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Stochastic Differential Dynamic Logic for Stochastic Hybrid Systems Andr e Platzer Logical Systems Lab Carnegie Mellon University, Pittsburgh, PA 0.5 0.4 0.3 0.2 1.0 0.1 0.8 0.6 0.4 0.2 Andr e Platzer (CMU) Stochastic

  1. Stochastic Differential Dynamic Logic for Stochastic Hybrid Systems Andr´ e Platzer Logical Systems Lab Carnegie Mellon University, Pittsburgh, PA 0.5 0.4 0.3 0.2 1.0 0.1 0.8 0.6 0.4 0.2 Andr´ e Platzer (CMU) Stochastic Differential Dynamic Logic for Stochastic Hybrid Systems CADE 1 / 22

  2. Outline Motivation 1 Stochastic Differential Dynamic Logic Sd L 2 Design Stochastic Differential Equations Syntax Semantics Well-definedness Stochastic Differential Dynamic Logic 3 Syntax Semantics Well-definedness Proof Calculus for Stochastic Hybrid Systems 4 Compositional Proof Calculus Soundness Conclusions 5 Andr´ e Platzer (CMU) Stochastic Differential Dynamic Logic for Stochastic Hybrid Systems CADE 1 / 22

  3. Cyber-Physical Systems: Q: I want to verify trains Challenge Andr´ e Platzer (CMU) Stochastic Differential Dynamic Logic for Stochastic Hybrid Systems CADE 2 / 22

  4. Cyber-Physical Systems: Hybrid Systems Q: I want to verify trains A: Hybrid systems Challenge (Hybrid Systems) Continuous dynamics (differential equations) Discrete dynamics (control decisions) z v a 6 3.0 2 5 2.5 1 4 2.0 3 1.5 4 t 1 2 3 2 1.0 � 1 1 0.5 4 t 4 t � 2 1 2 3 1 2 3 Andr´ e Platzer (CMU) Stochastic Differential Dynamic Logic for Stochastic Hybrid Systems CADE 2 / 22

  5. Cyber-Physical Systems: Hybrid Systems Q: I want to verify trains A: Hybrid systems Q: But there’s uncertainties! Challenge (Hybrid Systems) Continuous dynamics (differential equations) Discrete dynamics (control decisions) z v a 6 3.0 2 5 2.5 1 4 2.0 3 1.5 4 t 1 2 3 2 1.0 � 1 1 0.5 4 t 4 t � 2 1 2 3 1 2 3 Andr´ e Platzer (CMU) Stochastic Differential Dynamic Logic for Stochastic Hybrid Systems CADE 2 / 22

  6. Cyber-Physical Systems: Q: I want to verify uncertain trains Challenge Andr´ e Platzer (CMU) Stochastic Differential Dynamic Logic for Stochastic Hybrid Systems CADE 3 / 22

  7. Cyber-Physical Systems: Probabilistic Systems Q: I want to verify uncertain trains A: Markov chains Challenge (Probabilistic Systems) Directed graph (Countable state space) Weighted edges (Transition probabilities) 0.4 0.3 0.3 1.0 0.2 0.3 0.1 0.4 1.0 0.3 0.3 0.3 0.5 0.5 1.0 1.0 0.1 0.9 0.6 1.0 0.5 0.3 1.0 0.3 0.7 0.5 0.5 0.2 0.8 1.0 0.4 0.5 0.2 1.0 0.4 0.7 1.0 0.8 0.7 1.0 Andr´ e Platzer (CMU) Stochastic Differential Dynamic Logic for Stochastic Hybrid Systems CADE 3 / 22

