Alur & Dill’94] [ x ≤ 4 ] [ x ≤ 5 ] x ≤ 4; c 1 x > 3; u x := 0 ℓ 0 ℓ 1 Bad c 2 c 3 ; x := 0 ℓ 2 x < 2; u [ x ≤ 5 ] Runs = sequence of discrete and time steps 1 . 55 c 1 1 . 67 u ρ 1 : ( ℓ 0 , 0 ) − − − → ( ℓ 0 , 1 . 55 ) − − → ( ℓ 1 , 1 . 55 ) − − − → ( ℓ 1 , 3 . 22 ) − → ( Bad , 3 . 22 ) 1 . 1 c 1 2 . 1 c 2 ρ 2 : ( ℓ 0 , 0 ) − − → ( ℓ 0 , 1 . 1 ) − − → ( ℓ 1 , 1 . 1 ) − − → ( ℓ 1 , 3 . 2 ) − − → ( ℓ 2 , 3 . 2 ) 0 . 1 u − − → ( ℓ 2 , 3 . 3 ) − → ( ℓ 0 , 0 ) · · · · · · · · · MSR’05 (Autrans, France) Control of Timed Systems 14 / 32

  38. Verification & Control Discrete Games Timed Games Symbolic Algorithms Conclusion Timed Automata [Alur & Dill’94] [ x ≤ 4 ] [ x ≤ 5 ] x ≤ 4; c 1 x > 3; u x := 0 ℓ 0 ℓ 1 Bad c 2 c 3 ; x := 0 ℓ 2 x < 2; u [ x ≤ 5 ] Runs = sequence of discrete and time steps 1 . 55 c 1 1 . 67 u ρ 1 : ( ℓ 0 , 0 ) − − − → ( ℓ 0 , 1 . 55 ) − − → ( ℓ 1 , 1 . 55 ) − − − → ( ℓ 1 , 3 . 22 ) − → ( Bad , 3 . 22 ) 1 . 1 c 1 2 . 1 c 2 ρ 2 : ( ℓ 0 , 0 ) − − → ( ℓ 0 , 1 . 1 ) − − → ( ℓ 1 , 1 . 1 ) − − → ( ℓ 1 , 3 . 2 ) − − → ( ℓ 2 , 3 . 2 ) 0 . 1 u − − → ( ℓ 2 , 3 . 3 ) − → ( ℓ 0 , 0 ) · · · · · · · · · MSR’05 (Autrans, France) Control of Timed Systems 14 / 32

  39. Verification & Control Discrete Games Timed Games Symbolic Algorithms Conclusion Timed Automata [Alur & Dill’94] [ x ≤ 4 ] [ x ≤ 5 ] x ≤ 4; c 1 x > 3; u x := 0 ℓ 0 ℓ 1 Bad c 2 c 3 ; x := 0 ℓ 2 x < 2; u [ x ≤ 5 ] Runs = sequence of discrete and time steps 1 . 55 c 1 1 . 67 u ρ 1 : ( ℓ 0 , 0 ) − − − → ( ℓ 0 , 1 . 55 ) − − → ( ℓ 1 , 1 . 55 ) − − − → ( ℓ 1 , 3 . 22 ) − → ( Bad , 3 . 22 ) 1 . 1 c 1 2 . 1 c 2 ρ 2 : ( ℓ 0 , 0 ) − − → ( ℓ 0 , 1 . 1 ) − − → ( ℓ 1 , 1 . 1 ) − − → ( ℓ 1 , 3 . 2 ) − − → ( ℓ 2 , 3 . 2 ) 0 . 1 u − − → ( ℓ 2 , 3 . 3 ) − → ( ℓ 0 , 0 ) · · · · · · · · · MSR’05 (Autrans, France) Control of Timed Systems 14 / 32

  40. Verification & Control Discrete Games Timed Games Symbolic Algorithms Conclusion Timed Automata [Alur & Dill’94] [ x ≤ 4 ] [ x ≤ 5 ] x ≤ 4; c 1 x > 3; u x := 0 ℓ 0 ℓ 1 Bad c 2 c 3 ; x := 0 ℓ 2 x < 2; u [ x ≤ 5 ] Runs = sequence of discrete and time steps 1 . 55 c 1 1 . 67 u ρ 1 : ( ℓ 0 , 0 ) − − − → ( ℓ 0 , 1 . 55 ) − − → ( ℓ 1 , 1 . 55 ) − − − → ( ℓ 1 , 3 . 22 ) − → ( Bad , 3 . 22 ) 1 . 1 c 1 2 . 1 c 2 ρ 2 : ( ℓ 0 , 0 ) − − → ( ℓ 0 , 1 . 1 ) − − → ( ℓ 1 , 1 . 1 ) − − → ( ℓ 1 , 3 . 2 ) − − → ( ℓ 2 , 3 . 2 ) 0 . 1 u − − → ( ℓ 2 , 3 . 3 ) − → ( ℓ 0 , 0 ) · · · · · · · · · MSR’05 (Autrans, France) Control of Timed Systems 14 / 32

