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A Temporal Dynamic Logic for Verifying Hybrid System Invariants e Platzer 1 , 2 Andr 1 University of Oldenburg, Department of Computing Science, Germany 2 Carnegie Mellon University, Computer Science Department, Pittsburgh, PA, USA LFCS07

  1. A Temporal Dynamic Logic for Verifying Hybrid System Invariants e Platzer 1 , 2 Andr´ 1 University of Oldenburg, Department of Computing Science, Germany 2 Carnegie Mellon University, Computer Science Department, Pittsburgh, PA, USA LFCS’07 Andr´ e Platzer (University of Oldenburg) Temporal Dynamic Logic for Hybrid Systems LFCS’07 1 / 16

  2. Outline Motivation 1 Temporal Dynamic Logic dTL 2 Syntax Trace Semantics Conservative Extension Safety Invariants in Train Control Verification Calculus for dTL 3 Sequent Calculus Verifying Safety Invariants in Train Control Soundness Conclusions & Future Work 4 Andr´ e Platzer (University of Oldenburg) Temporal Dynamic Logic for Hybrid Systems LFCS’07 1 / 16

  3. Verifying Hybrid Systems RBC ST MA negot SB far corr Andr´ e Platzer (University of Oldenburg) Temporal Dynamic Logic for Hybrid Systems LFCS’07 2 / 16

  4. Verifying Hybrid Systems RBC ST MA negot SB far corr Hybrid Systems continuous evolution along differential equations + discrete change Andr´ e Platzer (University of Oldenburg) Temporal Dynamic Logic for Hybrid Systems LFCS’07 2 / 16

  5. Verifying Hybrid Systems RBC ST MA negot SB far corr Hybrid Systems continuous evolution along differential equations + discrete change z v t Andr´ e Platzer (University of Oldenburg) Temporal Dynamic Logic for Hybrid Systems LFCS’07 2 / 16

  6. Verifying Hybrid Systems RBC ST MA negot SB far corr problem technique OP PAR T closed ETCS | = z < MA TL-MC � × � × Andr´ e Platzer (University of Oldenburg) Temporal Dynamic Logic for Hybrid Systems LFCS’07 2 / 16

  7. Verifying Hybrid Systems RBC ST MA negot SB far corr problem technique OP PAR T closed ETCS | = z < MA TL-MC � × � × × no free parameters like ST, SB × no finite-state bisimulation for HS Andr´ e Platzer (University of Oldenburg) Temporal Dynamic Logic for Hybrid Systems LFCS’07 2 / 16

  8. Verifying Hybrid Systems RBC ST MA negot SB far corr problem technique OP PAR T closed ETCS | = z < MA TL-MC � × � × | = (Ax( ETCS ) → z < MA ) TL-calculus × . . . . . . � Andr´ e Platzer (University of Oldenburg) Temporal Dynamic Logic for Hybrid Systems LFCS’07 2 / 16

  9. Verifying Hybrid Systems RBC ST MA negot SB far corr problem technique OP PAR T closed ETCS | = z < MA TL-MC � × � × | = (Ax( ETCS ) → z < MA ) TL-calculus × . . . . . . � × declaratively axiomatise operational model Andr´ e Platzer (University of Oldenburg) Temporal Dynamic Logic for Hybrid Systems LFCS’07 2 / 16

  10. Verifying Hybrid Systems RBC ST MA negot SB far corr problem technique OP PAR T closed ETCS | = z < MA TL-MC � × � × | = (Ax( ETCS ) → z < MA ) TL-calculus × . . . . . . � | = [ ETCS ] z < MA DL-calculus � � × � Andr´ e Platzer (University of Oldenburg) Temporal Dynamic Logic for Hybrid Systems LFCS’07 2 / 16

  11. Verifying Hybrid Systems RBC ST MA negot SB far corr problem technique OP PAR T closed ETCS | = z < MA TL-MC � × � × | = (Ax( ETCS ) → z < MA ) TL-calculus × . . . . . . � | = [ ETCS ] z < MA DL-calculus � � × � � [RBC]partitioned → � Train � [RBC]safe × no intermediate states Andr´ e Platzer (University of Oldenburg) Temporal Dynamic Logic for Hybrid Systems LFCS’07 2 / 16

  12. Verifying Hybrid Systems RBC ST MA negot SB far corr problem technique OP PAR T closed ETCS | = z < MA TL-MC � × � × | = (Ax( ETCS ) → z < MA ) TL-calculus × . . . . . . � | = [ ETCS ] z < MA DL-calculus � � × � | = [ ETCS ] � z < MA DTL-calculus � � � � Andr´ e Platzer (University of Oldenburg) Temporal Dynamic Logic for Hybrid Systems LFCS’07 2 / 16

  13. Verifying Hybrid Systems RBC ST MA negot SB far corr problem technique OP PAR T closed ETCS | = z < MA TL-MC � × � × | = (Ax( ETCS ) → z < MA ) TL-calculus × . . . . . . � | = [ ETCS ] z < MA DL-calculus � � × � | = [ ETCS ] � z < MA DTL-calculus � � � � differential temporal dynamic logic dTL = TL + DL + HP Andr´ e Platzer (University of Oldenburg) Temporal Dynamic Logic for Hybrid Systems LFCS’07 2 / 16

  14. Outline Motivation 1 Temporal Dynamic Logic dTL 2 Syntax Trace Semantics Conservative Extension Safety Invariants in Train Control Verification Calculus for dTL 3 Sequent Calculus Verifying Safety Invariants in Train Control Soundness Conclusions & Future Work 4 Andr´ e Platzer (University of Oldenburg) Temporal Dynamic Logic for Hybrid Systems LFCS’07 2 / 16

