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Differential Dynamic Logic for Verifying Parametric Hybrid Systems e Platzer 1 , 2 Andr´ 1 University of Oldenburg, Department of Computing Science, Germany 2 Carnegie Mellon University, Computer Science Department, Pittsburgh, PA, USA Tableaux’07 Andr´ e Platzer (University of Oldenburg) Differential Dynamic Logic for Parametric Hybrid Systems Tableaux’07 1 / 18

Outline Motivation 1 Differential Logic d L 2 Design Motives Syntax Transition Semantics Speed Supervision in Train Control Verification Calculus for d L 3 Sequent Calculus Modular Combination by Side Deduction Verifying Speed Supervision in Train Control Soundness Conclusions & Future Work 4 Andr´ e Platzer (University of Oldenburg) Differential Dynamic Logic for Parametric Hybrid Systems Tableaux’07 1 / 18

Verifying Parametric Hybrid Systems Andr´ e Platzer (University of Oldenburg) Differential Dynamic Logic for Parametric Hybrid Systems Tableaux’07 2 / 18

Verifying Parametric Hybrid Systems Hybrid Systems continuous evolution along differential equations + discrete change z v t Andr´ e Platzer (University of Oldenburg) Differential Dynamic Logic for Parametric Hybrid Systems Tableaux’07 2 / 18

Verifying Parametric Hybrid Systems Parametric Hybrid Systems continuous evolution along differential equations + discrete change Fix parameter SB = 10000 and hope? z Handle SB as free symbolic parameter? Which constraints for SB ? v t ∀ MA ∃ SB all ( train-runs )safe Andr´ e Platzer (University of Oldenburg) Differential Dynamic Logic for Parametric Hybrid Systems Tableaux’07 2 / 18

Verifying Parametric Hybrid Systems Parametric Hybrid Systems continuous evolution along differential equations + discrete change differential dynamic logic d L = DL + HP Andr´ e Platzer (University of Oldenburg) Differential Dynamic Logic for Parametric Hybrid Systems Tableaux’07 2 / 18

J. M. Davoren and A. Nerode. Logics for hybrid systems. Proceedings of the IEEE , 88(7):985–1010, July 2000. M. R¨ onkk¨ o, A. P. Ravn, and K. Sere. Hybrid action systems. Theor. Comput. Sci. , 290(1):937–973, 2003. W. C. Rounds. A spatial logic for the hybrid π -calculus. In R. Alur and G. J. Pappas, editors, HSCC , volume 2993 of LNCS , pages 508–522. Springer, 2004. C. Zhou, A. P. Ravn, and M. R. Hansen. An extended duration calculus for hybrid real-time systems. In R. L. Grossman, A. Nerode, A. P. Ravn, and H. Rischel, editors, Hybrid Systems , volume 736 of LNCS , pages 36–59. Springer, 1992. Andr´ e Platzer (University of Oldenburg) Differential Dynamic Logic for Parametric Hybrid Systems Tableaux’07 2 / 18

Outline Motivation 1 Differential Logic d L 2 Design Motives Syntax Transition Semantics Speed Supervision in Train Control Verification Calculus for d L 3 Sequent Calculus Modular Combination by Side Deduction Verifying Speed Supervision in Train Control Soundness Conclusions & Future Work 4 Andr´ e Platzer (University of Oldenburg) Differential Dynamic Logic for Parametric Hybrid Systems Tableaux’07 2 / 18

Outline Motivation 1 Differential Logic d L 2 Design Motives Syntax Transition Semantics Speed Supervision in Train Control Verification Calculus for d L 3 Sequent Calculus Modular Combination by Side Deduction Verifying Speed Supervision in Train Control Soundness Conclusions & Future Work 4 Andr´ e Platzer (University of Oldenburg) Differential Dynamic Logic for Parametric Hybrid Systems Tableaux’07 2 / 18

d L Motives: differential dynamic logic d L = DL + HP Andr´ e Platzer (University of Oldenburg) Differential Dynamic Logic for Parametric Hybrid Systems Tableaux’07 3 / 18

d L Motives: Regions in First-order Logic differential dynamic logic d L = FOL R v v 2 ≤ 2 b ( MA − z ) MA − z MA Andr´ e Platzer (University of Oldenburg) Differential Dynamic Logic for Parametric Hybrid Systems Tableaux’07 3 / 18

d L Motives: Regions in First-order Logic differential dynamic logic d L = FOL R v v 2 ≤ 2 b ( MA − z ) MA − z MA ∀ t after (train-runs( t ))( v 2 ≤ 2 b ( MA − z )) Andr´ e Platzer (University of Oldenburg) Differential Dynamic Logic for Parametric Hybrid Systems Tableaux’07 3 / 18

d L Motives: State Transitions in Dynamic Logic differential dynamic logic d L = FOL R + DL v v 2 ≤ 2 b ( MA − z ) MA − z MA ∀ t after (train-runs( t ))( v 2 ≤ 2 b ( MA − z )) [train-runs] v 2 ≤ 2 b ( MA − z ) Andr´ e Platzer (University of Oldenburg) Differential Dynamic Logic for Parametric Hybrid Systems Tableaux’07 3 / 18