  41. Verification & Control Discrete Games Timed Games Symbolic Algorithms Conclusion Timed Automata [Alur & Dill’94] [ x ≤ 4 ] [ x ≤ 5 ] x ≤ 4; c 1 x > 3; u x := 0 ℓ 0 ℓ 1 Bad c 2 c 3 ; x := 0 ℓ 2 x < 2; u [ x ≤ 5 ] Runs = sequence of discrete and time steps 1 . 55 c 1 1 . 67 u ρ 1 : ( ℓ 0 , 0 ) − − − → ( ℓ 0 , 1 . 55 ) − − → ( ℓ 1 , 1 . 55 ) − − − → ( ℓ 1 , 3 . 22 ) − → ( Bad , 3 . 22 ) 1 . 1 c 1 2 . 1 c 2 ρ 2 : ( ℓ 0 , 0 ) − − → ( ℓ 0 , 1 . 1 ) − − → ( ℓ 1 , 1 . 1 ) − − → ( ℓ 1 , 3 . 2 ) − − → ( ℓ 2 , 3 . 2 ) 0 . 1 u − − → ( ℓ 2 , 3 . 3 ) − → ( ℓ 0 , 0 ) · · · · · · · · · MSR’05 (Autrans, France) Control of Timed Systems 14 / 32

  42. Verification & Control Discrete Games Timed Games Symbolic Algorithms Conclusion Timed Automata [Alur & Dill’94] [ x ≤ 4 ] [ x ≤ 5 ] x ≤ 4; c 1 x > 3; u x := 0 ℓ 0 ℓ 1 Bad c 2 c 3 ; x := 0 ℓ 2 x < 2; u [ x ≤ 5 ] Runs = sequence of discrete and time steps 1 . 55 c 1 1 . 67 u ρ 1 : ( ℓ 0 , 0 ) − − − → ( ℓ 0 , 1 . 55 ) − − → ( ℓ 1 , 1 . 55 ) − − − → ( ℓ 1 , 3 . 22 ) − → ( Bad , 3 . 22 ) 1 . 1 c 1 2 . 1 c 2 ρ 2 : ( ℓ 0 , 0 ) − − → ( ℓ 0 , 1 . 1 ) − − → ( ℓ 1 , 1 . 1 ) − − → ( ℓ 1 , 3 . 2 ) − − → ( ℓ 2 , 3 . 2 ) 0 . 1 u − − → ( ℓ 2 , 3 . 3 ) − → ( ℓ 0 , 0 ) · · · · · · · · · MSR’05 (Autrans, France) Control of Timed Systems 14 / 32

  43. Verification & Control Discrete Games Timed Games Symbolic Algorithms Conclusion Timed Automata [Alur & Dill’94] [ x ≤ 4 ] [ x ≤ 5 ] x ≤ 4; c 1 x > 3; u x := 0 ℓ 0 ℓ 1 Bad c 2 c 3 ; x := 0 ℓ 2 x < 2; u [ x ≤ 5 ] Runs = sequence of discrete and time steps 1 . 55 c 1 1 . 67 u ρ 1 : ( ℓ 0 , 0 ) − − − → ( ℓ 0 , 1 . 55 ) − − → ( ℓ 1 , 1 . 55 ) − − − → ( ℓ 1 , 3 . 22 ) − → ( Bad , 3 . 22 ) 1 . 1 c 1 2 . 1 c 2 ρ 2 : ( ℓ 0 , 0 ) − − → ( ℓ 0 , 1 . 1 ) − − → ( ℓ 1 , 1 . 1 ) − − → ( ℓ 1 , 3 . 2 ) − − → ( ℓ 2 , 3 . 2 ) 0 . 1 u − − → ( ℓ 2 , 3 . 3 ) − → ( ℓ 0 , 0 ) · · · · · · · · · MSR’05 (Autrans, France) Control of Timed Systems 14 / 32

  44. Verification & Control Discrete Games Timed Games Symbolic Algorithms Conclusion Timed Automata [Alur & Dill’94] [ x ≤ 4 ] [ x ≤ 5 ] x ≤ 4; c 1 x > 3; u x := 0 ℓ 0 ℓ 1 Bad c 2 c 3 ; x := 0 ℓ 2 x < 2; u [ x ≤ 5 ] Runs = sequence of discrete and time steps 1 . 55 c 1 1 . 67 u ρ 1 : ( ℓ 0 , 0 ) − − − → ( ℓ 0 , 1 . 55 ) − − → ( ℓ 1 , 1 . 55 ) − − − → ( ℓ 1 , 3 . 22 ) − → ( Bad , 3 . 22 ) 1 . 1 c 1 2 . 1 c 2 ρ 2 : ( ℓ 0 , 0 ) − − → ( ℓ 0 , 1 . 1 ) − − → ( ℓ 1 , 1 . 1 ) − − → ( ℓ 1 , 3 . 2 ) − − → ( ℓ 2 , 3 . 2 ) 0 . 1 u − − → ( ℓ 2 , 3 . 3 ) − → ( ℓ 0 , 0 ) · · · · · · · · · MSR’05 (Autrans, France) Control of Timed Systems 14 / 32