  15. Outline Motivation 1 Temporal Dynamic Logic dTL 2 Syntax Trace Semantics Conservative Extension Safety Invariants in Train Control Verification Calculus for dTL 3 Sequent Calculus Verifying Safety Invariants in Train Control Soundness Conclusions & Future Work 4 Andr´ e Platzer (University of Oldenburg) Temporal Dynamic Logic for Hybrid Systems LFCS’07 2 / 16

  16. Temporal Dynamic Logic dTL: Syntax Definition (Hybrid program α ) x ′ = f ( x ) (continuous evolution) x := θ (discrete jump) ? χ (conditional execution) α ; β (seq. composition) α ∪ β (nondet. choice) α ∗ (nondet. repetition) Andr´ e Platzer (University of Oldenburg) Temporal Dynamic Logic for Hybrid Systems LFCS’07 3 / 16

  17. Temporal Dynamic Logic dTL: Syntax Definition (Hybrid program α ) x ′ = f ( x ) (continuous evolution) x := θ (discrete jump) ? χ (conditional execution) α ; β (seq. composition) α ∪ β (nondet. choice) α ∗ (nondet. repetition) ETCS ≡ negot ; corr ; z ′′ = a RBC negot ≡ z ′ = v , ℓ ′ = 1 corr ≡ (? MA − z < SB ; a := − b ) ST SB MA far negot corr ∪ (? MA − z ≥ SB ; a := . . . ) Andr´ e Platzer (University of Oldenburg) Temporal Dynamic Logic for Hybrid Systems LFCS’07 3 / 16

  18. Temporal Dynamic Logic dTL: Syntax Definition (Formulas / state formulas φ ) ¬ , ∧ , ∨ , → , ∀ x , ∃ x , = , ≤ , + , · (first-order part) [ α ] π, � α � π (dynamic part) Definition (Trace formulas π ) φ (non-temporal part) � φ, ♦ φ (temporal part) Andr´ e Platzer (University of Oldenburg) Temporal Dynamic Logic for Hybrid Systems LFCS’07 3 / 16

  19. Temporal Dynamic Logic dTL: Syntax Definition (Formulas / state formulas φ ) ¬ , ∧ , ∨ , → , ∀ x , ∃ x , = , ≤ , + , · (first-order part) [ α ] π, � α � π (dynamic part) Definition (Trace formulas π ) φ (non-temporal part) � φ, ♦ φ (temporal part) RBC [ ETCS ] � ( ℓ ≤ L → z < MA ) ETCS ≡ negot ; corr ; z ′′ = a negot ≡ z ′ = v , ℓ ′ = 1 ST far negot SB corr MA Andr´ e Platzer (University of Oldenburg) Temporal Dynamic Logic for Hybrid Systems LFCS’07 3 / 16

  20. Temporal Dynamic Logic dTL: Trace Semantics Definition (Hybrid trace) Hybrid trace is sequence of continuous functions σ i : [0 , r i ] → Sta V x t Semantics of hybrid program: set of all its hybrid traces σ Andr´ e Platzer (University of Oldenburg) Temporal Dynamic Logic for Hybrid Systems LFCS’07 4 / 16

  21. Temporal Dynamic Logic dTL: Trace Semantics Definition (Hybrid trace) Hybrid trace is sequence of continuous functions σ i : [0 , r i ] → Sta V x t Semantics of hybrid program: set of all its hybrid traces σ Andr´ e Platzer (University of Oldenburg) Temporal Dynamic Logic for Hybrid Systems LFCS’07 4 / 16

  22. Temporal Dynamic Logic dTL: Trace Semantics Definition (Hybrid programs α : trace semantics) x := θ v w x . = val ( v , θ ) Andr´ e Platzer (University of Oldenburg) Temporal Dynamic Logic for Hybrid Systems LFCS’07 5 / 16

  23. Temporal Dynamic Logic dTL: Trace Semantics Definition (Hybrid programs α : trace semantics) x ′ = f ( x ) v w x ϕ ( t ) w v t x ′ = f ( x ) Andr´ e Platzer (University of Oldenburg) Temporal Dynamic Logic for Hybrid Systems LFCS’07 5 / 16

  24. Temporal Dynamic Logic dTL: Trace Semantics Definition (Hybrid programs α : trace semantics) ? χ if v | = χ v ? χ if v �| = χ limbo Andr´ e Platzer (University of Oldenburg) Temporal Dynamic Logic for Hybrid Systems LFCS’07 5 / 16

  25. Temporal Dynamic Logic dTL: Trace Semantics Definition (Hybrid programs α : trace semantics) α ; β v s w α β α ; β ≡ α Andr´ e Platzer (University of Oldenburg) Temporal Dynamic Logic for Hybrid Systems LFCS’07 5 / 16

  26. Temporal Dynamic Logic dTL: Trace Semantics Definition (Hybrid programs α : trace semantics) α ∗ v s 1 s 2 s n w α α α Andr´ e Platzer (University of Oldenburg) Temporal Dynamic Logic for Hybrid Systems LFCS’07 5 / 16

  27. Temporal Dynamic Logic dTL: Trace Semantics Definition (Hybrid programs α : trace semantics) w 1 α v α ∪ β β w 2 Andr´ e Platzer (University of Oldenburg) Temporal Dynamic Logic for Hybrid Systems LFCS’07 5 / 16

  28. Temporal Dynamic Logic dTL: Trace Semantics Definition (State formulas φ ) π π v [ α ] π π Andr´ e Platzer (University of Oldenburg) Temporal Dynamic Logic for Hybrid Systems LFCS’07 6 / 16

  29. Temporal Dynamic Logic dTL: Trace Semantics Definition (State formulas φ ) π v � α � π Andr´ e Platzer (University of Oldenburg) Temporal Dynamic Logic for Hybrid Systems LFCS’07 6 / 16

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