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Quantified Differential Dynamic Logic for Distributed Hybrid Systems Andr e Platzer 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) Quantified Differential Dynamic Logic

  1. Quantified Differential Dynamic Logic for Distributed Hybrid Systems Andr´ e Platzer 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) Quantified Differential Dynamic Logic for Distributed Hybrid Systems CSL’10 1 / 16

  2. Outline Motivation 1 Quantified Differential Dynamic Logic Qd L 2 Design Syntax Semantics Proof Calculus for Distributed Hybrid Systems 3 Compositional Verification Calculus Deduction Modulo with Free Variables & Skolemization Actual Existence and Creation Soundness and Completeness Conclusions 4 Andr´ e Platzer (CMU) Quantified Differential Dynamic Logic for Distributed Hybrid Systems CSL’10 1 / 16

  3. Complex Physical Systems: Q: I want to verify my car Challenge Andr´ e Platzer (CMU) Quantified Differential Dynamic Logic for Distributed Hybrid Systems CSL’10 2 / 16

  4. Complex Physical Systems: Hybrid Systems Q: I want to verify my car A: Hybrid systems Challenge (Hybrid Systems) Continuous dynamics (differential equations) Discrete dynamics (control decisions) z v 6 a 3.0 2 5 2.5 1 4 2.0 3 1.5 4 t 1 2 3 1.0 2 � 1 1 0.5 4 t 4 t � 2 1 2 3 1 2 3 Andr´ e Platzer (CMU) Quantified Differential Dynamic Logic for Distributed Hybrid Systems CSL’10 2 / 16

  5. Complex Physical Systems: Hybrid Systems Q: I want to verify my car A: Hybrid systems Q: But there’s a lot of cars! Challenge (Hybrid Systems) Continuous dynamics (differential equations) Discrete dynamics (control decisions) z v 6 a 3.0 2 5 2.5 1 4 2.0 3 1.5 4 t 1 2 3 1.0 2 � 1 1 0.5 4 t 4 t � 2 1 2 3 1 2 3 Andr´ e Platzer (CMU) Quantified Differential Dynamic Logic for Distributed Hybrid Systems CSL’10 2 / 16

  6. Complex Physical Systems: Q: I want to verify a lot of cars Challenge Andr´ e Platzer (CMU) Quantified Differential Dynamic Logic for Distributed Hybrid Systems CSL’10 3 / 16

  7. Complex Physical Systems: Distributed Systems Q: I want to verify a lot of cars A: Distributed systems Challenge (Distributed Systems) Local computation (finite state automaton) Remote communication (network graph) Andr´ e Platzer (CMU) Quantified Differential Dynamic Logic for Distributed Hybrid Systems CSL’10 3 / 16

  8. Complex Physical Systems: Distributed Systems Q: I want to verify a lot of cars A: Distributed systems Q: But they move! Challenge (Distributed Systems) Local computation (finite state automaton) Remote communication (network graph) Andr´ e Platzer (CMU) Quantified Differential Dynamic Logic for Distributed Hybrid Systems CSL’10 3 / 16

  9. Complex Physical Systems: Q: I want to verify lots of moving cars Challenge Andr´ e Platzer (CMU) Quantified Differential Dynamic Logic for Distributed Hybrid Systems CSL’10 4 / 16

  10. Complex Physical Systems: Distributed Hybrid Systems Q: I want to verify lots of moving cars A: Distributed hybrid systems Challenge (Distributed Hybrid Systems) Continuous dynamics (differential equations) Discrete dynamics (control decisions) Structural dynamics (remote communication) Andr´ e Platzer (CMU) Quantified Differential Dynamic Logic for Distributed Hybrid Systems CSL’10 4 / 16

  11. Complex Physical Systems: Distributed Hybrid Systems Q: I want to verify lots of moving cars A: Distributed hybrid systems Challenge (Distributed Hybrid Systems) Continuous dynamics (differential equations) Discrete dynamics (control decisions) Structural dynamics (remote communication) Dimensional dynamics (appearance) Andr´ e Platzer (CMU) Quantified Differential Dynamic Logic for Distributed Hybrid Systems CSL’10 4 / 16

  12. Complex Physical Systems: Distributed Hybrid Systems Q: I want to verify lots of moving cars A: Distributed hybrid systems Q: How? Challenge (Distributed Hybrid Systems) Continuous dynamics (differential equations) Discrete dynamics (control decisions) Structural dynamics (remote communication) Dimensional dynamics (appearance) Andr´ e Platzer (CMU) Quantified Differential Dynamic Logic for Distributed Hybrid Systems CSL’10 4 / 16

  13. State of the Art: Shift [DGV96] The Hybrid System R-Charon [KSPL06] Modeling Simulation Programming Language for Reconfigurable Language Hybrid Systems Hybrid CSP [CJR95] Semantics in Φ-calculus [Rou04] Semantics in rich Extended Duration Calculus set theory ACP srt HyPA [CR05] Translate fragment hs [BM05] Modeling language into normal form. proposal χ process algebra [vBMR + 06] OBSHS [MS06] Partial random Simulation, translation of simulation of objects fragments to PHAVER, UPPAAL Andr´ e Platzer (CMU) Quantified Differential Dynamic Logic for Distributed Hybrid Systems CSL’10 5 / 16