  45. Verification & Control Discrete Games Timed Games Symbolic Algorithms Conclusion Timed Automata [Alur & Dill’94] [ x ≤ 4 ] [ x ≤ 5 ] x ≤ 4; c 1 x > 3; u x := 0 ℓ 0 ℓ 1 Bad c 2 c 3 ; x := 0 ℓ 2 x < 2; u [ x ≤ 5 ] Runs = sequence of discrete and time steps 1 . 55 c 1 1 . 67 u ρ 1 : ( ℓ 0 , 0 ) − − − → ( ℓ 0 , 1 . 55 ) − − → ( ℓ 1 , 1 . 55 ) − − − → ( ℓ 1 , 3 . 22 ) − → ( Bad , 3 . 22 ) 1 . 1 c 1 2 . 1 c 2 ρ 2 : ( ℓ 0 , 0 ) − − → ( ℓ 0 , 1 . 1 ) − − → ( ℓ 1 , 1 . 1 ) − − → ( ℓ 1 , 3 . 2 ) − − → ( ℓ 2 , 3 . 2 ) 0 . 1 u − − → ( ℓ 2 , 3 . 3 ) − → ( ℓ 0 , 0 ) · · · · · · · · · MSR’05 (Autrans, France) Control of Timed Systems 14 / 32

  46. Verification & Control Discrete Games Timed Games Symbolic Algorithms Conclusion Timed Automata [Alur & Dill’94] [ x ≤ 4 ] [ x ≤ 5 ] x ≤ 4; c 1 x > 3; u x := 0 ℓ 0 ℓ 1 Bad c 2 c 3 ; x := 0 ℓ 2 x < 2; u [ x ≤ 5 ] Runs = sequence of discrete and time steps 1 . 55 c 1 1 . 67 u ρ 1 : ( ℓ 0 , 0 ) − − − → ( ℓ 0 , 1 . 55 ) − − → ( ℓ 1 , 1 . 55 ) − − − → ( ℓ 1 , 3 . 22 ) − → ( Bad , 3 . 22 ) 1 . 1 c 1 2 . 1 c 2 ρ 2 : ( ℓ 0 , 0 ) − − → ( ℓ 0 , 1 . 1 ) − − → ( ℓ 1 , 1 . 1 ) − − → ( ℓ 1 , 3 . 2 ) − − → ( ℓ 2 , 3 . 2 ) 0 . 1 u − − → ( ℓ 2 , 3 . 3 ) − → ( ℓ 0 , 0 ) · · · · · · · · · MSR’05 (Autrans, France) Control of Timed Systems 14 / 32

  47. Verification & Control Discrete Games Timed Games Symbolic Algorithms Conclusion Timed Game Automata [ x ≤ 4 ] [ x ≤ 5 ] x ≤ 4; c 1 x > 3; u x := 0 ℓ 0 ℓ 1 Bad c 2 c 3 ; x := 0 ℓ 2 x < 2; u ◮ Introduced by Maler, Pnueli, Sifakis [Maler, 95] ◮ The controller continuously observes the system time elapsing and discrete moves are observable ◮ It has the choice between two types of moves: ◮ “do nothing” ◮ “do a controllable action” (among the ones that are possible) ◮ It can stop time from elapsing by taking a controllable move MSR’05 (Autrans, France) Control of Timed Systems 15 / 32

  48. Verification & Control Discrete Games Timed Games Symbolic Algorithms Conclusion Timed Game Automata [ x ≤ 4 ] [ x ≤ 5 ] x ≤ 4; c 1 x > 3; u x := 0 ℓ 0 ℓ 1 Bad c 2 c 3 ; x := 0 ℓ 2 x < 2; u ◮ Introduced by Maler, Pnueli, Sifakis [Maler, 95] ◮ The controller continuously observes the system time elapsing and discrete moves are observable ◮ It has the choice between two types of moves: ◮ “do nothing” ◮ “do a controllable action” (among the ones that are possible) ◮ It can stop time from elapsing by taking a controllable move MSR’05 (Autrans, France) Control of Timed Systems 15 / 32