  8. Cyber-Physical Systems: Probabilistic Systems Q: I want to verify uncertain trains A: Markov chains Q: But trains move! Challenge (Probabilistic Systems) Directed graph (Countable state space) Weighted edges (Transition probabilities) 0.4 0.3 0.3 1.0 0.2 0.3 0.1 0.4 1.0 0.3 0.3 0.3 0.5 0.5 1.0 1.0 0.1 0.9 0.6 1.0 0.5 0.3 1.0 0.3 0.7 0.5 0.5 0.2 0.8 1.0 0.4 0.5 0.2 1.0 0.4 0.7 1.0 0.8 0.7 1.0 Andr´ e Platzer (CMU) Stochastic Differential Dynamic Logic for Stochastic Hybrid Systems CADE 3 / 22

  9. Cyber-Physical Systems: Q: I want to verify uncertain trains Challenge Andr´ e Platzer (CMU) Stochastic Differential Dynamic Logic for Stochastic Hybrid Systems CADE 4 / 22

  10. Cyber-Physical Systems: Stochastic Hybrid Systems Q: I want to verify uncertain trains A: Stochastic hybrid systems Challenge (Stochastic Hybrid Systems) Continuous dynamics (differential equations) Discrete dynamics (control decisions) Stochastic dynamics (uncertainty) Andr´ e Platzer (CMU) Stochastic Differential Dynamic Logic for Stochastic Hybrid Systems CADE 4 / 22

  11. Cyber-Physical Systems: Stochastic Hybrid Systems Q: I want to verify uncertain trains A: Stochastic hybrid systems Challenge (Stochastic Hybrid Systems) Continuous dynamics (differential equations) Discrete dynamics (control decisions) Stochastic dynamics (uncertainty) v Discrete stochastic (lossy communication) Continuous stochastic (wind, track) z m Andr´ e Platzer (CMU) Stochastic Differential Dynamic Logic for Stochastic Hybrid Systems CADE 4 / 22

  12. Cyber-Physical Systems: Stochastic Hybrid Systems Q: I want to verify uncertain trains A: Stochastic hybrid systems Q: How? Challenge (Stochastic Hybrid Systems) Continuous dynamics (differential equations) Discrete dynamics (control decisions) Stochastic dynamics (uncertainty) v Discrete stochastic (lossy communication) Continuous stochastic (wind, track) z m Andr´ e Platzer (CMU) Stochastic Differential Dynamic Logic for Stochastic Hybrid Systems CADE 4 / 22

  13. Contributions 1 System model and semantics for stochastic hybrid systems: SHP 2 Prove semantic processes are adapted and a.s. c` adl` ag 3 Prove natural process stopping times are Markov times 4 Specification and verification logic: Sd L 5 Prove measurability of Sd L semantics ⇒ probabilities well-defined 6 Proof rules for Sd L 7 Sound Dynkin use of infinitesimal generators of SDEs 8 First compositional verification for stochastic hybrid systems 9 Logical foundation for analysis of stochastic hybrid systems Andr´ e Platzer (CMU) Stochastic Differential Dynamic Logic for Stochastic Hybrid Systems CADE 5 / 22

  14. Outline Motivation 1 Stochastic Differential Dynamic Logic Sd L 2 Design Stochastic Differential Equations Syntax Semantics Well-definedness Stochastic Differential Dynamic Logic 3 Syntax Semantics Well-definedness Proof Calculus for Stochastic Hybrid Systems 4 Compositional Proof Calculus Soundness Conclusions 5 Andr´ e Platzer (CMU) Stochastic Differential Dynamic Logic for Stochastic Hybrid Systems CADE 5 / 22

  15. Outline (Conceptual Approach) Motivation 1 Stochastic Differential Dynamic Logic Sd L 2 Design Stochastic Differential Equations Syntax Semantics Well-definedness Stochastic Differential Dynamic Logic 3 Syntax Semantics Well-definedness Proof Calculus for Stochastic Hybrid Systems 4 Compositional Proof Calculus Soundness Conclusions 5 Andr´ e Platzer (CMU) Stochastic Differential Dynamic Logic for Stochastic Hybrid Systems CADE 5 / 22

  16. Model for Stochastic Hybrid Systems a := − b c r e s t i e d Andr´ e Platzer (CMU) Stochastic Differential Dynamic Logic for Stochastic Hybrid Systems CADE 6 / 22