  14. State of the Art: Modeling and Simulation Shift [DGV96] The Hybrid System R-Charon [KSPL06] Modeling Simulation Programming Language for Reconfigurable Language Hybrid Systems Hybrid CSP [CJR95] Semantics in Φ-calculus [Rou04] Semantics in rich Extended Duration Calculus set theory ACP srt HyPA [CR05] Translate fragment hs [BM05] Modeling language into normal form. proposal χ process algebra [vBMR + 06] OBSHS [MS06] Partial random Simulation, translation of simulation of objects fragments to PHAVER, UPPAAL Andr´ e Platzer (CMU) Quantified Differential Dynamic Logic for Distributed Hybrid Systems CSL’10 5 / 16

  15. State of the Art: Modeling and Simulation No formal verification of distributed hybrid systems Shift [DGV96] The Hybrid System R-Charon [KSPL06] Modeling Simulation Programming Language for Reconfigurable Language Hybrid Systems Hybrid CSP [CJR95] Semantics in Φ-calculus [Rou04] Semantics in rich Extended Duration Calculus set theory ACP srt HyPA [CR05] Translate fragment hs [BM05] Modeling language into normal form. proposal χ process algebra [vBMR + 06] OBSHS [MS06] Partial random Simulation, translation of simulation of objects fragments to PHAVER, UPPAAL Andr´ e Platzer (CMU) Quantified Differential Dynamic Logic for Distributed Hybrid Systems CSL’10 5 / 16

  16. Contributions 1 System model and semantics for distributed hybrid systems: QHP 2 Specification and verification logic: Qd L 3 Proof calculus for Qd L 4 First verification approach for distributed hybrid systems 5 Sound and complete axiomatization relative to differential equations 6 Prove collision freedom in a (simple) distributed car control system, where new cars may appear dynamically on the road 7 Logical foundation for analysis of distributed hybrid systems 8 Fundamental extension: first-order x ( i ) versus primitive x Andr´ e Platzer (CMU) Quantified Differential Dynamic Logic for Distributed Hybrid Systems CSL’10 6 / 16

  17. Outline Motivation 1 Quantified Differential Dynamic Logic Qd L 2 Design Syntax Semantics Proof Calculus for Distributed Hybrid Systems 3 Compositional Verification Calculus Deduction Modulo with Free Variables & Skolemization Actual Existence and Creation Soundness and Completeness Conclusions 4 Andr´ e Platzer (CMU) Quantified Differential Dynamic Logic for Distributed Hybrid Systems CSL’10 6 / 16

  18. Outline (Conceptual Approach) Motivation 1 Quantified Differential Dynamic Logic Qd L 2 Design Syntax Semantics Proof Calculus for Distributed Hybrid Systems 3 Compositional Verification Calculus Deduction Modulo with Free Variables & Skolemization Actual Existence and Creation Soundness and Completeness Conclusions 4 Andr´ e Platzer (CMU) Quantified Differential Dynamic Logic for Distributed Hybrid Systems CSL’10 6 / 16

  19. Model for Distributed Hybrid Systems Q: How to model distributed hybrid systems Model (Distributed Hybrid Systems) Continuous dynamics (differential equations) Discrete dynamics (control decisions) Structural dynamics (communication/coupling) Andr´ e Platzer (CMU) Quantified Differential Dynamic Logic for Distributed Hybrid Systems CSL’10 7 / 16

  20. Model for Distributed Hybrid Systems Q: How to model distributed hybrid systems Model (Distributed Hybrid Systems) Continuous dynamics (differential equations) x ′′ = a Discrete dynamics (control decisions) Structural dynamics (communication/coupling) Andr´ e Platzer (CMU) Quantified Differential Dynamic Logic for Distributed Hybrid Systems CSL’10 7 / 16

  21. Model for Distributed Hybrid Systems Q: How to model distributed hybrid systems Model (Distributed Hybrid Systems) Continuous dynamics (differential equations) x ′′ = a Discrete dynamics (control decisions) a := if .. then A else − b Structural dynamics (communication/coupling) Andr´ e Platzer (CMU) Quantified Differential Dynamic Logic for Distributed Hybrid Systems CSL’10 7 / 16

  22. Model for Distributed Hybrid Systems Q: How to model distributed hybrid systems Model (Distributed Hybrid Systems) Continuous dynamics (differential equations) x ′′ = a Discrete dynamics (control decisions) a := if .. then A else − b Structural dynamics (communication/coupling) Andr´ e Platzer (CMU) Quantified Differential Dynamic Logic for Distributed Hybrid Systems CSL’10 7 / 16

  23. Model for Distributed Hybrid Systems Q: How to model distributed hybrid systems Model (Distributed Hybrid Systems) Continuous dynamics (differential equations) (2) (2) (3) (3) (4) (1) (1) (4) x ′′ = a Discrete dynamics (control decisions) a := if .. then A else − b Structural dynamics (communication/coupling) Andr´ e Platzer (CMU) Quantified Differential Dynamic Logic for Distributed Hybrid Systems CSL’10 7 / 16

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