  49. Verification & Control Discrete Games Timed Games Symbolic Algorithms Conclusion Timed Game Automata [ x ≤ 4 ] [ x ≤ 5 ] x ≤ 4; c 1 x > 3; u x := 0 ℓ 0 ℓ 1 Bad c 2 c 3 ; x := 0 ℓ 2 x < 2; u ◮ Introduced by Maler, Pnueli, Sifakis [Maler, 95] ◮ The controller continuously observes the system time elapsing and discrete moves are observable ◮ It has the choice between two types of moves: ◮ “do nothing” ◮ “do a controllable action” (among the ones that are possible) ◮ It can stop time from elapsing by taking a controllable move MSR’05 (Autrans, France) Control of Timed Systems 15 / 32

  50. Verification & Control Discrete Games Timed Games Symbolic Algorithms Conclusion Timed Game Automata [ x ≤ 4 ] [ x ≤ 5 ] x ≤ 4; c 1 x > 3; u x := 0 ℓ 0 ℓ 1 Bad c 2 c 3 ; x := 0 ℓ 2 x < 2; u ◮ Introduced by Maler, Pnueli, Sifakis [Maler, 95] ◮ The controller continuously observes the system time elapsing and discrete moves are observable ◮ It has the choice between two types of moves: ◮ “do nothing” ◮ “do a controllable action” (among the ones that are possible) ◮ It can stop time from elapsing by taking a controllable move MSR’05 (Autrans, France) Control of Timed Systems 15 / 32

  51. Verification & Control Discrete Games Timed Games Symbolic Algorithms Conclusion Timed Game Automata [ x ≤ 4 ] [ x ≤ 5 ] x ≤ 4; c 1 x > 3; u x := 0 ℓ 0 ℓ 1 Bad c 2 c 3 ; x := 0 ℓ 2 x < 2; u ◮ Introduced by Maler, Pnueli, Sifakis [Maler, 95] ◮ The controller continuously observes the system time elapsing and discrete moves are observable ◮ It has the choice between two types of moves: ◮ “do nothing” ◮ “do a controllable action” (among the ones that are possible) ◮ It can stop time from elapsing by taking a controllable move MSR’05 (Autrans, France) Control of Timed Systems 15 / 32

  52. Verification & Control Discrete Games Timed Games Symbolic Algorithms Conclusion Timed Game Automata [ x ≤ 4 ] [ x ≤ 5 ] x ≤ 4; c 1 x > 3; u x := 0 ℓ 0 ℓ 1 Bad c 2 c 3 ; x := 0 ℓ 2 x < 2; u ◮ Introduced by Maler, Pnueli, Sifakis [Maler, 95] ◮ The controller continuously observes the system time elapsing and discrete moves are observable ◮ It has the choice between two types of moves: ◮ “do nothing” ◮ “do a controllable action” (among the ones that are possible) ◮ It can stop time from elapsing by taking a controllable move MSR’05 (Autrans, France) Control of Timed Systems 15 / 32

  53. Verification & Control Discrete Games Timed Games Symbolic Algorithms Conclusion Timed Game Automata [ x ≤ 4 ] [ x ≤ 5 ] x ≤ 4; c 1 x > 3; u x := 0 ℓ 0 ℓ 1 Bad c 2 c 3 ; x := 0 ℓ 2 x < 2; u ◮ Introduced by Maler, Pnueli, Sifakis [Maler, 95] ◮ The controller continuously observes the system time elapsing and discrete moves are observable ◮ It has the choice between two types of moves: ◮ “do nothing” ◮ “do a controllable action” (among the ones that are possible) ◮ It can stop time from elapsing by taking a controllable move MSR’05 (Autrans, France) Control of Timed Systems 15 / 32

  54. Verification & Control Discrete Games Timed Games Symbolic Algorithms Conclusion How to Deal with Dense-Time ? ◮ Infinite state systems Symbolic representation of states ◮ A strategy (or controller) can choose to wait Add a special wait action ◮ Dense time · · · the controller can be unfair ◮ block time ◮ do infinitely many actions in a bounded time ◮ do arbitrarily closed (in time) discrete actions Implementation Issues MSR’05 (Autrans, France) Control of Timed Systems 16 / 32

  55. Verification & Control Discrete Games Timed Games Symbolic Algorithms Conclusion How to Deal with Dense-Time ? ◮ Infinite state systems Symbolic representation of states ◮ A strategy (or controller) can choose to wait Add a special wait action ◮ Dense time · · · the controller can be unfair ◮ block time ◮ do infinitely many actions in a bounded time ◮ do arbitrarily closed (in time) discrete actions Implementation Issues MSR’05 (Autrans, France) Control of Timed Systems 16 / 32

  56. Verification & Control Discrete Games Timed Games Symbolic Algorithms Conclusion How to Deal with Dense-Time ? ◮ Infinite state systems Symbolic representation of states ◮ A strategy (or controller) can choose to wait Add a special wait action ◮ Dense time · · · the controller can be unfair ◮ block time ◮ do infinitely many actions in a bounded time ◮ do arbitrarily closed (in time) discrete actions Implementation Issues MSR’05 (Autrans, France) Control of Timed Systems 16 / 32