  17. Model for Stochastic Hybrid Systems a := − b c r e s t i e d c o n t i n u o u s d 2 x dt 2 = a Andr´ e Platzer (CMU) Stochastic Differential Dynamic Logic for Stochastic Hybrid Systems CADE 6 / 22

  18. Model for Stochastic Hybrid Systems a := − b c r e s t i e d c o n t i c n i u t s o a u h s c o t s 1 3 a := − b ⊕ 2 d 2 x 3 a := a + 1 dt 2 = a Andr´ e Platzer (CMU) Stochastic Differential Dynamic Logic for Stochastic Hybrid Systems CADE 6 / 22

  19. Model for Stochastic Hybrid Systems a := − b c r e s t i e d a := − b ; d 2 x dt 2 = a c o n t i c n i u t s o a u h s c o t s 1 3 a := − b ⊕ 2 d 2 x 3 a := a + 1 dt 2 = a Andr´ e Platzer (CMU) Stochastic Differential Dynamic Logic for Stochastic Hybrid Systems CADE 6 / 22

  20. Model for Stochastic Hybrid Systems a := − b c r e s t i e d a := − b ; d 2 x dt 2 = a a := ∗ c o n t i c n i u t s o a u h s c o t s 1 3 a := − b ⊕ 2 d 2 x 3 a := a + 1 dt 2 = a Andr´ e Platzer (CMU) Stochastic Differential Dynamic Logic for Stochastic Hybrid Systems CADE 6 / 22

  21. Model for Stochastic Hybrid Systems a := − b c r e s t i e d a := − b ; d 2 x dt 2 = a a := ∗ c o n t i c n i u t s o a u h s c o t s 1 3 a := − b ⊕ 2 d 2 x 3 a := a + 1 dt 2 = a dX = bdt + σ dW Andr´ e Platzer (CMU) Stochastic Differential Dynamic Logic for Stochastic Hybrid Systems CADE 6 / 22

  22. Model for Stochastic Hybrid Systems a := − b c r e s t i e d a := − b ; d 2 x dt 2 = a a := ∗ SHS c o n t i c n i u t s o a u h s c o t s 1 3 a := − b ⊕ 2 d 2 x 3 a := a + 1 dt 2 = a dX = bdt + σ dW Andr´ e Platzer (CMU) Stochastic Differential Dynamic Logic for Stochastic Hybrid Systems CADE 6 / 22

  23. Model for Stochastic Hybrid Systems Q: How to model stochastic hybrid systems Model (Stochastic Hybrid Systems) Andr´ e Platzer (CMU) Stochastic Differential Dynamic Logic for Stochastic Hybrid Systems CADE 7 / 22

  24. Model for Stochastic Hybrid Systems Q: How to model stochastic hybrid systems Model (Stochastic Hybrid Systems) Discrete dynamics (control decisions) a := − b Continuous dynamics (differential equations) Stochastic dynamics (structural) Andr´ e Platzer (CMU) Stochastic Differential Dynamic Logic for Stochastic Hybrid Systems CADE 7 / 22

  25. Model for Stochastic Hybrid Systems Q: How to model stochastic hybrid systems Model (Stochastic Hybrid Systems) Discrete dynamics (control decisions) a := − b Continuous dynamics (differential equations) x ′′ = a Stochastic dynamics (structural) Andr´ e Platzer (CMU) Stochastic Differential Dynamic Logic for Stochastic Hybrid Systems CADE 7 / 22

  26. Model for Stochastic Hybrid Systems Q: How to model stochastic hybrid systems Model (Stochastic Hybrid Systems) Discrete dynamics (control decisions) a := − b Continuous dynamics (differential equations) x ′′ = a Stochastic dynamics (structural) 1 3 a := − b ⊕ 2 3 a := a + 1 Andr´ e Platzer (CMU) Stochastic Differential Dynamic Logic for Stochastic Hybrid Systems CADE 7 / 22

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