Differential Dynamic Logic for Hybrid Systems e Platzer 1 , 2 Andr 1 - PowerPoint PPT Presentation

Differential Dynamic Logic for 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 KeY’07 Andr´ e Platzer (University of Oldenburg) Differential Dynamic Logic for Hybrid Systems KeY’07 1 / 16

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 Hybrid Systems KeY’07 1 / 16

Verifying Parametric Hybrid Systems RBC ST MA negot SB far corr Andr´ e Platzer (University of Oldenburg) Differential Dynamic Logic for Hybrid Systems KeY’07 2 / 16

Verifying Parametric 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) Differential Dynamic Logic for Hybrid Systems KeY’07 2 / 16

Verifying Parametric Hybrid Systems RBC ST MA negot SB far corr 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 [ Train ]safe Andr´ e Platzer (University of Oldenburg) Differential Dynamic Logic for Hybrid Systems KeY’07 2 / 16

Verifying Parametric Hybrid Systems RBC ST MA negot SB far corr 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 Hybrid Systems KeY’07 2 / 16

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 Hybrid Systems KeY’07 2 / 16

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 Hybrid Systems KeY’07 2 / 16

d L Motives: Regions in First-order Logic RBC differential dynamic logic d L = DL + HP ST MA negot SB far corr Andr´ e Platzer (University of Oldenburg) Differential Dynamic Logic for Hybrid Systems KeY’07 3 / 16

d L Motives: Regions in First-order Logic RBC differential dynamic logic d L = FOL ST MA negot SB far corr v v 2 ≤ 2 b ( MA − z ) MA − z MA Andr´ e Platzer (University of Oldenburg) Differential Dynamic Logic for Hybrid Systems KeY’07 3 / 16

d L Motives: Regions in First-order Logic RBC differential dynamic logic d L = FOL ST MA negot SB far corr 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 Hybrid Systems KeY’07 3 / 16

d L Motives: State Transitions in Dynamic Logic RBC differential dynamic logic d L = FOL + DL ST MA negot SB far corr 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 Hybrid Systems KeY’07 3 / 16

d L Motives: Hybrid Programs as Uniform Model RBC differential dynamic logic d L = FOL + DL + HP ST MA negot SB far corr [train-runs] v 2 ≤ 2 b ( MA − z ) Andr´ e Platzer (University of Oldenburg) Differential Dynamic Logic for Hybrid Systems KeY’07 3 / 16

d L Motives: Hybrid Programs as Uniform Model RBC differential dynamic logic d L = FOL + DL + HP ST MA negot SB far corr ] v 2 ≤ 2 b ( MA − z ) [ Andr´ e Platzer (University of Oldenburg) Differential Dynamic Logic for Hybrid Systems KeY’07 3 / 16

d L Motives: Hybrid Programs as Uniform Model RBC differential dynamic logic d L = FOL + DL + HP ST MA negot SB far corr far neg cor rec fsa Andr´ e Platzer (University of Oldenburg) Differential Dynamic Logic for Hybrid Systems KeY’07 3 / 16

d L Motives: Hybrid Programs as Uniform Model RBC differential dynamic logic d L = FOL + DL + HP ST MA negot SB far corr far neg cor rec fsa Andr´ e Platzer (University of Oldenburg) Differential Dynamic Logic for Hybrid Systems KeY’07 3 / 16

d L Motives: Hybrid Programs as Uniform Model RBC differential dynamic logic d L = FOL + DL + HP ST MA negot SB far corr far neg cor not compositional rec fsa Andr´ e Platzer (University of Oldenburg) Differential Dynamic Logic for Hybrid Systems KeY’07 3 / 16

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 Hybrid Systems KeY’07 4 / 16

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 ) ∗ RBC cor ≡ (? MA − z < SB ; a := − b ) ∪ (? MA − z ≥ SB ; a := 0) drive ≡ τ := 0; z ′′ = a ST MA far negot SB corr & v ≥ 0 ∧ τ ≤ ε Andr´ e Platzer (University of Oldenburg) Differential Dynamic Logic for Hybrid Systems KeY’07 4 / 16

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 ) ∗ RBC cor ≡ (? MA − z < SB ; a := − b ) ∪ (? MA − z ≥ SB ; a ≤ a max ) drive ≡ τ := 0; z ′′ = a ST MA far negot SB corr & v ≥ 0 ∧ τ ≤ ε Andr´ e Platzer (University of Oldenburg) Differential Dynamic Logic for Hybrid Systems KeY’07 4 / 16

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 ) ∗ RBC cor ≡ (? MA − z < SB ; a := − b ) ∪ (? MA − z ≥ SB ; a ≤ a max ) drive ≡ τ := 0; z ′ = v , v ′ = a , τ ′ = 1 ST MA far negot SB corr & v ≥ 0 ∧ τ ≤ ε Andr´ e Platzer (University of Oldenburg) Differential Dynamic Logic for Hybrid Systems KeY’07 4 / 16

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 ) ∗ RBC cor ≡ (? MA − z < SB ; a := − b ) ∪ (? MA − z ≥ SB ; a ≤ a max ) drive ≡ τ := 0; z ′ = v , v ′ = a , τ ′ = 1 ST MA far negot SB corr & v ≥ 0 ∧ τ ≤ ε Andr´ e Platzer (University of Oldenburg) Differential Dynamic Logic for Hybrid Systems KeY’07 4 / 16

Differential Logic d L : Syntax Definition (Formulas φ ) ¬ , ∧ , ∨ , → , ∀ x , ∃ x , = , ≤ , + , · (first-order part) [ α ] φ, � α � φ (dynamic part) ψ → [( cor ; drive ) ∗ ] z ≤ MA RBC All trains respect MA ST SB MA far negot corr ⇒ system safe Andr´ e Platzer (University of Oldenburg) Differential Dynamic Logic for Hybrid Systems KeY’07 4 / 16

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 Hybrid Systems KeY’07 5 / 16

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 Hybrid Systems KeY’07 5 / 16

Differential Logic d L : Transition Semantics Definition (Hybrid programs α : transition semantics) x ′ = f ( x ) v w & χ x ϕ ( t ) w χ v t x ′ = f ( x ) Andr´ e Platzer (University of Oldenburg) Differential Dynamic Logic for Hybrid Systems KeY’07 5 / 16

Differential Logic d L : Transition Semantics Definition (Hybrid programs α : transition semantics) x ′ = f ( x ) v w & χ x ϕ ( t ) w χ v t x ′ = f ( x ) Andr´ e Platzer (University of Oldenburg) Differential Dynamic Logic for Hybrid Systems KeY’07 5 / 16