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Hybrid Systems Verification and Robotics Andr e Platzer aplatzer@cs.cmu.edu Computer Science Department Carnegie Mellon University, Pittsburgh, PA http://symbolaris.com/ 0.5 0.4 0.3 0.2 1.0 0.1 0.8 0.6 0.4 0.2 Andr e Platzer

  1. Hybrid Systems Verification and Robotics Andr´ e Platzer aplatzer@cs.cmu.edu Computer Science Department Carnegie Mellon University, Pittsburgh, PA http://symbolaris.com/ 0.5 0.4 0.3 0.2 1.0 0.1 0.8 0.6 0.4 0.2 Andr´ e Platzer (CMU) Hybrid Systems Verification and Robotics RSS-FMRA 1 / 25

  2. Outline Hybrid Systems Applications 1 Logic for Hybrid Systems 2 Model Checking 3 Successive Image Computation Image Computation in Hybrid Systems Approximation Refinement Model Checking Summary Proofs for Hybrid Systems 4 Proof Rules Soundness and Completeness Survey 5 Summary 6 Andr´ e Platzer (CMU) Hybrid Systems Verification and Robotics RSS-FMRA 1 / 25

  3. Can you trust a computer to control physics? Andr´ e Platzer (CMU) Hybrid Systems Verification and Robotics RSS-FMRA 2 / 25

  4. Outline Hybrid Systems Applications 1 Logic for Hybrid Systems 2 Model Checking 3 Successive Image Computation Image Computation in Hybrid Systems Approximation Refinement Model Checking Summary Proofs for Hybrid Systems 4 Proof Rules Soundness and Completeness Survey 5 Summary 6 Andr´ e Platzer (CMU) Hybrid Systems Verification and Robotics RSS-FMRA 2 / 25

  5. Hybrid Systems Analysis Challenge (Hybrid Systems) Fixed rule describing state 3.5 evolution with both 3.0 2.5 Discrete dynamics 2.0 (control decisions) 1.5 1.0 Continuous dynamics 0.5 (differential equations) 0.0 0 1 2 3 4 5 6 0.2 a v p 2.5 p x 0.8 0.1 2.0 10 t 0.6 2 4 6 8 1.5 0.4 � 0.1 1.0 p y � 0.2 0.2 0.5 � 0.3 10 t 10 t 2 4 6 8 2 4 6 8 Andr´ e Platzer (CMU) Hybrid Systems Verification and Robotics RSS-FMRA 3 / 25

  6. Hybrid Systems Analysis Challenge (Hybrid Systems) Fixed rule describing state 3.5 evolution with both 3.0 2.5 Discrete dynamics 2.0 (control decisions) 1.5 1.0 Continuous dynamics 0.5 (differential equations) 0.0 0 1 2 3 4 5 6 0.2 a d Ω 1.0 d x 0.00008 0.1 0.8 0.00006 10 t 0.6 2 4 6 8 0.00004 � 0.1 0.4 0.00002 � 0.2 0.2 10 t 2 4 6 8 � 0.3 d y 10 t 2 4 6 8 Andr´ e Platzer (CMU) Hybrid Systems Verification and Robotics RSS-FMRA 3 / 25

  7. Hybrid Systems Analysis Challenge (Hybrid Systems) Fixed rule describing state 3.5 evolution with both 3.0 2.5 Discrete dynamics 2.0 (control decisions) 1.5 1.0 Continuous dynamics 0.5 (differential equations) 0.0 0 1 2 3 4 5 6 a 1.0 v p 0.2 8 0.8 10 t 2 4 6 8 6 0.6 � 0.2 p x 4 0.4 � 0.4 � 0.6 0.2 2 p y � 0.8 10 t 10 t 2 4 6 8 2 4 6 8 Andr´ e Platzer (CMU) Hybrid Systems Verification and Robotics RSS-FMRA 4 / 25

  8. Hybrid Systems Analysis Challenge (Hybrid Systems) Fixed rule describing state 3.5 evolution with both 3.0 2.5 Discrete dynamics 2.0 (control decisions) 1.5 1.0 Continuous dynamics 0.5 (differential equations) 0.0 0 1 2 3 4 5 6 a d Ω 1.0 d x 0.2 0.5 10 t 2 4 6 8 0.5 10 t 2 4 6 8 � 0.2 � 0.5 10 t � 0.4 2 4 6 8 d y � 0.6 � 1.0 � 0.5 � 0.8 Andr´ e Platzer (CMU) Hybrid Systems Verification and Robotics RSS-FMRA 4 / 25