  57. Verification & Control Discrete Games Timed Games Symbolic Algorithms Conclusion How to Deal with Dense-Time ? ◮ Infinite state systems Symbolic representation of states ◮ A strategy (or controller) can choose to wait Add a special wait action ◮ Dense time · · · the controller can be unfair ◮ block time ◮ do infinitely many actions in a bounded time ◮ do arbitrarily closed (in time) discrete actions Implementation Issues MSR’05 (Autrans, France) Control of Timed Systems 16 / 32

  58. Verification & Control Discrete Games Timed Games Symbolic Algorithms Conclusion How to Deal with Dense-Time ? ◮ Infinite state systems Symbolic representation of states ◮ A strategy (or controller) can choose to wait Add a special wait action ◮ Dense time · · · the controller can be unfair ◮ block time ◮ do infinitely many actions in a bounded time ◮ do arbitrarily closed (in time) discrete actions Implementation Issues MSR’05 (Autrans, France) Control of Timed Systems 16 / 32

  59. Verification & Control Discrete Games Timed Games Symbolic Algorithms Conclusion How to Deal with Dense-Time ? ◮ Infinite state systems Symbolic representation of states ◮ A strategy (or controller) can choose to wait Add a special wait action ◮ Dense time · · · the controller can be unfair ◮ block time ◮ do infinitely many actions in a bounded time ◮ do arbitrarily closed (in time) discrete actions Implementation Issues MSR’05 (Autrans, France) Control of Timed Systems 16 / 32

  60. Verification & Control Discrete Games Timed Games Symbolic Algorithms Conclusion How to Deal with Dense-Time ? ◮ Infinite state systems Symbolic representation of states ◮ A strategy (or controller) can choose to wait Add a special wait action ◮ Dense time · · · the controller can be unfair ◮ block time ◮ do infinitely many actions in a bounded time ◮ do arbitrarily closed (in time) discrete actions Implementation Issues MSR’05 (Autrans, France) Control of Timed Systems 16 / 32

  61. Verification & Control Discrete Games Timed Games Symbolic Algorithms Conclusion How to Deal with Dense-Time ? ◮ Infinite state systems Symbolic representation of states ◮ A strategy (or controller) can choose to wait Add a special wait action ◮ Dense time · · · the controller can be unfair ◮ block time ◮ do infinitely many actions in a bounded time ◮ do arbitrarily closed (in time) discrete actions Implementation Issues MSR’05 (Autrans, France) Control of Timed Systems 16 / 32

  62. Verification & Control Discrete Games Timed Games Symbolic Algorithms Conclusion How to Deal with Dense-Time ? ◮ Infinite state systems Symbolic representation of states ◮ A strategy (or controller) can choose to wait Add a special wait action ◮ Dense time · · · the controller can be unfair ◮ block time ◮ do infinitely many actions in a bounded time ◮ do arbitrarily closed (in time) discrete actions Implementation Issues MSR’05 (Autrans, France) Control of Timed Systems 16 / 32

  63. Verification & Control Discrete Games Timed Games Symbolic Algorithms Conclusion Strategies and Winning States [ x ≤ 4 ] [ x ≤ 5 ] x ≤ 4; c 1 x > 3; u x := 0 ℓ 0 ℓ 1 Bad c 2 c 3 ; x := 0 ℓ 2 x < 2; u [ x ≤ 5 ] MSR’05 (Autrans, France) Control of Timed Systems 17 / 32

  64. Verification & Control Discrete Games Timed Games Symbolic Algorithms Conclusion Strategies and Winning States [ x ≤ 4 ] [ x ≤ 5 ] x ≤ 4; c 1 x > 3; u x := 0 ℓ 0 ℓ 1 Bad c 2 c 3 ; x := 0 ℓ 2 x < 2; u [ x ≤ 5 ] The strategy f : “Always wait as long as the system permits” 4 c 1 0 . 5 u ρ 1 : ( ℓ 0 , 0 ) − → ( ℓ 0 , 4 ) − − → ( ℓ 1 , 4 ) − − → ( ℓ 1 , 4 . 5 ) − → ( Bad , 4 . 5 ) 4 c 1 1 . 0 c 2 c 3 − → ( ℓ 0 , 4 ) − − → ( ℓ 1 , 4 ) − − → ( ℓ 1 , 5 ) − − → ( ℓ 2 , 5 ) − − → ( ℓ 0 , 0 ) · · · ρ 2 : ( ℓ 0 , 0 ) MSR’05 (Autrans, France) Control of Timed Systems 17 / 32