d L Motives: Hybrid Programs as Uniform Model differential dynamic logic d L = FOL R + DL + HP [train-runs] v 2 ≤ 2 b ( MA − z ) Andr´ e Platzer (University of Oldenburg) Differential Dynamic Logic for Parametric Hybrid Systems Tableaux’07 3 / 18

d L Motives: Hybrid Programs as Uniform Model differential dynamic logic d L = FOL R + DL + HP ] v 2 ≤ 2 b ( MA − z ) [ Andr´ e Platzer (University of Oldenburg) Differential Dynamic Logic for Parametric Hybrid Systems Tableaux’07 3 / 18

d L Motives: Hybrid Programs as Uniform Model differential dynamic logic d L = FOL R + DL + HP far neg cor rec fsa Andr´ e Platzer (University of Oldenburg) Differential Dynamic Logic for Parametric Hybrid Systems Tableaux’07 3 / 18

d L Motives: Hybrid Programs as Uniform Model differential dynamic logic d L = FOL R + DL + HP far neg neg cor rec fsa Andr´ e Platzer (University of Oldenburg) Differential Dynamic Logic for Parametric Hybrid Systems Tableaux’07 3 / 18

d L Motives: Hybrid Programs as Uniform Model differential dynamic logic d L = FOL R + DL + HP far neg neg cor not compositional rec fsa Andr´ e Platzer (University of Oldenburg) Differential Dynamic Logic for Parametric Hybrid Systems Tableaux’07 3 / 18

Differential Logic d L : 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) Differential Dynamic Logic for Parametric Hybrid Systems Tableaux’07 4 / 18

Differential Logic d L : 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 ≡ ( cor ; drive ) ∗ cor ≡ (? MA − z ≤ SB ; a := − b ) ∪ (? MA − z ≥ SB ; a := 0) drive ≡ τ := 0; z ′′ = a & v ≥ 0 ∧ τ ≤ ε Andr´ e Platzer (University of Oldenburg) Differential Dynamic Logic for Parametric Hybrid Systems Tableaux’07 4 / 18

Differential Logic d L : 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 ≡ ( cor ; drive ) ∗ cor ≡ (? MA − z ≤ SB ; a := − b ) ∪ (? MA − z ≥ SB ; a ≤ a max ) drive ≡ τ := 0; z ′′ = a & v ≥ 0 ∧ τ ≤ ε Andr´ e Platzer (University of Oldenburg) Differential Dynamic Logic for Parametric Hybrid Systems Tableaux’07 4 / 18

Differential Logic d L : 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 ≡ ( cor ; drive ) ∗ cor ≡ (? MA − z ≤ SB ; a := − b ) ∪ (? MA − z ≥ SB ; a ≤ a max ) drive ≡ τ := 0; z ′ = v , v ′ = a , τ ′ = 1 & v ≥ 0 ∧ τ ≤ ε Andr´ e Platzer (University of Oldenburg) Differential Dynamic Logic for Parametric Hybrid Systems Tableaux’07 4 / 18

Differential Logic d L : Syntax Definition (Hybrid program α ) x ′ = f ( x ) & χ (continuous evolution within invariant region) x := θ (discrete jump) ? χ (conditional execution) α ; β (seq. composition) α ∪ β (nondet. choice) α ∗ (nondet. repetition) ETCS ≡ ( cor ; drive ) ∗ cor ≡ (? MA − z ≤ SB ; a := − b ) ∪ (? MA − z ≥ SB ; a ≤ a max ) drive ≡ τ := 0; z ′ = v , v ′ = a , τ ′ = 1 & v ≥ 0 ∧ τ ≤ ε Andr´ e Platzer (University of Oldenburg) Differential Dynamic Logic for Parametric Hybrid Systems Tableaux’07 4 / 18

Differential Logic d L : Syntax Definition (Formulas φ ) ¬ , ∧ , ∨ , → , ∀ x , ∃ x , = , ≤ , + , · ( R -first-order part) [ α ] φ, � α � φ (dynamic part) ψ → [( cor ; drive ) ∗ ] z ≤ MA All trains respect MA ⇒ system safe Andr´ e Platzer (University of Oldenburg) Differential Dynamic Logic for Parametric Hybrid Systems Tableaux’07 4 / 18

Differential Logic d L : Transition Semantics Definition (Hybrid programs α : transition semantics) x := θ v w x . = val ( v , θ ) Andr´ e Platzer (University of Oldenburg) Differential Dynamic Logic for Parametric Hybrid Systems Tableaux’07 5 / 18

Differential Logic d L : Transition Semantics Definition (Hybrid programs α : transition semantics) x ′ = f ( x ) v w x ϕ ( t ) w v t x ′ = f ( x ) Andr´ e Platzer (University of Oldenburg) Differential Dynamic Logic for Parametric Hybrid Systems Tableaux’07 5 / 18