  9. Hybrid Systems Analysis Challenge (Hybrid Systems) Fixed rule describing state 3.5 evolution with both 3.0 2.5 Discrete dynamics 2.0 (control decisions) 1.5 1.0 Continuous dynamics 0.5 (differential equations) 0.0 0 1 2 3 4 5 6 a 1.0 v p 10 t p x 2 4 6 8 4 0.8 � 1 3 0.6 � 2 2 0.4 � 3 1 0.2 p y 10 t � 4 10 t 2 4 6 8 2 4 6 8 Andr´ e Platzer (CMU) Hybrid Systems Verification and Robotics RSS-FMRA 5 / 25

  10. Hybrid Systems Analysis Challenge (Hybrid Systems) Fixed rule describing state 3.5 evolution with both 3.0 2.5 Discrete dynamics 2.0 (control decisions) 1.5 1.0 Continuous dynamics 0.5 (differential equations) 0.0 0 1 2 3 4 5 6 a d Ω 10 t 1.0 2 4 6 8 d x 0.5 � 1 0.5 10 t 2 4 6 8 � 2 10 t � 0.5 2 4 6 8 � 3 d y � 1.0 � 0.5 � 4 Andr´ e Platzer (CMU) Hybrid Systems Verification and Robotics RSS-FMRA 5 / 25

  11. Hybrid Systems Analysis Challenge (Hybrid Systems) Fixed rule describing state 3.5 evolution with both 3.0 2.5 Discrete dynamics 2.0 (control decisions) 1.5 1.0 Continuous dynamics 0.5 (differential equations) 0.0 0 1 2 3 4 5 6 a 1.2 v 7 p 0.4 6 1.0 0.2 5 0.8 10 t 4 2 4 6 8 0.6 p x � 0.2 3 0.4 � 0.4 2 0.2 1 � 0.6 p y 10 t 10 t 2 4 6 8 2 4 6 8 Andr´ e Platzer (CMU) Hybrid Systems Verification and Robotics RSS-FMRA 6 / 25

  12. Hybrid Systems Analysis Challenge (Hybrid Systems) Fixed rule describing state 3.5 evolution with both 3.0 2.5 Discrete dynamics 2.0 (control decisions) 1.5 1.0 Continuous dynamics 0.5 (differential equations) 0.0 0 1 2 3 4 5 6 a d Ω 1.0 d x 0.4 0.5 0.2 0.5 10 t 10 t d y 2 4 6 8 2 4 6 8 � 0.2 � 0.5 10 t 2 4 6 8 � 0.4 � 1.0 � 0.6 � 0.5 Andr´ e Platzer (CMU) Hybrid Systems Verification and Robotics RSS-FMRA 6 / 25

  13. Outline Hybrid Systems Applications 1 Logic for Hybrid Systems 2 Model Checking 3 Successive Image Computation Image Computation in Hybrid Systems Approximation Refinement Model Checking Summary Proofs for Hybrid Systems 4 Proof Rules Soundness and Completeness Survey 5 Summary 6 Andr´ e Platzer (CMU) Hybrid Systems Verification and Robotics RSS-FMRA 6 / 25

  14. Logic for Hybrid Systems 3.5 3.0 2.5 differential dynamic logic 2.0 1.5 d L = DL + HP 1.0 0.5 0.0 0 1 2 3 4 6 5 Andr´ e Platzer (CMU) Hybrid Systems Verification and Robotics RSS-FMRA 7 / 25

  15. Logic for Hybrid Systems 3.5 3.0 2.5 differential dynamic logic 2.0 1.5 d L = FOL R 1.0 0.5 0.0 0 1 2 3 4 6 5 v v 2 ≤ 2 b ( M − z ) z M Andr´ e Platzer (CMU) Hybrid Systems Verification and Robotics RSS-FMRA 7 / 25

  16. Logic for Hybrid Systems 3.5 3.0 2.5 differential dynamic logic 2.0 1.5 d L = FOL R 1.0 0.5 0.0 0 1 2 3 4 6 5 v v ≤ 1 z M Andr´ e Platzer (CMU) Hybrid Systems Verification and Robotics RSS-FMRA 7 / 25

  17. Logic for Hybrid Systems 3.5 3.0 2.5 differential dynamic logic 2.0 1.5 d L = FOL R 1.0 0.5 0.0 0 1 2 3 4 6 5 v v ≤ 1 ∧ v 2 ≤ 2 b ( M − z ) z M Andr´ e Platzer (CMU) Hybrid Systems Verification and Robotics RSS-FMRA 7 / 25