  65. Verification & Control Discrete Games Timed Games Symbolic Algorithms Conclusion Strategies and Winning States [ x ≤ 4 ] [ x ≤ 5 ] x ≤ 4; c 1 x > 3; u x := 0 ℓ 0 ℓ 1 Bad c 2 c 3 ; x := 0 ℓ 2 x < 2; u [ x ≤ 5 ] The strategy f : “Always wait as long as the system permits” 4 c 1 0 . 5 u ρ 1 : ( ℓ 0 , 0 ) − → ( ℓ 0 , 4 ) − − → ( ℓ 1 , 4 ) − − → ( ℓ 1 , 4 . 5 ) − → ( Bad , 4 . 5 ) 4 c 1 1 . 0 c 2 c 3 − → ( ℓ 0 , 4 ) − − → ( ℓ 1 , 4 ) − − → ( ℓ 1 , 5 ) − − → ( ℓ 2 , 5 ) − − → ( ℓ 0 , 0 ) · · · ρ 2 : ( ℓ 0 , 0 ) MSR’05 (Autrans, France) Control of Timed Systems 17 / 32

  66. Verification & Control Discrete Games Timed Games Symbolic Algorithms Conclusion Strategies and Winning States [ x ≤ 4 ] [ x ≤ 5 ] x ≤ 4; c 1 x > 3; u x := 0 ℓ 0 ℓ 1 Bad c 2 c 3 ; x := 0 ℓ 2 x < 2; u [ x ≤ 5 ] The strategy f : “Always wait as long as the system permits” 4 c 1 0 . 5 u ρ 1 : ( ℓ 0 , 0 ) − → ( ℓ 0 , 4 ) − − → ( ℓ 1 , 4 ) − − → ( ℓ 1 , 4 . 5 ) − → ( Bad , 4 . 5 ) 4 c 1 1 . 0 c 2 c 3 − → ( ℓ 0 , 4 ) − − → ( ℓ 1 , 4 ) − − → ( ℓ 1 , 5 ) − − → ( ℓ 2 , 5 ) − − → ( ℓ 0 , 0 ) · · · ρ 2 : ( ℓ 0 , 0 ) MSR’05 (Autrans, France) Control of Timed Systems 17 / 32

  67. Verification & Control Discrete Games Timed Games Symbolic Algorithms Conclusion Strategies and Winning States [ x ≤ 4 ] [ x ≤ 5 ] x ≤ 4; c 1 x > 3; u x := 0 ℓ 0 ℓ 1 Bad c 2 c 3 ; x := 0 ℓ 2 x < 2; u [ x ≤ 5 ] The strategy f : “Always wait as long as the system permits” 4 c 1 0 . 5 u ρ 1 : ( ℓ 0 , 0 ) − → ( ℓ 0 , 4 ) − − → ( ℓ 1 , 4 ) − − → ( ℓ 1 , 4 . 5 ) − → ( Bad , 4 . 5 ) 4 c 1 1 . 0 c 2 c 3 − → ( ℓ 0 , 4 ) − − → ( ℓ 1 , 4 ) − − → ( ℓ 1 , 5 ) − − → ( ℓ 2 , 5 ) − − → ( ℓ 0 , 0 ) · · · ρ 2 : ( ℓ 0 , 0 ) MSR’05 (Autrans, France) Control of Timed Systems 17 / 32

  68. Verification & Control Discrete Games Timed Games Symbolic Algorithms Conclusion Strategies and Winning States [ x ≤ 4 ] [ x ≤ 5 ] x ≤ 4; c 1 x > 3; u x := 0 ℓ 0 ℓ 1 Bad c 2 c 3 ; x := 0 ℓ 2 x < 2; u [ x ≤ 5 ] The strategy f : “Always wait as long as the system permits” 4 c 1 0 . 5 u ρ 1 : ( ℓ 0 , 0 ) − → ( ℓ 0 , 4 ) − − → ( ℓ 1 , 4 ) − − → ( ℓ 1 , 4 . 5 ) − → ( Bad , 4 . 5 ) 4 c 1 1 . 0 c 2 c 3 − → ( ℓ 0 , 4 ) − − → ( ℓ 1 , 4 ) − − → ( ℓ 1 , 5 ) − − → ( ℓ 2 , 5 ) − − → ( ℓ 0 , 0 ) · · · ρ 2 : ( ℓ 0 , 0 ) MSR’05 (Autrans, France) Control of Timed Systems 17 / 32

  69. Verification & Control Discrete Games Timed Games Symbolic Algorithms Conclusion Strategies and Winning States [ x ≤ 4 ] [ x ≤ 5 ] x ≤ 4; c 1 x > 3; u x := 0 ℓ 0 ℓ 1 Bad c 2 c 3 ; x := 0 ℓ 2 x < 2; u [ x ≤ 5 ] The strategy f : “Always wait as long as the system permits” 4 c 1 0 . 5 u ρ 1 : ( ℓ 0 , 0 ) − → ( ℓ 0 , 4 ) − − → ( ℓ 1 , 4 ) − − → ( ℓ 1 , 4 . 5 ) − → ( Bad , 4 . 5 ) 4 c 1 1 . 0 c 2 c 3 − → ( ℓ 0 , 4 ) − − → ( ℓ 1 , 4 ) − − → ( ℓ 1 , 5 ) − − → ( ℓ 2 , 5 ) − − → ( ℓ 0 , 0 ) · · · ρ 2 : ( ℓ 0 , 0 ) MSR’05 (Autrans, France) Control of Timed Systems 17 / 32

  70. Verification & Control Discrete Games Timed Games Symbolic Algorithms Conclusion Strategies and Winning States [ x ≤ 4 ] [ x ≤ 5 ] x ≤ 4; c 1 x > 3; u x := 0 ℓ 0 ℓ 1 Bad c 2 c 3 ; x := 0 ℓ 2 x < 2; u [ x ≤ 5 ] The strategy f : “Always wait as long as the system permits” 4 c 1 0 . 5 u ρ 1 : ( ℓ 0 , 0 ) − → ( ℓ 0 , 4 ) − − → ( ℓ 1 , 4 ) − − → ( ℓ 1 , 4 . 5 ) − → ( Bad , 4 . 5 ) 4 c 1 1 . 0 c 2 c 3 − → ( ℓ 0 , 4 ) − − → ( ℓ 1 , 4 ) − − → ( ℓ 1 , 5 ) − − → ( ℓ 2 , 5 ) − − → ( ℓ 0 , 0 ) · · · ρ 2 : ( ℓ 0 , 0 ) MSR’05 (Autrans, France) Control of Timed Systems 17 / 32

  71. Verification & Control Discrete Games Timed Games Symbolic Algorithms Conclusion Strategies and Winning States [ x ≤ 4 ] [ x ≤ 5 ] x ≤ 4; c 1 x > 3; u x := 0 ℓ 0 ℓ 1 Bad c 2 c 3 ; x := 0 ℓ 2 x < 2; u [ x ≤ 5 ] The strategy f : “Always wait as long as the system permits” 4 c 1 0 . 5 u ρ 1 : ( ℓ 0 , 0 ) − → ( ℓ 0 , 4 ) − − → ( ℓ 1 , 4 ) − − → ( ℓ 1 , 4 . 5 ) − → ( Bad , 4 . 5 ) 4 c 1 1 . 0 c 2 c 3 − → ( ℓ 0 , 4 ) − − → ( ℓ 1 , 4 ) − − → ( ℓ 1 , 5 ) − − → ( ℓ 2 , 5 ) − − → ( ℓ 0 , 0 ) · · · ρ 2 : ( ℓ 0 , 0 ) MSR’05 (Autrans, France) Control of Timed Systems 17 / 32

  72. Verification & Control Discrete Games Timed Games Symbolic Algorithms Conclusion Strategies and Winning States [ x ≤ 4 ] [ x ≤ 5 ] x ≤ 4; c 1 x > 3; u x := 0 ℓ 0 ℓ 1 Bad c 2 c 3 ; x := 0 ℓ 2 x < 2; u [ x ≤ 5 ] The strategy f : “Always wait as long as the system permits” 4 c 1 0 . 5 u ρ 1 : ( ℓ 0 , 0 ) − → ( ℓ 0 , 4 ) − − → ( ℓ 1 , 4 ) − − → ( ℓ 1 , 4 . 5 ) − → ( Bad , 4 . 5 ) 4 c 1 1 . 0 c 2 c 3 − → ( ℓ 0 , 4 ) − − → ( ℓ 1 , 4 ) − − → ( ℓ 1 , 5 ) − − → ( ℓ 2 , 5 ) − − → ( ℓ 0 , 0 ) · · · ρ 2 : ( ℓ 0 , 0 ) MSR’05 (Autrans, France) Control of Timed Systems 17 / 32

  73. Verification & Control Discrete Games Timed Games Symbolic Algorithms Conclusion Strategies and Winning States [ x ≤ 4 ] [ x ≤ 5 ] x ≤ 4; c 1 x > 3; u x := 0 ℓ 0 ℓ 1 Bad c 2 c 3 ; x := 0 ℓ 2 x < 2; u [ x ≤ 5 ] The strategy f : “Always wait as long as the system permits” 4 c 1 0 . 5 u ρ 1 : ( ℓ 0 , 0 ) − → ( ℓ 0 , 4 ) − − → ( ℓ 1 , 4 ) − − → ( ℓ 1 , 4 . 5 ) − → ( Bad , 4 . 5 ) 4 c 1 1 . 0 c 2 c 3 − → ( ℓ 0 , 4 ) − − → ( ℓ 1 , 4 ) − − → ( ℓ 1 , 5 ) − − → ( ℓ 2 , 5 ) − − → ( ℓ 0 , 0 ) · · · ρ 2 : ( ℓ 0 , 0 ) MSR’05 (Autrans, France) Control of Timed Systems 17 / 32