  18. Logic for Hybrid Systems 3.5 3.0 2.5 differential dynamic logic 2.0 1.5 d L = FOL R 1.0 0.5 0.0 0 1 2 3 4 6 5 v v ≤ 1 ∨ v 2 ≤ 2 b ( M − z ) z M Andr´ e Platzer (CMU) Hybrid Systems Verification and Robotics RSS-FMRA 7 / 25

  19. Logic for Hybrid Systems 3.5 3.0 2.5 differential dynamic logic 2.0 1.5 d L = FOL R 1.0 0.5 0.0 0 1 2 3 4 6 5 v v ≤ 1 ∨ v 2 ≤ 2 b ( M − z ) ∀ M ∃ SB . . . ∀ t ≥ 0 . . . z M Andr´ e Platzer (CMU) Hybrid Systems Verification and Robotics RSS-FMRA 7 / 25

  20. Logic for Hybrid Systems 3.5 3.0 2.5 differential dynamic logic 2.0 1.5 d L = FOL R + 1.0 0.5 0.0 0 1 2 3 4 6 5 v 2 ≤ 2 b Andr´ e Platzer (CMU) Hybrid Systems Verification and Robotics RSS-FMRA 7 / 25

  21. Logic for Hybrid Systems 3.5 3.0 2.5 differential dynamic logic 2.0 1.5 d L = FOL R + ML 1.0 0.5 0.0 0 1 2 3 4 6 5 v 2 ≤ 2 b v 2 ≤ 2 b � v 2 ≤ 2 b v 2 ≤ 2 b Andr´ e Platzer (CMU) Hybrid Systems Verification and Robotics RSS-FMRA 7 / 25

  22. Logic for Hybrid Systems 3.5 3.0 2.5 differential dynamic logic 2.0 1.5 d L = FOL R + DL 1.0 0.5 0.0 0 1 2 3 4 6 5 v 2 ≤ 2 b v 2 ≤ 2 b ] v 2 ≤ 2 b [ v 2 ≤ 2 b Andr´ e Platzer (CMU) Hybrid Systems Verification and Robotics RSS-FMRA 7 / 25

  23. Logic for Hybrid Systems 3.5 3.0 2.5 differential dynamic logic 2.0 1.5 d L = FOL R + DL + HP 1.0 0.5 0.0 0 1 2 3 4 6 5 v 2 ≤ 2 b v 2 ≤ 2 b [ z ′′ = a ] v 2 ≤ 2 b v 2 ≤ 2 b Andr´ e Platzer (CMU) Hybrid Systems Verification and Robotics RSS-FMRA 7 / 25

  24. Logic for Hybrid Systems 3.5 3.0 2.5 differential dynamic logic 2.0 1.5 d L = FOL R + DL + HP 1.0 0.5 0.0 0 1 2 3 4 6 5 v 2 ≤ 2 b v 2 ≤ 2 b [ if ( z > SB ) a := − b ; z ′′ = a ] v 2 ≤ 2 b v 2 ≤ 2 b Andr´ e Platzer (CMU) Hybrid Systems Verification and Robotics RSS-FMRA 7 / 25

  25. Logic for Hybrid Systems 3.5 3.0 2.5 differential dynamic logic 2.0 1.5 d L = FOL R + DL + HP 1.0 0.5 0.0 0 1 2 3 4 6 5 v 2 ≤ 2 b v 2 ≤ 2 b [ if ( z > SB ) a := − b ; z ′′ = a ] v 2 ≤ 2 b � �� � v 2 ≤ 2 b hybrid program Andr´ e Platzer (CMU) Hybrid Systems Verification and Robotics RSS-FMRA 7 / 25

  26. Logic for Hybrid Systems 3.5 3.0 2.5 differential dynamic logic 2.0 1.5 d L = FOL R + DL + HP 1.0 0.5 0.0 0 1 2 3 4 6 5 v 2 ≤ 2 b v 2 ≤ 2 b C → [ if ( z > SB ) a := − b ; z ′′ = a ] v 2 ≤ 2 b � �� � v 2 ≤ 2 b hybrid program Andr´ e Platzer (CMU) Hybrid Systems Verification and Robotics RSS-FMRA 7 / 25

  27. Logic for Hybrid Systems 3.5 3.0 2.5 differential dynamic logic 2.0 1.5 d L = FOL R + DL + HP 1.0 0.5 0.0 0 1 2 3 4 6 5 v 2 ≤ 2 b v 2 ≤ 2 b C → [ if ( z > SB ) a := − b ; z ′′ = a ] v 2 ≤ 2 b � �� � v 2 ≤ 2 b hybrid program Initial condition Andr´ e Platzer (CMU) Hybrid Systems Verification and Robotics RSS-FMRA 7 / 25

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