  74. Verification & Control Discrete Games Timed Games Symbolic Algorithms Conclusion Strategies and Winning States [ x ≤ 4 ] [ x ≤ 5 ] x ≤ 4; c 1 x > 3; u x := 0 ℓ 0 ℓ 1 Bad c 2 c 3 ; x := 0 ℓ 2 x < 2; u [ x ≤ 5 ] The strategy f : “Always wait as long as the system permits” 4 c 1 0 . 5 u ρ 1 : ( ℓ 0 , 0 ) − → ( ℓ 0 , 4 ) − − → ( ℓ 1 , 4 ) − − → ( ℓ 1 , 4 . 5 ) − → ( Bad , 4 . 5 ) 4 c 1 1 . 0 c 2 c 3 − → ( ℓ 0 , 4 ) − − → ( ℓ 1 , 4 ) − − → ( ℓ 1 , 5 ) − − → ( ℓ 2 , 5 ) − − → ( ℓ 0 , 0 ) · · · ρ 2 : ( ℓ 0 , 0 ) MSR’05 (Autrans, France) Control of Timed Systems 17 / 32

  75. Verification & Control Discrete Games Timed Games Symbolic Algorithms Conclusion Strategies and Winning States [ x ≤ 4 ] [ x ≤ 5 ] x ≤ 4; c 1 x > 3; u x := 0 ℓ 0 ℓ 1 Bad c 2 c 3 ; x := 0 ℓ 2 x < 2; u [ x ≤ 5 ] The strategy f : “Always wait as long as the system permits” 4 c 1 0 . 5 u ρ 1 : ( ℓ 0 , 0 ) − → ( ℓ 0 , 4 ) − − → ( ℓ 1 , 4 ) − − → ( ℓ 1 , 4 . 5 ) − → ( Bad , 4 . 5 ) 4 c 1 1 . 0 c 2 c 3 − → ( ℓ 0 , 4 ) − − → ( ℓ 1 , 4 ) − − → ( ℓ 1 , 5 ) − − → ( ℓ 2 , 5 ) − − → ( ℓ 0 , 0 ) · · · ρ 2 : ( ℓ 0 , 0 ) MSR’05 (Autrans, France) Control of Timed Systems 17 / 32

  76. Verification & Control Discrete Games Timed Games Symbolic Algorithms Conclusion Strategies and Winning States [ x ≤ 4 ] [ x ≤ 5 ] x ≤ 4; c 1 x > 3; u x := 0 ℓ 0 ℓ 1 Bad c 2 c 3 ; x := 0 ℓ 2 x < 2; u [ x ≤ 5 ] The strategy f : “Always wait as long as the system permits” 4 c 1 0 . 5 u ρ 1 : ( ℓ 0 , 0 ) − → ( ℓ 0 , 4 ) − − → ( ℓ 1 , 4 ) − − → ( ℓ 1 , 4 . 5 ) − → ( Bad , 4 . 5 ) 4 c 1 1 . 0 c 2 c 3 − → ( ℓ 0 , 4 ) − − → ( ℓ 1 , 4 ) − − → ( ℓ 1 , 5 ) − − → ( ℓ 2 , 5 ) − − → ( ℓ 0 , 0 ) · · · ρ 2 : ( ℓ 0 , 0 ) MSR’05 (Autrans, France) Control of Timed Systems 17 / 32

  77. Verification & Control Discrete Games Timed Games Symbolic Algorithms Conclusion Strategies and Winning States [ x ≤ 4 ] [ x ≤ 5 ] x ≤ 4; c 1 x > 3; u x := 0 ℓ 0 ℓ 1 Bad c 2 c 3 ; x := 0 ℓ 2 x < 2; u [ x ≤ 5 ] A winning strategy f ′ in ℓ 0 at x = 2 do c 1 ; in ℓ 1 at x = 2 . 5 do c 2 ; in ℓ 2 at x = 4 do c 3 2 c 1 ( ℓ 0 , 0 ) − → ( ℓ 0 , 2 ) − − → ( ℓ 1 , 2 ) ρ : MSR’05 (Autrans, France) Control of Timed Systems 17 / 32

  78. Verification & Control Discrete Games Timed Games Symbolic Algorithms Conclusion Strategies and Winning States [ x ≤ 4 ] [ x ≤ 5 ] x ≤ 4; c 1 x > 3; u x := 0 ℓ 0 ℓ 1 Bad c 2 c 3 ; x := 0 ℓ 2 x < 2; u [ x ≤ 5 ] A winning strategy f ′ in ℓ 0 at x = 2 do c 1 ; in ℓ 1 at x = 2 . 5 do c 2 ; in ℓ 2 at x = 4 do c 3 2 c 1 ( ℓ 0 , 0 ) − → ( ℓ 0 , 2 ) − − → ( ℓ 1 , 2 ) ρ : MSR’05 (Autrans, France) Control of Timed Systems 17 / 32

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